# Infinitesimal reference frames suffice to determine the asymmetry properties
of a quantum system

Rhea Alexander<EMAIL_ADDRESS>Department of Physics, Imperial College
London, London SW7 2AZ, UK  School of Physics and Astronomy, University of
Leeds, Leeds, LS2 9JT, UK  Si Gvirtz-Chen  School of Physics and Astronomy,
University of Leeds, Leeds, LS2 9JT, UK  David Jennings  School of Physics and
Astronomy, University of Leeds, Leeds, LS2 9JT, UK  Department of Physics,
Imperial College London, London SW7 2AZ, UK

###### Abstract

Symmetry principles are fundamental in physics, and while they are well
understood within Lagrangian mechanics, their impact on quantum channels has a
range of open questions. The theory of asymmetry grew out of information-
theoretic work on entanglement and quantum reference frames, and allows us to
quantify the degree to which a quantum system encodes coordinates of a
symmetry group. Recently, a complete set of entropic conditions was found for
asymmetry in terms of correlations relative to infinitely many quantum
reference frames. However, these conditions are difficult to use in practice
and their physical implications unclear. In the present theoretical work, we
show that this set of conditions has extensive redundancy, and one can
restrict to reference frames forming any closed surface in the state space
that has the maximally mixed state in its interior. This in turn implies that
asymmetry can be reduced to just a single entropic condition evaluated at the
maximally mixed state. Contrary to intuition, this shows that we do not need
macroscopic, classical reference frames to determine the asymmetry properties
of a quantum system, but instead infinitesimally small frames suffice.
Building on this analysis, we provide simple, closed conditions to estimate
the minimal depolarization needed to make a given quantum state accessible
under channels covariant with any given symmetry group.

##  I Introduction

Symmetry principles have been extensively studied both in classical and
quantum theory, and in particular for Lagrangian dynamics of a quantum system
[1]. However, such evolution is a strict subset of the most general kind of
physical transformation that quantum theory permits – quantum channels [2].
This more general setting not only includes unitary dynamics, open system
dynamics, and the ability to vary the system dimension as it transforms, but
it also interpolates between deterministic unitary dynamics and measurements
that sharply collapse a quantum system. How symmetry principles constrain
quantum channels is therefore a crucial question.

The study of quantum entanglement [3, 4] lead to a much broader conception of
physical properties in terms of ‘resources’ relative to a set of quantum
channels. This gave a precise way to quantify other fundamental features such
as quantum coherence [5, 6, 7, 8], thermodynamics [9, 10, 11], non-Gaussianity
[12, 13], magic states for quantum computing [14, 15], and many more [16]. In
particular, the theory of asymmetry provides an information-theoretic means to
quantify the degree to which a quantum system breaks a symmetry [6, 17, 18].

Asymmetry sits at the crossroads between abstract quantum information theory
and physical laws, and quantifies what has been called ‘unspeakable quantum
information’ [19, 20]. This information cannot be transcribed into a data
string on paper or in an email, but instead requires the transfer of a system
that carries a non-trivial action of the symmetry group, via a symmetric
(covariant) quantum channel. Given this, such concepts find application in
quantum metrology [21], symmetry-constrained dynamics [17, 22, 23], quantum
reference frames [24, 25, 26, 19, 27, 28], thermodynamics [29, 30, 31],
measurement theory [32, 33, 34, 35, 36], macroscopic coherence [37], and
quantum speed-limits [38]. More recent work has seen a renewed interest in
quantum reference frames in a relativistic setting and the problem of time in
quantum physics [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50], as well as
applications in quantum computing and covariant quantum error-correcting codes
[51, 52, 53, 54, 55, 56, 57] where the Eastin-Knill theorem provides an
obstacle to transversal gate-sets forming a continuous unitary sub-group [51].

The central question considered in this paper concerns transforming from a
quantum state $\rho$ to another quantum state $\sigma$ under a symmetry
constraint. More precisely, we address the following fundamental question:

Core Question: When is it possible to transform $\rho_{A}$ to $\sigma_{B}$
under a quantum channel between systems $A$ and $B$ that is covariant with
respect to a symmetry group $G$?

This question addresses the fundamental way in which symmetries constrain
quantum theory, and turns out to be surprisingly non-trivial. One might
initially conjecture that if we consider the generators $\\{J_{k}\\}$ of the
group representation $U(g)=\exp[i\sum_{k}g_{k}J_{k}]$ and compute their
moments $\langle J^{n}_{k}\rangle:=\mathrm{tr}[J_{k}^{n}\rho]$ for a given
state that this provides an answer to the above question. However, while this
intuition is correct in the case of unitary dynamics on pure states, it is
false more generally [17]. In the case of the rotation group, for example, it
is possible for $\langle J_{k}\rangle$ to both increase and decrease under
rotationally symmetric operations [17, 22].

Given that a disconnect occurs between the symmetry principle and the
generators of the group as observables for mixed quantum states, one might
therefore conjecture that the problem requires an additional entropic
accounting, and we must supplement our analysis with the von Neumann entropy
$S(\rho)$ of the quantum state (or any general entropies that are a function
of the spectrum of the state, such as the Rényi entropies) to determine the
solution. Again, this turns out to still be insufficient, and it has been
shown that even if we consider all moments of the generators and the entire
spectrum of the quantum state $\rho$, this still does not answer the above
question [17]. The missing asymmetry ingredient is instead a non-trivial
combination of quantum information aspects and physics specific to the
symmetry group.

Recent work [58] has provided a complete set of necessary and sufficient
conditions for asymmetry which fully determine the state interconversion
structure with respect to a symmetry group $G$. However, this set of
conditions turns out to not be particularly intuitive and moreover forms an
infinite set of conditions that must be checked. The present work unpacks
these conditions, determines the minimal set of conditions needed and obtains
conditions that could be used in practical situations.

###  I.1 Main results of the paper

The complete set of asymmetry conditions in [58] are framed in terms of
correlations between the given quantum system $A$ and quantum frame systems
$R$ that is locally in a state $\eta_{R}$ that transforms non-trivially under
the group action. These correlations are measured via the single-shot
conditional entropy measure $H_{\rm min}(R|A)$, which is a central quantity in
quantum encryption [59]. However, the problem is that the complete set of
conditions requires this entropy to be computed for all possible reference
frame states $\eta_{R}$, and so the question is whether one can reduce to a
much simpler set of conditions for asymmetry theory.

The main results of this work are as follows:

  1. 1.

We prove that it suffices to consider any closed surface $\partial{\cal D}$ of
reference frame states $\eta_{R}$ that contains the maximally mixed state in
its interior.

  2. 2.

We prove that under $\varepsilon$–smoothing a finite number of reference
states suffice to determine those states accessible under $G$–covariant
channels.

  3. 3.

We prove that infinitesimal reference frames suffice to specify asymmetry, and
interconversion under $G$–covariant channels is equivalent to a single
entropic minimality condition at the maximally mixed state.

  4. 4.

We derive $O(d^{2})$ closed conditions to estimate the minimal depolarization
noise needed to make any given output state $\sigma$ accessible from a state
$\rho$ under $G$–covariant channels. These essentially take the form

$\displaystyle\log||\sigma^{\lambda}_{j}||_{1}-D_{2}(\rho^{\lambda}_{j}||{\cal
G}(\rho))\leq f(d,p),$ (1)

where $\sigma^{\lambda}_{j},\rho^{\lambda}_{j}$ are asymmetry modes with
respect to the group [60], and $f(d,p)$ is a function depending on the output
system dimension, its irrep structure, and the level of depolarization. The
function $D_{2}(X||Y)$ is a generalization of the Sandwiched $\alpha=2$ Rényi
divergence [61, 62].

Results (1–3) show that the structure of reference frame states that determine
the asymmetry properties of a system has a range of freedoms. In particular,
result (3) is surprising because it is contrary to what is expected from
previous work on this topic. Previously, it was natural to conjecture that in
order to specify the asymmetry of a state one should make use of reference
frame states that encode a group element as distinguishably as possible.
Finally, result (4) exploits the general structure analysed to provide
conditions, and could find application in describing symmetry-constrained
quantum information in concrete settings.

##  II Symmetry constraints and relational physics

Quantum entanglement [4] is usually understood as associated with a pre-order
$\succ_{e}$ on quantum states, defined by a class of quantum channels called
Local Operations and Classical Communications (LOCC). The set of states that
can be generated under LOCC are called separable states, and any other state
is then said to have non-trivial entanglement. The pre-order is defined as
$\rho\succ_{e}\sigma$ if and only if we can transform from $\rho$ into
$\sigma$ via an LOCC channel. This provides the resource-theoretic formulation
of entanglement.

This general perspective on properties of quantum systems can be used in the
above problem on transforming between quantum states under a symmetry
constraint. Specifically, we can identify a symmetry ordering $\succ$ on
quantum states, defined now by $\rho\succ\sigma$ whenever it is possible to
transform from $\rho$ into $\sigma$ via a quantum channel that respects a
given symmetry group $G$. The symmetry pre-order then defines what it means
for one quantum state to be more asymmetric than another with respect to the
group $G$.

We can make this precise in the following way. Given a quantum system $A$,
with associated Hilbert space ${\cal H}_{A}$, we denote by ${\cal B}({\cal
H}_{A}$) the space of bounded linear operators on ${\cal H}_{A}$. A symmetry
group $G$ acts on the system via a unitary representation $U(g)$ on ${\cal
H}_{A}$. States of $A$ are positive, trace-one operators $\rho\in{\cal
B}({\cal H}_{A})$, and at the level of the density operator $\rho$ the
symmetry group acts as ${\cal U}_{g}(\rho):=U(g)\rho U(g)^{\dagger}$. A
quantum channel ${\cal E}:{\cal B}({\cal H}_{A})\rightarrow{\cal B}({\cal
H}_{B})$ is a completely-positive, trace-preserving map [2] that sends states
of an input system $A$ to states of some output system $B$. A quantum channel
${\cal E}$ is then said to be symmetric, or $G$–covariant, with respect to a
group action if ${\cal E}(U(g)\rho U(g)^{\dagger})=U(g){\cal
E}(\rho)U(g)^{\dagger}$ for all $g\in G$ and all states $\rho$ of the input
system. Expressed purely in terms of composition of channels this amounts to

$[{\cal U}_{g},{\cal E}]=0\mbox{ for all }g\in G.$ (2)

We then have that $\rho\succ\sigma$ when there is a $G$–covariant channel
${\cal E}$ such that $\sigma={\cal E}(\rho)$. Moreover, a measure of the
system’s asymmetry is any real-valued function ${\cal M}$ on quantum states
such that if $\rho\succ\sigma$ then it must be the case that ${\cal
M}(\rho)\geq{\cal M}(\sigma)$.

A number of measures of asymmetry have been developed, such as relative
entropy measures [28, 63], the skew-Fisher information [64, 17, 18], and the
purity of coherence [31]. Any such monotone ${\cal M}(\rho)$ provides a
necessary condition for a transformation to be possible. However, what is a
harder question is whether one can determine a sufficient set of monotones.
Any such set of measures would encode all the features of the quantum system
that relate to the symmetry constraint.

Very recently [58] just such a complete set of measures has been found, in
terms of single-shot entropies. The monotones appearing in these relations are
the quantum conditional min-entropies [59], which are defined, for some state
$\Omega_{RA}$ on a bipartite system $RA$, as:

$\displaystyle H_{\mathrm{min}}(R|A)_{\Omega}:=-\log\inf_{X_{A}\geq
0}\\{\mathrm{tr}[X_{A}]:\mathbbm{1}_{R}\otimes X_{A}\geq\Omega_{RA}\\},$

where the infimum ranges over all positive semidefinite operators $X_{A}$ on
Hilbert space ${\cal H}_{A}$. For any state $\rho_{A}$ a complete set of
measures is then given by

$\displaystyle H_{\eta}(\rho)\coloneqq H_{\mathrm{min}}(R|A)_{{\cal
G}(\eta\otimes\rho)}.$ (3)

where $\eta_{R}$ is an arbitrary quantum state on an external reference frame
system $R$, and the single-shot entropy is evaluated on the bipartite state

${\cal G}(\eta_{R}\otimes\rho_{A}):=\int\\!\\!dg\,\,{\cal
U}_{g}(\eta_{R})\otimes{\cal U}_{g}(\rho_{A}).$ (4)

In terms of transformations between quantum states under a $G$–covariant
channel, we now have the following result.

######  Theorem 1.

[58] Let $A$, $B$, and $R$ be three quantum systems with respective
Hamiltonians $H_{A}$, $H_{B}$ and $H_{R}$ and dimensions $d_{A}$, $d_{B}$ and
$d_{R}$. Furthermore, let the reference system $R$ be such that $d_{R}=d_{B}$
and $H_{R}=-(H_{B})^{T}$. The state transformation
$\rho_{A}\rightarrow\sigma_{B}$ is possible under a $G$–covariant operation if
and only if

$\Delta H_{\eta}\geq 0,$ (5)

for all states $\eta_{R}$ on ${\cal H}_{R}$, where we have defined

$\displaystyle\Delta H_{\eta}=H_{\eta}(\sigma)-H_{\eta}(\rho),$ (6)

as the difference in entropy between input and output systems.

As shown in the original paper, the infinite set of entropic conditions
outlined in Theorem 1 can be reformulated as a semi-definite program that can
be solved efficiently for sufficiently low-dimensional quantum systems.
However, for larger system sizes it quickly becomes computationally intensive.
Moreover, without simplification, working with these expressions analytically
is not an option, since we have an infinite set of conditions and the physics
involved remains hidden.

###  II.1 Appearance of relativistic features in the quantum-information
framework

In quantum gravity, one has the Wheeler-de Witt equation [65] that provides a
Hamiltonian constraint ${H|\Psi\rangle=0}$ for a global wavefunction
$|\Psi\rangle$. This in particular implies global time-translation covariance,
and raised the question of how observed dynamics are consistent with this
condition. One answer to this question was presented by Page and Wootters
[66], who argued that time-evolution of subsystems should be properly viewed
in terms of relational correlations between subsystems.

The above complete set of entropic conditions for general covariant
transformations has links with this formalism. In particular, the following
features appear from the quantum information-theoretic treatment when
specialised to $G$ being the time-translation group [58]:

  * •

External reference frame systems automatically appear in the information-
theoretic analysis.

  * •

The Hamiltonians on $R$ and $B$ obey $H_{R}+H_{B}^{T}=0$ as matrices.

  * •

The properties of any system $A$ that transform non-trivially under the
symmetry group are fully described by correlations between $R$ and $A$.

  * •

A single-shot Page–Wootters condition emerges in the classical reference frame
limit in terms of optimal guessing-probabilities of the time parameter.

  * •

Local gauge symmetries can be formulated with a causal structure on asymmetry
resources [67].

The appearance of a Page–Wootters condition is surprising. The classical limit
here is when reference frame $\eta$ acts as a good clock, in the sense that
one can encode the classical information $t$ into it in such a way that one
can discriminate between two different values $t_{1}$ and $t_{2}$ with high
probability.

For this regime, the state $\Omega_{RA}={\cal G}(\eta_{R}\otimes\rho_{A})$
tends to a classical-quantum state, with $R$ behaving as a classical
‘register’ for $t$. However, it can be shown that for a classical quantum
state $\Omega_{RA}$, where $R$ is classical, the single-shot entropy
corresponds exactly to an optimal guessing probability [68]. More precisely,
it can be proved that

$-\log H_{\rm min}(R|A)_{\Omega}=p_{\rm opt}(t),$ (7)

where $p_{\rm opt}(t)$ is the optimal guessing probability for the value of
$t$, over all generalised POVM measurements, given the state $U(t)\rho
U(t)^{\dagger}$. This provides a refinement of the Page–Wootters formalism.

This interpretation of the $H_{\rm min}(R|A)_{\Omega}$ terms extends to
arbitrary states on $R$. In the fully general case it quantifies the optimal
singlet fraction [68], the degree to which the state $\Omega_{RA}$ can be
transformed to a maximally entangled (perfectly correlated) state through
action on $A$ alone. It is also possible to include thermodynamics into this
setting without much complication. For this extension, if we consider varying
the state $\eta_{R}$ we also smoothly interpolate between free energy–like
conditions and clock conditions [58].

The above features come solely from the single-shot quantum-information
formalism of the problem, and show that these aspects are fundamental. It
therefore motivates a deeper analysis of the complete set of entropic
conditions with the aim of unpacking the physical content and determining the
minimal information-theoretic conditions that describe fully general symmetric
transformations of quantum systems.

###  II.2 Warm-up example: a curious dependence on reference frame states

It is useful to first illustrate special cases of the $H_{\rm min}(R|A)$
conditions for the elementary case of channels sending a single qubit to a
single qubit under a $U(1)$ covariant symmetry constraint. For concreteness we
take this to be time-translation $U(t)=e^{itH_{A}}$ under a qubit Hamiltonian
$H_{A}=\sigma_{z}$, where $(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli
matrices for the qubit system.

We must therefore consider an auxiliary qubit reference frame $R$ in a state
$\eta_{R}$ with $H_{R}=-\sigma_{z}$, and compute the conditional min-entropy
on the joint state

${\cal G}(\eta_{R}\otimes\rho_{A})=\int_{0}^{2\pi}\\!\\!\frac{dt}{2\pi}\,{\cal
U}_{t}(\eta_{R})\otimes{\cal U}_{t}(\rho_{A}).$ (8)

We also choose
$\rho_{A}=\frac{1}{2}(\mathbbm{1}+\frac{1}{2}\sigma_{x}+\frac{1}{2}\sigma_{z})$
and look at how each choice of reference frame state $\eta_{R}$ constrains the
region of quantum states accessible under time-covariant quantum channels. For
fixed input state $\rho$, we define the set

${\cal T}_{\eta}:=\\{\sigma_{A}:H_{\eta}(\sigma)\geq H_{\eta}(\rho)\\}.$ (9)

In other words, ${\cal T}_{\eta}$ is the region of quantum states the
reference frame state $\eta_{R}$ classes as admissible for time-covariant
transformations. As such, $\sigma\in{\cal T}_{\eta}$ constitutes a necessary,
but not sufficient, condition on the pre-order $\rho\succ\sigma$.

The natural first choice for a reference frame state is
$\eta_{R}=|+\rangle\langle+|$; with a uniform superposition over energy
eigenstates, this is in a sense the ‘best’ clock state one can find for the
qubit in that it can encode a single bit of data about the parameter $t$,
which is the maximum allowed by the Holevo bound [2]. The region $S_{\eta}$ in
this case is plotted in Fig. (1)(c).

Figure 1: “Less is more”: near–symmetric reference frame states are optimal.
We consider a qubit system under $G=U(1)$ symmetry. Given a reference frame
state $\eta_{R}$ (blue dot), the shaded region ${\cal T}_{\eta}$ corresponds
to potential states in the Bloch sphere which $\eta_{R}$ classes as accessible
from the state $\rho$ (the black dot) under a covariant transformation. The
black curve marks the boundary of all states that are covariantly accessible
from $\rho$, and is obtained from the intersection of all regions ${\cal
T}_{\eta}$. Suprisingly, the high coherence state
$\eta_{R}=|+\rangle\langle+|$ gives a weak bound (figure (c)), while in
contrast taking $\eta_{R}$ very close to $|0\rangle\langle 0|$ or
$|1\rangle\langle 1|$ provides complete constraints (the combination of
figures (a) and (d)).

We now consider other choices of reference frame states, and find a surprising
result. If we take $\eta_{R}=|\varphi\rangle\langle\varphi|$ with
$|\varphi\rangle$ approaching either $|0\rangle$ or $|1\rangle$, the region
${\cal T}_{\eta}$ provides a better approximation to the actual region of
quantum states accessible under time-covariant channels. This is shown in Fig.
(1).

We also find the following striking result: the accessible region is exactly
recovered if we consider just two reference frame states
$\eta_{0}=|\varphi_{0}\rangle\langle\varphi_{0}|$ and
$\eta_{1}=|\varphi_{1}\rangle\langle\varphi_{1}|$, where $\eta_{0}$ is
infinitesimally close to $|0\rangle\langle 0|$ and $\eta_{1}$ infinitesimally
close to $|1\rangle\langle 1|$ (a proof may be found in Appendix G.1.1).
However, if we took the reference frame states to be exactly equal to these
pure symmetric states, then ${\cal T}_{\eta}$ is the entire Bloch sphere, and
the constraints determined by the reference frame state disappear completely!

This suggests that the constraints coming from the reference frame, via the
correlations in the state $\Omega(\rho_{A})$, have a non-trivial and counter-
intuitive dependence on the state $\eta_{R}$. However, this simple example
also illustrates that there are significant redundancies in the entropic set
of conditions – we have reduced from having to compute infinitely many
conditions to just two conditions. The question then becomes whether such
features carry over to more general situations, and to what degree can we
reduce the set of reference frames so as to determine the minimal relational
data needed to specify the asymmetry of a quantum system with respect to a
general group $G$.

##  III Sufficient surfaces of reference frame states for a general group $G$

We shall begin our analysis by reducing the set of reference frames needed
substantially and establishing high-level results. These shed light on the
structure of the problem and lead to our tractable set of conditions in
Section V.

###  III.1 Basic reference frame redundancies

The entropic conditions in Theorem 1 are over-complete, and contain a large
number of redundancies. Firstly, given two reference states $\eta_{0}$ and
$\eta_{1}$ on $R$ such that

$\displaystyle\eta_{1}={\cal V}(\eta_{0}),$ (10)

for some unitary channel ${\cal V}_{R}$ such that

$[{\cal V}_{R}\otimes\text{id}_{A},{\cal G}]=0,$ (11)

then it can be shown (see Appendix B.1) that ${\Delta H_{\eta_{0}}=\Delta
H_{\eta_{1}}}$ for all possible input and output states. This invariance is a
special case of the following lemma, which we prove in Appendix B.1:

######  Lemma 2.

Let ${\cal V}_{R}:{\cal B}({\cal H}_{R})\rightarrow{\cal B}({\cal
H}_{R^{\prime}})$ and ${\cal W}_{A}:{\cal B}({\cal H}_{A})\rightarrow{\cal
B}(H_{A^{\prime}})$ be local isometries that jointly commute with the
$G$-twirl, i.e.

$\displaystyle[{\cal V}_{R}\otimes{\cal W}_{A},{\cal G}]=0.$ (12)

We then have:

$\displaystyle H_{{\cal V}(\eta)}({\cal W}(\rho))=H_{\eta}(\rho)$ (13)

for any pair of quantum states $\eta$ and $\rho$ on ${\cal H}_{R}$ and ${\cal
H}_{A}$ respectively.

A second set of redundancies comes from considering the asymmetric modes of
the input state. Let the asymmetric modes of a state $\rho$ be denoted
modes($\rho$). Then it is known [60] that

$\displaystyle\text{ modes}(\sigma)\subseteq\text{modes}(\rho),$ (14)

is a necessary condition for a $G$–covariant transition from $\rho$ to
$\sigma$. Moreover, if we assume that the condition Eq. (14) holds, then it
suffices to range only over reference frame states $\eta$ such that
modes($\eta$) = modes($\rho$) (a proof is given in Appendix C.2).

###  III.2 Necessary and sufficient surfaces of reference frame states

It turns out that the entropic set of conditions have a more non-obvious kind
of redundancy. Any reference state $\eta$ can be written in an orthogonal
basis of Hermitian operators
$\\{\frac{\mathbbm{1}}{d},X_{1},\dots,X_{d^{2}-1}\\}$ as

$\displaystyle\eta(\bm{x})\coloneqq\frac{\mathbbm{1}}{d}+\sum_{k=1}^{d^{2}-1}x_{k}X_{k},$
(15)

where $x_{k}\in\mathbb{R}$ and $\norm{X_{k}}_{\infty}=\frac{1}{d}$ for all
$k\in\\{1,\dots,d^{2}-1\\}$ provide coordinates for the state. With this in
mind, we now have the following result:

######  Theorem 3.

(Sufficient surfaces of states). Let all states and systems be defined as in
Theorem 1. Let $\partial{\cal D}$ be any closed $d^{2}-2$ dimensional surface
in the state space of $R$ that has $\frac{\mathbbm{1}}{d}$ in its interior.
The state transformation $\rho\rightarrow\sigma$ is possible under a
$G$–covariant channel if and only if

$\Delta H_{\eta}\geq 0,$ (16)

for all reference frame states $\eta\in\partial{\cal D}$.

A proof is given in Appendix D.2, and follows from fact that the conditional
min-entropies behave particularly nicely under the application of a partial
depolarizing channel on the reference system (see Lemma 18).

Combined with the redundancies of Sec. III.1, we have that only a subset of
$\partial{\cal D}$ will produce non-trivial constraints – namely the
intersection of $\partial{\cal D}$ with states having asymmetric modes,
quotiented by the action of the unitary sub-group of channels ${\cal V}_{R}$
obeying Equation (11).

Figure 2: (Sufficient surfaces of reference frames). There is extensive
freedom in the choice of sufficient reference states. According to Theorem 3,
any surface inside the set of all reference states ${\cal S}_{R}$ that
encloses the maximally mixed state is a sufficient set of reference frames –
the three surfaces shown all provide the same information. The blue region is
the set of symmetric states.

###  III.3 A finite set of reference states under $\varepsilon$–smoothing

It is natural to consider a ‘smoothed’ version of the asymmetry conditions in
which we are limited to some $\varepsilon$-ball resolution around states [59,
69], where an $\varepsilon$-ball around a state $\tau$ is defined by

${\cal B}_{\varepsilon}(\tau)\coloneqq\\{\tilde{\tau}\in{\cal B}({\cal
H}):D(\tau,\tilde{\tau})\leq\varepsilon\\},$ (17)

and $D$ is the generalized trace distance [70]. This is physically motivated
by the fact that in any practical experimental scenario, two states that are
in a sense close cannot be distinguished up to some finite precision in the
measurement apparatus. Here we prove that, if we allow for some
$\varepsilon$-probability of error in the transformation, then we can restrict
to a finite set of reference frame states.

To perform the $\varepsilon$-smoothing over the reference system, we need the
following lemma, which states that the entropic relations are continuous
functions of the reference frame state.

######  Lemma 4.

For any $\tilde{\eta}\in{\cal B}_{\varepsilon}(\eta)$, we have

$\displaystyle|\Delta H_{\tilde{\eta}}-\Delta
H_{\eta}|\leq\frac{2d_{R}^{2}\varepsilon}{\ln 2(1-2\varepsilon)}.$ (18)

Note that if we further restrict $\tilde{\eta}$ to normalised states and
define $r(\varepsilon)\coloneqq\frac{2d_{R}^{2}\varepsilon}{\ln 2}$ this
simplifies to

$\displaystyle|\Delta H_{\tilde{\eta}}-\Delta H_{\eta}|\leq r(\varepsilon).$
(19)

###### Proof.

A proof is given in Appendix E.1. ∎

Therefore, an $\varepsilon$-variation in the choice of the reference state
$\eta$ corresponds to an $\varepsilon$-small variation in the entropy
difference $\Delta H_{\eta}$.

A corollary of Theorem 3 is that it is sufficient to consider only reference
frame states of the form Eq. (15) that live on the surface of a sphere about
the maximally mixed state. It can be shown [71, 72] that there exists a finite
$\varepsilon$-net $\\{\eta_{1},\eta_{2},\dots,\eta_{N}\\}$ covering this set
of reference states. This gives rise to the following result:

######  Theorem 5.

Given any $\varepsilon>0$ resolving scale, there is a finite set of reference
frame states ${\cal N}\coloneqq\\{\eta_{k}\\}_{k=1}^{N}$ with
$N=O\left((1+\frac{1}{\varepsilon})^{d^{2}-1}\right)$ that leads to the
following cases:

  * •

if $\Delta H_{\eta_{k}}<0$ for any $\eta_{k}\in{\cal N}$ then
$\rho\rightarrow\sigma$ is forbidden under all $G$–covariant quantum channels.

  * •

if $\Delta H_{\eta_{k}}\geq r(\varepsilon)$ for all $\eta_{k}\in{\cal N}$ then
$\rho\rightarrow\sigma$ under a $G$–covariant quantum channel.

  * •

For each $\eta_{k}\in{\cal N}$ that has $0\leq\Delta
H_{\eta_{k}}<r(\varepsilon)$ we can obtain $O(\varepsilon)$ upper bound
estimates of the minimal asymmetry resources needed to realise the
transformation.

A proof is provided in Appendix E.2.

This implies that the entropic relation can be checked on a finite number of
reference frames states, and furthermore each individual reference frame state
can give us some information.

The above result is potentially of interest in numerical studies of low-
dimensional systems. However, it does not shed much additional light on the
structure of covariant state transformations. Therefore, instead of developing
this line further here, we look at a limiting regime of reference frame states
that make things clearer. This in turn lead to a more user-friendly set of
conditions for ‘smoothed’ interconversions in Section V.

##  IV Infinitesimal reference frames and a single minimality condition for
asymmetry

While we have a finite number of conditions for smoothed asymmetry, these are,
by construction, of only approximate validity, and are not very physically
informative. The surface condition of Theorem 3 reduces the problem
significantly, but still leaves us with an infinite set of reference frames to
check.

However we can always take the region ${\cal D}$ in Theorem 3 to be an
arbitrarily small region around the maximally mixed state, and so restrict to
reference frame states that are arbitrarily close to being trivial, and this
does not affect the completeness of the set of reference frames. This then
shows that the counter-intuitive features we highlighted in the qubit case are
in fact generic and appear for any dimension and any group action.

A statement of this is as follows.

######  Theorem 6.

Given systems $A$ and $B$, it is possible to transform a quantum state
$\rho_{A}$ of $A$ into a state $\sigma_{B}$ of $B$ under a $G$–covariant
quantum channel if and only if $\Delta H_{\eta}$ has a local minimum at
$\eta_{R}=\frac{\mathbbm{1}}{d}$.

###### Proof.

We first note that $\Delta H_{\eta}=0$ whenever $\eta_{R}$ is symmetric (see
Lemma 11 in the appendices), and therefore $\Delta H_{\eta}=0$ when
$\eta_{R}=\frac{\mathbbm{1}}{d}$. If we assume
$\rho_{A}\xrightarrow{G}\sigma_{B}$, namely $\rho_{A}$ can be transformed into
$\sigma_{B}$ via a $G$–covariant channel, then Theorem 1 implies that $\Delta
H_{\eta}$ has a global minimum at $\eta_{R}=\frac{\mathbbm{1}}{d}$, which must
therefore be a local minimum as well. Conversely, if we assume $\Delta
H_{\eta}$ has a local minimum at $\eta_{R}=\frac{\mathbbm{1}}{d}$, then there
exists a neighbourhood ${\cal D}$ around $\eta_{R}=\frac{\mathbbm{1}}{d}$ in
which $\Delta H_{\eta}\geq\Delta H_{\mathbbm{1}/d}=0$. The conditional
entropies are continuous in $\eta_{R}$, so we have $\Delta H_{\eta}\geq 0$ on
$\partial{\cal D}$ as well. We conclude by Theorem 3 that
$\rho_{A}\xrightarrow{G}\sigma_{B}$. Therefore,
$\rho_{A}\xrightarrow{G}\sigma_{B}$ if and only if
$\eta_{R}=\frac{\mathbbm{1}}{d}$ is a local minimum of $\Delta H_{\eta}$. ∎

This result is surprising, since we would expect that ‘optimal’ information
would be obtained by evaluating the entropic relations on reference states
that are closest to being a “classical” reference frame [19], namely a state
$|\phi\rangle$ whose orbit $|\phi(g)\rangle:=U(g)|\psi\rangle$ under $G$
encodes all group elements completely distinguishably in the sense that

$\displaystyle\innerproduct{\phi(g)}{\phi(g^{\prime})}=\delta(g^{-1}g^{\prime}),$
(20)

for all $g,g^{\prime}\in G$.

The use of such a reference frame $\ket{\phi}$ allows us to ‘relativise’ all
symmetries and construct covariant versions of every aspect of quantum theory
[19, 73, 64, 74, 75, 76] This is done via a relativising map

$\displaystyle A\rightarrow\tilde{A}\coloneqq\int dg\ {\cal
U}_{g}(A)\otimes\outerproduct{\phi(g)}{\phi(g)},$ (21)

which can be viewed as a quantum to classical-quantum channel. In the limit of
a classical reference frame with
$\bra{\phi(g)}\ket{\phi(g^{\prime})}=\delta(g^{-1}g^{\prime})$, the mapping
becomes reversible via a readout from the classical register. However, for
reference frame states that are not classical, the encoding is fundamentally
noisy and so it is expected that the asymmetry features of a quantum state are
not properly described within the encoding. Theorem 6, however, tells us this
is not the case.

This ability to restrict relational data to the case of infinitesimally small
reference frames suggests that asymmetry theory admits a differential geometry
description in terms of tangent space of operators at the maximally mixed
state. Given that the interconversion of states under $G$–covariant channels
corresponds to a local minimum condition we might also expect that asymmetry
is described by information geometry [77, 78], and a single curvature computed
from the $H_{\rm min}(R|A)$ entropy. This would imply that the asymmetry
properties of a system are fully described by a form of quantum Fisher
information [64].

We find that, for our warm-up example of $G=U(1)$ on a qubit, something like
this does indeed occur. We show in Appendix G.1.1 that

$\rho\xrightarrow{U(1)}\sigma\mbox{ if and only if
}\partial_{\theta}^{2}(\Delta H_{\eta})\geq 0,\mbox{ at }\theta=0,\pi,$ (22)

where $\theta$ is the angle the Bloch vector of $\eta$ makes with the
$Z$–axis. Therefore we reduce the problem down to checking just two
conditions, framed as a curvature term in the angular direction. However, in
the radial direction one does not have a smooth variation. Instead, we
conjecture that under $\varepsilon$–smoothing a complete curvature condition
exists in all directions with the angular directions providing the non-trivial
constraint. In the next section we give explicit details on this troublesome
radial behaviour.

###  IV.1 Conical behaviour at the maximally mixed state

We now consider the behaviour of the $H_{\eta}$ entropies in the neighbourhood
of the maximally mixed state. Once again, we characterise reference frame
states using the co-ordinate system in Equation 15 as $\eta(\bm{x})$, where
the maximally mixed state is located at $\bm{x}=\mathbf{0}$. We further define

$\displaystyle\Phi_{\tau}(\bm{x})\coloneqq 2^{-H_{\eta(\bm{x})}(\tau)}.$ (23)

and
$\tilde{\Phi}_{\tau}(\bm{x})\coloneqq\Phi_{\tau}(\bm{x})-\Phi_{\tau}(\mathbf{0})$,
which gives the difference in $\Phi_{\tau}$ between the maximally mixed state
to the reference state at $\bm{x}$.

As a result of Lemma 19, $\tilde{\Phi}_{\tau}(\bm{x})$ has the following
properties:

######  Lemma 7.

Let $\lambda\geq 0$. Then for all $\bm{x},\lambda\bm{x}\in{\cal S}$, where
${\cal S}$ is the set of all co-ordinates corresponding to reference states:

$\displaystyle\tilde{\Phi}_{\tau}(\lambda\bm{x})=\lambda\tilde{\Phi}_{\tau}(\bm{x})$
(24)

Furthermore,

$\displaystyle\tilde{\Phi}_{\tau}(\bm{x})\geq 0.$ (25)

###### Proof.

A proof can be found at Appendix F. ∎

We conclude from the above lemma that $\tilde{\Phi}_{\tau}(\bm{x})$, and
consequently $\Phi_{\tau}(\bm{x})$, is linearly non-decreasing in every
direction out of the maximally mixed state. This means $\Phi_{\tau}(\bm{x})$
will, in general, have a conical form at the maximally mixed state; as a
result, unless $\Delta\Phi(\bm{x})$ is completely linear, it too will have a
conical form at the maximally mixed state. Using the defining relationship
between $\Phi_{\tau}(\bm{x})$ and the min–entropy, we further derive from
Lemma 7 that, for sufficiently small $\varepsilon\geq 0$,

$\displaystyle\Delta
H(\varepsilon\bm{x})=d\varepsilon\Delta\Phi(\bm{x})+O(\varepsilon^{2}),$ (26)

where $\Delta\Phi(\bm{x})\coloneqq\Phi_{\rho}(\bm{x})-\Phi_{\sigma}(\bm{x})$.
In the neighbourhood of the maximally mixed state, the behaviour of $\Delta
H_{\eta}$ is thus given by that of $\Delta\Phi$, and so in this single-shot
regime we do not in general have smooth behaviour.

###  IV.2 Structure of $\Phi_{\tau}(x)$ for simple cases

To illustrate this in practice, we now provide two examples on a qubit system
for $G=U(1)$ and $G=SU(2)$.

####  IV.2.1 The case of time–covariant $U(1)$

We first present $\Phi_{\tau}$ for time-covariant transformations in a non-
degenerate qubit. Consider a qubit with the Hamiltonian $\sigma_{z}$. The
states of this qubit are restricted to transforming among each other
exclusively via channels that commute with all time translations $\\{{\cal
U}_{t}:t\in[0,2\pi]\\}$, where ${\cal
U}_{t}(\cdot)=e^{-i\sigma_{z}t}(\cdot)e^{i\sigma_{z}t}$. This set of time-
translations form a unitary representation of the group $U(1)$.

We parameterise an arbitrary state $\tau$ of this qubit in its energy
eigenbasis as:

$\displaystyle\tau\coloneqq\begin{pmatrix}p_{\tau}&c_{\tau}\\\
c^{*}_{\tau}&1-p_{\tau}\end{pmatrix}$ (27)

We further use the (scaled) Pauli operators
$\\{\frac{\mathbbm{1}}{2},\frac{\sigma_{x}}{2},\frac{\sigma_{y}}{2},\frac{\sigma_{z}}{2}\\}$
as our basis for characterising reference frame states according to Equation
15. The Bloch vector of a state, $(x,y,z)$, gives its co-ordinates in this
basis. A direct computation (see Appendix G.1) gives

$\Phi_{\tau}(x,y,z)=\frac{\absolutevalue{c_{\tau}}^{2}}{1-p_{\tau}}\frac{x^{2}+y^{2}}{4z}+\frac{z}{2}+\frac{1}{2}$
(28a) for the region
$0\leq\frac{\sqrt{x^{2}+y^{2}}}{2z}\leq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}$,
and
$\Phi_{\tau}(x,y,z)=\left(p_{\tau}-\frac{1}{2}\right)z+\absolutevalue{c_{\tau}}\sqrt{x^{2}+y^{2}}+\frac{1}{2}$
(28b) for the region
$\frac{\sqrt{x^{2}+y^{2}}}{2z}\geq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}$
and
$\frac{\sqrt{x^{2}+y^{2}}}{2z}\leq-\frac{p_{\tau}}{\absolutevalue{c_{\tau}}}$,
and finally
$\Phi_{\tau}(x,y,z)=-\frac{\absolutevalue{c_{\tau}}^{2}}{p_{\tau}}\frac{x^{2}+y^{2}}{4z}-\frac{z}{2}+\frac{1}{2}$
(28c)

for the region
$0\geq\frac{\sqrt{x^{2}+y^{2}}}{2z}\geq-\frac{p_{\tau}}{\absolutevalue{c_{\tau}}}$.
When neither $\rho$ nor $\sigma$ is symmetric, one can find neighbourhoods
around the poles of the Bloch sphere in which $\Delta\Phi(\bm{x})$ is not
completely linear if and only if
$\frac{\absolutevalue{c_{\rho}}^{2}}{1-p_{\rho}}=\frac{\absolutevalue{c_{\sigma}}^{2}}{1-p_{\sigma}}$
and
$\frac{\absolutevalue{c_{\rho}}^{2}}{p_{\rho}}=\frac{\absolutevalue{c_{\sigma}}^{2}}{p_{\sigma}}$.
Since these conditions are equivalent to $\rho={\cal U}_{t}(\sigma)$ for some
$t$, for arbitrary choices of $\rho$ and $\sigma$ we almost always expect a
conical singularity in $\Delta H_{\eta}$ at $\eta=\frac{\mathbbm{1}}{d}$.

Figure 3: Conical structure time-covariance. Shown here is
$\Phi_{\tau}(\bm{x})$ for time-covariant transformations in a non-degenerate
qubit from the state $\tau$ with $c_{\tau}=\frac{1}{3}$ and
$p_{\tau}=\frac{1}{3}$, plotted for $0\leq r\leq 1$, where
$r\coloneqq\sqrt{x^{2}+y^{2}}$, and $-1\leq z\leq 1$. The non-negative linear
behaviour of $\Phi_{\tau}(\bm{x})$ in every direction out of the maximally
mixed state (i.e. along any straight line out of $r=0,z=0$) is readily seen in
(a) and generates the conical behaviour discussed in the text. Figure (b)
gives a contour plot of $\Phi_{\tau}(\bm{x})$ as a function of the reference
frame state Bloch vector.

This analysis illustrates how the original complete set of entropic conditions
has many redundancies. We explicitly see the conical behaviour as
$(x,y,z)\rightarrow(0,0,0)$ in that

$\displaystyle\Phi_{\tau}(\lambda(x,y,z))=\lambda\left(\Phi_{\tau}(x,y,z)-\frac{1}{2}\right)+\frac{1}{2},$
(29)

where $\lambda$ is a positive scaling factor. Furthermore, because $U(1)$ is
Abelian, we have that $[{\cal U}_{t}\otimes\mathbbm{1},{\cal G}]=0$ for all
$t$, so Lemma 2 implies, for any $r\geq 0$, that

$\displaystyle\Phi_{\tau}(r\cos(t),r\sin(t),z)=\Phi_{\tau}(r,0,z),$ (30)

and so $\Phi_{\tau}$ has cylindrical symmetry around the $z$-axis. More non-
trivially, Lemma 2 may also be applied to ${{\cal X}\otimes{\cal X}}$, where
${\cal X}(\cdot)\coloneqq\sigma_{x}(\cdot)\sigma_{x}$, since $[{\cal
X}\otimes{\cal X},{\cal G}]=0$. This means

$\displaystyle\Phi_{{\cal X}(\tau)}(x,y,z)=\Phi_{\tau}(x,-y,-z)$ (31)

According to the parameterisation of $\tau$ we have chosen, ${\cal X}(\tau)$
means $c_{\tau}\rightarrow c^{*}_{\tau}$ and $p_{\tau}\rightarrow 1-p_{\tau}$.
In this way, $\Phi_{\tau}$ for reference states in the bottom half of the
Bloch sphere (i.e. $z\leq 0$) can be calculated from $\Phi_{{\cal X}(\tau)}$
for reference states in the top half ($z\geq 0$).

Given any $\rho$ and $\sigma$, we can look at the minimality condition at
$(0,0,0)$ and obtain reference frame independent conditions that recover known
results [79] on necessary and sufficient conditions for time-covariant
transitions in a non-degenerate qubit:

$\displaystyle\frac{\absolutevalue{c_{\rho}}^{2}}{1-p_{\rho}}\geq\frac{\absolutevalue{c_{\sigma}}^{2}}{1-p_{\sigma}}$
$\displaystyle\text{ for }p_{\sigma}\geq p_{\rho},$ (32)

and

$\displaystyle\frac{\absolutevalue{c_{\rho}}^{2}}{p_{\rho}}\geq\frac{\absolutevalue{c_{\sigma}}^{2}}{p_{\sigma}}$
$\displaystyle\text{ for }p_{\sigma}\leq p_{\rho}.$ (33)

Comparing with Eq. (28), this means when $p_{\sigma}\geq p_{\rho}$, checking
$\Delta\Phi_{\eta}\geq 0$ for a single reference state with co-ordinates in
the range

$0\leq\frac{\sqrt{x^{2}+y^{2}}}{2z}\leq\min\left\\{\frac{1-p_{\rho}}{\absolutevalue{c_{\rho}}},\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\right\\}$
(34)

is sufficient to determine whether a covariant transition can occur.
Similarly, when $p_{\sigma}\leq p_{\rho}$, checking $\Delta\Phi_{\eta}$ for a
single reference state with co-ordinates in the range

$0\geq\frac{\sqrt{x^{2}+y^{2}}}{2z}\geq\max\left\\{-\frac{p_{\rho}}{\absolutevalue{c_{\rho}}},-\frac{p_{\sigma}}{\absolutevalue{c_{\sigma}}}\right\\}$
(35)

is sufficient.

####  IV.2.2 The case of $SU(2)$–covariant transformations.

We now consider the case of $G=SU(2)$ on a qubit. In this case, $G$–covariant
channels partially depolarise and may additionally invert the input state
about the maximally mixed state [22]. We will continue to write $\eta$ in its
Bloch basis as in the $U(1)$ example, and will parameterise $\tau$ as before.
Using the simplifying abbreviations
$\bm{\sigma}\coloneqq(\sigma_{x},\sigma_{y},\sigma_{z})^{T}$ and
$\bm{x}=(x,y,z)^{T}$, we have

$\displaystyle\eta(\bm{x})$
$\displaystyle\coloneqq\frac{1}{2}(\mathbbm{1}+\bm{x}\cdot\bm{\sigma}),$ (36)
$\displaystyle\rho(\bm{r})$
$\displaystyle\coloneqq\frac{1}{2}(\mathbbm{1}+\bm{r}\cdot\bm{\sigma}),$ (37)

Given $\bm{x}=(x,y,z)$ we define $\overline{\bm{x}}=(x,-y,z)$.

The form of $\Phi_{\tau}(\bm{x})$ is then given by (see Appendix G.2):

$\displaystyle\Phi_{\tau}(\bm{x})$
$\displaystyle=\begin{cases}\frac{1}{2}(1+\overline{\bm{x}}\cdot\bm{r})&\text{if
}\overline{\bm{x}}\cdot\bm{r}\geq 0,\\\
\frac{1}{2}(1-\frac{1}{3}\overline{\bm{x}})\cdot\bm{r}&\mbox{
otherwise}.\end{cases}$ (38)

We see that $\Phi_{\tau}(\bm{x})$ is piecewise linear with the plane
${\overline{\bm{x}}\cdot\bm{r}=0}$ distinguishing the two regions.

Consider an input state $\rho$ and an output state $\sigma$ that are not
maximally mixed. Letting $\bm{r}$ and $\bm{s}$ be the Bloch vectors of $\rho$
and $\sigma$ respectively, this means $\bm{r},\bm{s}\neq\bm{0}$. Let us
further restrict ourselves to the case where $\rho$ and $\sigma$ are not
located along the same diameter in the Bloch sphere. This implies both
$\bm{r}-\bm{s}\neq\bm{0}$ and that $\bm{r}-\bm{s}$ cannot be anti-parallel to
$\bm{s}$. Therefore, it is always possible to find
$\overline{\bm{x}^{\prime}}\neq\bm{0}$ such that $\overline{\bm{x}^{\prime}}$
lies strictly above both the plane $\bm{x}\cdot\bm{s}=0$ and the plane
$\bm{x}\cdot(\bm{r}-\bm{s})=0$. This means

$\displaystyle\overline{\bm{x}^{\prime}}\cdot\bm{r}>\overline{\bm{x}^{\prime}}\cdot\bm{s}>0,$
(39)

so $\Delta\Phi(\overline{\bm{x}^{\prime}})$ must be calculated from the top
solution in Eq. (38) as

$\displaystyle\Delta\Phi(\overline{\bm{x}^{\prime}})=\frac{1}{2}\bm{x}^{\prime}\cdot(\bm{r}-\bm{s})$
(40)

Conversely, $-\overline{\bm{x}^{\prime}}$ must lie strictly below both the
plane $\bm{x}\cdot\bm{s}=0$ and the plane $\bm{x}\cdot(\bm{r}-\bm{s})=0$, so
$\Delta\Phi(-\overline{\bm{x}^{\prime}})$ must be calculated from the bottom
solution in Eq. (38), which means

$\displaystyle\Delta\Phi(-\overline{\bm{x}^{\prime}})=\frac{1}{6}\bm{x}^{\prime}\cdot(\bm{r}-\bm{s}).$
(41)

We must therefore conclude that if $\rho$ and $\sigma$ are neither maximally
mixed nor located along the same diameter of the Bloch sphere, then there
exists $\overline{\bm{x}^{\prime}}$ such that:

$\displaystyle\Delta\Phi(\bm{0})=0$
$\displaystyle\neq\frac{2}{3}\bm{x}^{\prime}\cdot(\bm{r}-\bm{s})$ (42)
$\displaystyle=\Delta\Phi(\overline{\bm{x}^{\prime}})+\Delta\Phi(-\overline{\bm{x}^{\prime}})$
(43)

Since most choices of $\rho$ and $\sigma$ satisfy these requirements, we see
that $\Delta\Phi(\bm{x})$ is almost never completely linear. In this example,
$\Delta H_{\eta}$ also almost always has a conical singularity at
$\eta=\frac{\mathbbm{1}}{d}$.

##  V Robust symmetric transformations of general states with minimal
depolarization

Figure 4: Constructing state interconversion conditions. If, for a complete
set of reference frame states $\\{\eta\\}$, we construct some family of
covariant protocols that transform from a state $\rho$ to ${\cal
E}^{\eta}(\rho)$ that has a higher overlap with $\eta$ than $\sigma$ has with
$\eta$ then it is possible to transform from $\rho$ to $\sigma$ under a
$G$–covariant channel.

In principle the condition given in Theorem 6 gives a complete description of
the asymmetry properties of quantum states. However, as the preceding examples
have shown, standard tests for local minima are typically not applicable for
the functional $\Delta H_{\eta}$, and thus computing this necessary and
sufficient condition presents a technical challenge which we must leave for
future study. Instead, we can adopt a more physical perspective on the problem
and look for a complete set of conditions where we weaken the assumptions for
the interconversion. For example instead of $\rho\xrightarrow{G}\sigma$ we
could ask the question:

What is the minimal amount of depolarization noise we need to add to $\sigma$
so as to make it accessible from the initial state $\rho$ via a $G$–covariant
channel?

Since the maximally mixed state is invariant for any group action this form
applies to all symmetry groups $G$. It also incorporates robustness. Suppose,
for example, that $\sigma$ was essentially identical to $\rho$ except it has a
very small, but non-zero, $O(\varepsilon)$ mode that does not appear in
$\rho$. The strict conditions would say that it is impossible to transform
from $\rho$ to $\sigma$, yet it is clear that we only require $O(\varepsilon)$
amount of depolarising noise in order to make the transformation possible.
Therefore the above question is more physically relevant than the simple
‘yes/no’ question of exact interconversion.

The formulation of the problem therefore involves smoothing our output state
with the maximally mixed state:

$\displaystyle\sigma\rightarrow\sigma_{p}:=(1-p)\sigma+p\frac{\mathbbm{1}}{d},$
(44)

where $p$ is an error probability, and we wish to estimate how small $p$ can
be so as to make $\rho\xrightarrow{G}\sigma_{p}$ possible via a covariant
quantum channel. As we will see, this set of sufficient conditions has the
benefit of being straightforward to compute.

We make use of two core ingredients for our results. First, note that any
state $\rho$ can be decomposed into independent modes of asymmetry [60]
labelled by $(\lambda,j)$:

$\displaystyle\rho=\sum_{\lambda,j}\rho^{\lambda}_{j}:\,\rho^{\lambda}_{j}=\sum_{\alpha}\mathrm{tr}\left[X^{(\lambda,\alpha)\dagger}_{j}\rho\right]X^{(\lambda,\alpha)}_{j},$
(45)

where $\lambda$ labels an irreducible representation (irrep) of $G$, $j$
labels the basis vector of the given irrep $\lambda$, $\alpha$ labels any
multiplicity degrees of freedom, and the set
$\left\\{X^{(\lambda,\alpha)}_{j}\right\\}$ form an orthonormal irreducible
tensor operator (ITO) basis for ${\cal B}({\cal H})$ (see Appendix A for
details). We denote the trivial irrep of the group by $\lambda=0$. It was
shown Ref. [60] that every $G$–covariant operation ${\cal E}:{\cal
E}(\rho)=\sigma$ acts independently on the different modes of the input state
such that

$\displaystyle{\cal E}(\rho^{\lambda}_{j})=\sigma^{\lambda}_{j},$ (46)

for any $(\lambda,j)$. In other words, a $G$–covariant quantum channel always
maps any given mode of the input state to the very same mode of the output
state, with no “mixing” between the different modes.

Secondly, we have the Sandwiched $\alpha$–Rényi divergence
$D_{\alpha}(\rho||\sigma)$ for two states $\rho,\sigma$ of a quantum system,
which is defined as [61, 62]

$D_{\alpha}(\rho||\sigma):=\frac{1}{\alpha-1}\log\mathrm{tr}\left[\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right]^{\alpha},$
(47)

whenever the support of $\rho$ lies in the support of $\sigma$, and is
infinite otherwise. Our results turn out to be most compactly expressed in
terms of the following generalization of the $\alpha=2$ Sandwiched Rényi
divergence, which extends the domain of the first argument to all linear
operators in the support of $\sigma$, and reproduces the standard definition
when that first argument is Hermitian:

$D_{2}(X||\sigma):=\log\mathrm{tr}\left(\left[\sigma^{-\frac{1}{4}}X\sigma^{-\frac{1}{4}}\right]^{\dagger}\left[\sigma^{-\frac{1}{4}}X\sigma^{-\frac{1}{4}}\right]\right).$
(48)

We now have the following theorem, which gives an estimate of the minimal
amount of depolarization needed in order to make a transformation possible
under covariant channels.

######  Theorem 8.

Let $0\leq p\leq 1$ be a probability. There exists a $G$–covariant channel
transforming $\rho$ into
$\sigma_{p}\coloneqq(1-p)\sigma+p\frac{\mathbbm{1}}{d}$ if

$\displaystyle D_{2}(\rho^{\lambda}_{j}||{\cal G}(\rho))\geq\log
g^{\lambda}_{j}(\sigma)-\log n^{-1}\left(\lambda_{\rm
min}+\frac{p}{d(1-p)}\right),$ (49)

for all $\lambda\neq 0,j$, where $\lambda_{\mathrm{min}}$ is the smallest non-
zero eigenvalue of ${\cal G}(\sigma)$, and
$g^{\lambda}_{j}(\sigma)\coloneqq\sum_{\alpha}\absolutevalue{\mathrm{tr}[X^{(\lambda,\alpha)\dagger}_{j}\sigma]}$.
The operators $X^{(\lambda,\alpha)}_{j}$ form an ITO basis for ${\cal B}({\cal
H}_{B})$, where ${\cal H}_{B}$ is the Hilbert space of the output system, and
$n$ is the sum of the dimensions of all distinct non-trivial irreps appearing
in the representation of $G$ on ${\cal B}({\cal H}_{B})$.

A proof is given in Appendix H, and exploits the SDP duality structure for
covariant interconversion to determine an admissible range of values for $p$.
This analysis is done by using a family of Pretty Good Measurement schemes
[80] that attempt to generate as large a fidelity with the set of all
reference frame states $\eta_{R}$ as possible. By modifying this general
strategy, we anticipate that the results presented here can almost certainly
be improved upon, and it would be of interest to study how well similar
families perform relative to the exact SDP solution to the covariant
interconversion problem.

Figure 5: Depolarization conditions for a qubit system. An exact treatment of
how states transform under $G$–covariant channels requires non-trivial
computations, however the closed, robust conditions given in Theorem 9 provide
a simple means to estimate the interconversion structure. Here we demonstrate
this for the case of $d=2$ and $G=U(1)$. Left: The black dot shows an initial
qubit state $\rho$ with Bloch vector
$\bm{r}\coloneqq(\frac{1}{2},0,\frac{1}{2})$. The large blue shaded region
(SDP) defines the full set of output qubit states that can be reached under
covariant maps for the group $G=U(1)$ of time-translations generated by
$H=\sigma_{z}$, computed via the exact interconversion conditions. The smaller
pink shaded region (SC) overlapping this shows the region given by the
conditions given in Theorem 9. Right: Here the black dot now represents the
output qubit state $\sigma$, and the shaded regions correspond to the set of
input states $\\{\rho\\}$ that can be transformed into $\sigma$ via a
$G$–covariant channel for the full set of conditions (SDP) and the robust
approximation conditions (SC). (Note that despite appearances the boundary of
the SDP region is not linear.)

When the input and output systems are the same, we can provide a strengthening
of the above conditions to the following form:

######  Theorem 9.

Consider transformations from a quantum system $A$ to itself. Assume for
simplicity that ${\cal G}(\sigma)$ is full-rank. There exists a $G$–covariant
channel transforming $\rho$ into
$\sigma_{p}\coloneqq(1-p)\sigma+p\frac{\mathbbm{1}}{d}$ if $\rho=\sigma_{p}$
or if for any $q\in(q^{*},1]$ we have

$\displaystyle D_{2}(\rho^{\lambda}_{j}||{\cal G}(\rho))\geq\log
g^{\lambda}_{j}(\sigma_{p}(q))-\log n^{-1}\lambda_{\mathrm{min}}({\cal
G}[\sigma_{p}(q)]),$ (50)

for all $\lambda\neq 0,j$, where we have
$\sigma_{p}(q)\coloneqq\sigma_{p}-(1-q)\rho$,
$q^{*}\coloneqq\min\\{q\in\mathbb{R}^{+}:{\cal G}(\sigma_{p}(q))\geq 0\\}$,
$\lambda_{\mathrm{min}}({\cal G}[\sigma_{p}(q)])$ is the smallest eigenvalue
of ${\cal G}[\sigma_{p}(q)]$, and all other terms are as in Theorem 8.

Given that the analysis is built on Pretty Good Measurement schemes for
resolving group elements, it is expected that a measure-and-prepare strategy
(such as above, or a slightly modified version) will behave well when $\rho$
has many large modes of asymmetry for the group. While this can be achieved
for systems with a large dimension, we find that even for low dimensional
systems the conditions perform well. For example, in Fig. (5), we plot the
performance of the sufficient condition for the group $G=U(1)$ of time
translations generated by the Hamiltonian $H=\sigma_{z}$ for a qubit system in
initial state
$\rho=\frac{1}{2}(\mathbbm{1}+\frac{1}{2}\sigma_{x}+\frac{1}{2}\sigma_{z})$.
On the left, we plot the set of output states $\\{\sigma\\}$ for which our
sufficient condition tells us are accessible from $\rho$ (pink shaded region)
relative to the full set of accessible output states granted by the complete
set of conditions stated in Theorem 1 (blue shaded region).

We also note that we can recast our sufficient conditions in terms of familiar
norms. We first note that we always have
$g^{\lambda}_{j}(\sigma)\leq\norm{\sigma_{j}^{\lambda}}_{1}$, where
$\norm{X}_{1}\coloneqq\mathrm{tr}\left[\sqrt{X^{\dagger}X}\right]$ is the
trace norm. Therefore it follows from Theorem 8 that exists a $G$–covariant
operation transforming $\rho$ into $\sigma$ if

$\displaystyle
n^{-1}\norm{\overline{\rho}^{\lambda}_{j}}_{2}^{2}\geq\norm{\tilde{\sigma}_{j}^{\lambda}}_{1},\quad\forall\lambda\neq
0,j,$ (51)

where the notation $\overline{\rho}^{\lambda}_{j}\coloneqq{{\cal
G}(\rho)}^{-\frac{1}{4}}\rho^{\lambda}_{j}{{\cal G}(\rho)}^{-\frac{1}{4}}$ and
$\tilde{\sigma}=\sigma\lambda_{\mathrm{min}}^{-1}$, and
$\norm{X}_{2}\coloneqq\sqrt{\mathrm{tr}[X^{\dagger}X]}$ is the Frobenius norm.
We note that $\norm{\sigma_{j}^{\lambda}}_{1}$ for $\lambda\neq 0$ is a known
asymmetry monotone that measures the the degree of asymmetry in the
$(\lambda,j)$–mode of $\sigma$ [60].

##  VI Outlook

In this work we have shown that the recent complete set of entropic conditions
for asymmetry can be greatly simplified, and more importantly, can be
converted into useful forms. The fact that the reference frames that are
needed to describe asymmetry can be taken to have arbitrarily small modes of
asymmetry suggest that a deeper analysis should be possible in terms of
differential geometry, as opposed to quantities such as the degree to which a
quantum state encodes group data. We expect that this should take the form of
a Fisher-like information [64, 17, 18], and in particular it is of interest to
see if it is possible to replace the $H_{\rm min}(R|A)$ entropy with the
conditional von-Neumann entropy $H(R|A)$, which would allow explicit analytic
computations.

Beyond this, a range of other interesting questions exist. For example we have
not exploited the duality relations [68, 70] between $H_{\rm min}(R|A)$ and
$H_{\rm max}(R|S)$ where $S$ is a purifying system for the state
$\Omega_{RA}$. For example, for the case of time-translation symmetry the
joint purified state admits two notable forms. The first is an energetic form:

$\Omega_{RAS}=\sum_{E}\sqrt{p(E)}|\varphi(E)\rangle_{RA}\otimes|E\rangle_{S}$
(52)

obtained from considering $\Omega_{RA}={\cal
G}(\eta_{R}\otimes\rho_{A})=\sum_{E}\Pi(E)(\eta_{R}\otimes\rho_{A})\Pi(E)$ as
an ensemble of states over energy sectors, $\Pi(E)$ being the projector onto
the energy $E$ subspace of $RA$. While the second is a temporal form, given by

$\Omega_{RAS}=\int\\!\\!dt\,\,|\phi(t)\rangle_{RAS},$ (53)

with $|\phi(t)\rangle_{RAS}$ being a purification of ${\cal
U}_{t}(\eta_{R})\otimes{\cal U}_{t}(\rho_{A})$. It would be of interest to
explore these two forms and also their connection to entropic uncertainty
relations.

Finally, it would also be valuable to see how the explicit conditions given by
Theorem 8 and Theorem 9 could be used in concrete settings, such as for
covariant quantum error-correcting codes [52, 53, 54], thermodynamics [58] or
metrology [21]. Moreover, the method of constructing these conditions can
certainly be improved upon by using more detailed covariant protocols.

##  VII Acknowledgements

We would like to thank Iman Marvian for helpful and insightful discussions,
and in particular for pointing out that our depolarization result is compactly
expressed in terms of a Sandwiched Rényi divergence. RA is supported by the
EPSRC Centre for Doctoral Training in Controlled Quantum Dynamics. SGC is
supported by the Bell Burnell Graduate Scholarship Fund and the University of
Leeds. DJ is supported by the Royal Society and also a University Academic
Fellowship.

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##  Appendix A Notation and background details

To any quantum system we have an associated Hilbert space ${\cal H}$, and the
set of bounded linear operators on this space denoted by ${\cal B}({\cal H})$.
Given a group $G$ we denote its representation on ${\cal H}$ by $U(g)$ and on
${\cal B}({\cal H})$ as ${\cal U}_{g}$ where ${\cal
U}_{g}(X)=U(g)XU(g)^{\dagger}$ for any $X\in{\cal B}({\cal H})$ and any $g\in
G$.

We define
$D(\rho,\sigma)\coloneqq\frac{1}{2}\norm{\rho-\sigma}_{1}+\frac{1}{2}\absolutevalue{\mathrm{tr}\rho-\mathrm{tr}\sigma}$
to be the generalized trace distance between any two $\rho,\sigma\in{\cal
B}({\cal H})$. We also define ${\cal S}_{\leq}({\cal H})$ to be the set of all
normalized and sub-normalized quantum states on ${\cal H}$.

A quantum channel ${\cal E}:{\cal B}({\cal H}_{A})\rightarrow{\cal B}({\cal
H}_{B})$, from a quantum system $A$ to a quantum system $B$, is a
superoperator that is both trace-preserving and completely positive [2]. A
quantum channel is covariant with respect to the group action if we have that

${\cal E}({\cal U}_{g}(\rho))={\cal U}_{g}({\cal E}(\rho)),$ (54)

for all quantum states $\rho\in{\cal B}({\cal H}_{A})$ and all $g\in G$. Note
here that the group representation of the input and output systems are
generally different, and we should strictly write ${\cal U}_{g}^{A}$ and
${\cal U}_{g}^{B}$ for each action. However, to simplify notation, we shall
use ${\cal U}_{g}$ throughout, as it does not cause ambiguity in practice. The
above condition can also be compactly written as $[{\cal U}_{g},{\cal E}]=0$
for all $g\in G$.

The reference system $R$ for $G$–covariant transformations from $A$ to $B$ is
chosen such that the representation of $G$ on ${\cal H}_{R}$, the Hilbert
space of system $R$, is dual to its representation on ${\cal H}_{B}$.

We also make use of an irreducible tensor operator (ITO) basis. An ITO
consists of a basis of operators ${X^{(\lambda,\alpha)}_{j}\in{\cal B}({\cal
H})}$ that have the property that

${\cal
U}_{g}\left(X^{(\lambda,\alpha)}_{j}\right)=\sum_{i}v^{\lambda}_{ij}(g)X^{(\lambda,\alpha)}_{i},$
(55)

where $(v^{\lambda}_{ji}(g))$ are the matrix components of the $\lambda$–irrep
of the group $G$ on ${\cal H}$. The irrep $\lambda$ may occur with
multiplicities, and so we also denote this as $(\lambda,\alpha)$ where
$\alpha$ is a multiplicity label for the irrep. Since
$\\{X^{(\lambda,\alpha)}_{j}\\}$ form a basis for ${\cal B}({\cal H})$ we may
decompose any operator, and in particular any quantum state as

$\rho=\sum_{\lambda,j}\left(\sum_{\alpha}\mathrm{tr}\left(X^{(\lambda,\alpha)\dagger}_{j}\rho\right)X^{(\lambda,\alpha)}_{j}\right)\coloneqq\sum_{\lambda,j}\rho^{\lambda}_{j}$
(56)

Since each $\rho^{\lambda}_{j}$ transforms irreducibly under the group action
${\cal U}_{g}(\cdot)$, this defines a decomposition of $\rho$ into modes of
asymmetry. If ${\cal E}$ is a covariant channel, then it does not mix modes,
and so

${\cal E}(\rho^{\lambda}_{j})={\cal E}(\rho)^{\lambda}_{j},$ (57)

for all $\lambda,j$. See [64, 60] for more details.

##  Appendix B Properties of min-entropies

In this section we review some useful properties of the conditional min-
entropies $H_{\rm min}(R|A)_{\Omega}$ and properties that relate to the case
of $\Omega_{RA}$ being a $G$-twirled bipartite quantum state.

It proves useful to define the functional $\Phi(M_{RA}):M_{RA}\mapsto
2^{-H_{\mathrm{min}}(R|A)_{M}}$ defined on bipartite Hermitian operators
$M_{RA}$:

######  Definition 10.

Let $M_{RA}$ be a Hermitian operator on ${\cal H}_{RA}$. Then we define the
argument of the conditional min-entropy of $M_{RA}$, $\Phi(M_{RA})$, via

$\displaystyle\Phi(M_{RA})\coloneqq\inf_{X_{A}\geq 0}\\{\mathrm{tr}[X_{A}]\ :\
\mathbbm{1}_{R}\otimes X_{A}-M_{RA}\geq 0\\}.$ (58)

We note the following known properties of the functions $\Phi(\cdot)$, proofs
of which can be found in [69] or are obvious from the definition.

  1. (P1)

(Scalar multiplication). $\Phi(\lambda M_{RA})=\lambda\Phi(M_{RA})$ for any
$\lambda\geq 0$.

  2. (P2)

(Convexity). $\Phi(pM_{RA}+(1-p)N_{RA})\leq p\Phi(M_{RA})+(1-p)\Phi(N_{RA})$
for any $p\in[0,1]$.

  3. (P3)

(Invariance under local isometries). Let $\mathcal{U}_{R}\coloneqq
U_{R}\otimes\mathbbm{1}_{A}(\cdot)U_{R}^{\dagger}\otimes\mathbbm{1}_{A}$ and
$\mathcal{V}_{A}\coloneqq\mathbbm{1}_{R}\otimes
V_{A}(\cdot)\mathbbm{1}_{R}\otimes V_{A}^{\dagger}$ be isometries on
subsystems $R$ and $A$ respectively. Then $\Phi({\cal U}_{R}\circ{\cal
V}_{A}(M_{RA}))=\Phi(M_{RA})$.

  4. (P4)

(Local data processing inequality). Let ${\cal E}_{R}:{\cal B}({\cal
H}_{R})\rightarrow{\cal B}({\cal H}_{R^{\prime}})$ be a unital CPTP map,
${\cal N}_{A}:{\cal B}({\cal H}_{A})\rightarrow{\cal B}({\cal H}_{B})$ be a
CPTP map, and
$\mathcal{I}_{X}(\cdot)\coloneqq\mathbbm{1}_{X}(\cdot)\mathbbm{1}_{X}$ denote
the identity channel on system $X$. Then $\Phi\left(({\cal
E}_{R}\otimes\mathcal{I}_{A})(M_{RA})\right)\leq\Phi(M_{RA})$ and
$\Phi((\mathcal{I}_{R}\otimes{\cal N}_{A})(M_{RA}))\leq\Phi(M_{RA})$.

We also introduce the following simplifying notation for the bipartite ${\cal
G}$–twirled states:

$\displaystyle\Phi_{\eta}(\tau)\coloneqq\Phi({\cal
G}(\eta_{R}\otimes\tau_{A})),$ (59)

and also $H_{\eta}(\tau):=-\log\Phi_{\eta}(\tau)$.

###  B.1 Invariance under local isometries that commute with ${\cal G}$

Here we prove the following lemma from the main text, which specializes
property (P4) to the particular form of the conditional min-entropies
appearing in Theorem 1, which are instead equivalent up to local isometries
that jointly commute with the $G$-twirl on the global system.

See 2

###### Proof.

By a straightforward appeal to Property (P3), we have

$\displaystyle\Phi({\cal G}[{\cal U}_{R}(\eta)\otimes{\cal
V}_{A}(\tau)])=\Phi({\cal U}_{R}\otimes{\cal V}_{A}\circ{\cal
G}[\eta\otimes\tau])=\Phi({\cal G}[\eta\otimes\tau]).$ (60)

Since $H_{\eta}(\rho)\coloneqq-\log\Phi({\cal G}[\eta\otimes\rho])$, this
implies

$\displaystyle H_{{\cal U}(\eta)}({\cal V}(\rho))=H_{\eta}(\rho),$ (61)

as claimed. ∎

###  B.2 Symmetric input states

######  Lemma 11.

For any input state $\rho$ on system $A$ and reference state $\eta$ on system
$R$, the following identity holds

$\displaystyle H_{{\cal G}(\eta)}(\rho)$ $\displaystyle=H_{\eta}({\cal
G}(\rho))=H_{{\cal G}(\eta)}({\cal G}(\rho))$ (62)
$\displaystyle=-\log\norm{{\cal G}(\eta)}_{\infty}.$ (63)

###### Proof.

The first two equalities in Eq. (62) straightforwardly follow from the fact
that ${\cal G}({\cal G}(\eta)\otimes\rho)={\cal G}(\eta)\otimes{\cal
G}(\rho)={\cal G}(\eta\otimes{\cal G}(\rho))={\cal G}({\cal
G}(\eta)\otimes{\cal G}(\rho))$. To show the final equality, we first examine

$\displaystyle\Phi({\cal G}(\eta)\otimes{\cal G}(\rho))$
$\displaystyle=\inf_{X\geq 0}\\{\mathrm{tr}[X]\ :\ \mathbbm{1}\otimes X-{\cal
G}(\eta)\otimes{\cal G}(\rho)\geq 0\\}$ (64)

Since ${\cal G}(\eta)$ is Hermitian, it can be diagonalised as ${\cal
G}(\eta)\coloneqq\sum_{i}\lambda_{i}\outerproduct{i}{i}$ for some basis
$\\{\ket{i}\\}$ of the reference system. Working in this basis, we obtain

$\displaystyle\mathbbm{1}\otimes X-{\cal G}(\eta)\otimes{\cal
G}(\rho)=\left(\sum_{i}\outerproduct{i}{i}\otimes
X\right)-\left(\sum_{i}\lambda_{i}\outerproduct{i}{i}\otimes{\cal
G}(\rho)\right)=\sum_{i}\outerproduct{i}{i}\otimes(X-\lambda_{i}{\cal
G}(\rho))$ (65)

Therefore, $\mathbbm{1}\otimes X-{\cal G}(\eta)\otimes{\cal G}(\rho)\geq 0$ if
and only if $X-\lambda_{i}{\cal G}(\rho)\geq 0$ for all $i$, which in turn is
true if and only if $X-\lambda_{\mathrm{max}}{\cal G}(\rho)\geq 0$, where
$\lambda_{\mathrm{max}}$ denotes the largest eigenvalue of ${\cal G}(\rho)$.
We can lower-bound the $\mathrm{tr}[X]$ needed to achieve this by

$\displaystyle\mathrm{tr}[X]\geq\lambda_{\mathrm{max}}\mathrm{tr}[{\cal
G}(\rho)]=\lambda_{\mathrm{max}},$ (66)

and this minimum can be attained simply by choosing
$X\coloneqq\lambda_{\mathrm{max}}{\cal G}(\rho)$. Therefore,

$\displaystyle\Phi({\cal G}(\eta)\otimes{\cal
G}(\rho))=\lambda_{\mathrm{max}}=\norm{{\cal G}(\rho_{A})}_{\infty}\Rightarrow
H_{{\cal G}(\eta)}({\cal G}(\rho))=-\log\norm{{\cal G}(\rho)}_{\infty},$ (67)

which completes the proof.∎

###  B.3 Dual Formulation

From Eq. (59), we see that $\Phi_{\eta}(\tau)$ is defined via a semidefinite
programme (SDP). In this subsection, we prove a lemma stating what the dual
form of this SDP is, which is convenient for proving several results of this
paper, including the depolarization conditions for $G$–covariant channels.

######  Lemma 12.

Let $\mathcal{O}_{\rm cov}$ be the set of $G$–covariant channels from input
system $A$ to output system $B$. Let $\eta$ be state of the reference system
$R$ for this transformation, and $\tau$ be a state of input system $A$. Then
the dual formulation of $\Phi_{\eta}(\tau)$ is

$\displaystyle\Phi_{\eta}(\tau)=\max_{\mathcal{E}\in\mathcal{O}_{\text{\rm
cov}}}\mathrm{tr}(\eta^{T}{\cal E}(\tau)),$ (68)

###### Proof.

From Lemma 3 in [58], we see that

$\displaystyle\Phi_{\eta}(\tau)=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\bra{\phi_{+}}\mathbbm{1}\otimes{\cal E}[{\cal
G}(\eta\otimes\tau)]\ket{\phi_{+}},$ (69)

where $\ket{\phi_{+}}\coloneqq\sum_{i}\ket{ii}$ for local computational bases
$\\{\ket{i}\\}$ for the reference $R$ and output system $B$. We first note
that $\ket{\phi_{+}}=\ket{vec(\mathbbm{1})}$, which means [2] that

$\displaystyle U^{*}(g)\otimes U(g)\ket{\phi_{+}}=U^{*}(g)\otimes
U(g)\ket{vec(\mathbbm{1})}=\ket{vec\left(U^{*}(g)\mathbbm{1}U^{T}(g)\right)}=\ket{vec(\mathbbm{1})}=\ket{\phi_{+}},$
(70)

where $U(g)$ is the representation of the group element $g$ on ${\cal H}_{B}$,
the Hilbert space of $B$. We therefore see that ${{\cal
G}(\outerproduct{\phi_{+}}{\phi_{+}})=\outerproduct{\phi_{+}}{\phi_{+}}}$,
which allows us to derive:

$\displaystyle\Phi_{\eta}(\tau)$ $\displaystyle=\max_{{\cal
E}\in\mathcal{O}_{\rm
cov}}\mathrm{tr}(\outerproduct{\phi_{+}}{\phi_{+}}\mathbbm{1}\otimes{\cal
E}[{\cal G}(\eta\otimes\tau)])$ (71) $\displaystyle=\max_{{\cal
E}\in\mathcal{O}_{\rm cov}}\mathrm{tr}(\outerproduct{\phi_{+}}{\phi_{+}}{\cal
G}(\eta\otimes{\cal E}[\tau]))$ (72) $\displaystyle=\max_{{\cal
E}\in\mathcal{O}_{\rm
cov}}\mathrm{tr}\left(\outerproduct{\phi_{+}}{\phi_{+}}\left[\int
dgU^{*}(g)\otimes U(g)(\eta\otimes{\cal E}[\tau])U^{T}(g)\otimes
U^{\dagger}(g)\right]\right)$ (73) $\displaystyle=\max_{{\cal
E}\in\mathcal{O}_{\rm cov}}\mathrm{tr}\left(\left[\int dgU^{T}(g)\otimes
U^{\dagger}(g)\outerproduct{\phi_{+}}{\phi_{+}}U^{*}(g)\otimes
U(g)\right]\eta\otimes{\cal E}[\tau]\right)$ (74) $\displaystyle=\max_{{\cal
E}\in\mathcal{O}_{\rm cov}}\mathrm{tr}\left(\left[\int dg\ {\cal
U}^{*}_{g^{-1}}\otimes{\cal
U}_{g^{-1}}(\outerproduct{\phi_{+}}{\phi_{+}})\right]\eta\otimes{\cal
E}[\tau]\right)$ (75) $\displaystyle=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\mathrm{tr}({\cal G}(\outerproduct{\phi_{+}}{\phi_{+}})\eta\otimes{\cal
E}[\tau])$ (76) $\displaystyle=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\mathrm{tr}(\outerproduct{\phi_{+}}{\phi_{+}}\eta\otimes{\cal E}[\tau])$
(77) $\displaystyle=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\sum_{ij}\bra{j}\eta\ket{i}\bra{j}{\cal E}[\tau]\ket{i}$ (78)
$\displaystyle=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\sum_{ij}\bra{i}\eta^{T}\ket{j}\bra{j}{\cal E}[\tau]\ket{i}$ (79)
$\displaystyle=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\mathrm{tr}(\eta^{T}{\cal E}[\tau]),$ (80)

as claimed. ∎

###  B.4 Truncation of Output System

The following two Lemmas detail when one can truncate the Hilbert space of the
output system without affecting the possibility of interconversion to a
particular output state $\sigma$. This is of use in our analysis of state
interconversion with partial depolarization.

######  Lemma 13.

Let $\rho$ be a state of the input system $A$, associated to the Hilbert space
${\cal H}_{A}$. Let $\sigma$ be a state of the output system $B$, associated
to the Hilbert space ${\cal H}_{B}$.

Let ${\cal H}_{S}$ be any subspace of ${\cal H}_{B}$ with the following two
properties:

  1. 1.

${\cal H}_{S}$ carries its own representation of $G$, i.e. ${\cal H}_{B}$ can
be decomposed into ${\cal H}_{B}={\cal H}_{S}\oplus{\cal H}_{A}$ such that
$U_{B}(g)=U_{S}(g)\oplus U_{A}(g)$.

  2. 2.

The support of $\sigma$ is contained entirely within ${\cal H}_{S}$, i.e.
letting $\Pi_{S}$ be the projector onto ${\cal H}_{S}$,
$\Pi_{S}\sigma\Pi_{S}=\sigma$.

Let ${\cal H}_{S}$ be the Hilbert space appropriate to a new output system $S$
truncated from $B$. Then there exists a $G$–covariant operation from $A$ to
$B$ that takes $\rho$ to $\sigma$ if and only if there exists a $G$–covariant
operation from $A$ to $S$ that takes $\rho$ to $\sigma$.

###### Proof.

Let us first assume that there exists a $G$–covariant operation ${\cal E}_{\rm
cov}:{\cal B}({\cal H}_{A})\rightarrow{\cal B}({\cal H}_{B})$ such that
${{\cal E}_{\rm cov}(\rho)=\sigma}$. We then observe that

$\displaystyle\Pi_{S}U_{B}(g)=\left(\mathbbm{1}_{S}\oplus
0_{A}\right)\left(U_{S}(g)\oplus U_{A}(g)\right)=U_{S}(g)\oplus
0_{A}=(U_{S}(g)\mathbbm{1}_{S})\oplus 0_{A}=U_{S}(g)\Pi_{S}$ (81)

Therefore, $\Pi_{S}(\cdot)\Pi_{S}$ is a covariant map from ${\cal B}({\cal
H}_{B})$ to ${\cal B}({\cal H}_{S})$. As a result, $\Pi_{S}[{\cal E}_{\rm
cov}(\cdot)]\Pi_{S}$ is a covariant operation from ${\cal B}({\cal H}_{A})$ to
${\cal B}({\cal H}_{S})$ such that

$\displaystyle\Pi_{S}{\cal E}_{\rm
cov}(\rho)\Pi_{S}=\Pi_{S}\sigma\Pi_{S}=\sigma.$ (82)

Conversely, let us now assume that that there exists a covariant
transformation ${{\cal F}_{\rm cov}:{\cal B}({\cal H}_{A})\rightarrow{\cal
B}({\cal H}_{S})}$ such that ${\cal F}_{\rm cov}(\rho)=\sigma$. We then extend
${\cal H}_{S}$ into the bigger Hilbert space ${\cal H}_{B}={\cal
H}_{S}\oplus{\cal H}_{A}$ such that ${\cal H}_{S}$ still forms its own
representation of $G$, i.e. $U_{B}(g)=U_{S}(g)\oplus U_{A}(g)$. Then ${\cal
F}_{\rm cov}$ can be reinterpreted as a covariant channel from ${\cal B}({\cal
H}_{A})$ to ${\cal B}({\cal H}_{B})$.

We therefore conclude that $G$–covariant interconversion from $\rho$ to
$\sigma$ is unaffected by treating $\sigma$ as a state of $B$ or as a state of
$S$. ∎

######  Lemma 14.

Given any particular output state $\sigma$, it is always possible to truncate
the Hilbert space of the output system, ${\cal H}_{B}$, to the support of
${\cal G}(\sigma)$ without affecting the possibility of $G$–covariant
interconversion.

###### Proof.

The representation of $G$ on ${\cal H}_{B}$ splits up in the following way
[19]:

$\displaystyle{\cal H}_{B}=\bigoplus_{q}{\cal H}_{q}.$ (83)

The ${\cal H}_{q}$ are known as the charge sectors of ${\cal H}_{B}$, and they
each carry an inequivalent representation of $G$. Each ${\cal H}_{q}$ can be
further decomposed into a tensor product

$\displaystyle{\cal H}_{q}={\cal M}_{q}\otimes{\cal N}_{q}.$ (84)

The ${\cal M}_{q}$ carry inequivalent irreps of $G$, while the ${\cal N}_{q}$
carry trivial representations of $G$. This means every element $g$ is
represented on ${\cal H}_{q}$ in the form $U_{{\cal
M}_{q}}(g)\otimes\mathbbm{1}_{{\cal N}_{q}}$. As a result, given any pure
state $\ket{\psi_{q}}$ in ${\cal N}_{q}$, ${\cal
M}_{q}\otimes\operatorname{Span}(\ket{\psi_{q}})$ is an irrep of $G$.
Projectors onto irreps of $G$ thus take the form $\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}$.

We note the following properties about the projector $\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}$. Because ${\cal
M}_{q}\otimes\operatorname{Span}\left(\ket{\psi_{q}}\right)$ is a subspace of
${\cal H}_{q}$,

$\displaystyle\left[\Pi_{q},\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right]=0$ (85)

For the same reason, $\Pi_{q}$ is identity on ${\cal
M}_{q}\otimes\operatorname{Span}\left(\ket{\psi_{q}}\right)$, which means

$\displaystyle\mathbbm{1}_{M_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}=\Pi_{q}\left(\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right).$ (86)

A subspace ${\cal H}$ of ${\cal H}_{B}$ lies inside the kernel of $\sigma$ if
and only if $\mathrm{tr}(\Pi\sigma)=0$, where $\Pi$ is the projector onto
${\cal H}$. Therefore, the irrep ${\cal
M}_{q}\otimes\operatorname{Span}\left(\ket{\psi_{q}}\right)$ lies in the
kernel of $\sigma$ if and only if

$\displaystyle\mathrm{tr}(\left(\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right)\sigma)$
$\displaystyle=\mathrm{tr}(\Pi_{q}\left(\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right)\sigma)$ (87)
$\displaystyle=\mathrm{tr}(\Pi_{q}\Pi_{q}\left(\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right)\sigma)$ (88)
$\displaystyle=\mathrm{tr}(\Pi_{q}\left(\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right)\sigma\Pi_{q})$ (89)
$\displaystyle=\mathrm{tr}(\left(\mathbbm{1}_{{\cal
M}_{q}}\otimes\outerproduct{\psi_{q}}{\psi_{q}}\right)\Pi_{q}\sigma\Pi_{q})$
(90) $\displaystyle=\bra{\psi_{q}}\mathrm{tr}_{{\cal
M}_{q}}(\Pi_{q}\sigma\Pi_{q})\ket{\psi_{q}}=0$ (91)

where in the first equality we made use of Eq. (86), and in the fourth
equality we made use of Eq. (85). This short calculation means ${\cal
M}_{q}\otimes\operatorname{Span}(\ket{\psi_{q}})$ lies inside the kernel of
$\sigma$ if and only if $\ket{\psi_{q}}$ lies inside the kernel of
$\mathrm{tr}_{{\cal M}_{q}}(\Pi_{q}\sigma\Pi_{q})$.

Let $\left\\{\ket{\psi_{q,i}}\right\\}$ be an orthonormal basis for ${\cal
N}_{q}$ in which $\mathrm{tr}_{{\cal M}_{q}}(\Pi_{q}\sigma\Pi_{q})$ is
diagonalised. One possible irrep decomposition for ${\cal H}_{B}$ is then

$\displaystyle{\cal H}_{B}=\bigoplus_{q,i}{\cal
M}_{q}\otimes\operatorname{Span}\left(\ket{\psi_{q,i}}\right).$ (92)

An irrep in this decomposition lies inside the kernel of $\sigma$ if and only
if $\ket{\psi_{q,i}}$ is a basis element for the kernel of $\mathrm{tr}_{{\cal
M}_{q}}(\Pi_{q}\sigma\Pi_{q})$. This means

$\displaystyle{\cal H}_{S}^{\perp}\coloneqq\bigoplus_{q,\
\ket{\psi_{q,i}}\in\ker[\mathrm{tr}_{{\cal
M}_{q}}(\Pi_{q}\sigma\Pi_{q})]}{\cal
M}_{q}\otimes\operatorname{Span}\left(\ket{\psi_{q,i}}\right)=\bigoplus_{q}{\cal
M}_{q}\otimes\ker(\mathrm{tr}_{{\cal M}_{q}}(\Pi_{q}\sigma\Pi_{q}))$ (93)

must lie inside the kernel of $\sigma$ on ${\cal H}_{B}$. Conversely, the
support of $\sigma$ must lie inside the subspace of ${\cal H}_{B}$ that is
orthogonal to ${\cal H}_{S}^{\perp}$, i.e.

$\displaystyle{\cal H}_{S}=\bigoplus_{q}{\cal
M}_{q}\otimes\operatorname{supp}[\mathrm{tr}_{{\cal
M}_{q}}(\Pi_{q}\sigma\Pi_{q})].$ (94)

As we see from the above equation, ${\cal H}_{S}$ is also a direct sum over
irreps of $G$ and so carries its own representation of $G$. Thus by Lemma 13,
the possibility of interconversion is unaffected if we truncate ${\cal H}_{B}$
to ${\cal H}_{S}$.

The action of the $G$-twirl is given by [19]:

$\displaystyle{\cal G}=\sum_{q}({\cal D}_{{\cal
M}_{q}}\otimes\mathcal{I}_{{\cal N}_{q}})\circ{\cal P}_{q},$ (95)

where ${\cal P}_{q}\coloneqq\Pi_{q}(\cdot)\Pi_{q}$ is the projector onto the
charge sector ${\cal H}_{q}$, ${\cal D}_{{\cal M}_{q}}$ is the completely
depolarising channel on ${\cal M}_{q}$ and $I_{{\cal N}_{q}}$ is the identity
channel on ${\cal N}_{q}$. Therefore,

$\displaystyle{\cal G}(\sigma)=\sum_{q}\frac{\mathbbm{1}}{d_{{\cal
M}_{q}}}\otimes\mathrm{tr}_{{\cal M}_{q}}(\Pi_{q}\sigma\Pi_{q}),$ (96)

where $d_{{\cal M}_{q}}$ is the dimension of ${\cal M}_{q}$. Looking back at
Equation (94), we see that ${\cal H}_{S}=\operatorname{supp}[{\cal
G}(\sigma)]$. It is therefore always possible to truncate the output Hilbert
space to the support of ${\cal G}(\sigma)$ without affecting possibility of
interconversion. ∎

##  Appendix C Redundancies in the entropic relations

Here we consider a few basic redundencies in the infinite set of conditions
appearing in Theorem 1. We first note that $H_{\eta}(\rho)\leq
H_{\eta}(\sigma)$ and $\Phi_{\eta}(\rho)\geq\Phi_{\eta}(\sigma)$ are
equivalent conditions, since we have that $\Phi(X)>0$ for any $X$ that is
positive-semidefinite with at least one non-zero eigenvalue [69], and the fact
that $-\log(x)$ is monotonic decreasing in $x$ for $x>0$.

###  C.1 Unitaries on the reference

Lemma 2 immediately gives rise to the following corollary.

######  Corollary 15.

If $\eta_{1}={\cal V}(\eta_{0})$ for any pair of quantum states
$\eta_{0},\eta_{1}$ on ${\cal H}_{R}$ and for any unitary ${\cal V}:{\cal
B}({\cal H}_{R})\rightarrow{\cal B}({\cal H}_{R})$ such that $[{\cal
V}\otimes\mathbbm{1},{\cal G}]=0$, then $\Delta H_{\eta_{0}}\geq 0$ if and
only if $\Delta H_{\eta_{1}}\geq 0$.

As an example of this redundency, we can consider the group $G=U(1)$ of time-
translations generated by the Hamiltonian $H$, $U(t)=e^{-iHt}$. Here we have
that any reference state $\eta$ drawn from the set $\\{U(t)\eta
U(t)^{\dagger}:\forall t\in[0,2\pi)\\}$ will provide an equivalent constraint.

###  C.2 Modes of asymmetry

Given an ITO basis $\left\\{X^{(\lambda,\alpha)}_{j}\right\\}$, which in the
following we always take to be orthonormal such that

$\displaystyle\left\langle
X^{(\lambda,\alpha)}_{j},X^{(\mu,\beta)}_{k}\right\rangle=\delta_{\lambda,\mu}\delta_{\alpha,\beta}\delta_{j,k},$
(97)

where $\langle A,B\rangle=\mathrm{tr}[A^{\dagger}B]$ is the Hilbert-Schmidt
inner product on ${\cal B}({\cal H})$, we denote by $A^{\lambda}_{j}$ the
$(\lambda,j)$ mode of the operator $A$

$A^{\lambda}_{j}\coloneqq\sum_{\alpha}\left\langle
X^{(\lambda,\alpha)}_{j},A\right\rangle X^{(\lambda,\alpha)}_{j}.$ (98)

The following gives some basic properties for handling inner products
involving modes of asymmetry.

######  Lemma 16.

Let $A$ and $B$ be any two linear operators on ${\cal H}$. Then we have that

$\displaystyle\langle
A^{\lambda}_{j},B^{\mu}_{k}\rangle=\delta_{\lambda,\mu}\delta_{j,k}\langle
A^{\lambda}_{j},B^{\mu}_{k}\rangle$ (99)

from which it follows that:

$\displaystyle\langle A,B\rangle=\sum_{\lambda,j}\langle
A^{\lambda}_{j},B^{\lambda}_{j}\rangle$ (100)

###### Proof.

Writing out the mode decompositions of $A$ and $B$ in the ITO basis explicitly
in the trace product, from the orthonormality of the basis operators
$\\{X^{(\lambda,\alpha)}_{j}\\}$ we obtain

$\displaystyle\langle A^{\lambda}_{j},B^{\mu}_{k}\rangle$
$\displaystyle=\left\langle\sum_{\alpha}\langle
X^{(\lambda,\alpha)}_{j},A\rangle X^{(\lambda,\alpha)}_{j},\sum_{\beta}\langle
X^{(\mu,\beta)}_{k},B\rangle X^{(\mu,\beta)}_{k}\right\rangle$ (101)
$\displaystyle=\sum_{\alpha,\beta}\langle
X^{(\lambda,\alpha)}_{j},A\rangle\langle X^{(\mu,\beta)}_{k},B\rangle\langle
X^{(\lambda,\alpha)}_{j},X^{(\mu,\beta)}_{k}\rangle$ (102)
$\displaystyle=\sum_{\alpha,\beta}\langle
X^{(\lambda,\alpha)}_{j},A\rangle\langle
X^{(\mu,\beta)}_{k},B\rangle\delta_{\lambda,\mu}\delta_{j,k}\langle
X^{(\lambda,\alpha)}_{j},X^{(\mu,\beta)}_{k}\rangle$ (103)
$\displaystyle=\delta_{\lambda,\mu}\delta_{j,k}\left\langle\sum_{\alpha}\langle
X^{(\lambda,\alpha)}_{j},A\rangle X^{(\lambda,\alpha)}_{j},\sum_{\beta}\langle
X^{(\mu,\beta)}_{k},B\rangle X^{(\mu,\beta)}_{k}\right\rangle$ (104)
$\displaystyle=\delta_{\lambda,\mu}\delta_{j,k}\langle
A^{\lambda}_{j},B^{\mu}_{k}\rangle,$ (105)

as required. ∎

An immediate consequence of such a mode decomposition is that if $\text{
modes}(\sigma)\subseteq\text{modes}(\rho)$, then it suffices to range only
over reference frame states $\eta$ such that modes($\eta$) = modes($\rho$) in
Theorem 1. The reasoning is as follows. Let $\eta$ have an irrep mode $\mu$
that does not occur in $\rho$. By hermiticity, it also has the irrep mode
$\mu^{*}$. As seen in the following lemma, when computing ${\cal
G}[\eta\otimes\rho]$, the only mode terms that survive this G-twirl are of the
form $\sum_{i}\eta^{\mu^{*}}_{i}\otimes\rho^{\mu}_{i}$, which is the unique
way to form a singlet from a given irrep:

######  Lemma 17.

$\displaystyle{\cal
G}(\eta\otimes\rho)=\sum_{i,\mu}\eta^{\mu^{*}}_{i}\otimes\rho^{\mu}_{i}.$
(106)

###### Proof.

Equation 4.3 of [60] states that

$\displaystyle\int dg\
v^{\lambda}_{i^{\prime}i}(g)v^{\mu}_{j^{\prime}j}=\frac{1}{d_{\mu}}\delta_{i^{\prime},i}\delta_{j^{\prime},j}\delta_{\lambda^{*},\mu}.$
(107)

Making use of this result, we can then demonstrate that

$\displaystyle{\cal G}(\eta\otimes\rho)$ $\displaystyle=\int dg\
\sum_{\lambda,i,\mu,j}\mathcal{U}_{g}(\eta^{\lambda}_{i})\otimes\mathcal{U}_{g}(\rho^{\mu}_{j})$
(108) $\displaystyle=\int dg\
\sum_{\lambda,i,\mu,j}\sum_{i^{\prime},j^{\prime}}v^{\lambda}_{i^{\prime}i}(g)v^{\mu}_{j^{\prime}j}\eta^{\lambda}_{i^{\prime}}\otimes\rho^{\mu}_{j^{\prime}}$
(109)
$\displaystyle=\sum_{\lambda,i,\mu,j}\sum_{i^{\prime}j^{\prime}}\frac{1}{d_{\mu}}\delta_{i^{\prime},j^{\prime}}\delta_{i,j}\delta_{\lambda^{*},\mu}\eta^{\lambda}_{i^{\prime}}\otimes\rho^{\mu}_{j^{\prime}}$
(110)
$\displaystyle=\sum_{\mu,i^{\prime}}\left(\sum_{j}\frac{1}{d_{\mu}}\right)\eta^{\mu^{*}}_{i^{\prime}}\otimes\rho^{\mu}_{i^{\prime}}$
(111) $\displaystyle=\sum_{\mu,i}\eta^{\mu^{*}}_{j}\otimes\rho^{\mu}_{j}$
(112)

∎

We conclude from this lemma that if $\rho$ does not contain a $\mu$ mode, then
$\eta^{\mu^{*}}_{i}$ does not contribute to the state ${\cal
G}[\eta\otimes\rho]$.

If we range over all $\eta$ contained within a small surface around
$\mathbbm{1}/d$, then we obtain a complete set of conditions. We know that
$\eta^{\mu^{*}}_{j}$ does not contribute to $\Omega_{RA}$. Now if the region
is chosen sufficiently small, we claim that
$\eta^{\prime}=\eta-\left[\sum_{j}\left(\eta^{\mu}_{j}+\eta^{\mu^{*}}_{j}\right)\right]$
is still a valid state for $\mu\neq 0$, but with the $\mu$ mode removed. To
see this, firstly note that, by orthonormality, the term in the brackets is
traceless and so the net result still has trace one. Secondly, by choosing the
surface appropriately, the eigenvalues of $\eta$ can be chosen arbitrarily
close to the uniform distribution, and those of the term in bracket made
arbitrarily small. Therefore the eigenvalues of the resultant operator
$\eta^{\prime}$ are all non-negative. Therefore we have a reference frame
state $\eta^{\prime}$ with the $\mu$ mode entirely removed, and gives the
state joint state ${\cal G}[\eta\otimes\rho]$ as did $\eta$. This implies it
suffices to range over reference frame states $\eta$ with modes the same as
$\rho$.

##  Appendix D A sufficient surface of reference frames

###  D.1 Depolarizing the reference state

Let us define the partially depolarizing channel for some fixed probability
$p$:

$\displaystyle\Lambda_{p}[\rho]\coloneqq p\rho+(1-p)\frac{\mathbbm{1}}{d}.$
(113)

In general, from (P2) we know that the functional $\Phi_{\eta}(\rho)$ is
convex in the reference system, i.e., $\eta=p\eta_{0}+(1-p)\eta_{1}$ implies

$\displaystyle\Phi_{\eta}(\rho)\leq
p\Phi_{\eta_{0}}(\rho)+(1-p)\Phi_{\eta_{1}}.$ (114)

The following lemma shows that the functional $\Phi_{\eta}(\rho)$ behaves
linearly when we take convex combinations of the reference state with the
maximally mixed state.

######  Lemma 18.

Let all states and systems be defined as in Theorem 1. For any reference state
$\eta_{R}$ and input state $\tau_{A}$, we have

$\displaystyle\Phi_{\Lambda_{p}(\eta)}(\tau)$
$\displaystyle=p\Phi_{\eta}(\tau)+(1-p)\Phi_{\mathbbm{1}/d}(\tau)$
$\displaystyle=p\Phi_{\eta}(\tau)+\frac{1-p}{d}.$ (115)

###### Proof.

Since $\mathbbm{1}$ is symmetric for any group $G$, we have ${\cal
G}[\Lambda_{p}(\eta_{R})\otimes\tau_{A}]=p{\cal
G}[\eta_{R}\otimes\tau_{A}]+\frac{1-p}{d}\mathbbm{1}\otimes{\cal
G}[\tau_{A}]$. Substituting this into Eq. (59) and rearranging terms gives

$\displaystyle\Phi_{\Lambda_{p}[\eta]}\left(\tau\right)=\inf_{X_{A}\geq
0}\left\\{\mathrm{tr}[X_{A}]\ :\
\mathbbm{1}_{R}\otimes\left(X_{A}-\frac{1-p}{d}{\cal
G}[\tau_{A}]\right)-p{\cal G}\left[\eta_{R}\otimes\tau_{A}\right]\geq
0\right\\}.$ (116)

For any positive semidefinite operators $T_{A}$ and $Z_{RA}$, we have

$\\{X_{A}\ :\ X_{A}\geq 0,\ \mathbbm{1}_{R}\otimes[X_{A}-T_{A}]-Z_{RA}\geq
0\\}=\\{X_{A}\ :\ X_{A}-Y_{A}\geq 0,\
\mathbbm{1}_{R}\otimes[X_{A}-YTA]-Z_{RA}\geq 0\\}.$ (117)

Therefore we can rewrite the feasible set over which we perform the
optimization in Eq. (116) as follows

$\displaystyle\Phi_{\Lambda_{p}[\eta]}\left(\tau\right)$
$\displaystyle=\inf_{X_{A}-\frac{1-p}{d}{\cal G}[\tau_{A}]\geq
0}\left\\{\mathrm{tr}[X_{A}]\ :\
\mathbbm{1}_{R}\otimes\left(X_{A}-\frac{1-p}{d}{\cal
G}[\tau_{A}]\right)-p{\cal G}\left[\eta_{R}\otimes\tau_{A}\right]\geq
0\right\\}$ (118) $\displaystyle=\inf_{Y_{A}\geq
0}\left\\{\mathrm{tr}\left[Y_{A}+\frac{1-p}{d}{\cal G}[\tau_{A}]\right]\ :\
\mathbbm{1}_{R}\otimes Y_{A}-p{\cal G}\left[\eta_{R}\otimes\tau_{A}\right]\geq
0\right\\},$ (119)

where we have defined $Y_{A}\coloneqq X_{A}-\frac{1-p}{d}{\cal G}(\tau_{A})$.
Now since $\mathrm{tr}[{\cal G}(\tau_{A})]=1$, we can take the constant term
out of the infimum

$\displaystyle\Phi_{\Lambda_{p}[\eta]}\left(\tau\right)$
$\displaystyle=\inf_{Y_{A}\geq 0}\left\\{\mathrm{tr}[Y_{A}]\ :\
\mathbbm{1}_{R}\otimes Y_{A}-p{\cal G}\left[\eta_{R}\otimes\tau_{A}\right]\geq
0\right\\}+\frac{1-p}{d}.$ (120)

Finally we make use of property (P1) to arrive at

$\displaystyle\Phi_{\Lambda_{p}[\eta]}\left(\tau\right)$
$\displaystyle=p\Phi_{\eta}(\tau)+\frac{1-p}{d}.$ (121)

which concludes the proof. ∎

An automatic consequence of Lemma 18 is that taking a convex mixture of any
reference state with the maximally mixed state will not change the entropic
relation in Theorem 1, which we state in the following lemma:

######  Lemma 19.

Let all states and systems be defined as in Theorem 1 and let us further
define the partially depolarizing quantum channel $\Lambda_{p}[\rho]\coloneqq
p\rho+(1-p)\frac{\mathbbm{1}}{d}$, where $p$ is a probability and
$d\coloneqq\mathrm{dim}({\cal H}_{R})$. Then the following two statements are
equivalent for any $\eta$:

  1. 1.

$\Delta H_{\eta}\geq 0$.

  2. 2.

$\Delta H_{\Lambda_{p}[\eta]}\geq 0$ for any $p\in(0,1]$.

###### Proof.

Defining $\Delta\Phi_{\eta}\coloneqq\Phi_{\eta}(\rho)-\Phi_{\eta}(\sigma)$,
Lemma 18 implies that $\Delta\Phi_{\Lambda_{p}[\eta]}=p\Delta\Phi_{\eta}$ for
any $p$ probability. Therefore, $\Delta\Phi_{\eta}\geq 0$ if and only if
$\Delta\Phi_{\Lambda_{p}[\eta]}\geq 0$, for any $p\in[0,1]$. Since
$\Phi_{\eta}$ and $H_{\eta}$ are monotonically related, this then gives the
statement of the lemma. ∎

###  D.2 Proof of Theorem 3

We now present a proof of Theorem 3, which we restate here for clarity: See 3

###### Proof.

If the transformation is possible under a $G$–covariant channel then $\Delta
H_{\eta}\geq 0$ for all states $\eta$, and hence in particular for all $\eta$
restricted to $\partial{\cal D}$. Conversely, suppose $\Delta H_{\eta}\geq 0$
for all $\eta\in\partial{\cal D}$. Let $\eta^{\prime}$ be an arbitrary quantum
state of $R$ that is not the maximally mixed state, and consider the one-
parameter family of states $\eta^{\prime}(p):=\Lambda_{p}(\eta^{\prime})$ for
$p\in[0,1]$. This defines a continuous line of states connecting
$\eta^{\prime}$ to the maximally mixed state $\mathbbm{1}/d$. Since
$\partial{\cal D}$ encloses the maximally mixed state the set
$\\{\eta^{\prime}(p):0\leq p\leq 1\\}$ must either intersect $\partial{\cal
D}$ for some value $p_{\star}$ with $0<p_{\star}\leq 1$ or lie entirely within
the interior of ${\cal D}$. If the set is entirely inside ${\cal D}$ then we
can find a quantum state $\eta^{\prime\prime}\in\partial{\cal D}$ such that
$\Delta_{q}[\eta^{\prime\prime}]=\eta^{\prime}$ for some $q\in(0,1)$. However
from lemma 19 we have that $\eta^{\prime}$ and $\eta^{\prime}(p_{\star})$ (or
$\eta^{\prime}$ and $\eta^{\prime\prime}$ for the second case) give equivalent
entropic constraints. Since $\eta^{\prime}$ was arbitrary it therefore
suffices to restrict to states lying solely on the surface $\partial{\cal D}$,
which completes the proof.

∎

##  Appendix E Smoothed asymmetry theory

###  E.1 Continuity of entropic relations under variations of the reference
state

In this section, we consider the following definition of an $\varepsilon$-ball
of operators on ${\cal H}$ around some $\rho\in{\cal S}_{\leq}({\cal H})$

$\displaystyle B^{\epsilon}(\rho)\coloneqq\\{\tilde{\rho}\in S_{\leq}({\cal
H}):D(\tilde{\rho},\rho)\leq\epsilon\\},$ (122)

but we note that all the results derived in this section also apply if we use
the purified distance $P(\cdot,\cdot)$ as our distance measure instead, due to
the property $D(\rho,\sigma)\leq P(\rho,\sigma)$ for all $\rho$, $\sigma$
[70].

We also make use of the following theorem, which was proven in Ref. [69].

######  Theorem 20.

(Continuity of min-entropy). Let $\rho,\sigma\in{\cal S}_{\leq}({\cal
H}_{RA})$. Then

$|H_{\mathrm{min}}(R|A)_{\rho}-H_{\mathrm{min}}(R|A)_{\sigma}|\leq\frac{d_{R}\min\\{d_{R},d_{A}\\}}{\ln
2\,\min\\{\mathrm{tr}\rho,\mathrm{tr}\sigma\\}}D(\rho,\sigma),$ (123)

where
$D(\rho,\sigma)\coloneqq\frac{1}{2}\norm{\rho-\sigma}_{1}+\frac{1}{2}\absolutevalue{\mathrm{tr}\rho-\mathrm{tr}\sigma}$
is the generalized trace distance.

######  Lemma 21.

Consider the following bipartite quantum states on ${\cal H}_{RA}$:

$\displaystyle\Omega^{RA}\coloneqq\eta_{R}\otimes\rho_{A},\quad\tilde{\Omega}^{RA}\coloneqq\tilde{\eta}_{R}\otimes\rho_{A}.$
(124)

If $\tilde{\eta}\in{\cal B}_{\varepsilon}(\eta)$, then

$\displaystyle D({\cal G}[\Omega],{\cal G}[\tilde{\Omega}])$
$\displaystyle\leq\varepsilon.$ (125)

###### Proof.

The trace distance is contractive under quantum operations, and thus

$\displaystyle\frac{1}{2}\norm{{\cal G}[\Omega]-{\cal G}[\tilde{\Omega}]}_{1}$
$\displaystyle\leq\frac{1}{2}\norm{\Omega-\tilde{\Omega}}_{1}$ (126)
$\displaystyle=\frac{1}{2}\norm{(\eta-\tilde{\eta})\otimes\rho}_{1}$ (127)
$\displaystyle=\frac{1}{2}\norm{(\eta-\tilde{\eta})}_{1},$ (128)

where in the second equality we have used the identity $\norm{A\otimes
B}_{1}=\norm{A}_{1}\norm{B}_{1}$. Similarly, since ${\cal G}$ is trace-
preserving and $\mathrm{tr}[A\otimes B]=\mathrm{tr}[A]\mathrm{tr}[B]$, we have

$\frac{1}{2}\left|\mathrm{tr}\left({\cal
G}[\Omega]\right)-\mathrm{tr}\left({\cal
G}[\tilde{\Omega}]\right)\right|=\frac{1}{2}\left|\mathrm{tr}(\Omega)-\mathrm{tr}(\tilde{\Omega})\right|=\frac{1}{2}\left|\mathrm{tr}(\eta)-\mathrm{tr}(\tilde{\eta})\right|.$
(129)

Thus, combining results from Eqs. (128) and (129) we find the generalized
trace distance between ${\cal G}[\Omega]$ and ${\cal G}[\tilde{\Omega}]$ is
upper bounded as

$\displaystyle D({\cal G}[\Omega],{\cal G}[\tilde{\Omega}])$
$\displaystyle\leq\left(\frac{1}{2}\norm{(\eta-\tilde{\eta})}_{1}+\frac{1}{2}\left|\mathrm{tr}(\eta)-\mathrm{tr}(\tilde{\eta})\right|\right)=D(\eta,\tilde{\eta}).$
(130)

Therefore, if $\tilde{\eta}\in{\cal B}_{\varepsilon}(\eta)$ then
$D(\Omega,\tilde{\Omega})\leq\varepsilon$ immediately follows from Eq. (130),
which concludes the proof of Lemma 21. ∎

######  Lemma 22.

If $\tilde{\eta}\in{\cal B}_{\varepsilon}(\eta)$, then

$|H_{\eta}(\tau)-H_{\tilde{\eta}}(\tau)|\leq\frac{d_{R}^{2}}{\ln
2}\left(\frac{\varepsilon}{1-2\varepsilon}\right),$ (131)

where $H_{\eta}(\tau)\coloneqq H_{\mathrm{min}}(R|A)_{{\cal G}[\Omega]}$.

###### Proof.

Combining Lemma 21 and Theorem 20 (and using the fact that the $G$-twirl is a
trace-preserving map) we immediately have that

$|H_{\eta}(\tau)-H_{\tilde{\eta}}(\tau)|\leq\frac{d_{R}\min\\{d_{R},d_{A}\\}}{\ln
2\,\min\\{\mathrm{tr}\Omega,\mathrm{tr}\tilde{\Omega}\\}}\,\varepsilon,$ (132)

for any $\tilde{\eta}\in{\cal B}_{\varepsilon}(\eta)$. To get the simplified
form as stated, we note that
$\min\\{\mathrm{tr}(\Omega),\mathrm{tr}(\tilde{\Omega})\\}=\mathrm{tr}(\tilde{\Omega})$
since $\Omega$ is normalised and $\tilde{\Omega}$ is sub-normalized. This then
evaluates to

$\displaystyle\mathrm{tr}(\tilde{\Omega})=\mathrm{tr}(\tau)\mathrm{tr}(\tilde{\eta})=\mathrm{tr}(\tilde{\eta}).$
(133)

Now $D(\eta,\tilde{\eta})\leq\varepsilon$ implies
$|1-\mathrm{tr}(\tilde{\eta})|\leq 2\varepsilon$, and therefore

$\displaystyle\mathrm{tr}(\tilde{\eta})\geq 1-2\varepsilon.$ (134)

Substituting Eq. (134) into Eq. (133) thus gives

$\min\\{\mathrm{tr}(\Omega),\mathrm{tr}(\tilde{\Omega})\\}\geq
1-2\varepsilon.$ (135)

Also, clearly $d_{R}\geq\min\\{d_{R},d_{A}\\}$. Substituting this and Eq.
(135) into Eq. (132) gives

$|H_{\eta}(\tau)-H_{\tilde{\eta}}(\tau)|\leq\frac{d_{R}^{2}}{\ln
2}\frac{\varepsilon}{1-2\varepsilon},$ (136)

as claimed. ∎

We are now able to prove the result presented in the main text:

See 4

###### Proof.

First note:

$\displaystyle\Delta H_{\tilde{\eta}}-\Delta H_{\eta}$
$\displaystyle=[H_{\tilde{\eta}}(\sigma)-H_{\tilde{\eta}}(\rho)]-[H_{\eta}(\sigma)-H_{\eta}(\rho)]$
(137)
$\displaystyle=[H_{\tilde{\eta}}(\sigma)-H_{\eta}(\sigma)]-[H_{\tilde{\eta}}(\rho)-H_{\eta}(\rho)]\eqqcolon
x.$ (138)

It then follows immediately from Lemma 22 that for any $\tilde{\eta}\in{\cal
B}_{\varepsilon}(\eta)$ we have $\absolutevalue{x}\leq\frac{2d_{R}^{2}}{\ln
2}\frac{\varepsilon}{1-2\varepsilon}$, which concludes the proof. ∎

###  E.2 Proof of Theorem 5

First we need the following theorem (e.g. see [71, 72]):

######  Theorem 23.

Let $\norm{\cdot}$ be a norm on $\mathbb{R}^{d}$. Then for every $\delta>0$,
the unit sphere $\\{x\in\mathbb{R}^{d},\norm{x}=1\\}$ admits a $\delta$-net,
$\mathcal{N}$ with respect to the distance measured by $\norm{\cdot}$, of
cardinality $|\mathcal{N}|$ such that

$\displaystyle|\mathcal{N}|\leq\left(1+\frac{2}{\delta}\right)^{d}.$ (139)

This theorem implies that there exists an $\varepsilon$-covering of a unit-
sphere, with a finite number of elements, and which can be applied to a sphere
of reference frame states around the maximally mixed state. We restate the
theorem which we seek to prove: See 5

###### Proof.

Given Theorem 3, we choose as our sufficient set of reference frame states the
surface

$\partial{\cal D}=\left\\{\eta:\eta=\frac{1}{d}(\mathbbm{1}+A),\mbox{ where
}||A||_{\infty}=1\mbox{ and }\mathrm{tr}(A)=0\right\\}.$ (140)

It is seen by inspection that this gives a closed surface of quantum states
that contain the maximally mixed state. We can describe $\partial{\cal D}$
entirely in the space of traceless $d\times d$ Hermitian matrices, which is
embedded in $\mathbb{R}^{d^{2}-1}$. In this embedding space the surface is the
unit ball defined by $||A||_{\infty}=1$, and so by the above theorem admits an
$\delta$–net covering in the $||\cdot||_{\infty}$ norm. The cardinality of
this covering $\mathcal{N}$ obeys

$|\mathcal{N}|\leq\left(1+\frac{2}{\delta}\right)^{d^{2}-1}.$ (141)

We want an $\varepsilon$-net in the $||\cdot||_{1}$ norm, so that for any
$\eta=\frac{1}{d}(\mathbbm{1}+A)$ in $\partial{\cal D}$ there is an
$\eta_{k}=\frac{1}{d}(\mathbbm{1}+A_{k})$ in the net such that

$D(\eta,\eta_{k})=\frac{1}{2}||\eta-\eta_{k}||_{1}-\frac{1}{2}\absolutevalue{\mathrm{tr}\eta-\mathrm{tr}\eta_{k}}=\frac{1}{2d}||A-A_{k}||_{1}\leq\varepsilon.$
(142)

However in finite dimensions all matrix norms are equivalent, and we have that
[81]

$||A-A_{k}||_{1}\leq\mbox{rank}(A-A_{k})||A-A_{k}||_{\infty}\leq d\delta.$
(143)

Therefore choosing $\delta=2\varepsilon$ ensures that
$D(\eta,\eta_{k})\leq\varepsilon$ as required, and we can always find a
sufficient set of reference frame states
$\mathcal{N}=\\{\eta_{k}\\}_{k=1}^{N}$ such that
$N\leq(1+1/\varepsilon)^{d^{2}-1}$.

We now check the $\Delta H_{\eta}$ condition on each $\eta=\eta_{k}$ in the
$\varepsilon$–net. If $\Delta H_{\eta_{k}}<0$ for one of these states then the
transformation is impossible. If however we find that

$\Delta H_{\eta_{k}}\geq r(\varepsilon)\mbox{ for all
}\eta_{k}\in\mathcal{N},$ (144)

then we know from the continuity of the function $\Delta H_{\eta}$, Lemma 4,
that this implies that $\Delta H_{\eta}\geq 0$ for all reference frame states
$\eta\in\partial{\cal D}$. However this is a sufficient set of states (Theorem
3) and therefore we deduce from these $N$ conditions that the transformation
is possible covariantly. The final case of at least one of the $\eta_{k}$
conditions giving $0\leq\Delta H_{\eta_{k}}\leq r(\varepsilon)$ can be handled
as follows: we can supplement the state $\rho_{A}$ with an additional
reference frame state $\chi_{A^{\prime}}$. If we take $\chi_{A^{\prime}}$ to
be large and approximating a perfect reference frame state (perfectly encoding
the group element) then it is possible to transform to any quantum state
covariantly. This does so by reducing the entropy $H_{\rm
min}(R|AA^{\prime})$. This can be used to increase $\Delta H_{\eta_{k}}$ and
ensure that $\Delta H_{\eta_{k}}\geq r(\varepsilon)$, from which we can deduce
that the transformation is now possible. The state $\chi_{A^{\prime}}$
therefore provides an upper bound estimate on the minimal additional asymmetry
required to realise the transformation. Since $r(\varepsilon)=O(\varepsilon)$
the requirement on $\chi$ is to provide $O(\varepsilon)$ resources as measured
by the single-shot entropy. ∎

##  Appendix F The conical structure of $\Phi_{\tau}(\mathbf{x})$ (proof of
Lemma 7)

See 7

###### Proof.

Let ${\cal P}$ be the completely depolarising channel on the reference system,
and define $A(\bm{x})\coloneqq\sum_{k=1}^{d^{2}-1}x_{k}X_{k}$. Then by
Property (P4) of $\Phi_{\tau}(\mathbf{x})$, we have:

$\displaystyle\Phi\left({\cal
G}\left[\left(\frac{\mathbbm{1}}{d}+A(\mathbf{x})\right)\otimes\tau\right]\right)$
$\displaystyle\geq\Phi\left({\cal P}\otimes\mathbbm{1}\circ{\cal
G}\left[\left(\frac{\mathbbm{1}}{d}+A(\mathbf{x})\right)\otimes\tau\right]\right)$
(145) $\displaystyle=\Phi\left({\cal G}\left[{\cal
P}\left(\left(\frac{\mathbbm{1}}{d}+A(\mathbf{x})\right)\right)\otimes\tau\right]\right)$
(146) $\displaystyle=\Phi\left({\cal
G}\left[\frac{\mathbbm{1}}{d}\otimes\tau\right]\right)\equiv\Phi_{\tau}(\mathbf{0}).$
(147)

Therefore, $\tilde{\Phi}_{\tau}(\mathbf{x})\geq 0$, as claimed.

Let $p$ be a probability. Then

$\displaystyle
p\eta(\mathbf{x})+(1-p)\frac{\mathbbm{1}}{d}=p\frac{\mathbbm{1}}{d}+pA(\mathbf{x})+(1-p)\frac{\mathbbm{1}}{d}=\frac{\mathbbm{1}}{d}+A(p\mathbf{x})$
(148)

Using this equation, we can rewrite Lemma 18 as:

######  Lemma 24.

Let $p$ be a probability; i.e. $0\leq p\leq 1$. Then for all
$\mathbf{x}\in{\cal S}$:

$\displaystyle\Phi_{\tau}\left(p\mathbf{x}\right)=p\Phi_{\tau}(\mathbf{x})+\frac{1-p}{d}$
(149)

We then immediately have

$\displaystyle\tilde{\Phi}_{\tau}(p\mathbf{x})=p\Phi_{\tau}(\mathbf{x})+\frac{1-p}{d}-\frac{1}{d}=p\Phi_{\tau}(\mathbf{x})-\frac{p}{d}=p\tilde{\Phi}_{\tau}(\mathbf{x}),\
\forall\mathbf{x}\in{\cal S}.$ (150)

Making the change of variables $\mathbf{x}^{\prime}\coloneqq p\mathbf{x}$, we
find that the equation above is equivalent to:

$\displaystyle\tilde{\Phi}_{\tau}(\mathbf{x}^{\prime})=p\tilde{\Phi}\left(\frac{1}{p}\mathbf{x}^{\prime}\right)\Rightarrow\frac{1}{p}\tilde{\Phi}_{\tau}(\mathbf{x}^{\prime})=\tilde{\Phi}_{\tau}\left(\frac{1}{p}\mathbf{x}^{\prime}\right),\forall\
\frac{1}{p}\mathbf{x^{\prime}}\in{\cal S}.$ (151)

Since $1\leq\frac{1}{p}\leq\infty$, we can combine the facts above and
conclude that, for all $\lambda\geq 0$ such that
$\mathbf{x},\lambda\mathbf{x}\in{\cal S}$:

$\displaystyle\tilde{\Phi}_{\tau}(\lambda\mathbf{x})=\lambda\tilde{\Phi}_{\tau}(\mathbf{x}),$
(152)

as claimed. ∎

##  Appendix G Calculating $\Phi_{\tau}(\mathbf{x})$ for covariant
transformations in a qubit

###  G.1 Time-Covariant Transformations ($U(1)$)

We will consider a qubit with the Hamiltonian $\sigma_{z}$, transformations
among whose states are limited to those that are symmetric under all time
translations $\\{e^{i\sigma_{z}t}\ |\ 0\leq t<2\pi\\}$. The reference system
is then a qubit with Hamiltonian $-\sigma_{z}$. We will calculate
$\Phi_{\tau}(\mathbf{x})$ for an arbitrary state $\tau$ of this qubit, while
reference states are given using the Bloch co-ordinates:

$\displaystyle\eta(x,y,z)=\frac{\mathbbm{1}}{2}+x\frac{\sigma_{x}}{2}+y\frac{\sigma_{y}}{2}+z\frac{\sigma_{z}}{2},$
(153)

where the $\sigma_{i}$ are the Pauli matrices, and $x^{2}+y^{2}+z^{2}\leq 1$.
The derivation will be conducted entirely in the energy eigenbasis.

Recall from Eq. (59):

$\displaystyle\Phi_{\tau}(x,y,z)\equiv\Phi_{\eta(x,y,z)}(\tau)\coloneqq\inf_{X\geq
0}\\{\mathrm{tr}(X):\mathbbm{1}\otimes X-{\cal G}(\eta(x,y,z)\otimes\tau)\geq
0\\}$ (154)

We first introduce this simplifying lemma that allows us to vastly reduce
which $X$ we must consider:

######  Lemma 25.

When calculating $\Phi_{\eta}(\tau)$, it is sufficient to minimise over $X$
such that $X={\cal G}(X)$.

###### Proof.

Since ${\cal U}_{g}$ can be regarded as an (active) change of basis, it does
not affect the eigenvalues of a Hermitian operator $K$. Therefore,

$\displaystyle K\geq 0\Rightarrow{\cal U}_{g}(K)\geq 0$ (155)
$\displaystyle\mathrm{tr}(K)=\mathrm{tr}[{\cal U}_{g}(K)]$ (156)

for all $g$ and Hermitian $K$. Averaging over the entire group $G$, we arrive
at

$\displaystyle K\geq 0\Rightarrow{\cal G}(K)\geq 0$ (157)
$\displaystyle\mathrm{tr}(K)=\mathrm{tr}[{\cal G}(K)]$ (158)

If $X$ is a feasible solution, then it obeys the two conic constraints in the
SDP defining $\Phi_{\eta}(\tau)$: $X\geq 0$ and ${\mathbbm{1}\otimes X-{\cal
G}(\eta\otimes\tau)\geq 0}$. In this case, we see immediately from Equation
157 that ${\cal G}(X)$ also obeys the first constraint. Furthermore, since
${\cal G}[\mathbbm{1}\otimes X-{\cal
G}(\eta\otimes\tau)]=\mathbbm{1}\otimes{\cal G}(X)-{\cal G}(\eta\otimes\tau)$,
applying Equation 157 to $\mathbbm{1}\otimes X-{\cal G}(\eta\otimes\tau)$
shows that ${\cal G}(X)$ also obeys the second constraint. Therefore, if $X$
is a feasible solution, then so is ${\cal G}(X)$.

By Equation 158, ${\cal G}(X)$ produces the same value as $X$ on the objective
function to be minimised in the SDP defining $\Phi_{\eta}(\tau)$. We can thus
further conclude that if $X$ is a feasible solution, then ${\cal G}(X)$ is an
equally good feasible solution. It is then sufficient to only minimise over
$X$ such that $X={\cal G}(X)$ when calculating $\Phi_{\eta}(\tau)$. ∎

This lemma means we can take $X$ to be diagonal without loss of generality,
i.e.

$\displaystyle X=\begin{pmatrix}x_{1}&0\\\
0&x_{2}\end{pmatrix}\Rightarrow\mathbbm{1}\otimes
X=\begin{pmatrix}x_{1}&0&0&0\\\ 0&x_{2}&0&0\\\ 0&0&x_{1}&0\\\
0&0&0&x_{2}\end{pmatrix}$ (159)

We have already seen from the main text (Eq. (30)) that $\Phi_{\tau}(x,y,z)$
is cylindrically symmetric about the $z$-axis. This means we can restrict our
calculation to $x\geq 0,y=0$, and then equate

$\displaystyle\Phi_{\tau}(x,y,z)=\Phi_{\tau}(\sqrt{x^{2}+y^{2}},0,z)$ (160)

Having applied this restriction, we further see from Eq. (31) that

$\displaystyle\Phi_{\sigma_{x}\tau\sigma_{x}}(x,0,z)=\Phi_{\tau}(x,0,-z),$
(161)

which means we can additionally restrict our attention to $z\geq 0$. We
therefore only need to consider reference states of the form:

$\displaystyle\eta=\frac{1}{2}\begin{pmatrix}1+z&x\\\ x&1-z\end{pmatrix},\
x,z\geq 0.$ (162)

Recalling the parameterisation (Eq. (27))

$\displaystyle\tau\coloneqq\begin{pmatrix}p_{\tau}&c_{\tau}\\\
c^{*}_{\tau}&1-p_{\tau}\end{pmatrix},$ (163)

we calculate

$\displaystyle\eta\otimes\tau$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1+z&x\\\
x&1-z\end{pmatrix}\otimes\begin{pmatrix}p_{\tau}&c_{\tau}\\\
c^{*}_{\tau}&1-p_{\tau}\end{pmatrix}$ (164)
$\displaystyle=\frac{1}{2}\begin{pmatrix}p_{\tau}(1+z)&c_{\tau}(1+z)&p_{\tau}x&c_{\tau}x\\\
c^{*}_{\tau}(1+z)&(1-p_{\tau})(1+z)&c^{*}_{\tau}x&(1-p_{\tau})x\\\
p_{\tau}x&c_{\tau}x&p_{\tau}(1-z)&c_{\tau}(1-z)\\\
c^{*}_{\tau}x&(1-p_{\tau})x&c^{*}_{\tau}(1-z)&(1-p_{\tau})(1-z)\end{pmatrix}$
(165)

The $G$-twirling only leaves elements in
$\operatorname{span}(|00\rangle,|11\rangle),\operatorname{span}(|01\rangle)$
and $\operatorname{span}(|10\rangle)$ intact:

$\displaystyle{\cal
G}(\eta\otimes\tau)=\frac{1}{2}\begin{pmatrix}p_{\tau}(1+z)&0&0&c_{\tau}x\\\
0&(1-p_{\tau})(1+z)&0&0\\\ 0&0&p_{\tau}(1-z)&0\\\
c^{*}_{\tau}x&0&0&(1-p_{\tau})(1-z)\end{pmatrix}.$ (166)

Therefore:

$\displaystyle\mathbbm{1}\otimes
X-\mathcal{G}(\eta\otimes\tau)=\begin{pmatrix}x_{1}-p_{\tau}\frac{1+z}{2}&0&0&-c_{\tau}\frac{x}{2}\\\
0&x_{2}-(1-p_{\tau})\frac{1+z}{2}&0&0\\\ 0&0&x_{1}-p_{\tau}\frac{1-z}{2}&0\\\
-c^{*}_{\tau}\frac{x}{2}&0&0&x_{2}-(1-p_{\tau})\frac{1-z}{2}\end{pmatrix}.$
(167)

Making a (passive) change of basis so the matrix appears block-diagonal, we
obtain:

$\displaystyle\mathbbm{1}\otimes
X-\mathcal{G}(\eta\otimes\tau)=\begin{pmatrix}x_{1}-p_{\tau}\frac{1+z}{2}&-c_{\tau}\frac{x}{2}&0&0\\\
-c^{*}_{\tau}\frac{x}{2}&x_{2}-(1-p_{\tau})\frac{1-z}{2}&0&0\\\
0&0&x_{2}-(1-p_{\tau})\frac{1+z}{2}&0\\\
0&0&0&x_{1}-p_{\tau}\frac{1-z}{2}\end{pmatrix}.$ (168)

The Sylvester Criterion [81] states that a matrix is semi-definite positive if
and only if all its upper-left determinants are greater than or equal to 0.
This produces the following criteria:

  1. 1.

$x_{1}\geq p_{\tau}\frac{1+z}{2}$.

  2. 2.

$(x_{1}-p_{\tau}\frac{1+z}{2})(x_{2}-(1-p_{\tau})\frac{1-z}{2})\geq\frac{\absolutevalue{c_{\tau}}^{2}x^{2}}{4}$.

  3. 3.

$x_{2}\geq(1-p_{\tau})\frac{1+z}{2}$.

  4. 4.

$x_{1}\geq p_{\tau}\frac{1-z}{2}$.

Since we have restricted ourselves to $x\geq 0$, condition 4 is redundant
given condition 1. Furthermore, given any value of $x_{2}$ that satisfies
condition 3, the smallest value of $x_{1}$ satisfying condition 1 is

$\displaystyle
x_{1}-p_{\tau}\frac{1+z}{2}=\frac{\absolutevalue{c_{\tau}}^{2}x^{2}}{4}\frac{1}{x_{2}-(1-p_{\tau})\frac{1-z}{2}}.$
(169)

We are therefore looking to minimise

$\displaystyle
x_{1}+x_{2}=x_{2}+\frac{\absolutevalue{c_{\tau}}^{2}x^{2}}{4}\frac{1}{x_{2}-(1-p_{\tau})\frac{1-z}{2}}+p_{\tau}\frac{1+z}{2}.$
(170)

This occurs at

$\displaystyle
x_{2}=(1-p_{\tau})\frac{1-z}{2}\pm\absolutevalue{c_{\tau}}\frac{x}{2}.$ (171)

However, $x_{2}$ also has to satisfy condition 3. Therefore, the value $x_{2}$
should take is whichever of the following two

$\displaystyle x_{2}=\begin{cases}(1-p_{\tau})\frac{1+z}{2}\\\
(1-p_{\tau})\frac{1-z}{2}+\absolutevalue{c_{\tau}}\frac{x}{2}\end{cases}$
(172)

is bigger, which results in

$\displaystyle\Phi_{\tau}(x\geq 0,y=0,z\geq
0)=\begin{cases}\frac{\absolutevalue{c_{\tau}}^{2}}{1-p_{\tau}}\frac{x^{2}}{4z}+\frac{z}{2}+\frac{1}{2}&\text{
for }\frac{x}{2z}\leq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\\\
\left(p_{\tau}-\frac{1}{2}\right)z+\absolutevalue{c_{\tau}}x+\frac{1}{2}&\text{
for }\frac{x}{2z}\geq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\end{cases}$
(173)

Finally, applying Equations 160 and 161, and noting that
$\sigma_{x}\tau\sigma_{x}$ translates to $p_{\tau}\rightarrow 1-p_{\tau}$ and
$c_{\tau}\rightarrow c^{*}_{\tau}$ in our parameterisation of $\tau$, we can
calculate $\Phi_{\tau}$ for all reference states from the above result as seen
in Eq. (28):

$\displaystyle\Phi_{\tau}(x,y,z)=\begin{cases}\frac{\absolutevalue{c_{\tau}}^{2}}{1-p_{\tau}}\frac{x^{2}+y^{2}}{4z}+\frac{z}{2}+\frac{1}{2}&\text{
for
}0\leq\frac{\sqrt{x^{2}+y^{2}}}{2z}\leq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\\\
\left(p_{\tau}-\frac{1}{2}\right)z+\absolutevalue{c_{\tau}}\sqrt{x^{2}+y^{2}}+\frac{1}{2}&\text{
for
}\frac{\sqrt{x^{2}+y^{2}}}{2z}\geq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\text{
and
}\frac{\sqrt{x^{2}+y^{2}}}{2z}\leq-\frac{p_{\tau}}{\absolutevalue{c_{\tau}}}\\\
-\frac{\absolutevalue{c_{\tau}}^{2}}{p_{\tau}}\frac{x^{2}+y^{2}}{4z}-\frac{z}{2}+\frac{1}{2}&\text{
for
}0\geq\frac{\sqrt{x^{2}+y^{2}}}{2z}\geq-\frac{p_{\tau}}{\absolutevalue{c_{\tau}}}.\end{cases}$
(174)

####  G.1.1 Two entropic conditions suffice to characterise time-covariant
transformations in a non-degenerate qubit

We can alternatively characterise each qubit reference frame state as
$\eta(r,\theta,\phi)$, where $(r,\theta,\phi)^{T}$ is state’s Bloch vector in
spherical polar co-ordinates (radial, polar and azimuthal respectively). Using
the standard conversion between spherical and Cartesian co-ordinates
$x=r\sin(\theta)\cos(\phi)$, $y=r\sin(\theta)\sin(\phi)$ and
$z=r\cos(\theta)$, we can rewrite Equation 174 as

$\displaystyle\Phi_{\tau}(x,y,z)$
$\displaystyle\equiv\Phi_{\tau}(r\sin(\theta)\cos(\phi),r\sin(\theta)\sin(\phi),r\cos(\theta))$
(175)
$\displaystyle=\begin{cases}r\frac{\absolutevalue{c_{\tau}}^{2}}{1-p_{\tau}}\frac{\tan(\theta)\sin(\theta)}{4}+r\frac{\cos(\theta)}{2}+\frac{1}{2}&\text{
for
}0\leq\frac{\tan(\theta)}{2}\leq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\\\
r\left(p_{\tau}-\frac{1}{2}\right)\cos(\theta)+r\absolutevalue{c_{\tau}}\sin(\theta)+\frac{1}{2}&\text{
for
}\frac{\tan(\theta)}{2}\geq\frac{1-p_{\tau}}{\absolutevalue{c_{\tau}}}\text{
and }\frac{\tan(\theta)}{2}\leq-\frac{p_{\tau}}{\absolutevalue{c_{\tau}}}\\\
-r\frac{\absolutevalue{c_{\tau}}^{2}}{p_{\tau}}\frac{\tan(\theta)\sin(\theta)}{4}-r\frac{\cos(\theta)}{2}+\frac{1}{2}&\text{
for
}0\geq\frac{\tan(\theta)}{2}\geq-\frac{p_{\tau}}{\absolutevalue{c_{\tau}}}.\end{cases}$
(176)

######  Lemma 26.

There exists a time-covariant transformation $\rho$ to $\sigma$ in a qubit
with Hamiltonian $\sigma_{z}$ if and only if

$\displaystyle\partial^{2}_{\theta}(\Delta\Phi_{\eta})|_{\theta=0}\geq 0\text{
and }\partial^{2}_{\theta}(\Delta\Phi_{\eta})|_{\theta=\pi}\geq 0$ (177)

for all $0<r\leq 1$ and $0\leq\phi<2\pi$, where we recall
$\Delta\Phi_{\eta}\coloneqq\Phi_{\eta(r,\theta,\phi)}(\rho)-\Phi_{\eta(r,\theta,\phi)}(\sigma)$
and $\partial_{\theta}\coloneqq\frac{\partial}{\partial\theta}$. Since
$\Phi_{\eta}(\tau)$ is monotonically decreasing in $H_{\eta}(\tau)$, this is
equivalent to

$\displaystyle\partial^{2}_{\theta}(\Delta H_{\eta})|_{\theta=0}\geq 0\text{
and }\partial^{2}_{\theta}(\Delta H_{\eta})|_{\theta=\pi}\geq 0,$ (178)

where we recall $\Delta H_{\eta}\coloneqq
H_{\eta(r,\theta,\phi)}(\sigma)-H_{\eta(r,\theta,\phi)}(\rho)$.

###### Proof.

From Eq. (176), we can straightforwardly evaluate:

$\displaystyle\partial^{2}_{\theta}\Phi_{\eta}(\tau)|_{\theta=0}=\frac{r}{2}\left(\frac{\absolutevalue{c_{\tau}}^{2}}{1-p_{\tau}}-1\right)\text{
and
}\partial^{2}_{\theta}\Phi_{\eta}(\tau)|_{\theta=\pi}=\frac{r}{2}\left(\frac{\absolutevalue{c_{\tau}}^{2}}{p_{\tau}}-1\right),$
(179)

from which it immediately follows that

$\displaystyle\partial^{2}_{\theta}(\Delta\Phi_{\eta})|_{\theta=0}\geq
0\Longleftrightarrow c_{\sigma}\leq
c_{\rho}\sqrt{\frac{1-p_{\sigma}}{1-p_{\rho}}}\text{ and
}\partial^{2}_{\theta}(\Delta\Phi_{\eta})|_{\theta=\pi}\geq
0\Longleftrightarrow c_{\sigma}\leq
c_{\rho}\sqrt{\frac{p_{\sigma}}{p_{\rho}}},$ (180)

which are known necessary and sufficient conditions for a time-covariant
transfromation from $\rho$ to $\sigma$ in a qubit with the Hamiltonian
$\sigma_{z}$ [79]. ∎

We note that $\partial_{\theta}(\Delta\Phi_{\eta})|_{\theta=0}=0$ and
$\partial_{\theta}(\Delta\Phi_{\eta})|_{\theta=\pi}=0$. At $r=1$, this Lemma
can be loosely interpreted as asserting that two pure reference states,
infinitesimally close to the energy eigenstates $\outerproduct{0}{0}$ and
$\outerproduct{1}{1}$ respectively, are sufficient for determining whether
time-covariant interconversion is possible between any input and output states
in a non-degenerate qubit.

###  G.2 Unitarily covariant transformations ($SU(2)$)

The only channels that are covariant with all unitary transformations on the
qubit are [22]:

$\displaystyle{\cal
E}_{\lambda}(\rho)=\frac{1}{2}(\mathbbm{1}+\lambda\bm{r}\cdot\bm{\sigma}),$
(181)

where $\lambda\in[-1/3,1]$, and $\bm{r}$ is the Bloch vector of $\rho$.
Therefore, writing $\bar{\bm{x}}=(x,-y,z)$ given the Bloch vector
$\bm{x}=(x,y,z)$ of $\eta$, we have that

$\displaystyle\mathrm{tr}[\eta^{T}{\cal
E}_{\lambda}(\rho)]=\frac{1}{2}(1+\lambda\bar{\bm{x}}\cdot\bm{r}).$ (182)

We now want to consider the quantity

$\displaystyle\Phi_{\rho}(\bm{x})=\max_{{\cal E}\in\mathcal{O}_{\rm
cov}}\mathrm{tr}[\eta^{T}{\cal
E}(\rho)]=\max_{\lambda\in\left[-\frac{1}{3},1\right]}\mathrm{tr}[\eta^{T}{\cal
E}_{\lambda}(\rho)]=\max_{\lambda\in\left[-\frac{1}{3},1\right]}\frac{1}{2}(1+\lambda\bar{\bm{x}}\cdot\bm{r}).$
(183)