# Quantum reference frames for general symmetry groups

Anne-Catherine de la Hamette<EMAIL_ADDRESS>Institute
for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich,
Switzerland  Perimeter Institute for Theoretical Physics, 31 Caroline St. N,
Waterloo, Ontario, N2L 2Y5, Canada  Thomas D. Galley
<EMAIL_ADDRESS>Perimeter Institute for Theoretical Physics, 31
Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada

(23 November 2020)

###### Abstract

A fully relational quantum theory necessarily requires an account of changes
of quantum reference frames, where quantum reference frames are quantum
systems relative to which other systems are described. By introducing a
relational formalism which identifies coordinate systems with elements of a
symmetry group $G$, we define a general operator for reversibly changing
between quantum reference frames associated to a group $G$. This generalises
the known operator for translations and boosts to arbitrary finite and locally
compact groups, including non-Abelian groups. We show under which conditions
one can uniquely assign coordinate choices to physical systems (to form
reference frames) and how to reversibly transform between them, providing
transformations between coordinate systems which are ‘in a superposition’ of
other coordinate systems. We obtain the change of quantum reference frame from
the principles of relational physics and of coherent change of reference
frame. We prove a theorem stating that the change of quantum reference frame
consistent with these principles is unitary if and only if the reference
systems carry the left and right regular representations of $G$. We also
define irreversible changes of reference frame for classical and quantum
systems in the case where the symmetry group $G$ is a semi-direct product
$G=N\rtimes P$ or a direct product $G=N\times P$, providing multiple examples
of both reversible and irreversible changes of quantum reference system along
the way. Finally, we apply the relational formalism and changes of reference
frame developed in this work to the Wigner’s friend scenario, finding similar
conclusions to those in relational quantum mechanics using an explicit change
of reference frame as opposed to indirect reasoning using measurement
operators.

##  1 Introduction

In quantum mechanics, physical systems are implicitly described relative to
some set of measurement devices. When writing down the quantum state of a
system of interest, say a spin-$1/2$ system in the state $\ket{\uparrow_{z}}$,
we mean that the state of the system is ‘up’ relative to a specified direction
$\hat{z}$ in the laboratory. In practice, this direction will be associated to
a macroscopic physical system in the lab. If we assume that quantum mechanics
is a universal theory and therefore applicable at all scales, the systems we
make reference to to describe quantum systems should eventually be treated
quantum mechanically as well. Reference systems that are themselves treated as
quantum systems are referred to as quantum reference frames. Following the
success of Einstein’s theory of relativity and its inherently relational
nature, one may seek to adopt a relational approach to quantum theory as well.
In such an approach, most physically meaningful quantities are relational,
i.e. they only take on well defined values once we agree on the reference
system (or the observer) relative to which they are described. In his papers
[1, 2], Rovelli suggested that quantum mechanics is a complete theory about
the description of physical systems relative to other physical systems. In his
Relational Quantum Mechanics (RQM) he rejected the idea of observer-
independent states of systems and values of observables. The importance of
changes of reference frame in special and general relativity suggests the
development of an account of changes of quantum reference frame in RQM. Such
an account is given in the present work.

Recently, there has been an increased interest in analysing spatial and
temporal quantum reference frames and in establishing a formalism that allows
to switch between different perspectives [3, 4, 5, 6, 7]. The present work is
partially based on these approaches which define changes of quantum reference
frames for systems that transform under the translation group (in space and
time) and the rotation group in three dimensional space. As opposed to other
more standard approaches [8, 9, 10, 11], this formalism stresses the lack of
an external reference frame from the outset and defines states of subsystems
relative to another subsystem. In standard approaches to quantum reference
frames [9, 10], one often starts from a description relative to an external
reference frame and removes any dependence on this reference frame by applying
a $G$ twirl (a group averaging over all possible configurations of the
external reference frame). In some cases, one can refactor the total Hilbert
space into global and relational subsystems and trace out the global degrees
of freedom [9, 10, 12, 13]. The main emphasis of these standard approaches is
often to obtain the physically meaningful (or reference frame independent)
quantities, in a similar fashion to identifying noise free subsystems in error
correction. In the work of [3, 4, 5, 6, 7, 14, 15] however emphasis is given
on the relational nature of the description (always starting from a state that
is given from the viewpoint of one of the subsystems) and the main object of
study is the relation between different accounts. We make a similar emphasis
in the present work. We abstract the formalism of [6] and introduce an
approach which makes heavy use of the inherently group theoretic nature of
quantum reference frames. This allows us to generalise the known results
beyond the translation and rotation groups to arbitrary finite and locally
compact groups (including non-Abelian groups).

In Section 2 we outline the relational approach to quantum theory embraced in
the present work as well as give a simple example of a change of reference
frame for classical bits and an example of a change of quantum reference frame
for qubits. In Section 3 we define the notion of a reference frame in terms of
reference systems and coordinate systems, as well as give a full account of
active and passive transformations as left and right regular group actions.
Combining these we define changes of reference frame under a group $G$ for
classical systems with configuration space $X\cong G$. In Section 4 we extend
the classical change of reference frame to quantum systems $L^{2}(G)$
following the principle of coherent change of reference system; and define a
general unitary operator which implements this change of reference system. We
prove a theorem stating that only systems carrying a regular representation of
$G$ can serve as reference frame, subject to the principle of coherent change
of reference system. Following this we extend the change of quantum reference
frame operator between $L^{2}(G)$ systems describing systems which do not
carry the right regular representation of $G$. In Section 5 we define
irreversible changes of reference frames for groups $G=N\rtimes P$ and
$G=N\times P$ via a truncation procedure. In Section 6 we extend this change
of reference frame to quantum reference frames using the principle of coherent
change of reference system once more. In Section 7 we apply the tools
developed in the preceding sections to the Wigner’s friend thought experiment,
providing an explicit change of reference frame from Wigner’s description to
the friend’s. We discuss related work in Section 8 and discuss implications of
the present work as well as suggestions for future work. In Section 9 we give
some concluding remarks.

##  2 Relational approach to quantum theory

In the construction of a relational formalism of quantum mechanics, an
essential task is to write quantum states of systems relative to a specified
reference system. We introduce the following notation: $\ket{\psi}_{\sf
B}^{\sf A}$ indicates the state of system ${\sf B}$ _relative_ to system ${\sf
A}$. In contrast to the approach of [6], we assign a Hilbert space to the
system whose perspective is adopted and assign to it the trivial state,
corresponding to the identity element of the group. Hence, by convention,
system ${\sf A}$ is in a default ‘zero-state’ relative to itself. Once we
introduce the notion of symmetry groups and how they enter into the formalism,
we will see that this default zero-state corresponds to the identity element
of the group that describes the transformations of the system. Thus, to be
more precise, one can write

$\ket{0}_{\sf A}^{\sf A}\otimes\ket{\psi}_{\sf B}^{\sf A}.$ (1)

The upper index refers to the system relative to which the state is given
while the lower index refers to the system that is being described (similarly
to the perspectival approach of [16]). This description does not make use of
any external abstract reference frame nor does it assume the existence of
absolute, observer-dependent values of physical observables. We observe that
since system ${\sf A}$ can only ever assign itself a single state there are no
state self-assignment paradoxes [17, 18].

A natural question to address on the relational approach to quantum theory is
how to change reference systems. Namely if the state of ${\sf B}$ relative to
${\sf A}$ is $\ket{\psi}^{\sf A}_{{\sf B}}=\ket{0}_{\sf A}^{\sf
A}\otimes\ket{\psi}_{\sf B}^{\sf A}$, what is the state $\ket{\psi}^{\sf
B}_{{\sf A}}$ of ${\sf A}$ relative to ${\sf B}$? This is the problem which
will be addressed in the present work.

Before introducing the general framework we will be using, we give two simple
examples of changes of reference frame for relational states. The first is
classical and the second its quantum generalisation. These should hopefully
provide the reader with an intuitive picture of the general mechanisms at
play.

######  Example 1 ($\mathbb{Z}_{2}$ change of classical reference frame).

Let us consider the case where systems can be in two states $\uparrow$ or
$\downarrow$. Every system considers themselves to be in the state $\uparrow$
(for example an observer free floating in empty space would always consider
the up direction to be aligned from their feet to their head). Consider
classical systems where the state relative to ${\sf A}$ is $\uparrow_{\sf
A}^{\sf A}\uparrow_{\sf B}^{\sf A}\downarrow_{\sf C}^{\sf A}$. Since ${\sf A}$
sees ${\sf B}$ in the state $\uparrow$ relative to itself, ${\sf B}$ also sees
${\sf A}$ in the state $\uparrow$ relative to itself. The state relative to
${\sf B}$ is $\uparrow_{\sf B}^{\sf B}\uparrow_{\sf A}^{\sf B}\downarrow_{\sf
C}^{\sf B}$. If the state relative to ${\sf A}$ was instead $\uparrow_{\sf
A}^{\sf A}\downarrow_{\sf B}^{\sf A}\downarrow_{\sf C}^{\sf A}$ then since
${\sf A}$ views ${\sf B}$ in the $\downarrow$ state, this implies that ${\sf
B}$ views things ‘upside down’ relative to ${\sf A}$. The change of
perspective would give $\uparrow_{\sf B}^{\sf B}\downarrow_{\sf A}^{\sf
B}\uparrow_{\sf C}^{\sf B}$.

In the next example we give a quantum generalisation of the $\mathbb{Z}_{2}$
change of reference frame. This is a specific instance of the general changes
of quantum reference frame defined in this work.

######  Example 2 ($\mathbb{Z}_{2}$ change of quantum reference frame).

Let us consider the case with quantum systems ${\mathbb{C}}^{2}$ with basis
$\\{\ket{\uparrow},\ket{\downarrow}\\}$. Every system considers themselves to
be in the state $\ket{\uparrow}$. By embedding the classical scenario above
with the map $\uparrow\ \mapsto\ket{\uparrow}$ and $\downarrow\
\mapsto\ket{\downarrow}$ we can reconstruct the classical example: if the
state relative to ${\sf A}$ is $\ket{\psi}^{\sf A}_{{\sf B}{\sf
C}}=\ket{\uparrow}_{\sf A}^{\sf A}\ket{\uparrow}_{\sf B}^{\sf
A}\ket{\downarrow}_{\sf C}^{\sf A}$ then the state relative to ${\sf B}$ is
$\ket{\psi}^{\sf B}_{{\sf A}{\sf C}}=\ket{\uparrow}_{\sf B}^{\sf
B}\ket{\uparrow}_{\sf A}^{\sf B}\ket{\downarrow}_{\sf C}^{\sf B}$. If the
state was $\ket{\phi}^{\sf A}_{{\sf B}{\sf C}}=\ket{\uparrow}_{\sf A}^{\sf
A}\ket{\downarrow}_{\sf B}^{\sf A}\ket{\downarrow}_{\sf C}^{\sf A}$ then the
change of perspective would give $\ket{\phi}^{\sf B}_{{\sf A}{\sf
C}}=\ket{\uparrow}_{\sf B}^{\sf B}\ket{\downarrow}_{\sf A}^{\sf
B}\ket{\uparrow}_{\sf C}^{\sf B}$.

Let us move to the quantum case with an initial state $\ket{\tau}^{\sf
A}_{{\sf B}{\sf C}}=\ket{\uparrow}_{\sf A}^{\sf A}\left(\ket{\uparrow}_{\sf
B}^{\sf A}+\ket{\downarrow}_{\sf B}^{\sf A}\right)\ket{\downarrow}_{\sf
C}^{\sf A}$. What is the state $\ket{\tau}^{\sf B}_{{\sf A}{\sf C}}$? First
let us observe that $\ket{\tau}^{\sf A}_{{\sf B}{\sf C}}=\ket{\psi}^{\sf
A}_{{\sf B}{\sf C}}+\ket{\phi}^{\sf A}_{{\sf B}{\sf C}}$, and let us assume
that changes of quantum reference frame are coherent (they observe the
superposition principle). Then the state $\ket{\tau}^{\sf B}_{{\sf A}{\sf
C}}=\ket{\uparrow}_{\sf B}^{\sf B}\left(\ket{\uparrow}_{\sf A}^{\sf
B}\ket{\downarrow}_{\sf C}^{\sf B}+\ket{\downarrow}_{\sf A}^{\sf
B}\ket{\uparrow}_{\sf C}^{\sf B}\right)$ which is an entangled state of ${\sf
A}$ and ${\sf C}$.

The above example made use of the two guiding principles of this work: the
principle of _relational physics_ and the principle of _coherent change of
reference system_. These are defined in Section 4.3.

Whenever we use phrases such as ‘from the viewpoint of’ or ‘from the
perspective of’, we simply mean ‘relative to’. Although these expressions
might imply that the state $\ket{\psi}_{\sf B}^{\sf A}$ indicates how system
${\sf A}$ _perceives_ system ${\sf B}$, we do not make this interpretation
here. System ${\sf A}$ acts as the observer in this description but we should
note that there is nothing special about an observer system. No interpretation
is made as to what the system _sees_. Rather a change of reference system
${\sf A}\to{\sf B}$ is a change of description from one where ${\sf A}$ is at
the origin to one where ${\sf B}$ is at the origin.

##  3 Classical changes of reference frames associated to symmetry groups

###  3.1 Reference systems, coordinate choices and changes of reference frame

A coordinate system is a purely mathematical object, and need not in general
be associated to a physical system. A reference frame consists of a physical
system (known as a reference system), and a choice of coordinates such that
the reference system is at the origin in that coordinate system. For full
definitions we refer the reader to Appendix A. In this section we define
changes of reference frames for classical systems where the configuration
space is itself a group $G$. In Section 5 we will consider cases where this is
no longer holds.

We begin by a simple example which illustrates changes of reference frames and
the use of group elements for relative coordinates.

######  Example 3 (Three particles on a line).

Consider three classical particles ${\sf A},{\sf B}$ and ${\sf C}$ on a line,
with state $s=(x_{\sf A},x_{\sf B},x_{\sf C})$ in some Cartesian coordinate
system (here we omit the velocities since we are just interested in
translations in space). The coordinate system $x^{\prime}$ such that
$x^{\prime}_{A}=0$ is said to be associated to ${\sf A}$. In this coordinate
system the particles have state $s=(x^{\prime}_{\sf A}=0,x^{\prime}_{\sf
B}=x_{\sf B}-x_{\sf A},x^{\prime}_{\sf C}=x_{\sf C}-x_{\sf A})$. We observe
here that the relative coordinates (to ${\sf A}$) $x^{\prime}_{\sf
A},x^{\prime}_{\sf B}$ and $x^{\prime}_{\sf C}$ uniquely identify the
translation which maps system ${\sf A}$ to systems ${\sf A}$, ${\sf B}$ and
${\sf C}$. Namely the relative distance $x^{\prime}_{\sf B}=x_{\sf B}-x_{\sf
A}$ is the distance needed to translate ${\sf A}$ to ${\sf B}$. The relative
coordinates $x^{\prime}_{\sf A},x^{\prime}_{\sf B}$ and $x^{\prime}_{\sf C}$
correspond to the symmetry group transformations relating ${\sf A}$, ${\sf B}$
and ${\sf C}$ to ${\sf A}$. If we label a translation of distance $d$ by
$t_{L}(d)$, where $t_{L}(d)x=d+x$, we have the state relative to ${\sf A}$ as
$s^{\sf A}=(t_{L}(0),t_{L}(x_{\sf B}-x_{\sf A}),t_{L}(x_{\sf C}-x_{\sf A}))$.
The state relative to ${\sf B}$ is $s^{\sf B}=(t_{L}(x_{\sf A}-x_{\sf
B}),t_{L}(0),t_{L}(x_{\sf C}-x_{\sf B}))$. These two relative states are
themselves related by the transformation $s^{\sf B}=s^{\sf A}-(x_{\sf
B}-x_{\sf A})$. We define the right regular action of the translation group
$T_{R}(d)=x-d$. The change of reference frame ${\sf A}$ to ${\sf B}$ is given
by the right regular action $T_{R}(x_{\sf B}-x_{\sf A})$ of the group element
$x_{\sf B}-x_{\sf A}$ mapping ${\sf A}$ to ${\sf B}$.

In the above example the configuration space $\mathbb{R}$ and the symmetry
group $T=(\mathbb{R},+)$ acting on it are equivalent as manifolds. This
equivalence is essential for the existence of a well defined reversible change
of quantum reference frame. In Sections 5 and 6 we study scenarios where this
is no longer the case, and the changes of reference frame are irreversible.
Since the results in this paper also apply to finite groups we cover a simple
example.

######  Example 4 ($\mathbb{Z}_{2}$).

Let us consider systems with configuration space
$X=\\{\uparrow,\downarrow\\}$. The symmetry group $G=\\{I,F\\}=\mathbb{Z}_{2}$
consisting of the identity $I(\uparrow)=\ \uparrow$ and the flip
$F(\uparrow)=\ \downarrow$ is the symmetry group of $X$. A state of four
systems of the form $s=\uparrow,\uparrow,\downarrow,\downarrow$ can be
expressed as $s=I(\uparrow),I(\uparrow),F(\uparrow),F(\uparrow)$. By
considering $\uparrow$ as the ‘coordinate system’ we have that the state $s$
has coefficients $(I,I,F,F)$. In the ‘coordinate system’ $\downarrow$ the
state $s$ would have coefficients $(F,F,I,I)$.

We observe that in the above examples _group elements_ of the global symmetry
groups serve as _relative coordinates_. In the next subsection we make this
link more explicit. We also note the importance of the one to one
correspondence between states and coordinate transformations. We observe that
there is always some conventionality in changes of coordinates: one considers
only translations on $\mathbb{R}$ for instance, and not all diffeomorphisms of
$\mathbb{R}$ as relating different coordinates. In Appendix A we give explicit
definitions of coordinate systems on a manifold $X$, emphasising that a
coordinate system is different to a coordinate chart (typically used in
general relativity), a distinction made in [19]. Roughly speaking a
_coordinate system_ on a manifold $X$ is an isomorphism $f:X\to Y$ (where $Y$
a known mathematical object used to describe $X$), whereas a _coordinate
chart_ is a map from the mathematical object $Y$ to the physical object $X$
which need not be an isomorphism (for instance multiple charts
$\mathbb{R}^{n}$ are used to describe a curved $n$-dimensional manifold $X$ in
general relativity but there is no isomorphism from $X$ to $\mathbb{R}^{n}$).

###  3.2 General treatment of reversible changes of reference frame

Let us extract the general features of the above scenario which allow for well
defined reference frames and reversible changes of reference frame. Consider a
configuration space $X$ (which is typically a set with a manifold structure)
and a group $G$ acting on $X$ such that there is a unique transformation $g\in
G$ relating any pair of points (the action is transitive and free). This
implies $X\cong G$ (as sets/manifolds), and the action of $G$ on $X\cong G$ is
the group multiplication on itself: $G\times G\to G$. This space is a
_principle homogeneous space for $G$_, sometimes called a _$G$ torsor_. We
assume $G$ locally compact and thus equipped with a left Haar measure, denoted
$dg$. Many groups of interest in physics, such as the Poincaré group, the
symmetric group, $\mathrm{SU}(d)$ and $\mathrm{SO}(d)$ are locally compact
(compact and finite groups are instances of locally compact groups). One
exception is the diffeomorphism group of some space-time manifold $M$, which
in general is not locally compact. We observe that $X\cong G$ follows from the
requirement that there exists a unique transformation $g\in G$ relating any
two points in $X$. Note that we will later go beyond such perfect reference
frames and consider cases for which the configuration space $X$ differs from
the group $G$.

We use the following example from [20] to introduce active and passive
transformations on $G$ torsors.

######  Example 5 (Single observer and system on $X\cong G$).

Consider an observer $0$ at location $x_{0}$ on $X\cong G$ and an object $1$
at location $x_{1}$. Then the unique transformation $g$ such that
$gx_{0}=x_{1}$ is the relative location of $1$ relative to $0$.

An active transformation is a transformation on the object 1. A transformation
$h$ on the object 1 is given by the left regular action $x_{1}\mapsto
x_{1}^{\prime}=hx_{1}$. The relative location is now $k$ where
$kx_{0}=x_{1}^{\prime}$. Using $gx_{0}=x_{1}$ and $hx_{1}=x_{1}^{\prime}$ we
find that $k=hg$: $hgx_{0}=hx_{1}=x_{1}^{\prime}$. Therefore an active
transformation by $h$ corresponds to the left regular action of $h$ on the
relative location $g$: $g\mapsto hg$.

A passive transformation is a transformation on the observer $x_{0}\mapsto
hx_{0}$. This induces a transformation on the relative location of $1$ to $0$
which we now outline. Consider the case where $gx_{0}=x_{1}$ and a passive
transformation $h$ on $0$ is applied while $1$ is left unchanged (at position
$x_{1})$. We have $x_{0}\mapsto x_{0}^{\prime}=hx_{0}$. Then the relative
location of $1$ to $0$ is $k$ where $kx_{0}^{\prime}=x_{1}$. Substituting in
$gx_{0}=x_{1}$ and $hx_{0}=x_{0}^{\prime}$ gives $khx_{0}=gx_{0}$ implying
that that $kh=g$, and therefore $k=gh^{-1}$ (where we remember that since
$X\cong G$ there is always a unique $g\in G$ mapping a pair of points in $X$).
Writing in full $gh^{-1}hx_{0}=gh^{-1}x_{0}^{\prime}=x_{1}$ and so the
relative location of $1$ relative to $0$ is now $gh^{-1}$. A passive
transformation $h$ on $0$ corresponds to the right regular action of $h$ on
the relative location $g$: $g\mapsto gh^{-1}$.

The left regular action and right regular action on $G\cong X$ are defined as
follows:

$\displaystyle\phi_{L}(g,x)$ $\displaystyle=gx\ ,$ (2)
$\displaystyle\phi_{R}(g,x)$ $\displaystyle=xg^{-1}.$ (3)

Both are defined using the group multiplication, where $x\in G$. These two
actions naturally commute, and hence $X\cong G$ carries an action of $G\times
G$, with one factor typically being understood as the active and the second as
the passive transformations [21]. Although $\phi_{R}$ acts ‘to the right’, it
is a left group action:
$\phi(gh,x)=x(gh)^{-1}=xh^{-1}g^{-1}=\phi(h,x)g^{-1}=\phi(g,\phi(h,x))$. Here
we take $\phi_{L}$ as active and $\phi_{R}$ as passive.111Active and passive
transformations are typically defined as either left actions on different
spaces (states and coordinates) or a left and a right action on the same space
(typically coordinates). In this case ($X\cong G$) they can be defined as left
actions on the same space.

A given state of $n$ systems is $s=(x_{0},x_{1},...,x_{n-1})$, where we omit
the velocities $\dot{x_{i}}$ since we are defining changes of reference frame
for the ‘translation’ group $G$ on $X\cong G$. This can be expressed as:

$\displaystyle
s=\left(g_{0}^{0}x_{0},g^{0}_{1}x_{0},g^{0}_{2}x_{0},...,g^{0}_{n-1}x_{0}\right)\
,$ (4)

where $g^{i}_{j}$ is the unique $g\in G$ such that $g^{i}_{j}x_{i}=x_{j}$, and
$e=g^{i}_{i}$ the identity element. We observe that
$g^{j}_{k}g^{i}_{j}=g^{i}_{k}$ and $(g_{i}^{j})^{-1}=g_{j}^{i}$. Then _the
state $s^{0}$ of the $n$ systems relative to system $0$ is:_

$\displaystyle
s^{0}=\left(g_{0}^{0},g^{0}_{1},g^{0}_{2},...,g^{0}_{n-1}\right).$ (5)

The state relative to the system $i$ is:

$\displaystyle
s^{i}=\left(g^{i}_{0},g^{i}_{1},g^{i}_{2},...,g^{i}_{n-1}\right).$ (6)

We observe that we can also describe the state relative to hypothetical
systems (i.e. relative to a point $x\in X$ which is not occupied by a system).
For instance in the above consider a point $x_{n}\in X$ such that $x_{i}\neq
x_{n}\ \forall i\in\\{0,...,n-1\\}$. Then we can write:

$\displaystyle
s^{n}=\left(g^{n}_{0},g^{n}_{1},g^{n}_{2},...,g^{n}_{n-1}\right).$ (7)

As in the examples given above, we see that _a relative state $s^{i}$ is given
by all the symmetry transformations $g^{i}_{j}$ from system $i$ to system $j$
for all $j\in\\{0,...,n-1\\}$_. There is a unique relative state $s^{i}$,
which is such that particle $i$ is in state $e$.

A change of reference frame from system $0$ to system $i$ is a map $s^{0}\to
s^{i}$. Let us extend the left and right regular actions of $G$ on itself to
states:

$\displaystyle\phi_{L}(g,s)$ $\displaystyle=(gx_{0},gx_{1},...,gx_{n-1})\ ,$
(8) $\displaystyle\phi_{R}(g,s)$
$\displaystyle=(x_{0}g^{-1},x_{1}g^{-1},...,x_{n-1}g^{-1}).$ (9)

The transformation $s^{0}\to s^{i}$ is given by the _right regular action_ of
$g^{0}_{i}$:

$\displaystyle\phi_{R}(g^{0}_{i},s^{0})$
$\displaystyle=\left(eg_{0}^{i},g^{0}_{1}g^{i}_{0},g^{0}_{2}g^{i}_{0},...,g^{0}_{n-1}g^{i}_{0}\right)$
$\displaystyle=\left(g^{i}_{0},g^{i}_{1},g^{i}_{2},...,g^{i}_{n-1}\right)=s^{i}.$
(10)

We observe that this transformation cannot be achieved using the left regular
action: there is no elements $g\in G$ such that $\phi_{L}(g,s^{0})=s^{i}$
(unless $G$ is Abelian). The transformation $s^{0}\to s^{i}$ is a _passive
transformation_.

${1}$${2}$${0}$$\scriptstyle{g_{0}^{1}}$$\scriptstyle{g^{1}_{2}=g_{2}^{0}g_{0}^{1}}$$\scriptstyle{g^{2}_{1}=g_{1}^{0}g_{0}^{2}}$$\scriptstyle{g_{0}^{2}}$$\scriptstyle{g_{2}^{0}}$$\scriptstyle{g_{1}^{0}}$
Figure 1: Diagram capturing the relational states between three systems. Each
system $i$ assigns the relative state along the arrow point from $i$ to $j$ to
system $j$ (and the identity to themselves). For instance system $0$ assigns
state $e,g^{0}_{1},g^{0}_{2}$. By the right regular action of $g^{0}_{1}$, we
obtain $g^{1}_{0},e,g^{0}_{2}g^{1}_{0}=g^{1}_{2}$ which is the relative state
assigned by system $1$.

##  4 Quantum reference frames associated to symmetry groups

We begin this section by reviewing changes of quantum reference frame for
three particles on the line. We then define a quantum change of reference
frame operator for $n$ identical systems $L^{2}(G)$ for arbitrary $G$. This
generalises the change of reference frame in [6] beyond one parameter
subgroups of the Galilean group. Furthermore we show that it is only the
$L^{2}(G)$ system described so far for which a unitary reversible change of
reference system is possible. Finally, we define a change of reference frame
operator for $m$ identical $L^{2}(G)$ systems serving as reference frames and
$n-m$ systems which are not.

###  4.1 Comment on finite groups and notation

All our results apply for finite groups. In this case $L^{2}(G)$ should be
replaced by ${\mathbb{C}}[G]\cong{\mathbb{C}}^{|G|}$ and integrals $\int_{g\in
G}\outerproduct{g}{g}dg$ by $\sum_{i}\outerproduct{g_{i}}{g_{i}}$.
${\mathbb{C}}[G]$ is the vector space freely generated by the elements of $G$,
i.e. for which the elements of $G$ form a basis.

In the cases where it is clear which system is the reference system we
sometimes omit the top label for ease of reading. For instance the state
$\ket{0}_{\sf A}^{\sf A}\ket{x_{1}}_{\sf B}^{\sf A}\ket{x_{2}}_{\sf C}^{\sf
A}$ is written as $\ket{0}_{\sf A}\ket{x_{1}}_{\sf B}\ket{x_{2}}_{\sf C}$.

###  4.2 The example of $L^{2}(\mathbb{R})\otimes L^{2}(\mathbb{R})\otimes
L^{2}(\mathbb{R})$

Let us first rephrase the known case of the translation group acting on three
particles on the line [6] in the formalism outlined above.

Take the translation group $T=(\mathbb{R},+)$ and three systems ${\sf A}$,
${\sf B}$ and ${\sf C}$ whose joint state space is $L^{2}(\mathbb{R})\otimes
L^{2}(\mathbb{R})\otimes L^{2}(\mathbb{R})$. Let us for instance consider the
state

$\ket{0}_{\sf A}\ket{x_{1}}_{\sf B}\ket{x_{3}}_{\sf C}\ ,$ (11)

which is the state of three perfectly localised systems, described using a
coordinate system centred on system ${\sf A}$. In standard quantum mechanics,
when changing from a classical, highly localised reference frame at the
position of ${\sf A}$ to another classical reference frame localised at ${\sf
B}$ translated by an amount $x_{1}$, one simply applies the translation
operator $\hat{T}(-x_{1})=e^{ix_{1}(\hat{p}_{\sf A}+\hat{p}_{\sf
B}+\hat{p}_{\sf C})}$ to the state of the three systems, where $\hat{p}_{\sf
A}$ is the momentum operator for system ${\sf A}$ and similarly for
$\hat{p}_{\sf B}$, $\hat{p}_{\sf C}$ and systems ${\sf B}$ and ${\sf C}$. This
shifts the state to:

$\ket{-x_{1}}_{\sf A}\ket{0}_{\sf B}\ket{x_{3}-x_{1}}_{\sf C}.$ (12)

In the previous language we have $g^{\sf A}_{\sf B}=x_{1}$ and $g^{\sf A}_{\sf
C}=x_{3}$. The action of $\hat{T}(-x_{1})$ corresponds to the right action of
$g^{\sf A}_{\sf B}=x_{1}$.

The next step is to begin with a state of the following form:

$\displaystyle\ket{0}_{\sf A}\frac{1}{\sqrt{2}}(\ket{x_{1}}+\ket{x_{2}})_{\sf
B}\ket{x_{3}}_{\sf C}\ ,$ (13)

which is described by a coordinate system localised at ${\sf A}$. What is the
change of reference frame ${\sf A}\to{\sf B}$ in this case? How can one
describe classical coordinates which assign state $\ket{0}_{\sf B}$, when
${\sf B}$ is not localised relative to ${\sf A}$? A standard translation of
all states will not work.

Following the reasoning presented in [6] we assume that the change of
perspective obeys the principle of superposition. Namely the state of Equation
(13) is an equally weighted superposition of the states $\ket{0}_{\sf
A}\ket{x_{1}}_{\sf B}\ket{x_{3}}_{\sf C}$ and $\ket{0}_{\sf A}\ket{x_{2}}_{\sf
B}\ket{x_{3}}_{\sf C}$. The change of reference frame for each of these states
individually is obtained by translating by $-x_{1}$ and $-x_{2}$ respectively.

Assuming that the superposition principle applies to changes of reference
systems, the state described in coordinates ‘localised’ at system ${\sf B}$ is
just the superposition of the classical states obtained by translation by
$-x_{1}$ and $-x_{2}$. This leads to the following state of the joint system
from the viewpoint of ${\sf B}$:

$\ket{0}_{\sf B}\frac{1}{\sqrt{2}}(\ket{-x_{1}}_{\sf A}\ket{x_{3}-x_{1}}_{\sf
C}+\ket{-x_{2}}_{\sf A}\ket{x_{3}-x_{2}}_{\sf C}).$ (14)

When changing between the viewpoints of _quantum_ systems, we apply a weighted
translation of the states of systems, dependent on the state of the new
reference frame whose viewpoint we are adopting. For the state given above,
this means applying a translation for the state of ${\sf B}$ being
$\ket{x_{1}}$ and one for it being $\ket{x_{2}}$.

Figure 2: Example for translation group: In the upper subfigure, the state of
the three systems is given from the perspective of system ${\sf A}$. The lower
subfigure shows the state relative to ${\sf B}$.

The states of ${\sf A}$ and ${\sf C}$ become entangled relative to ${\sf B}$.
We see that to perform this change of reference frame, the state of ${\sf A}$
is mapped to the inverse of the group element associated with the old state of
${\sf B}$. Also for each state of ${\sf B}$, the state of ${\sf C}$ is shifted
respectively. Hence, for the translation group on the real line, the reference
frame change operator is

$\displaystyle U^{{\sf A}\to{\sf B}}=$ $\displaystyle{\rm SWAP}_{{\sf A},{\sf
B}}\circ$ $\displaystyle\int dx_{i}dx_{j}\outerproduct{-x_{i}}{x_{i}}_{\sf
B}\otimes\mathbb{1}_{\sf A}\otimes\outerproduct{x_{j}-x_{i}}{x_{j}}_{\sf C}.$
(15)

This operator performs exactly the same reference frame change as the operator
given in [6]:

$\hat{S}^{{\sf A}\to{\sf
B}}=\hat{\mathcal{P}}_{AB}e^{i/\hbar\hat{x}_{B}\hat{p}_{C}}\ ,$ (16)

where $\hat{\mathcal{P}}_{AB}$ is the so-called parity-swap operator. It acts
as $\hat{\mathcal{P}}_{AB}\psi_{B}(x)=\psi_{A}(-x)$. The proof of this is
given in Appendix F.

###  4.3 $n$ identical systems $L^{2}(G)$

Consider a configuration space $X\cong G$ and $n$ systems each with associated
Hilbert space ${\mathcal{H}}_{i}\cong L^{2}(G)$ for $G$ continuous (or
${\mathbb{C}}^{|G|}$ for $G$ finite):

$\displaystyle G$ $\displaystyle\to L^{2}(G)\ ,$ $\displaystyle g_{i}$
$\displaystyle\mapsto\ket{g_{i}}.$ (17)

$L^{2}(G)$ is the space of square integrable functions $G\to{\mathbb{C}}$.

The left and right action of $G$ onto itself induces the _left regular_ and
_right regular_ representation of $G$ on each ${\mathcal{H}}_{i}$. For a given
${\mathcal{H}}_{i}$ this representation acts on the basis $\\{\ket{g}\\}$ as:

$\displaystyle U_{L}(g_{2}):\ket{g_{1}}$
$\displaystyle\mapsto\ket{g_{2}g_{1}}\ ,$ (18) $\displaystyle
U_{R}(g_{2}):\ket{g_{1}}$ $\displaystyle\mapsto\ket{g_{1}g_{2}^{-1}}.$ (19)

An arbitrary basis state of the $n$ systems is:

$\displaystyle\ket{\psi}=\ket{g_{0}}_{0}\ket{g_{1}}_{1}...\ket{g_{n-1}}_{n-1}.$
(20)

Following the classical case, the choice of coordinates on $G$ associated to
${\mathcal{H}}_{0}$ is:

$\displaystyle\ket{\psi}^{0}=\ket{e}_{0}\ket{g^{0}_{1}}_{1}...\ket{g_{n-1}^{0}}_{n-1}\
,$ (21)

where $g^{i}_{j}g_{i}=g_{j}$. For general ${\mathcal{H}}_{i}$ it is:

$\displaystyle\ket{\psi}^{i}=\ket{g^{i}_{0}}_{0}\ket{g^{i}_{1}}_{1}...\ket{e}_{i}...\ket{g_{n-1}^{i}}_{n-1}.$
(22)

The change of coordinate system $\ket{\psi}^{0}\to\ket{\psi}^{i}$ is given by
$U_{R}(g^{0}_{i})^{\otimes n}$, when considering orthogonal basis states
alone.

Let us observe that the left regular representation on the space of wave
functions acts as $\psi(x)\mapsto\psi(g^{-1}x)$ and the right regular
representation as $\psi(x)\mapsto\psi(xg)$. This follows from $\int_{x\in
G}\psi(x)\ket{gx}dx=\int_{x\in G}\psi(g^{-1}x)\ket{x}dx$ and $\int_{x\in
G}\psi(x)\ket{xg^{-1}}dx=\int_{x\in G}\psi(xg)\ket{x}dx$. We note that for Lie
groups $G$ the objects $\ket{g}$ are not in $L^{2}(G)$ and one should
typically prefer the representation acting on the wavefunctions. In the
following however we consider the representation acting on the elements
$\ket{g}$ in order to describe the continuous and discrete case
simultaneously.

Unlike some approaches to relational quantum dynamics [14, 3, 15] we do not
assign a global state $\ket{\psi}\in{\mathcal{H}}$ and then work out its
expression relative to a certain system. Rather, we begin from a state
relative to a system and define changes of reference frame to other systems.
We formalise this in the following principle:

######  Principle 1 (Relational physics).

Given $n$ systems, states are defined to be relative to one of the systems. A
state relative to system $i$ is a description of the other $n-1$ systems,
relative to $i$.

We observe that this principle does not preclude the existence of a well
defined global state of the $n$ systems.

A superposition state (relative to $0$ in the $G$ product basis) is of the
form:

$\displaystyle\ket{\psi}^{0}=$
$\displaystyle\ket{e}_{0}\ket{g_{1}^{0}}_{1}...\ket{g_{n-1}^{0}}_{n-1}+\ket{e}_{0}\ket{h_{1}^{0}}_{1}...\ket{h_{n-1}^{0}}_{n-1}.$
(23)

Since the state is in general not a basis state there is no a priori well
defined change given by an operator of the form $U_{R}(g^{0}_{i})^{\otimes
n}$. However, following the example of [6] one can define a _coherent change
of reference frame_ operator. We explicitly state this as a principle:

######  Principle 2 (Coherent change of reference system).

If $\ket{\psi}^{0}\mapsto\ket{\psi}^{i}$ and
$\ket{\phi}^{0}\mapsto\ket{\phi}^{i}$ then
$\alpha\ket{\psi}^{0}+\beta\ket{\phi}^{0}\mapsto\alpha\ket{\psi}^{i}+\beta\ket{\phi}^{i}$,
$\alpha,\beta\in{\mathbb{C}}$.

This implies that $\ket{\psi}^{0}$ defined above changes to:

$\displaystyle\ket{\psi}^{i}=$
$\displaystyle\ket{e}_{i}\ket{g_{0}^{i}}_{0}\ket{g_{1}^{0}g_{0}^{i}}_{1}...\ket{g_{n-1}^{0}g_{0}^{i}}_{n-1}$
$\displaystyle+\ket{e}_{i}\ket{h_{0}^{i}}_{0}\ket{h_{1}^{0}h_{0}^{i}}_{1}...\ket{h_{n-1}^{0}h_{0}^{i}}_{n-1}.$
(24)

The operator which implements the coherent change of reference systems $0\to
i$ is:

$\displaystyle U^{0\to i}=$ $\displaystyle{\rm SWAP}_{0,i}\circ$
$\displaystyle\int_{g^{0}_{i}\in
G}\outerproduct{g^{i}_{0}}{g_{i}^{0}}_{i}\otimes{\mathbb{1}}_{0}\otimes
U_{R}(g^{0}_{i})^{\otimes n-2}dg_{i}^{0}.$ (25)

The following lemmas are proven in Appendix B.

######  Lemma 1.

$U^{0\to i}$ is unitary.

######  Lemma 2.

$\left(U^{0\to i}\right)^{\dagger}=U^{i\to 0}$

######  Lemma 3.

$U^{i\to j}U^{k\to i}=U^{k\to j}$

####  4.3.1 Change of reference frame for observables

The change of reference frame operator also allows us to transform between
observables. Namely if system $0$ describes an observable of systems
$1,...,n-1$ as $Z^{0}_{0,1,...,n-1}={\mathbb{1}}_{0}\otimes Z^{0}_{1,...,n-1}$
then system $i$ describes the observable as $Z^{i}_{0,1,...,n-1}=U^{0\to
i}Z^{0}_{0,1,...,n-1}U^{i\to 0}$.

####  4.3.2 $L^{2}(\mathrm{U}(1))\otimes L^{2}(\mathrm{U}(1))\otimes
L^{2}(\mathrm{U}(1))$

To illustrate the changes of reference frame described previously we will give
an example. Let us consider the symmetry group $\mathrm{U}(1)$ and three
particles ${\sf A},{\sf B},{\sf C}$ on a circle with associated Hilbert space
$L^{2}(\mathrm{U}(1))\otimes L^{2}(\mathrm{U}(1))\otimes
L^{2}(\mathrm{U}(1))$. In this case, the map from group elements to elements
of the Hilbert space is

$\displaystyle\mathrm{U}(1)$ $\displaystyle\to L^{2}(\mathrm{U}(1))\ ,$
$\displaystyle\theta_{i}$ $\displaystyle\mapsto\ket{\theta_{i}}\ ,$ (26)

with $\theta_{i}\in[0,2\pi[$ and
$\innerproduct{\theta_{i}}{\theta_{j}}=\delta(\theta_{i}-\theta_{j})$.

Figure 3: Basis states $\ket{0},\ket{\theta_{1}}$ and $\ket{\theta_{2}}$ of
the state space for $L^{2}(U(1))$.

The states $\\{\ket{\theta_{i}}|\ \theta_{i}\in[0,2\pi[\\}$ are states at all
angular positions of the unit circle and form a basis of the Hilbert space
$L^{2}(\mathrm{U}(1))$. A system consisting of three particles on a circle
could for instance be in the product state $\ket{0}_{\sf
A}\ket{\frac{\pi}{2}}_{\sf B}\ket{\pi}_{\sf C}$ relative to ${\sf A}$.

An arbitrary state of the joint system relative to particle ${\sf A}$ is given
by

$\ket{0}_{\sf A}\otimes\int d\theta_{i}d\theta_{j}\
\psi(\theta_{i},\theta_{j})\ket{\theta_{i}}_{\sf
B}\otimes\ket{\theta_{j}}_{\sf C}\ ,$ (27)

where $\ket{e}=\ket{0}$ is the state associated to the identity element of
$\mathrm{U}(1)$. As a specific example, take the state

$\ket{0}_{\sf
A}\otimes\left(\sqrt{\frac{1}{3}}\ket{\theta_{1}}+\sqrt{\frac{2}{3}}\ket{\theta_{2}}\right)_{\sf
B}\otimes\ket{\theta_{3}}_{\sf C}.$ (28)

Relative to particle ${\sf B}$, the state assigned to the joint system would
be

$\ket{0}_{\sf B}\left(\sqrt{\frac{1}{3}}\ket{-\theta_{1}}_{\sf
A}\otimes\ket{\theta_{3}-\theta_{1}}_{\sf
C}+\sqrt{\frac{2}{3}}\ket{-\theta_{2}}_{\sf
A}\otimes\ket{\theta_{3}-\theta_{2}}_{\sf C}\right).$ (29)

We see that the state of ${\sf B}$ is mapped to the state corresponding to the
inverse group element assigned to the old state of ${\sf B}$ and the state of
${\sf C}$ is shifted respectively. In the end, the labels of ${\sf A}$ and
${\sf B}$ are swapped. The operator that performs this reference frame change
is

$\displaystyle U^{{\sf A}\to{\sf B}}=$ $\displaystyle{\rm SWAP}_{{\sf A},{\sf
B}}\circ$ $\displaystyle\int d\theta
d\theta^{\prime}\outerproduct{-\theta}{\theta}_{\sf B}\otimes\mathbb{1}_{\sf
A}\otimes\outerproduct{\theta^{\prime}-\theta}{\theta^{\prime}}_{\sf C}$
$\displaystyle=$ $\displaystyle{\rm SWAP}_{{\sf A},{\sf B}}\circ\int d\theta
d\theta^{\prime}\outerproduct{-\theta}{\theta}_{\sf B}\otimes\mathbb{1}_{\sf
A}\otimes U_{R}(\theta)_{\sf C}.$ (30)

Figure 4: Example of three particles on a circle.

###  4.4 Unitarity of change of reference frame operator

The change of reference frame defined previously is highly constrained: it
applies only to systems $L^{2}(G)$ with symmetry group $G$. One could ask
whether one could define similar changes of reference frame for systems
${\mathcal{H}}\not\cong L^{2}(G)$ with states $\ket{\psi(g)}$ carrying two
representations: $U_{L}(h)\ket{\psi(g)}=\ket{\psi(hg)}$ and
$U_{R}(h)\ket{\psi(g)}=\ket{\psi(gh^{-1})}$.

First consider the symmetry group $\mathrm{U}(1)$. Our results show that for a
change of reference frame to recreate our classical intuitions one needs
systems $L^{2}(\mathrm{U}(1))$ which carry the right regular representation.
However one may wonder whether one could use qubits with states along the
$X-Y$ plane $\ket{\theta}=\cos(\theta/2)\ket{0}+\sin(\theta/2)\ket{1}$ as
reference systems which transform in a manner which obeys the classical change
of reference frame.

We first provide an example to show that this breaks linearity of the change
of reference frame operator for the case of rebits before proving a general
result.

######  Example 6 (Three qubits with $\mathrm{U}(1)$ group action).

Consider three qubits with states restricted to real valued superpositions:
$\ket{\theta}=\cos(\theta/2)\ket{0}+\sin(\theta/2)\ket{1}$ (sometimes known as
rebits). The space of pure states of the three systems is
$\mathrm{U}(1)\otimes\mathrm{U}(1)\otimes\mathrm{U}(1)$. We apply our
classical intuition of how a reference frame change should act. Let the
initial state of the three systems relative to ${\sf A}$ be

$\ket{\psi(0)}_{\sf A}\otimes\ket{\psi(\theta)}_{\sf
B}\otimes\ket{\psi(\theta^{\prime})}_{\sf C}.$ (31)

The map $\psi$ takes the group element $\theta$ of $\mathrm{U}(1)$ to the
state in the two-dimensional Hilbert space:

$\displaystyle\psi:\ \mathrm{U}(1)$ $\displaystyle\to\mathcal{H}$
$\displaystyle\theta$
$\displaystyle\mapsto\cos(\theta/2)\ket{0}+\sin(\theta/2)\ket{1}.$ (32)

We want this state to be mapped to the final state relative to ${\sf B}$:

$\ket{\psi(-\theta)}_{\sf A}\otimes\ket{\psi(0)}_{\sf
B}\otimes\ket{\psi(\theta^{\prime}-\theta)}_{\sf C}.$ (33)

This corresponds to our intuition of what should happen when one changes from
the viewpoint of ${\sf A}$ to the viewpoint of ${\sf B}$. Writing out this
transformation in the rebit basis corresponds to the map:

$\displaystyle\ket{0}_{\sf
A}\otimes(\cos(\theta/2)\ket{0}+\sin(\theta/2)\ket{1})_{\sf B}$
$\displaystyle\otimes(\cos(\theta^{\prime}/2)\ket{0}+\sin(\theta^{\prime}/2)\ket{1})_{\sf
C}$ $\displaystyle\mapsto$
$\displaystyle(\cos(-\theta/2)\ket{0}+\sin(-\theta/2)\ket{1})_{\sf
A}\otimes\ket{0}_{\sf B}$
$\displaystyle\otimes(\cos((\theta^{\prime}-\theta)/2)\ket{0}+\sin((\theta^{\prime}-\theta)/2)\ket{1})_{\sf
C}.$ (34)

On the other hand, considering the basis states of the joint Hilbert spaces
and assuming the map is linear, the following should hold:

$\displaystyle\ket{000}_{{\sf A}{\sf B}{\sf C}}\mapsto$
$\displaystyle\ket{000}_{{\sf A}{\sf B}{\sf C}}$ $\displaystyle\
\theta=\theta^{\prime}=0\ ,$ $\displaystyle\ket{001}_{{\sf A}{\sf B}{\sf
C}}\mapsto$ $\displaystyle\ket{001}_{{\sf A}{\sf B}{\sf C}}$ $\displaystyle\
\theta=0,\ \theta^{\prime}=\pi\ ,$ $\displaystyle\ket{010}_{{\sf A}{\sf B}{\sf
C}}\mapsto$ $\displaystyle\ket{101}_{{\sf A}{\sf B}{\sf C}}$ $\displaystyle\
\theta=\pi,\ \theta^{\prime}=0\ ,$ $\displaystyle\ket{011}_{{\sf A}{\sf B}{\sf
C}}\mapsto$ $\displaystyle-\ket{100}_{{\sf A}{\sf B}{\sf C}}$ $\displaystyle\
\theta=\theta^{\prime}=\pi.$ (35)

When comparing the coefficients in the map (6), one sees that the reference
frame change cannot be linear. This means that the operator describing the
change from one rebit reference system to another one is non-linear. As this
non-linearity causes issues concerning the invariance of probabilities under
reference frame change we conclude that rebits cannot serve as reference
frames that allow to reversibly transform between each other.

Given $n$ classical systems with configuration space $X\cong G$ acted on by a
symmetry group $G$ we have shown how to define states relative to these
systems, and to transform between them using the left and right regular action
of $G$ on $X$.

The case $L^{2}(G)$ is a very specific ‘encoding’ of $G$ into a quantum
system. It is a natural choice, in that the classical states are embedded into
orthogonal states of the quantum system. However one could have an injection
$G\to{\mathcal{H}}$, with $g\mapsto\ket{\psi(g)}$ such that the states
$\ket{\psi(g)}$ are not all mutually orthogonal and ask whether a change of
reference system can be defined. We require ${\mathcal{H}}$ to carry two
unitary representations $U_{L}$ and $U_{R}$, corresponding to active and
passive transformations, such that $U_{L}(h)\ket{\psi(g)}=\ket{\psi(hg)}$ and
$U_{R}(h)\ket{\psi(g)}=\ket{\psi(gh^{-1})}$, where
$\ket{\psi(g)}=\ket{\psi(h)}\leftrightarrow g=h$. Although one would naturally
desire them to commute (since active and passive transformations as usually
defined act on different spaces and therefore trivially commute), we do not
impose this here. The following theorem tells us that the change of reference
frame which acts as expected on product states
$\ket{\psi(e)}_{i}\ket{\psi(g_{0}^{i})}_{0}...\ket{\psi(g_{j}^{i})}_{j}...\ket{\psi(g_{n-1}^{i})}_{n-1}$
and obeys the principle of coherent change of reference frame is unitary
exactly if the states $\ket{\psi(g)}$ it acts on form an orthonormal basis of
the Hilbert space.

######  Theorem 1.

Take $n$ identical systems with associated Hilbert spaces ${\mathcal{H}}_{i}$
each carrying two representations of $G$: $U_{L}$ and $U_{R}$ such that
$U_{L}(h)\ket{\psi(g)}=\ket{\psi(hg)}$ and
$U_{R}(h)\ket{\psi(g)}=\ket{\psi(gh^{-1})}$, where
$\ket{\psi(g)}=\ket{\psi(h)}\leftrightarrow g=h$. Then any operator $U$ which
performs the change
$\ket{\psi(e)}_{i}\ket{\psi(g_{0}^{i})}_{0}...\ket{\psi(g_{j}^{i})}_{j}...\ket{\psi(g_{n-1}^{i})}_{n-1}\mapsto\ket{\psi(e)}_{j}\ket{\psi(g_{0}^{j})}_{0}...\ket{\psi(g_{i}^{j})}_{i}...\ket{\psi(g_{n-1}^{j})}_{n-1}$
and obeys the principle of coherent change of reference system is unitary if
and only if the representations $U_{L}$ and $U_{R}$ are the left and right
regular representations acting on states $\ket{\psi(g)}$ which form an
orthonormal basis of ${\mathcal{H}}_{i}$ (or a subspace thereof).

The proof can be found in Appendix C.

###  4.5 $m$ $L^{2}(G)$ systems describing $n-m$ systems

Let us consider the case where reference systems $L^{2}(G)$ describe systems
of a different type. The total Hilbert space is $L^{2}(G)^{\otimes
m}\otimes{\mathcal{H}}^{\otimes n-m}$ where for simplicity we have assumed the
$n-m$ systems to be of the same type (but not $L^{2}(G)$). The systems
${\mathcal{H}}$ are such that there exists an injection $\phi$:

$\displaystyle\phi:$ $\displaystyle\ G\to{\mathcal{H}}\ ,$ $\displaystyle\
g\mapsto\ket{\psi(g)}\ ,$ (36)

and two representations $V_{L}$ and $V_{R}$ such that:

$\displaystyle V_{L}(g)\ket{\psi(h)}$ $\displaystyle=\ket{\psi(gh)}\ ,$ (37)
$\displaystyle V_{R}(g)\ket{\psi(h)}$ $\displaystyle=\ket{\psi(hg^{-1})}.$
(38)

To change from reference system $0$ to $i$, where both systems are assumed to
be of the type $L^{2}(G)$, we apply the operator:

$\displaystyle U^{0\to i}={\rm SWAP}_{0,i}\circ$
$\displaystyle\int_{g^{0}_{i}\in
G}\outerproduct{g^{i}_{0}}{g_{i}^{0}}_{i}\otimes{\mathbb{1}}_{0}\otimes
U_{R}(g^{0}_{i})^{\otimes m-2}\otimes V_{R}(g^{0}_{i})^{\otimes
n-m}dg_{i}^{0},$ (39)

where $U_{R}$ is the right regular representation acting on the first $m$
$L^{2}(G)$ systems.

We observe that not all systems ${\mathcal{H}}$ which carry a representation
of $G$ will be such that there exists an injective map
$\phi:g\mapsto\ket{\psi(g)}$. For instance the qubit carries a representation
of $\mathrm{SU}(2)$ but there is no injection of $\phi:\mathrm{SU}(2)\to{\rm
P}{\mathbb{C}}^{2}$ (where here we emphasise that the pure states of a
${\mathbb{C}}^{2}$ system form ${\rm P}{\mathbb{C}}^{2}$ the projective space
of rays). Observe that for $\mathrm{U}(1)$ there is an injection
$\phi:\mathrm{U}(1)\to{\rm P}{\mathbb{C}}^{2}$. We explore the example of two
$L^{2}(\mathrm{U}(1))$ systems describing a system ${\mathbb{C}}^{2}$ carrying
a representation of $\mathrm{U}(1)$.

####  4.5.1 $L^{2}(\mathrm{U}(1))\otimes
L^{2}(\mathrm{U}(1))\otimes{\mathbb{C}}^{2}$

Let us adapt the previous example of three particles on a circle to the case
in which the third system is a qubit ${\mathcal{H}}_{\sf
C}\cong{\mathbb{C}}^{2}$ giving a total Hilbert space of the joint system
$L^{2}(\mathrm{U}(1))\otimes L^{2}(\mathrm{U}(1))\otimes{\mathbb{C}}^{2}$.
${\mathcal{H}}_{\sf C}$ carries a representation of $\mathrm{U}(1)$ and an
injection $\phi:\mathrm{U}(1)\to{\mathbb{C}}^{2}$. The representation $V_{R}$
is given by:

$\displaystyle
V_{R}(\theta)=\begin{pmatrix}\cos(-\theta/2)&-\sin(-\theta/2)\\\
\sin(-\theta/2)&\cos(-\theta/2)\end{pmatrix}$ (40)

in the $\\{\ket{0},\ket{1}\\}$ basis and acts by matrix multiplication from
the left. The injection is a map

$\displaystyle\phi\ :$ $\displaystyle\ \mathrm{U}(1)\to{\mathbb{C}}^{2}\ ,$
$\displaystyle\ \theta\mapsto\cos(\theta/2)\ket{0}+\sin(\theta/2)\ket{1}.$
(41)

The operator that maps the state relative to ${\sf A}$ to the state relative
to ${\sf B}$ is

$\displaystyle U^{{\sf A}\to{\sf B}}={\rm SWAP}_{{\sf A},{\sf B}}\circ\int
d\theta\outerproduct{-\theta}{\theta}_{\sf B}\otimes\mathbb{1}_{\sf A}\otimes
V_{R}(\theta)_{\sf C}.$ (42)

As a specific example, consider the state

$\ket{0}_{\sf A}\ket{\pi}_{\sf B}\ket{\psi(\frac{\pi}{2})}_{\sf C}$ (43)

relative to system ${\sf A}$, where
$\ket{\psi(\frac{\pi}{2})}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$. From the
viewpoint of system ${\sf B}$, the state is

$\ket{0}_{\sf B}\ket{\pi}_{\sf A}\ket{\psi(-\frac{\pi}{2}))}_{\sf
C}=\ket{0}_{\sf B}\ket{\pi}_{\sf A}\frac{1}{\sqrt{2}}(\ket{0}-\ket{1})_{\sf
C}.$ (44)

###  4.6 Changes of reference frame for arbitrary identical systems

The above treatment shows that for any group $G$ one can define a change of
quantum reference frame between $n$ identical systems. However given $n$
identical systems can one always find a group allowing for a reversible change
of quantum reference frame? Namely for a configuration space $X$, can one
always find a binary operation turning it into a group $G\cong X$? In the case
of finite systems ${\mathbb{C}}^{d}$ one can pick an orthonormal basis
$\ket{x}$, $x\in\\{0,...,d-1\\}$ and choose the cyclic group $\mathbb{Z}_{d}$
acting on $\\{0,...,d-1\\}$. In the case where $X$ is a countable set one has
the group $\mathbb{Z}$. In the case where $X$ is uncountable, the existence of
a group $G$ such that $G\cong X$ is equivalent to the axiom of choice [22]. We
observe that if $X$ has some additional structure (such as being a manifold),
then one may not be able to find a group which is isomorphic as a manifold.

##  5 Irreversible changes of classical reference frame

In some cases one may not have access to a reference system which can
distinguish all elements of the symmetry group. Consider once more the case of
particles on $\mathbb{R}$ acted on by the translation group. Given a ruler
with a set of marks corresponding only to the subset of integers (i.e. with
configuration space $X\cong\mathbb{Z}$), one would not be able to distinguish
all possible different configurations of the particles and by extension all
possible translations. Such imperfect reference frames, with configuration
space $X$ which is a coarse-graining of the group $G$, will lead to
irreversible changes of reference frame as we will see in this section.

For a given configuration space $X$ all changes of coordinates are related by
a transformation $g\in G$. In the case $X\cong G$ there is a one to one
correspondence between points in $X$ and coordinate systems. As such one can
identify coordinate systems as systems with configuration space $X$. Namely if
system $i$ is in state $x_{i}\in X$ then one assigns it the unique coordinate
system $x_{i}^{\prime}$ which maps $x_{i}\mapsto 0$. However in situations
such as the one described previously one has a symmetry group $G$ which is
larger than $X$ and there is no unique element in $G$ mapping a point $x_{i}$
to $0$. We consider an explicit example of this in the following before
describing the general case.

###  5.1 $E^{+}(3)\cong\mathbb{R}^{3}\rtimes\mathrm{SO}(3)$

Let us consider $n$ particles in $\mathbb{R}^{3}$, with each particle $i$
having state $(x_{i},y_{i},z_{i})$ expressed in Cartesian coordinates
$(x,y,z)$. The set of Cartesian coordinates is acted on by the Euclidean group
$E^{+}(3)=\mathbb{R}^{3}\rtimes\mathrm{SO}(3)$. A choice of coordinates
$(x^{\prime},y^{\prime},z^{\prime})$ such that
$(x_{i}^{\prime},y_{i}^{\prime},z_{i}^{\prime})=(0,0,0)$ is said to be
associated to particle $i$ if and only if it is the unique set of coordinates
with this property. There are infinitely many such coordinate choices (for
instance all coordinate systems which are rotated relative to
$(x^{\prime},y^{\prime},z^{\prime})$ will also assign state $(0,0,0)$ to
particle $i$). In this case there is no obvious unique manner of associating a
coordinate system to a particle.

All Cartesian coordinate systems for $\mathbb{R}^{3}$ are related by an
element $g\in E^{+}(3)$ where $E^{+}(3)$ is the Euclidean group. The action of
$E^{+}(3)$ on the set of Cartesian coordinates is transitive and free. To put
it visually every element in $E^{+}(3)$ can be considered as a translation
followed by a rotation. Every choice of Cartesian coordinates is associated to
a set of orthogonal axes located at some point $r\in\mathbb{R}^{3}$ with a
given orientation. These are all related to the Cartesian coordinates at
$(0,0,0)$ in a given orientation by a rotation followed by a translation. We
cannot assign a unique coordinate system to each point in $\mathbb{R}^{3}$.
For instance take a coordinate system centred at the origin; then any other
coordinate system obtained by rotation about the origin will also assign the
state $(0,0,0)$ to the origin. See Figure 5.

Figure 5: Reference frames associated to the same point in $\mathbb{R}^{3}$.

####  5.1.1 Enlarging the space of states of the reference systems

There are multiple ways of addressing this issue. One can say that systems
with configuration spaces $\mathbb{R}^{3}$ (i.e. particles) are not good
reference systems for $E^{+}(3)$. Rather one should choose systems with a
larger configuration space. This is what is typically done, where we choose
solid bodies in $\mathbb{R}^{3}$ as reference systems. Since solid bodies have
an orientation (unlike points), which is to say that rotating a solid body
changes its state, they have configuration space $E^{+}(3)$. One can assign a
unique coordinate system to every state $x\in X\cong E^{+}(3)$ of a solid
body. An example of a solid body would be three physical orthogonal axes in
$\mathbb{R}^{3}$ labelled $1$, $2$ and $3$. For a given state $x$ of these
three physical axes one can associate the coordinate system which assigns
$+x,+y$ and $+z$ to the axes $1$, $2$ and $3$. Using this approach would allow
us to make use of the results of the previous section.

However one could also keep the reference systems as having configuration
space $X$ but rather assign to each state $x$ the equivalence class of
coordinate systems centred on $x$. One can either choose a _representative
member_ of the equivalence class (in the above case one can fix all coordinate
systems to have a given orientation as in Figure 6) or one could average over
the possible elements of the equivalence class.

Figure 6: For each point in $\mathbb{R}^{3}$ a representative member of all
reference frames centred at the point is chosen. Here the representative
member is chosen so that each representative member has the same orientation.
This ensures that the closure of the set of transformations relating the
different reference frames is $T(\mathbb{R}^{3})$ and not a larger group.

####  5.1.2 Representative element of each equivalence class

In the case where all the reference systems have configuration space
$\mathbb{R}^{3}$, it makes sense to assign to each point $x\in\mathbb{R}^{3}$
a unique coordinate system centred on that point (from the equivalence class
of coordinate systems centred on that point).

Take $K$ to be the subset of transformations which relates these coordinate
systems. We require that $K$ is a group in order for us to have a well defined
change of coordinate system. If $K$ is not a group, then by composing
different elements in $K$ we can obtain a group $G^{\prime}$ (which is larger
than $K$ as a set). This symmetry group will take coordinate systems we have
selected to coordinate systems which we have not chosen.

In order for all representative members (i.e. the coordinate system we chose
to be associated to each point) to be related by a group $K\subset E^{+}(3)$
and for the representative members to be closed under the action of $K$ one
can choose them to all have the same orientation (i.e. be related by just
translations). In this case the symmetry group relating coordinate choices
becomes $K\cong\mathbb{R}^{3}$ once more.

###  5.2 $G=N\rtimes P$ and $G=N\times P$: truncation

Let us consider the case where there are systems with configuration space $G$
and systems with configuration space $N$, where $G=N\rtimes P$ or $G=N\times
P$. In both cases $N$ is normal, and for every $g\in G$ there is a unique
$n\in N$ and $p\in P$ such that $g=np$.

The configuration space $N$ is embedded in $G$ via an embedding map $E:N\to
G$, $E:n\mapsto np_{C}$ for some constant $p_{C}$, where the choice of $p_{C}$
is conventional and is typically chosen to be the identity. For a choice
$p_{C}$, the points $np_{C}$ $\forall n\in N$ are related by transformations
$n\in N$ (acting to the left). As such the symmetry group of $E(N)$ is $K\cong
N$. If the map did not fix a unique convention (for instance $np\mapsto
np_{0}(n)$) where the image depends on which equivalence class is chosen, then
the set $K$ of transformations between the images $E(n)$ would typically not
be a group, and its closure would not be isomorphic to $N$ (in some cases it
would be the full group $G$).

Take $k$ systems $G$ and $l-k$ systems $N$. A general state of the $l$ systems
is:

$\displaystyle s=\left((g_{0},g_{1},...,g_{k-1}),(g_{k},...,g_{l-1})\right)\
,$ (45)

where $g_{i}\in G$ for $i\in\\{0,\ldots,k-1\\}$ and $g_{j}\in N$ for
$j\in\\{k,\ldots,l-1\\}$. Moreover there is a unique $n_{i}\in N$ and
$p_{i}\in P$ such that $g_{i}=n_{i}p_{i}$. Here $p_{j}=e$ for systems
$j\in\\{k,\ldots,l-1\\}$. The description relative to the first $k$ systems
and the transformations between them is just the case described in Section 3.
In the following we describe how to change reference system from a system with
configuration space $G$ to a system with configuration space $N$.

The embedding of $N\subset G$ is given by $n\mapsto ne$. We define the
truncation map:

$\displaystyle{\rm T}:\ $ $\displaystyle G\to N\ ,$ $\displaystyle g=np\mapsto
n\ ,$ (46)

and the map ${\rm R}_{G}^{i}$:

$\displaystyle{\rm
R}_{G}^{i}:(g_{0},...,g_{l-1})\mapsto(g^{i}_{0},...,g^{i}_{l-1})\ .$ (47)

Then, the relative state $s^{0}={\rm R}_{G}^{0}(s)$ is:

$\displaystyle s^{0}$
$\displaystyle=\left((e,g_{1}^{0},...,g_{k-1}^{0}),(g_{k}^{0},...,g_{l-1}^{0})\right)$
$\displaystyle=\left((e,n_{1}^{0}p_{1}^{0},...,n_{k-1}^{0}p_{k-1}^{0}),(n_{k}^{0}p_{k}^{0},...,n_{l-1}^{0}p_{l-1}^{0})\right),$
(48)

where $n_{i}^{j}p_{i}^{j}=g_{i}^{j}$.

Let us consider the case of $j\in\\{k,\ldots,l-1\\}$, i.e. where
$g_{j}=n_{j}$. Then we have that $g_{j}^{0}=g_{j}g_{0}^{-1}=n_{j}g_{0}^{-1}$.
Now let us observe that for all $g$ there is a unique decomposition into
$g=np$, and so we write $g_{j}^{0}=n_{j}^{0}p_{j}^{0}$. We now work out this
$n_{j}^{0}p_{j}^{0}$ in terms of $n_{j}$ and $g_{0}=n_{0}p_{0}$ using the two
following equalities:

$\displaystyle g_{j}^{0}$
$\displaystyle=n_{j}g_{0}^{-1}=n_{j}p_{0}^{-1}n_{0}^{-1}\ ,$ $\displaystyle
g_{j}^{0}$ $\displaystyle=n_{j}^{0}p_{j}^{0}\ .$

Combining these gives:

$\displaystyle n_{j}p_{0}^{-1}n_{0}^{-1}=n_{j}^{0}p_{j}^{0}.$

Now let us introduce an identity $p_{0}p_{0}^{-1}$ on the RHS:

$\displaystyle n_{j}p_{0}^{-1}n_{0}^{-1}p_{0}p_{0}^{-1}=n_{j}^{0}p_{j}^{0}\ ,$

and observe that $gng^{-1}\in N$ for all $g\in G$, which implies that
$p_{0}^{-1}n_{0}^{-1}p_{0}\in N$, in turn implying that
$n_{j}p_{0}^{-1}n_{0}^{-1}p_{0}\in N$. Since the decomposition of $g_{j}^{0}$
into $n_{j}^{0}p_{j}^{0}$ is unique, this implies that
$n_{j}^{0}=n_{j}p_{0}^{-1}n_{0}^{-1}p_{0}$ and $p_{j}^{0}=p_{0}^{-1}$ for all
$j\in\\{k,\ldots,l-1\\}$. Therefore

$\displaystyle s^{0}$
$\displaystyle=\left((e,g_{1}^{0},...,g_{k-1}^{0}),(g_{k}^{0},...,g_{l-1}^{0})\right)$
$\displaystyle=\left((e,n_{1}^{0}p_{1}^{0},...,n_{k-1}^{0}p_{k-1}^{0}),(n_{k}^{0}p_{C},...,n_{l-1}^{0}p_{C})\right),$
(49)

where $p_{C}=p_{0}^{-1}$.

For particle $j$ with configuration space $N$, the state $s^{j}$ is:

$\displaystyle
s^{j}=\left((n^{j}_{0},n_{1}^{j},...,n_{k-1}^{j}),(n_{k}^{j},...,n_{l-1}^{j})\right),$
(50)

which is obtained from $s^{0}$ by the map $(\Gamma_{R}(n^{0}_{j})\circ{\rm
T})$. Here $\Gamma_{R}(n)$ is just shorthand for the right regular action:
$\Gamma_{R}(n)(g_{0},...,g_{l-1})=\phi_{R}(n,(g_{0},...,g_{l-1}))=(g_{0}n^{-1},...,g_{l-1}n^{-1})$.
We have $n_{i}^{0}n^{j}_{0}=n^{j}_{i}$.

We observe that all states $s^{0}$ of the form
$\left((n_{0}^{0}p_{0}^{0},n_{1}^{0}p_{1}^{0},...,n_{k-1}^{0}p_{k-1}^{0}),(n_{k}^{0}p_{k}^{0},...,n_{l-1}^{0}p_{l-1}^{0})\right)$
for all $p_{i}^{0}\in P$ give the same $s^{j}$. The change of reference frame
$s^{0}\mapsto s^{j}$ is an irreversible change of reference frame.

####  5.2.1 $\mathbb{R}\cong\mathbb{Z}\rtimes\mathrm{U}(1)$

An example of such a truncation is the ‘modular truncation’ of the translation
group: $\mathbb{R}\cong\mathbb{Z}\rtimes\rm{U}(1)$. Instead of distinguishing
all position states on the real line, we consider a reference frame that
essentially consists of a classical ruler. All points that lie within an
interval of length $L$ are mapped to the same point on the ruler (for
simplicity, one can choose $L=1$). Hence, the map identifies a subset of
elements of $G=\mathbb{R}$ with the same element of $N=\mathbb{Z}$:

$\displaystyle T:\ \mathbb{R}$ $\displaystyle\to\mathbb{Z}\ ,$ $\displaystyle
x$ $\displaystyle\mapsto n\ ,$ (51)

where $x=nL+p,\ p\in[0,L[,\
n=\left\lfloor{\frac{x}{L}}\right\rfloor\in\mathbb{Z}$. Physically, this
truncation can be viewed as coarse-graining; in the sense of resolution, this
means we transform from a finer to a coarser resolution.

Figure 7: Modular encoding of the real line.

Consider now three particles on the real line at positions $x_{1}^{1}$,
$x_{2}^{1}$ and $x_{3}^{1}$ relative to particle $1$, where particle $1$ and
$2$ have configuration space $\mathbb{R}$ and particle $3$ has configuration
space $\mathbb{Z}$. Let us write the state
$s^{1}=(0,n_{2}^{1}L+p_{2}^{1},n_{3}^{1}L+p_{3}^{1})$, then the state $s^{3}$
is obtained by $\Lambda^{1\to 3}\left(T(s^{1})\right)=\Lambda^{1\to
3}(0,n_{2}^{1}L,n_{3}^{1}L)=(-n_{3}^{1}L,(n_{2}^{1}-n_{3}^{1})L,0)$. If for
instance $T(x_{1}^{1})=T(x_{2}^{1})$, i.e. $n_{1}^{1}=n_{2}^{1}$, particle $3$
assigns the same state to particles $1$ and $2$.

####  5.2.2 $\mathbb{R}^{3}\cong\mathbb{R}^{2}\times\mathbb{R}$

Another example is the truncation map $\mathbb{R}^{3}\to\mathbb{R}^{2}$. If we
have a symmetry group $\mathbb{R}^{2}$ in a three-dimensional space, all
reference frames along a one-dimensional line are identified with each other.
In this case, when applying the truncation map, all group elements in
$\mathbb{R}^{3}$ are projected to the associated group elements in
$\mathbb{R}^{2}$. Consider three systems of which the first two have
configuration space $\mathbb{R}^{3}$ and the last has configuration space
$\mathbb{R}^{2}$. A general state of the systems is $s=(g_{1},g_{2},g_{3})$
where $g_{1},g_{2}\in\mathbb{R}^{3}$ and $g_{3}\in\mathbb{R}^{2}$. Relative to
particle $1$, the state is
$s^{1}=(e,g_{2}^{1},g_{3}^{1})=(e,n_{2}^{1}p_{2}^{1},n_{3}^{1}p_{C})$ where
$p_{2}^{1},p_{C}\in\mathbb{R}$ and $n_{2}^{1},n_{3}^{1}\in\mathbb{R}^{2}$.
First, we truncate the state: $\tilde{s}=(e=n_{1}^{1},n_{2}^{1},n_{3}^{1})$.
Then, we change to the state relative to system $3$:
$s^{3}=(n_{1}^{3},n_{2}^{3},e)$. This can be understood as projecting the
points in $\mathbb{R}^{3}$ to a plane in $\mathbb{R}^{2}$. If all reference
frames along the $z$-axis are identified with each other, this corresponds to
projecting on the $x$-$y-$plane.

###  5.3 Inconsistency using the truncation method for $G=N\rtimes P$

In Appendix D we prove that the state $s^{i}$ for a system $i$ with
configuration space $N$ obtained from state $s^{0}$ by truncating and changing
reference system is not equivalent to the state $\tilde{s}^{i}$ obtained from
the state $s$ by first truncating to obtain $\tilde{s}$, then finding
$\tilde{s}^{0}$ and then changing reference system to obtain $\tilde{s}^{i}$.
Let us write the change of reference system $0\to i$ for a group $G$ as
$\Lambda_{G}^{0\to i}s^{0}=s^{i}$:

$\displaystyle\Lambda_{G}^{0\to
i}(g^{0}_{0},...,g_{l-1}^{0})=\Gamma_{R}(g^{0}_{i})(g^{0}_{0},...,g_{l-1}^{0})\
,$ (52)

where $\Gamma_{R}(g^{0}_{i})s^{0}$ is $\phi_{R}(g^{0}_{i},s^{0})$, i.e. the
right regular action of $g^{0}_{i}$ on $s^{0}$.

######  Theorem 2.

Let $s=\left((g_{0},g_{1},...,g_{k-1}),(g_{k},...,g_{l-1})\right)$ for
$G=N\rtimes P$ with $g_{j}=n_{j}$ for $j\in\\{k,\ldots,l-1\\}$ and
$g_{i}=n_{i}p_{i}$ otherwise. Then $\Lambda_{N}^{0\to i}\left(T({\rm
R}_{G}^{0}(s))\right)\neq\Lambda_{N}^{0\to i}\left({\rm
R}_{N}^{0}(T(s))\right)$. Let $\Lambda_{N}^{0\to i}\left(T({\rm
R}_{G}^{0}(s))\right)=\left((n_{0},n_{1},...,n_{k-1}),(n_{k},...,n_{l-1})\right)$
and $\Lambda_{N}^{0\to i}\left({\rm
R}_{N}^{0}(T(s))\right)=\left((m_{0},m_{1},...,m_{k-1}),(m_{k},...,m_{l-1})\right)$,
then $n_{j}=m_{j}$ for $j\in\\{k,\ldots,l-1\\}$. For $j\in\\{0,\ldots,k-1\\}$
this is not always the case.

The above theorem shows that depending on our prior commitment to a well
defined state $s$ and our interpretation of relative states $s^{i}$, the
truncation method of obtaining relative states may not be desirable. One may
prefer an averaging procedure or a method based on finding invariants. These
two approaches are described in Appendix D using the example of
$\mathrm{SO}(3)\rtimes T(\mathbb{R}^{3})$. We make use of the truncating
procedure in the present work since it can easily be extended to a quantum
version using the principle of coherent change of reference system. We
describe this quantum generalisation in the next section.

##  6 Irreversible changes of quantum reference frame

The irreversible change of reference frame of the previous section can be
extended to the quantum case by applying the principle of _coherent change of
reference frame_ , as in the reversible case. We describe this in more detail
in the following. We note that we only define this change of reference frame
for the specific cases $G=N\rtimes P$ and $G=N\times P$.

###  6.1 $G=N\rtimes P$ and $G=N\times P$

Let us consider the change of reference frame from system $i$ with Hilbert
space $L^{2}(G)$ to a system $j$ with Hilbert space $L^{2}(N)$. Let us
consider the first $k$ systems as being $L^{2}(G)$ and the last $l-k$ systems
as being $L^{2}(N)$.

A generic product basis state
$\ket{\psi^{0}}=\ket{e}_{0}\ket{n^{0}_{1}p^{0}_{1}}_{1}...\ket{n^{0}_{k-1}p^{0}_{k-1}}_{k-1}\ket{n^{0}_{k}p^{0}_{k}}_{k}...\ket{n^{0}_{l-1}p^{0}_{l-1}}_{l-1}$
(where $p_{j}^{0}=e$ for $j\in\\{k,\dots,l-1\\}$) maps to
$\ket{\psi^{i}}=\ket{e}_{i}\ket{n^{i}_{0}}_{0}\ket{n^{i}_{1}}_{1}...\ket{n^{i}_{l-1}}_{l-1}$
for $i\in\\{k,\dots,l-1\\}$. Applying the _coherent change of reference
system_ we have that the action on a superposition state:

$\displaystyle\ket{\psi^{0}}=\ket{e}_{0}\bigotimes_{r\neq
0}\ket{n^{0}_{r}p^{0}_{r}}_{r}$ $\displaystyle+\ket{e}_{0}\bigotimes_{s\neq
0}\ket{m^{0}_{s}q^{0}_{s}}_{s}\ ,$ (53)

with $m^{0}_{s}\in N$, $q^{0}_{s}\in P$ and $p^{0}_{r}=p^{0}_{s}$ for all
$r,s\in\\{k,...,l-1\\}$ (and similarly for $q^{0}_{s}$) maps to:

$\displaystyle\ket{\psi^{i}}$ $\displaystyle=\ket{e}_{i}\bigotimes_{r\neq
i}\ket{n^{0}_{r}n^{i}_{0}}_{r}+\ket{e}_{i}\bigotimes_{s\neq
i}\ket{m^{0}_{s}m^{i}_{0}}_{s}\ .$ (54)

First define the truncation map ${\rm T}:L^{2}(G)\to L^{2}(N)$.

$\displaystyle{\rm T}=\int_{n\in N}\int_{p\in P}\outerproduct{n}{np}dpdn\ ,$
(55)

and the map:

$\displaystyle U^{i\to j}_{N}=$ $\displaystyle{\rm SWAP}_{i,j}\circ$
$\displaystyle\int_{n^{i}_{j}\in
N}\outerproduct{n^{j}_{i}}{n_{j}^{i}}_{j}\otimes{\mathbb{1}}_{i}\otimes
U_{R}(n^{i}_{j})^{\otimes l-2}dn_{j}^{i}.$ (56)

The change of reference frame is given by:

$\displaystyle V^{i\to j}=U^{i\to j}_{N}\circ\left({\rm T}^{\otimes
k}\otimes{\mathbb{1}}^{\otimes l-k}\right).$ (57)

####  6.1.1 $\mathbb{R}\cong\mathbb{Z}\rtimes\mathrm{U}(1)$

Consider again the example of the modular truncation of the translation group:
$\mathbb{R}\cong\mathbb{Z}\rtimes\mathrm{U}(1)$. Take three (quantum) systems
${\sf A}$, ${\sf B}$ and ${\sf C}$ where ${\sf A}$ has configuration space
$\mathbb{R}$ but states relative to ${\sf C}$ are encoded in $L\mathbb{Z}$.
Changing reference frame from ${\sf A}$ to ${\sf C}$ requires

$\displaystyle\ket{0}_{\sf A}\ket{x_{0}}_{\sf B}\ket{x_{1}}_{\sf C}$
$\displaystyle=\ket{0}_{\sf A}\ket{n_{0}L+p_{0}}_{\sf B}\ket{n_{1}L}_{C}$
$\displaystyle\mapsto\ket{0}_{\sf C}\ket{-n_{1}L}_{\sf
A}\ket{(n_{0}-n_{1})L}_{\sf B}.$ (58)

The truncation operator is:

$\displaystyle{\rm
T}=\sum_{n\in\mathbb{Z}}\int_{p\in\mathrm{U}(1)}dp\outerproduct{nL}{nL+p},$
(59)

and the change of reference frame operator for $L^{2}(N)$ is:

$\displaystyle U^{{\sf A}\to{\sf C}}_{N}={\rm SWAP}_{{\sf A},{\sf
C}}\circ\int_{n\in N}\outerproduct{n^{-1}}{n}_{\sf C}\otimes{\mathbb{1}}_{\sf
A}\otimes U_{R}(n)_{\sf B}dn.$ (60)

The following operator performs the change ${\sf A}\to{\sf C}$:

$\displaystyle V^{{\sf A}\to{\sf C}}=U^{{\sf A}\to{\sf
C}}_{N}\circ({\mathbb{1}}_{\sf A}\otimes{\rm T}_{\sf B}\otimes{\rm T}_{\sf
C}).$ (61)

When applying the change of reference frame from ${\sf A}$ to ${\sf C}$ to a
state in which system ${\sf C}$ is in a superposition state relative to ${\sf
A}$, the state becomes entangled relative to ${\sf C}$:

$\displaystyle V^{{\sf A}\to{\sf C}}\ket{0}_{\sf A}\ket{x_{0}}_{\sf
B}\left(\ket{x_{1}}_{\sf C}+\ket{x_{1}^{\prime}}_{\sf C}\right)$
$\displaystyle=$ $\displaystyle V^{{\sf A}\to{\sf C}}\ket{0}_{\sf
A}\ket{n_{0}L+p_{0}}_{\sf B}\left(\ket{n_{1}L}_{\sf
C}+\ket{n_{1}^{\prime}L}_{\sf C}\right)$ $\displaystyle=$
$\displaystyle\ket{0}_{\sf C}\left(\ket{-n_{1}L}_{\sf
A}\ket{(n_{0}-n_{1})L}_{\sf B}+\ket{-n_{1}^{\prime}L}_{\sf
A}\ket{(n_{0}-n_{1}^{\prime})L}_{\sf B}\right).$ (62)

However, if $x_{1}$ and $x_{1}^{\prime}$ are located in the same interval
relative to ${\sf C}$ (with configuration space $L\mathbb{Z}$), the
entanglement vanishes. Here, we recognize the dependence of entanglement on
the reference frame relative to which it is described. This was already
pointed out in [6]. Note that we observe entanglement only if the uncertainty
in the position of ${\sf C}$ with respect to ${\sf A}$ (i.e. the difference
between $x_{1}$ and $x^{\prime}_{1}$) is larger than the resolution $L$ of the
configuration space of reference system ${\sf C}$.

####  6.1.2 $\mathbb{R}^{3}\cong\mathbb{R}^{2}\times\mathbb{R}$

As mentioned before, the truncation map $\mathbb{R}^{3}\to\mathbb{R}^{2}$
essentially consists of a projection from a three-dimensional configuration
space to a two-dimensional one. Here, we project all points in
$\mathbb{R}^{3}$ onto the $x$-$y$-plane.

$\displaystyle\mathbb{R}^{3}$ $\displaystyle\to\mathbb{R}^{2}$
$\displaystyle\vec{P}\coloneqq(x,y,z)$
$\displaystyle\mapsto\vec{p}\coloneqq(x,y)\ .$ (63)

On the level of Hilbert spaces, we assign the state $\ket{\vec{P}}$ relative
to a three-dimensional configuration space while $\ket{\vec{p}}$ denotes a
state in $\mathcal{H}\cong\mathbb{R}^{2}$. Hence, to change from the state
relative to ${\sf A}$ with configuration space $\mathbb{R}^{3}$ to the state
relative to ${\sf C}$ with configuration space $\mathbb{R}^{2}$, we apply the
operator

$\displaystyle V^{{\sf A}\to{\sf C}}=\textrm{SWAP}_{{\sf A},{\sf C}}\circ\iint
d\vec{P}d\vec{Q}\outerproduct{-\vec{p}}{\vec{P}}_{\sf
C}\otimes\outerproduct{\vec{q}-\vec{p}}{\vec{Q}}_{\sf B},$ (64)

where
$\vec{Q}\coloneqq(x^{\prime},y^{\prime},z^{\prime})\mapsto\vec{q}\coloneqq(x^{\prime},y^{\prime})$.

##  7 Wigner’s friend experiment

In this section we consider the Wigner’s friend experiment, introduced by E.
Wigner [23] in 1961. The thought experiment consists of two observers, Wigner
and his friend, and a two-level quantum system. While Wigner is an external
observer of the experiment, his friend is located inside an isolated box,
together with the system ${\sf S}$ in the quantum state $\ket{\psi}$. Once the
experiment is initiated, the friend measures the system in a certain basis.
According to the projection postulate of standard quantum theory, the state of
the system collapses to one of the eigenstates of the measurement operator. On
the other hand, Wigner describes this process from the outside and would
assign a unitary evolution. After the measurement of the system by the friend,
the outcome of which Wigner does not know, Wigner would assign an entangled
state to the joint system of the friend and the system. These two seemingly
contradicting prescriptions of standard quantum mechanics are at the core of
the so-called Wigner’s friend paradox.

Before proceeding we observe that there is no logical contradiction in the
above two descriptions. The state after applying the projection postulate is a
state of the system ${\sf S}$ alone, of the form $\ket{\psi}_{\sf S}$. The
entangled state assigned by Wigner is a state on ${\sf S}+{\sf F}$ (where
${\sf F}$ is the friend) of the form $\ket{\phi}_{{\sf S}{\sf F}}$. Since
these are states of two different objects (${\sf S}$ versus ${\sf S}+{\sf F}$)
there is no logical contradiction within the postulates of quantum theory.

We will now apply the relational formalism introduced previously to this
apparent paradox. Let us consider ${\sf W}$, short for Wigner, describing his
friend ${\sf F}$ who measures the system ${\sf S}$ in the
$\\{\ket{\uparrow},\ket{\downarrow}\\}$ basis. The friend is located inside a
perfectly isolated box. We model everything using two-dimensional systems,
since we are interested in two degrees of freedom alone. The change of
reference frame will therefore be for $\mathbb{Z}_{2}$, first described in
Example 2. We consider the ready state of the friend to just be
$\ket{\uparrow}$ and the state of the system before the measurement to be
$\ket{\psi}=\alpha\ket{\uparrow}+\beta\ket{\downarrow}$. We first describe the
measurement interaction from the point of view of ${\sf W}$ starting in the
case where ${\sf S}$ is in an eigenstate of the measurement operator:

$\displaystyle\ket{\uparrow}_{\sf F}^{\sf W}\ket{\uparrow}_{\sf S}^{\sf W}$
$\displaystyle\mapsto\ket{\uparrow}_{\sf F}^{\sf W}\ket{\uparrow}_{\sf S}^{\sf
W}\ ,$ (65) $\displaystyle\ket{\uparrow}_{\sf F}^{\sf W}\ket{\downarrow}_{\sf
S}^{\sf W}$ $\displaystyle\mapsto\ket{\downarrow}_{\sf F}^{\sf
W}\ket{\downarrow}_{\sf S}^{\sf W}\ .$ (66)

The state of the friend depends on the measurement outcome, hence the record
of the outcome can be seen as being stored in the state of the friend. The
state $\ket{\uparrow}_{\sf F}$ is the state ‘the friend sees up’, and
similarly for $\ket{\downarrow}_{\sf F}$ and ‘the friend sees down’.

The change of reference frame ${\sf W}\mapsto{\sf F}$ for the final states
gives:

$\displaystyle U^{{\sf W}\to{\sf F}}\ket{\uparrow}_{\sf F}^{\sf
W}\ket{\uparrow}_{\sf S}^{\sf W}=\ket{\uparrow}_{\sf W}^{\sf
F}\ket{\uparrow}_{\sf S}^{\sf F}\ ,$ (67) $\displaystyle U^{{\sf W}\to{\sf
F}}\ket{\downarrow}_{\sf F}^{\sf W}\ket{\downarrow}_{\sf S}^{\sf
W}=\ket{\downarrow}_{\sf W}^{\sf F}\ket{\uparrow}_{\sf S}^{\sf F}\ ,$ (68)

where we observe that in both cases the state of ${\sf S}$ relative to ${\sf
F}$ is $\ket{\uparrow}_{\sf S}^{\sf F}$. This state encodes the fact that the
friend and the system are perfectly correlated in both cases;
$\ket{\uparrow}_{\sf S}^{\sf F}$ tells us that the state of the friend and the
system are related by the identity element. For an arbitrary superposition
state of the system the unitary measurement interaction gives the following
evolution:

$\ket{\uparrow}_{\sf F}^{{\sf W}}(\alpha\ket{\uparrow}_{\sf S}^{\sf
W}+\beta\ket{\downarrow}_{\sf S}^{\sf W})\mapsto\alpha\ket{\uparrow}_{\sf
F}^{{\sf W}}\ket{\uparrow}_{\sf S}^{{\sf W}}+\beta\ket{\downarrow}_{\sf
F}^{{\sf W}}\ket{\downarrow}_{\sf S}^{{\sf W}}\ .$ (69)

Now, we want to apply the change of reference frame to switch to the
perspective of ${\sf F}$. If we apply the operator $U^{{\sf W}\to{\sf F}}$ to
the initial state, using the principle of _coherent change of reference
system_ , we obtain:

$U^{{\sf W}\to{\sf F}}\ket{\uparrow}_{\sf F}^{{\sf
W}}(\alpha\ket{\uparrow}_{\sf S}^{\sf W}+\beta\ket{\downarrow}_{\sf S}^{\sf
W})=\ket{\uparrow}_{\sf W}^{\sf F}(\alpha\ket{\uparrow}_{\sf S}^{\sf
F}+\beta\ket{\downarrow}_{\sf S}^{\sf F})\ .$ (70)

If we apply the change of reference frame $U^{{\sf W}\to{\sf F}}$ to the final
state we get:

$\displaystyle U^{{\sf W}\to{\sf F}}\alpha\ket{\uparrow}_{\sf F}^{{\sf
W}}\ket{\uparrow}_{\sf S}^{{\sf W}}+\beta\ket{\downarrow}_{\sf F}^{{\sf
W}}\ket{\downarrow}_{\sf S}^{{\sf W}}=(\alpha\ket{\uparrow}_{\sf W}^{{\sf
F}}+\beta\ket{\downarrow}_{\sf W}^{{\sf F}})\ket{\uparrow}_{\sf S}^{\sf F}.$
(71)

If we interpret this result as the state seen by the friend, this would imply
that the friend always sees the system as correlated with herself. This is
consistent with the initial description from Wigner’s perspective, where the
state of the friend and system are correlated in both terms of the entangled
state $\alpha\ket{\uparrow}_{\sf F}^{{\sf W}}\ket{\uparrow}_{\sf S}^{{\sf
W}}+\beta\ket{\downarrow}_{\sf F}^{{\sf W}}\ket{\downarrow}_{\sf S}^{{\sf
W}}$.

Now, we want to compare this result to the state we get if we simply start
from the perspective of the friend. She describes the initial state as:

$\ket{\uparrow}_{\sf W}^{\sf F}(\alpha\ket{\uparrow}_{\sf S}^{\sf
F}+\beta\ket{\downarrow}_{\sf S}^{\sf F})\ ,$ (72)

where Wigner is in the ready state. Applying the projection postulate to the
system gives:

$\displaystyle p_{\uparrow}$ $\displaystyle=|\alpha|^{2}:\ket{\uparrow}_{\sf
W}^{\sf F}\ket{\uparrow}_{\sf S}^{\sf F}\ ,$ (73) $\displaystyle
p_{\downarrow}$ $\displaystyle=|\beta|^{2}:\ket{\uparrow}_{\sf W}^{\sf
F}\ket{\downarrow}_{\sf S}^{\sf F}\ .$ (74)

Whatever outcome the friend observes, she would always describe the system
definitely being in either one of the basis states and Wigner in the state
$\ket{\uparrow}_{\sf W}^{\sf F}$. This seems very different to the description
we obtain by taking Wigner’s perspective and changing to the friend’s
reference frame. Note however that this does not necessarily imply a
contradiction. More precisely, one should be careful when interpreting the
state in Equation (71) as the final state seen by the friend. Rather, one
should interpret it as the state Wigner _infers_ (concludes) the friend would
see. In fact, starting from the state from Wigner’s perspective, we only take
into account the information that Wigner has at his disposal. Simply changing
reference frames by applying a unitary operator can by no means introduce new
information, such as the actual outcome observed by the friend.

Hence, it is not a trivial assumption that the state actually seen by the
friend should be the same as the state of the system relative to the friend,
obtained by changing perspective from Wigner’s viewpoint. In fact, this is
assumption $\mathcal{C}$ (consistency assumption) of the Frauchiger-Renner no-
go theorem [24]. In the framework of relational quantum mechanics, one should
not assume the consistency condition to hold _a priori_. Indeed in this work
we show that what Wigner infers about the friend’s state assignments is given
by the change of reference frame $U^{{\sf W}\to{\sf F}}$ and not the seemingly
straightforward assumption $\mathcal{C}$.

We observe that the conclusion that what Wigner can infer about the state of
the system relative to the friend is that they are correlated is the same as
Rovelli’s treatment of the measurement process in relational quantum mechanics
[2]. Here instead of reasoning using Wigner’s measurement operators we used an
explicit change of reference frame from Wigner to the friend.

One may think that introducing an additional reference system ${\sf R}$ into
the box containing ${\sf F}$ and ${\sf S}$ could help resolve the issue with
Wigner’s friend. We show in Appendix E that this is not the case.

##  8 Discussion

###  8.1 Related work

The underlying approach of this work is based on the relational quantum
mechanics of [2]. Other work formulating a fully relational quantum theory
include [3, 4, 5, 6, 7, 14, 15], the toy model of [25] and the systematic
treatment of quantum reference frames in [26, 11]. The main difference between
[26, 11] and the present treatment is that we begin from an explicitly
relational state and emphasise the notion of changing between reference frames
as opposed to deriving relational states from an external non-relational
state. Related approaches to relational quantum mechanics include the
perspectival quantum mechanics of [16] and the Ithaca interpretation [27].

The notion of quantum reference frames first appeared as part of the debate on
the existence of charge superselection rules [28]. Later, Aharonov and
Kaufherr gave the first explicit study of quantum reference frames [8].
Typically a description of a quantum system is given relative to a classical
measuring device of infinite mass. It was shown that there is a consistent
description of quantum systems relative to quantum reference frames of finite
mass. This addresses the issue of _universality_ of quantum theory; namely
quantum systems are usually described relative to an implicit classical
reference frame. If quantum theory is universal, then one would expect that
reference frames should correspond to quantum systems.

Given a quantum reference frame ${\sf R}$ and a quantum system ${\sf S}$ the
relational observables are observables of ${\sf R}+{\sf S}$ which are
invariant under the symmetry group. Depending on the state of ${\sf R}$ one
may recover a relational description of ${\sf S}$ which is equivalent to the
standard absolute description of ${\sf S}$ (i.e. relative to an implicit
classical reference frame). Equivalently, for any absolute description of the
state or the observables on ${\sf S}$, one can find a system ${\sf R}$ such
that there is a gauge invariant description on ${\sf S}+{\sf R}$. This
observation was important for resolving the ‘optical coherence controversy’
[29]. Explicit maps between the absolute and relativised descriptions can be
found in [9, 11], where an emphasis on the localisation of the reference
system is placed in [26, 11]. Other works discussing relational observables
include [3, 4, 5, 14, 15].

In [30, 10] a description of quantum systems relative to other quantum systems
is given, in particular for the translation group. The initial description of
${\sf R}$ and ${\sf S}$ is given relative to an external classical reference
frame (in the position basis), and the description relative to ${\sf R}$ is
obtained by refactoring into center of mass and relative position. Tracing out
the center of mass partition then allows to remove all global degrees of
freedom and leaves us with a relational description of the systems. Moreover
the description of systems is shown to be dependent on the mass of the quantum
reference system. Note that such global degrees of freedom never enter into
our formalism in the first place. Instead, we give the states of systems
relative to a specified system from the outset.

The change of reference frame in this work generalises the known changes of
reference frame $L^{2}(\mathbb{R})$ and $L^{2}(\mathbb{R}^{3})$ for spatial
position of [6, 14], and $L^{2}(\mathbb{R})$ for temporal degrees of freedom
of [4, 5, 7, 15]. Reference frames for rotational degrees of freedom are
studied in the perspective-neutral approach in [14]. Brukner and Mikusch are
independently working on a treatment of rotational degrees of freedom of
quantum reference frames using large spin coherent states.

We observe that in both [6] and the present work the consistency of the fully
relational account with a description which is initially given externally (and
from which the relative description is then obtained) is not proven. However
in [3] it is shown that there is a ‘perspective-neutral’ framework which
encompasses all perspectives for [6]. Note that this framework does not
describe an external structure as it only encodes relative information.
Extending this perspective-neutral approach to the general cases in the
present paper could constitute an interesting direction for future work.

In [31] an emphasis is placed on the notion that a reference system in a
superposition relative to another system gives new coordinates, which are not
related by a classical coordinate transformation to the initial ones. This is
conceptually closer to the perspective in the present work than that of [6].
Namely we do not make the claim that the relative descriptions are
‘operational’ as in [6].

Prior work with an emphasis on changing perspectives between quantum reference
frames is found in [32] where this change of reference frame is mediated by an
external description, and uses tools such as $G$-twirling.

In recent years there has been a significant interest in the study of quantum
reference frames as resources for measurements, communication tasks and
thermodynamic exchanges amongst others [33, 34, 35, 36, 37, 38, 39]. This
approach is not motivated by the same considerations as the present paper and
the link between the two approaches is not fully clear to the authors. It is
possible that the present exposition, carried out explicitly in a group
theoretic language, could be used to relate the two. For instance the ‘perfect
reference frames’ of [9] are of the form $L^{2}(G)$, which also plays a
prominent role here. The link between imperfect reference frames as standardly
defined [9, 36, 38] and the truncation based approach of the present work also
remains to be worked out.

###  8.2 General comments

####  8.2.1 Relational quantum mechanics

We have provided a formalism for relational quantum mechanics which captures
its spirit and recovers the same conclusions for Wigner’s friend and the
Frauchiger-Renner theorem. Future work could involve applying the formalism
developed in this work to other thought experiments, in order to provide an
explicit account of how relational quantum mechanics addresses various
apparent paradoxes.

####  8.2.2 Observer dependence of the symmetry group

One further contribution of this work is that symmetries are not only relative
to the system being described, but also to the system doing the describing. A
system with Hilbert space ${\mathcal{H}}$ has a symmetry group $G$ if it
carries a representation of $G$, however it can only be described as
transforming under $G$ by a system which carries a regular representation of
$G$. For example although a qubit carries a representation of
$\mathrm{SU}(2)$, two qubits ‘describing’ each other and a third qubit would
use the subgroup $\mathbb{Z}_{2}$, since they can only carry a regular
representation of $\mathbb{Z}_{2}$ and not the full $\mathrm{SU}(2)$. This is
reminiscent of the argument by Penrose in [40].

####  8.2.3 $G=N\rtimes P$ and $G=N\times P$

For the imperfect reference frames we describe a very specific case, though it
seems natural since it is the structure of well known space time groups. Group
extension tells us that for any group subgroup pair $(G,N)$ with $N$ normal,
the extension is of the form $G=N\rtimes P$ or $G=N\times P$. Hence our
approach fully covers imperfect reference frames for group subgroup pairs
$(G,N)$ with $N$ normal. However for a full account of imperfect reference
frames one would need to relax the assumption of $N$ normal and see whether
one can define meaningful changes of reference frame.

##  9 Conclusion

The construction of a relational quantum theory requires a description of
systems and states relative to other systems. In this work, we present a
formalism that allows to describe the relative states of systems and to change
between the descriptions of different reference systems. Starting from the
analysis of reference frames in the classical realm, we moved on to the
description of quantum states relative to quantum reference frames. Depending
on the system whose viewpoint is adopted, the description of physical
phenomena changes. We find that quantum properties such as entanglement and
superposition are not absolute but depend on the reference frame relative to
which they are described. As such the conclusions of [6] are shown to be
generic to quantum reference frames for arbitrary groups.

Previous work on quantum reference frames in the relational paradigm
restricted itself to the translation group (for time, space and momentum
translations) and the Euclidean group in three dimensions (including
rotations) [14]. Here we have extended this to arbitrary groups, including
finite groups and non-Abelian groups. Moreover we have highlighted the key
representation theoretic features needed for reversible changes of quantum
reference frame of a group $G$, namely that the quantum reference systems have
Hilbert space $L^{2}(G)$. Using this insight we have developed a more abstract
formulation directly in terms of the relevant symmetry group.

We also extended the study of reference frames in the relational approach to
imperfect reference frames using a ‘truncation’ approach, in contrast to group
averaging approaches more common in quantum information based approaches.
Future work involves exploring imperfect reference frames for different cases
than $(G,N)$ with $N$ a normal subgroup. One benefit of our approach is that
it formulates the relational approach of [6] in a group theoretic language
which is more common in the quantum information literature. As such it may
help in developing a fully rigorous account of the link between the two.
Recent work [15] in this direction has been exploring links between different
approaches in the case of the translation group and would be useful to pursue
in this more general case.

Our exploration of the Wigner’s friend scenario shows how to apply our
relational framework to an important discussion point in the literature. Using
our approach we reach the same conclusion as Rovelli in his relational
treatment of the measurement process in [2]: all that can be said by Wigner is
that the friend is correlated with the measurement outcome. The novel aspect
of this work is the use of an explicitly relational formalism (which embodies
the philosophical position of relational quantum mechanics) as well as
obtaining the friend’s perspective through an explicit change of reference
frame, as opposed to reasoning indirectly using measurement operators.

The authors thank Časlav Brukner, Esteban Castro-Ruiz, Flaminia Giacomini,
Matt Leifer, Leon Loveridge, Pierre Martin-Dussaud, Carlo Rovelli, David
Schmid and Rob Spekkens for helpful discussions. The authors thank Philipp
Höhn, Leon Loveridge and Markus Müller for helpful comments on a draft of this
paper. This research was supported by Perimeter Institute for Theoretical
Physics. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.

## References

  * Rovelli [1991] C. Rovelli, Quantum reference systems, Classical and Quantum Gravity 8, 317–331 (1991).
  * Rovelli [1996] C. Rovelli, Relational quantum mechanics, International Journal of Theoretical Physics 35, 1637–1678 (1996).
  * Vanrietvelde  _et al._ [2020] A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz, A change of perspective: switching quantum reference frames via a perspective-neutral framework, Quantum 4, 225 (2020).
  * Hoehn and Vanrietvelde [2020] P. A. Hoehn and A. Vanrietvelde, How to switch between relational quantum clocks (2020), arXiv:1810.04153 [gr-qc] .
  * Höhn [2019] P. A. Höhn, Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’, Universe 5, 116 (2019).
  * Giacomini  _et al._ [2019] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames, Nature Communications 10, 494 (2019).
  * Castro-Ruiz  _et al._ [2020] E. Castro-Ruiz, F. Giacomini, A. Belenchia, and Č. Brukner, Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems, Nature Communications 11, 2672 (2020).
  * Aharonov and Kaufherr [1984] Y. Aharonov and T. Kaufherr, Quantum frames of reference, Physical Review D 30, 368–385 (1984).
  * Bartlett  _et al._ [2007] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference frames, superselection rules, and quantum information, Reviews of Modern Physics 79, 555–609 (2007).
  * Angelo and Ribeiro [2012] R. M. Angelo and A. D. Ribeiro, Kinematics and dynamics in noninertial quantum frames of reference, Journal of Physics A: Mathematical and Theoretical 45, 465306 (2012).
  * Loveridge  _et al._ [2018] L. Loveridge, T. Miyadera, and P. Busch, Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics, Foundations of Physics 48, 135–198 (2018).
  * Pereira and Angelo [2015] S. T. Pereira and R. M. Angelo, Galilei covariance and Einstein’s equivalence principle in quantum reference frames, Phys. Rev. A 91, 022107 (2015).
  * Smith  _et al._ [2016] A. R. H. Smith, M. Piani, and R. B. Mann, Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass, Physical Review A 94, 012333 (2016).
  * Vanrietvelde  _et al._ [2018] A. Vanrietvelde, P. A. Hoehn, and F. Giacomini, Switching quantum reference frames in the n-body problem and the absence of global relational perspectives (2018), arXiv:1809.05093 [quant-ph] .
  * Hoehn  _et al._ [2019] P. A. Hoehn, A. R. H. Smith, and M. P. E. Lock, The trinity of relational quantum dynamics (2019), arXiv:1912.00033 [quant-ph] .
  * Bene and Dieks [2001] G. Bene and D. Dieks, A Perspectival Version of the Modal Interpretation of Quantum mechanics and the Origin of Macroscopic Behavior, Foundations of Physics 32, 645–671 (2001).
  * Dalla Chiara [1977] M. L. Dalla Chiara, Logical self reference, set theoretical paradoxes and the measurement problem in quantum mechanics, Journal of Philosophical Logic 6, 331–347 (1977).
  * Breuer [1995] T. Breuer, The Impossibility of Accurate State Self-Measurements, Philosophy of Science 62, 197–214 (1995).
  * Baez [2006] J. Baez, Dimensional analysis and coordinate systems, The n Category Cafe (2006).
  * Tao [2013] T. Tao,  _Compactness and Contradiction_, Miscellaneous Books (American Mathematical Society, 2013).
  * Kitaev  _et al._ [2004] A. Kitaev, D. Mayers, and J. Preskill, Superselection rules and quantum protocols, Physical Review A 69, 052326 (2004).
  * [22] A. Magidin, Does every set have a group structure?, Mathematics Stack Exchange, url: https://math.stackexchange.com/q/105440 (version: 2017-04-13).
  * Wigner [1995] E. P. Wigner, Remarks on the mind-body question, in  _Philosophical Reflections and Syntheses_, edited by J. Mehra (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995) pp. 247–260.
  * Frauchiger and Renner [2018] D. Frauchiger and R. Renner, Quantum theory cannot consistently describe the use of itself, Nature Communications 9, 3711 (2018).
  * Poulin [2006] D. Poulin, Toy Model for a Relational Formulation of Quantum Theory, International Journal of Theoretical Physics 45, 1189–1215 (2006).
  * Loveridge  _et al._ [2017] L. Loveridge, P. Busch, and T. Miyadera, Relativity of quantum states and observables, EPL (Europhysics Letters) 117, 40004 (2017).
  * Mermin [1998] N. D. Mermin, What is quantum mechanics trying to tell us?, American Journal of Physics 66, 753–767 (1998).
  * Aharonov and Susskind [1967] Y. Aharonov and L. Susskind, Charge superselection rule, Phys. Rev. 155, 1428–1431 (1967).
  * Bartlett  _et al._ [2006] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Dialogue Concerning Two Views on Quantum Coherence: Factist and Fictionist, International Journal of Quantum Information 04, 17–43 (2006).
  * Angelo  _et al._ [2011] R. M. Angelo, N. Brunner, S. Popescu, A. J. Short, and P. Skrzypczyk, Physics within a quantum reference frame, Journal of Physics A: Mathematical and Theoretical 44, 145304 (2011).
  * Zych  _et al._ [2018] M. Zych, F. Costa, and T. C. Ralph, Relativity of quantum superpositions (2018), arXiv:1809.04999 [quant-ph] .
  * Palmer  _et al._ [2014] M. C. Palmer, F. Girelli, and S. D. Bartlett, Changing quantum reference frames, Physical Review A 89, 052121 (2014).
  * Gour and Spekkens [2008] G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New Journal of Physics 10, 033023 (2008).
  * Popescu  _et al._ [2018] S. Popescu, A. B. Sainz, A. J. Short, and A. Winter, Quantum Reference Frames and Their Applications to Thermodynamics, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20180111 (2018).
  * Poulin and Yard [2007] D. Poulin and J. Yard, Dynamics of a quantum reference frame, New Journal of Physics 9, 156–156 (2007).
  * Bartlett  _et al._ [2009] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner, Quantum communication using a bounded-size quantum reference frame, New Journal of Physics 11, 063013 (2009).
  * Yunger Halpern  _et al._ [2016] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges, Nature Communications 7, 12051 (2016).
  * Miyadera  _et al._ [2016] T. Miyadera, L. Loveridge, and P. Busch, Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off, Journal of Physics A Mathematical General 49, 185301 (2016).
  * Skotiniotis  _et al._ [2017] M. Skotiniotis, W. Dür, and P. Sekatski, Macroscopic superpositions require tremendous measurement devices, Quantum 1, 34 (2017).
  * Penrose [1971] R. Penrose, Angular momentum: an approach to combinatorial spacetime, Bastin, T. (ed.), Quantum Theory and Beyond , 151–180 (1971).
  * Baez [2009] J. Baez, Torsors made easy, John Baez Stuff (2009).

##  Appendix A Background

###  A.1 Coordinate systems and coordinate charts

We distinguish two separate notions: coordinate systems and coordinate charts
following the presentation in [19]. A _coordinate system_ is an isomorphism
$f:X\to Y$ where $X$ is the object of interest and $Y$ is some known object we
are using to describe $X$. In the terminology of Korzybski, $X$ is the
territory and $Y$ is the map.

However it is often the case (as in general relativity) that there is no
isomorphism between the object (space-time) and the description of the object
(coordinate chart). A _coordinate chart_ is a map in the opposite direction:
$f:Y\to X$ where once more $X$ is the object of interest and $Y$ is the known
object being used to describe $X$, where the map $f$ is no longer an
isomorphism (for instance for $X$ a curved manifold one needs multiple
coordinate charts $\cong\mathbb{R}^{n}$ to describe $X$ fully).

In the present paper we consider coordinate systems of a manifold $X$ and not
coordinate charts.

###  A.2 Coordinate systems on G-torsors

Given a space $X\cong G$ for some group $G$ it is clear that every $g\in G$
gives rise to an isomorphism $f:X\to Y$ (where $Y\cong G$) via the map
$f:x\mapsto gx$. While $e$ is the identity element of the group $G$ it is the
_origin_ of the $G$-torsor. Different isomorphisms $f:X\to Y$ correspond to
different choices of origin. To quote [41]: ‘A torsor is like a group that has
forgotten its identity.’ Since every $g\in G$ leads to a different coordinate
system, the set of different choices of coordinate systems is also $G$. As
noted in the main body in the case where $X\cong G$ is a Lie group and we are
not concerned with the group structure but only the smooth differentiable
manifold structure one could find other coordinate systems via the isomorphism
$f:x\mapsto mx$ where $m$ is an element of ${\rm Diff}(X)$, the diffeomorphism
group on $X$. In the case where $X\cong\mathbb{R}$ this would consist of
considering coordinate systems which are related by more general
transformations than translations, for instance re-scaling by a real scalar.

###  A.3 Reference frames

A _reference frame_ is a coordinate system together with a physical system
whose configuration uniquely determines the coordinate system. Since every
choice of coordinate system is in correspondence with an element $g\in G$ for
a space $X\cong G$, a valid choice of physical system used to form a reference
frame is a physical system with configuration space $G$.

##  Appendix B Proofs of Lemmas 1, 2 and 3

###  B.1 Proof of Lemma 1

We show that $U^{i\to j}$ is unitary.

To change reference frame between systems $i$ and $j$, where $i$ and $j$ have
state space $L^{2}(G)$, and $n-2$ other systems carrying the right regular
representation
$U_{k}(g_{j}^{i})\ket{\psi(g_{k}^{i})}_{k}=\ket{\psi(g_{k}^{i}g_{i}^{j})}_{k}=\ket{\psi(g_{k}^{j})}_{k}$,
the general operator is

$\displaystyle U^{i\to j}={\rm SWAP}_{i,j}\circ\int_{g^{i}_{j}\in
G}dg_{j}^{i}\
\outerproduct{g^{j}_{i}}{g_{j}^{i}}_{j}\otimes{\mathbb{1}}_{i}\otimes$
$\displaystyle\bigotimes_{k\neq i,j}U_{k}(g_{j}^{i}).$ (75)

We can show that this operator is unitary:

$\displaystyle(U^{i\to j})^{\dagger}=\int_{h^{i}_{j}\in G}dh_{j}^{i}\
\outerproduct{h^{i}_{j}}{h_{i}^{j}}_{j}\otimes{\mathbb{1}}_{i}\otimes$
$\displaystyle\bigotimes_{k\neq i,j}U^{\dagger}_{k}(h_{j}^{i})\circ{\rm
SWAP}_{i,j}^{\dagger}.$ (76)

Hence,

$\displaystyle(U^{i\to j})^{\dagger}U^{i\to j}$ $\displaystyle=$
$\displaystyle\int_{h^{i}_{j}\in G}dh_{j}^{i}\
\outerproduct{h^{i}_{j}}{h_{i}^{j}}_{j}\otimes{\mathbb{1}}_{i}\otimes\bigotimes_{k\neq
i,j}U^{\dagger}_{k}(h_{j}^{i})\circ{\rm SWAP}_{i,j}^{\dagger}$
$\displaystyle\circ{\rm SWAP}_{i,j}\circ\int_{g^{i}_{j}\in G}dg_{j}^{i}\
\outerproduct{g^{j}_{i}}{g_{j}^{i}}_{j}\otimes{\mathbb{1}}_{i}\otimes\bigotimes_{k\neq
i,j}U_{k}(g_{j}^{i})$ $\displaystyle=$ $\displaystyle\int_{h^{i}_{j}\in
G}dh_{j}^{i}\int_{g^{i}_{j}\in G}dg_{j}^{i}\
\ket{h^{i}_{j}}\innerproduct{h_{i}^{j}}{g^{j}_{i}}\bra{g_{j}^{i}}_{j}\otimes{\mathbb{1}}_{i}\
\otimes$ $\displaystyle\bigotimes_{k\neq
i,j}U^{\dagger}_{k}(h_{j}^{i})U_{k}(g_{j}^{i})$
$\displaystyle=\int_{g^{i}_{j}\in G}dg_{j}^{i}\
\outerproduct{g^{i}_{j}}{g_{j}^{i}}_{j}\otimes{\mathbb{1}}_{i}\otimes\bigotimes_{k\neq
i,j}U^{\dagger}_{k}(g_{j}^{i})U_{k}(g_{j}^{i})$
$\displaystyle={\mathbb{1}}_{j}\otimes{\mathbb{1}}_{i}\otimes\bigotimes_{k\neq
i,j}U_{k}((g_{j}^{i})^{-1}g_{j}^{i})$
$\displaystyle={\mathbb{1}}_{j}\otimes{\mathbb{1}}_{i}\otimes\bigotimes_{k\neq
i,j}{\mathbb{1}}_{k}$ $\displaystyle={\mathbb{1}}.$

Here, we used $\innerproduct{h_{i}^{j}}{g_{i}^{j}}=\delta_{gh}$, $\int
dg_{j}^{i}\ \outerproduct{g^{i}_{j}}{g_{j}^{i}}_{j}={\mathbb{1}}_{j}$ and
$U_{k}(e)={\mathbb{1}}_{k}$.

###  B.2 Proof of Lemma 2

We show $\left(U^{0\to i}\right)^{\dagger}=U^{i\to 0}$.

$\displaystyle\left(U^{0\to i}\right)^{\dagger}$
$\displaystyle=\int_{g^{0}_{i}\in G}\
dg_{i}^{0}\outerproduct{g_{i}^{0}}{g^{i}_{0}}_{i}\otimes{\mathbb{1}}_{0}\otimes
U_{R}^{\dagger}(g^{0}_{i})^{\otimes n-2}\circ{\rm SWAP}_{0,i}^{\dagger}$
$\displaystyle={\rm SWAP}_{i,0}\circ\int_{g^{0}_{i}\in G}\
dg_{i}^{0}\outerproduct{g_{i}^{0}}{g^{i}_{0}}_{0}\otimes{\mathbb{1}}_{i}\otimes
U_{R}^{\dagger}(g^{0}_{i})^{\otimes n-2}$ $\displaystyle={\rm
SWAP}_{i,0}\circ\int_{g^{i}_{0}\in G}\
dg_{0}^{i}\outerproduct{g_{i}^{0}}{g^{i}_{0}}_{0}\otimes{\mathbb{1}}_{i}\otimes
U_{R}(g^{i}_{0})^{\otimes n-2}$ $\displaystyle=U^{i\to 0}$

From the second to the third line, we commuted the SWAP operator through to
the left by changing the labels of partitions $0$ and $i$. From the third to
the fourth line, we used the fact that the integral over $g_{i}^{0}$ is the
same as the integral over $g_{0}^{i}$. Moreover, it holds that
$U_{R}^{\dagger}(g^{0}_{i})=U_{R}(g^{i}_{0})$ because
$U_{R}(g_{0}^{i})U_{R}(g_{i}^{0})\ket{g}=\ket{gg_{0}^{i}g_{i}^{0}}=\ket{g}$,
hence $U_{R}(g_{0}^{i})=U_{R}^{\dagger}(g_{i}^{0})$.

###  B.3 Proof of Lemma 3

We prove that $U^{i\to j}U^{k\to i}=U^{k\to j}$.

For this, we show how the operators act on an arbitrary basis state:

$\displaystyle U^{i\to j}U^{k\to
i}\ket{e}_{k}\ket{g^{k}_{0}}_{0}...\ket{g^{k}_{i}}_{i}...\ket{g^{k}_{j}}_{j}...\ket{g_{n-1}^{k}}_{n-1}$
$\displaystyle=\\!U^{i\to j}\\!{\rm
SWAP}_{k,i}\\!\ket{e}_{k}\\!\ket{g^{k}_{0}\\!g_{k}^{i}}_{0}\\!...\\!\ket{g^{k}_{i}}_{i}\\!...\\!\ket{g^{k}_{j}\\!g_{k}^{i}}_{j}\\!...\\!\ket{g_{n-1}^{k}\\!g_{k}^{i}}_{n-1}$
$\displaystyle=U^{i\to
j}\ket{e}_{i}\ket{g^{k}_{0}g_{k}^{i}}_{0}...\ket{g^{k}_{i}}_{k}...\ket{g^{k}_{j}g_{k}^{i}}_{j}...\ket{g_{n-1}^{k}g_{k}^{i}}_{n-1}$
$\displaystyle={\rm
SWAP}_{i,j}\ket{e}_{i}\ket{g_{0}^{i}g_{i}^{j}}_{0}\\!...\\!\ket{g^{k}_{i}g_{i}^{j}}_{k}\\!...\\!\ket{g_{i}^{j}}_{j}\\!...\\!\ket{g_{n-1}^{i}g_{i}^{j}}_{n-1}$
$\displaystyle=\ket{e}_{j}\ket{g_{0}^{j}}_{0}...\ket{g_{k}^{j}}_{k}...\ket{g_{i}^{j}}_{i}...\ket{g_{n-1}^{j}}_{n-1}$
$\displaystyle=U^{k\to
j}\ket{e}_{k}\ket{g^{k}_{0}}_{0}...\ket{g^{k}_{i}}_{i}...\ket{g^{k}_{j}}_{j}...\ket{g_{n-1}^{k}}_{n-1}.$

##  Appendix C On the unitarity of the change of reference frame operator

Let us consider $\ket{\psi(g)}$, with $U_{L}(k)\ket{\psi(g)}=\ket{\psi(kg)}$.
Take two initial states defined relative to ${\sf A}$:

$\displaystyle\ket{\psi^{1}}=\ket{\psi(e)}_{\sf A}\ket{\psi(g_{1})}_{\sf
B}\ket{\psi(g_{2})}_{\sf C}\ ,$ (77)
$\displaystyle\ket{\psi^{2}}=\ket{\psi(e)}_{\sf A}\ket{\psi(h_{1})}_{\sf
B}\ket{\psi(g_{2})}_{\sf C}.$ (78)

The change of reference frame ${\sf A}\to{\sf B}$:
$\ket{\psi^{i}}\mapsto\ket{\phi^{i}}$ gives:

$\displaystyle\ket{\phi^{1}}=\ket{\psi(e)}_{\sf B}\ket{\psi(g_{1}^{-1})}_{\sf
A}\ket{\psi(g_{2}g_{1}^{-1})}_{\sf C}\ ,$ (79)
$\displaystyle\ket{\phi^{2}}=\ket{\psi(e)}_{\sf B}\ket{\psi(h_{1}^{-1})}_{\sf
A}\ket{\psi(g_{2}h_{1}^{-1})}_{\sf C}.$ (80)

For a superposition state
$\ket{\psi^{3}}=\alpha\ket{\psi^{1}}+\beta\ket{\psi^{2}}$ ($\alpha,\beta\neq
0$) the change of reference system gives:

$\displaystyle\ket{\psi^{3}}=\alpha\ket{\psi^{1}}+\beta\ket{\psi^{2}}\mapsto\ket{\phi^{3}}=\alpha\ket{\phi^{1}}+\beta\ket{\phi^{2}}\
,$ (81)

where

$\displaystyle\ket{\phi^{3}}$ $\displaystyle=\ket{\psi(e)}_{\sf
B}(\alpha\ket{\psi(g_{1}^{-1})}_{\sf A}\ket{\psi(g_{2}g_{1}^{-1})}_{\sf C}$
$\displaystyle+\beta\ket{\psi(h_{1}^{-1})}_{\sf
A}\ket{\psi(g_{2}h_{1}^{-1})}_{\sf C})\ .$ (82)

Let us show that unitarity of the change of reference frame implies that the
states $\ket{\psi(g)}$ are mutually orthogonal. We compute the inner product
of two initial states $\ket{\psi^{1}}$ and $\ket{\psi^{3}}$:

$\displaystyle\innerproduct{\psi^{1}}{\psi^{3}}=\alpha+\beta\innerproduct{\psi(g_{1})}{\psi(h_{1})}_{\sf
B}\ ,$ (83)

and the inner product of the two final states $\ket{\phi^{1}}$ and
$\ket{\phi^{3}}$:

$\displaystyle\innerproduct{\phi^{1}}{\phi^{3}}=\alpha+\beta\innerproduct{\phi^{1}}{\phi^{2}}\
,$ (84)

where

$\displaystyle\innerproduct{\phi^{1}}{\phi^{2}}=\innerproduct{\psi(g_{1}^{-1})}{\psi(h_{1}^{-1})}_{\sf
A}\innerproduct{\psi(g_{2}g_{1}^{-1})}{\psi(g_{2}h_{1}^{-1})}_{\sf C}\ .$ (85)

Since $\ket{\psi(g_{2}g_{1}^{-1})}=U(g_{2})\ket{\psi(g_{1}^{-1})}$ and
$\ket{\psi(g_{2}h_{1}^{-1})}=U(g_{2})\ket{\psi(h_{1}^{-1})}$ we have that
$\innerproduct{\psi(g_{2}g_{1}^{-1})}{\psi(g_{2}h_{1}^{-1})}_{\sf
C}=\innerproduct{\psi(g_{1}^{-1})}{\psi(h_{1}^{-1})}_{\sf C}$. Therefore:

$\displaystyle\innerproduct{\phi^{1}}{\phi^{3}}=\alpha+\beta(\innerproduct{\psi(g_{1}^{-1})}{\psi(h_{1}^{-1})}^{2}).$
(86)

By assumption there are two commuting actions:
$U_{L}(g)\ket{\psi(h)}=\ket{\psi(gh)}$ and
$U_{R}(g)\ket{\psi(h)}=\ket{\psi(hg^{-1})}$.

Now $\innerproduct{\psi^{1}}{\psi^{3}}=\innerproduct{\phi^{1}}{\phi^{3}}$
requires
$\innerproduct{\psi(g_{1})}{\psi(h_{1})}=\innerproduct{\psi(g_{1}^{-1})}{\psi(h_{1}^{-1})}^{2}$.
We assume this holds and show it implies that
$\innerproduct{\psi(g)}{\psi(h)}=\delta(h,g)$.

$\displaystyle\innerproduct{\psi(g_{1})}{\psi(h_{1})}=\innerproduct{\psi(g_{1}^{-1})}{\psi(h_{1}^{-1})}^{2}$
$\displaystyle\Leftrightarrow\innerproduct{\psi(g_{1})}{\psi(h_{1})}=\overline{\innerproduct{\psi(g_{1}^{-1})}{\psi(h_{1}^{-1})}}^{2}$
$\displaystyle\Leftrightarrow\innerproduct{U_{L}(g_{1}^{-1})\psi(g_{1})}{U_{L}(g_{1}^{-1})\psi(h_{1})}$
$\displaystyle=\overline{\innerproduct{U_{R}(h_{1}^{-1})\psi(h_{1}^{-1})}{U_{R}(h_{1}^{-1})\psi(g_{1}^{-1})}}^{2}$
$\displaystyle\Leftrightarrow\innerproduct{\psi(e)}{\psi(g_{1}^{-1}h_{1})}=\overline{\innerproduct{\psi(e)}{\psi(g_{1}^{-1}h_{1})}}^{2}$

Which only holds when $\ket{\psi(g_{1}^{-1}h_{1})}$ is orthogonal to
$\ket{\psi(e)}$ (when $g_{1}\neq h_{1}$). Therefore for all $g\in G$ it is the
case that $\innerproduct{\psi(e)}{\psi(g)}=\delta(g,e)$.

Now consider $\innerproduct{\psi(g)}{\psi(h)}$ for arbitrary $g,h\in G$. This
is equal to
$\innerproduct{U_{L}(g^{-1})\psi(g)}{U_{L}(g^{-1})\psi(h)}=\innerproduct{\psi(e)}{\psi(g^{-1}h)}=\delta(e,g^{-1}h)=\delta(g,h)$.
This entails that the states $\ket{\psi(g)}$ form an orthonormal basis for
${\mathcal{H}}_{i}$ or a subspace of it if they do not span the full space.

Hence, the operator which performs the required change of reference frame and
obeys the principle of coherent change of reference frame is unitary if and
only if the states $\ket{\psi(g)}$ form an orthonormal basis for
${\mathcal{H}}_{i}$ or a subspace thereof.

##  Appendix D Imperfect reference frames

###  D.1 Proof of Theorem 2

$\displaystyle s$
$\displaystyle=\left((g_{0},...,g_{k-1}),(g_{k},...,g_{l-1})\right)$ (87)
$\displaystyle=\left((n_{0}p_{0},...,n_{k-1}p_{k-1}),(n_{k},...,n_{l-1})\right)$
(88)

We first find the state $\tilde{s}^{j}=\Lambda^{0\to j}\left({\rm
R}_{N}^{0}(T(s))\right)$. Let us truncate $s$:

$\displaystyle\tilde{s}=T(s)=\left((n_{0},...,n_{k-1}),(n_{k},...,n_{l-1})\right).$
(89)

Then the state $\tilde{s}^{0}$ is:

$\displaystyle\tilde{s}^{0}={\rm
R}_{N}^{0}\left((n_{0}^{0},...,n_{k-1}^{0}),(n_{k}^{0},...,n_{l-1}^{0})\right).$
(90)

Then the state $\tilde{s}^{j}=\Lambda^{0\to j}\tilde{s}^{0}$ is:

$\displaystyle T(s)^{j}$
$\displaystyle=\left((n_{0}^{0}n_{0}^{j},...,n_{k-1}^{0}n_{0}^{j}),(n_{k}^{0}n_{0}^{j},...,n_{l-1}^{0}n_{0}^{j})\right)$
(91)
$\displaystyle=\left((n_{0}^{j},...,n_{k-1}^{j}),(n_{k}^{j},...,n_{l-1}^{j})\right)$
(92)

where $n^{i}_{j}n_{i}=n_{j}$ and $n^{r}_{i}n_{r}^{j}=n_{i}^{j}$.

Now we compute $\tilde{s}^{j}=\Lambda^{0\to j}\left(T({\rm
R}_{G}^{0}(s))\right)$. $s^{0}={\rm R}_{G}^{0}s$ is:

$\displaystyle s^{0}$
$\displaystyle=\left((g_{0}^{0},...,g_{k-1}^{0}),(g_{k}^{0},...,g_{l-1}^{0})\right)$
(93)
$\displaystyle=\left((m_{0}^{0}q_{0}^{0},...,m_{k-1}^{0}q_{k-1}^{0}),(m_{k}^{0}q_{k}^{0},...,m_{l-1}^{0}q_{l-1}^{0})\right)\
,$ (94)

where $g_{i}^{j}=m_{i}^{j}q_{i}^{j}$ is the decomposition into $NP$. Let us
consider the elements $m_{j}^{0}q_{j}^{0}$ for $j\in\\{k,\dots,l-1\\}$. We
have that $g^{0}_{j}g_{0}=n_{j}\rightarrow
m_{j}^{0}q_{j}^{0}n_{0}p_{0}=n_{j}$. Now observe that since $m_{j}^{0}$ and
$n_{j}$ in $N$ this implies that $q_{j}^{0}n_{0}p_{0}=n^{\prime}\in N$. Now
there is a unique $q_{j}^{0}$ such that this holds, and observe that
$p_{0}^{-1}n_{0}p_{0}=m^{\prime}$. There is a unique $m\in N$ such that
$mm^{\prime}=n^{\prime}$. Therefore $q_{j}^{0}=mp_{0}^{-1}$. However since
$q_{j}^{0}\in P$ this implies $m=e$ and therefore $q_{j}^{0}=p_{0}^{-1}$.

Let us truncate:

$\displaystyle
T(s^{0})=\left((m_{0}^{0},...,m_{k-1}^{0}),(m_{k}^{0},...,m_{l-1}^{0})\right).$
(95)

Then one can obtain $T(s^{0})^{j}=\Lambda^{0\to
j}T(s^{0})=\Gamma_{R}(m_{j}^{0})T(s^{0})$:

$\displaystyle
T(s^{0})^{j}=\left((m_{0}^{j},...,m_{k-1}^{0}m_{0}^{j}),(m_{k}^{0}m_{0}^{j},...,m_{l-1}^{0}m_{0}^{j})\right).$
(96)

Does $m_{i}^{0}m_{0}^{j}=n_{i}^{j}$, i.e. does
$m_{i}^{0}m_{0}^{j}n_{j}=n_{i}$. There are two cases: $i\in\\{0,\ldots,k-1\\}$
and $i\in\\{k,\ldots,l-1\\}$. Let us consider the second:

$\displaystyle m^{0}_{i}q^{0}_{i}n_{0}p_{0}=n_{i}$ (97) $\displaystyle
m^{0}_{i}=n_{i}(p_{0}^{-1}n_{0}p_{0})^{-1}$ (98)

since $q^{0}_{i}=p_{0}^{-1}$. Similarly
$m^{0}_{j}=n_{j}(p_{0}^{-1}n_{0}p_{0})^{-1}$. Therefore

$\displaystyle m_{i}^{0}m_{0}^{j}=m_{i}^{0}(m_{0}^{j})^{-1}$
$\displaystyle=n_{i}(p_{0}^{-1}n_{0}p_{0})^{-1}(p_{0}^{-1}n_{0}p_{0})n_{j}^{-1}\
,$ $\displaystyle=n_{i}n_{j}^{-1}.$ (99)

This is indeed $n_{i}^{j}$ since it maps $n_{j}$ to $n_{i}$.

Now let us consider the case $i\in\\{0,\dots,k-1\\}$:

$\displaystyle m^{0}_{i}q^{0}_{i}n_{0}p_{0}=n_{i}p_{i}$ (100)

where $q^{0}_{i}$ is not necessarily equal to $p_{0}^{-1}$. Then we obtain

$\displaystyle m_{i}^{0}m_{0}^{j}$
$\displaystyle=m_{i}^{0}(m_{0}^{j})^{-1}=n_{i}p_{i}(q^{0}_{i}n_{0}p_{0})^{-1}(p_{0}^{-1}n_{0}p_{0})n_{j}^{-1}$
(101)
$\displaystyle=n_{i}p_{i}p_{0}^{-1}n_{0}^{-1}(q^{0}_{i})^{-1}p_{0}^{-1}n_{0}p_{0}n_{j}^{-1}$
(102)

which is not equal to $n_{i}^{j}$ in general. For a specific example we can
look at $\mathbb{R}=\mathbb{Z}\rtimes S^{1}$. Consider

$\displaystyle s=(n_{0}L+x_{0},n_{1}L+x_{1},n_{2}L,n_{3}L).$ (103)

Then $T(s)=(n_{0}L,n_{1}L,n_{2}L,n_{3}L)$ and
$T(s)^{2}=(n_{0}-n_{2},n_{1}-n_{2},0,n_{3}-n_{2})L$. For the other order we
get
$s^{0}=(0,n_{1}L+x_{1}-(n_{0}L+x_{0}),n_{2}L-(n_{0}L+x_{0}),n_{3}L-(n_{0}L+x_{0}))$.
Let us assume that $x_{1}-x_{0}<0$, then
$T(s^{0})=(0,n_{1}-n_{0}-1,n_{2}-n_{0},n_{3}-n_{0})L$ and
$T(s^{0})^{2}=(-(n_{2}-n_{0}),n_{1}-n_{0}-1-(n_{2}-n_{0}),n_{2}-n_{0}-(n_{2}-n_{0}),n_{3}-n_{0}-(n_{2}-n_{0}))L$
which gives $T(s^{0})^{2}=(n_{0}-n_{2},n_{1}-n_{2}-1,0,n_{3}-n_{2})L$ and does
not equal $T(s)^{2}$.

###  D.2 Averaging and invariants for the example of $\mathrm{SO}(3)\rtimes
T(\mathbb{R}^{3})$

Now let us return to the case where the configuration space is
$\mathbb{R}^{3}$ but the transformation group is $E^{+}(3)\cong
T(\mathbb{R}^{3})\rtimes\mathrm{SO}(3)$ and the case of three particles. Here
$E^{+}(3)$ is the special Euclidean group consisting of translations followed
by a rotation. Let us say that particle $0$ uses a coordinate system
$(x^{0},y^{0},z^{0})$ such that it assigns itself the state $(0,0,0)=\vec{0}$.
There are infinitely many such coordinate systems corresponding to all
reference frames centred on particle $0$, related by rotations
$O\in\mathrm{SO}(3)$. We write
$\vec{v}^{0}_{i}=(x^{0}_{i},y^{0}_{i},z^{0}_{i})$ for the state of particle
$i$ in a reference frame centred at $0$. The coordinates corresponding to
Cartesian reference frames are in one to one correspondence with group
elements in $T(\mathbb{R}^{3})\rtimes\mathrm{SO}(3)$. Each state
$\vec{v}^{0}_{i}=(x^{0}_{i},y^{0}_{i},z^{0}_{i})$ is stabilized by a
$\mathrm{SO}(3)$ subgroup: $\vec{v}^{0}_{i}\rtimes O$ $\forall
O\in\mathrm{SO}(3)$ . We can write $\mathbb{R}^{3}\cong
E^{+}(3)/\mathrm{SO}(3)$.

The description of the state of the three particles is
$s^{0}_{0,1,2}=\left(\vec{0},\vec{v}^{0}_{1},\vec{v}^{0}_{2}\right)$. What can
particle $0$ infer about the description used by particle 1, knowing only that
the convention is such that particle 1 uses Cartesian coordinates placed at
its position? There are infinitely many such choices of coordinates, related
to the coordinates $(x^{0}_{i},y^{0}_{i},z^{0}_{i})$ by $t_{-v^{0}_{1}}\rtimes
O$ for all $O\in\mathrm{SO}(3)$. Hence the states
$s^{1,O}_{0,1,2}=\left(-O\vec{v^{0}_{1}},\vec{0},O\left(\vec{v}^{0}_{2}-\vec{v^{0}_{1}}\right)\right)$
for any $O\in\mathrm{SO}(3)$ correspond to coordinate choices centred on
particle $1$. In order to describe the ignorance about $O$ one needs a
probabilistic representation of states: a state is now a measure on
$\mathbb{R}^{3}$. The state $-\vec{a}$ is the Dirac measure
$\delta_{-\vec{a}}$ and the description that particle 0 assigns to particle 1
is:

$\displaystyle s^{1,av}_{0,1,2}$
$\displaystyle=\int_{O\in\mathrm{SO}(3)}\left(O\delta_{-\vec{v^{0}_{1}}},\delta_{\vec{0}},\delta_{\vec{v}^{0}_{2}-\vec{v^{0}_{1}}}\right)$
(104)
$\displaystyle=\left(\mu_{S^{2},|\vec{v^{0}_{1}}|},\delta_{\vec{0}},\mu_{S^{2},|\vec{v}^{0}_{2}-\vec{v^{0}_{1}}|}\right)\
,$ (105)

where $\mu_{S^{2},|\vec{a}|}$ is the normalised Haar measure on the sphere
$S^{2}$ of radius $|\vec{a}|$ centred at the origin (in particle $1$’s
coordinates). We observe that there is no reversible transformation from
$s^{1,{\rm average}}_{0,1}$ representing particle $0$’s knowledge of particle
$1$’s description back to the initial description of particle 0. Starting from
$s^{0}$ and mapping to $s^{2}$ one obtains:

$\displaystyle s^{2,av}_{0,1,2}$
$\displaystyle=\int_{O\in\mathrm{SO}(3)}\left(O\delta_{-\vec{v}^{0}_{2}},\delta_{\vec{v}^{0}_{1}-\vec{v}^{0}_{2}},\delta_{0}\right)$
(106)
$\displaystyle=\left(\mu_{S^{2},|\vec{v^{0}_{2}}|},\mu_{S^{2},|\vec{v}^{0}_{1}-\vec{v^{0}_{2}}|},\delta_{\vec{0}}\right).$
(107)

We leave to future work the development of a full account of changes of
reference frame in this probabilistic case (for instance it is not clear that
there is a well defined change $s^{1,av}_{0,1,2}\mapsto s^{2,av}_{0,1,2}$).
Here we are interested only in the single change from $s^{0}\mapsto s^{1}$.

We also note that when describing ignorance one cannot integrate over the pure
states, or configurations. In the above example this would result in
$s^{1,{\rm average}}_{0,1}$ being $\left(\vec{0},\vec{0}\right)$ which is
nonsensical. This is similar to the case in quantum theory where integrating
over a group action on pure states does not represent ignorance (rather it is
a projection) but integrating over mixed states corresponds to ignorance. One
must move to a probabilistic description of states in terms of measures on $X$
in order to infer states of other particles for these types of cases.
Alternatively one could stay at the pure state description, and instead of
finding the point of view that particle 0 infers particle 1 holds, we can
search for what particle 0 knows holds true for all possible descriptions
$s^{1}_{0,1}=\left(O\vec{-a},\vec{0}\right)$ for all $O\in\mathrm{SO}(3)$ from
the point of view of particle 1. In other words: what are the _invariants_?
Here it is straightforward to see that any $f(|\vec{a}|)$ is an invariant
quantity.

##  Appendix E Wigner’s friend with additional reference system

One could argue that introducing an additional reference system ${\sf R}$ into
the box containing ${\sf F}$ and ${\sf S}$ might help to resolve the paradox
of Wigner’s friend. In this case, we can describe the measurement interaction
from the point of view of ${\sf W}$ as follows:

$\ket{\uparrow}_{\sf R}^{{\sf W}}\ket{\uparrow}_{\sf F}^{{\sf
W}}\ket{\psi}_{\sf S}^{{\sf W}}\to\ket{\uparrow}_{\sf R}^{{\sf
W}}(\alpha\ket{\uparrow}_{\sf F}^{{\sf W}}\ket{\uparrow}_{\sf S}^{{\sf
W}}+\beta\ket{\downarrow}_{\sf F}^{{\sf W}}\ket{\downarrow}_{\sf S}^{{\sf
W}}).$ (108)

The reference system is not affected by the measurement. Now, we want to apply
the change of reference frame to switch to the perspective of ${\sf F}$. We
apply the operator $U^{{\sf W}\to{\sf F}}_{{\sf A}{\sf S}}$ to the initial
state:

$\displaystyle U^{{\sf W}\to{\sf F}}_{{\sf R}{\sf S}}\ket{\uparrow}_{\sf
R}^{{\sf W}}\ket{\uparrow}_{\sf F}^{{\sf W}}\ket{\psi}_{\sf S}^{{\sf
W}}=\ket{\uparrow}_{\sf W}^{\sf F}\ket{\uparrow}_{\sf R}^{\sf
F}\ket{\psi}_{\sf S}^{{\sf F}}\ ,$ (109)

and the final state:

$\displaystyle U^{{\sf W}\to{\sf F}}_{{\sf R}{\sf S}}\ket{\uparrow}_{\sf
R}^{{\sf W}}(\alpha\ket{\uparrow}_{\sf F}^{{\sf W}}\ket{\uparrow}_{\sf
S}^{{\sf W}}+\beta\ket{\downarrow}_{\sf F}^{{\sf W}}\ket{\downarrow}_{\sf
S}^{{\sf W}})$ $\displaystyle=(\alpha\ket{\uparrow}_{\sf W}^{{\sf
F}}\ket{\uparrow}_{\sf R}^{{\sf F}}+\beta\ket{\downarrow}_{\sf W}^{{\sf
F}}\ket{\downarrow}_{\sf R}^{{\sf F}})\ket{\uparrow}_{\sf S}^{\sf F}.$ (110)

We see that, again, the state of the system is always correlated with the
state of the friend. Wigner and the additional reference system end up being
entangled. We conclude that introducing an additional reference system into
the laboratory does not provide a resolution to the paradox.

##  Appendix F Comparison to RF change operator of [6]

This is the formal proof that the operator given in [6] for the change between
two reference frames ${\sf A}$ and ${\sf B}$ on the real line (one-dimensional
translation group) is equivalent to the operator given in Equation (4.2),
namely

$\displaystyle U^{{\sf A}\to{\sf B}}={\rm SWAP}_{{\sf A},{\sf B}}\circ\int
dxdy\ \outerproduct{-x}{x}_{\sf
B}\otimes\mathbb{1}_{A}\otimes\outerproduct{y-x}{y}_{\sf C}.$ (111)

Let us take the most general state of three particles on the real line
$\mathbb{R}$ relative to the first particle:

$\ket{\Psi}_{{\sf A}{\sf B}{\sf C}}=\ket{0}_{\sf A}\otimes\int dx\int dy\
\psi(x,y)\ket{x}_{\sf B}\otimes\int dy\ket{y}_{\sf C}.$ (112)

In the formalism of [6], the reference system is not explicitly included in
the state of the joint system. The equivalent state would be

$\ket{\Psi}_{{\sf B}{\sf C}}=\int dx\int dy\ \psi(x,y)\ket{x}_{\sf
B}\otimes\int dy\ket{y}_{\sf C}.$ (113)

When changing from reference system ${\sf A}$ to reference system ${\sf B}$,
we apply the operator of [6]:

$\hat{S}^{{\sf A}\to{\sf B}}=\hat{\mathcal{P}}_{{\sf A}{\sf
B}}e^{i/\hbar\hat{x}_{\sf B}\hat{p}_{\sf C}}.$ (114)

The final state is then

$\displaystyle\hat{S}^{{\sf A}\to{\sf B}}\ket{\Psi}_{{\sf B}{\sf C}}=\int
dx\int dy\ \psi(x,y)\ket{-x}_{\sf A}\otimes\ket{y-x}_{\sf C}.$ (115)

When applying the operator in Equation (111) to the initial state in Equation
(112), we get the final state

$\displaystyle U^{{\sf A}\to{\sf B}}\ket{\Psi}_{{\sf A}{\sf B}{\sf C}}$
$\displaystyle=\ $ $\displaystyle{\rm SWAP}_{{\sf A},{\sf B}}\ \circ$
$\displaystyle\left(\ket{0}_{\sf A}\int dx^{\prime}dy^{\prime}dxdy\
\psi(x,y)\outerproduct{-x^{\prime}}{x^{\prime}}\ket{x}\outerproduct{y^{\prime}-x^{\prime}}{y^{\prime\prime}}\ket{y}\right)$
$\displaystyle=\ $ $\displaystyle\ket{0}_{\sf B}\int dx\int dy\
\psi(x,y)\ket{-x}_{\sf A}\otimes\ket{y-x}_{\sf C}.$ (116)

Hence, we see that both operators act in the same way on the most general
states of particles on the real line.