# Quantum measurements and equilibration:
the emergence of objective reality via entropy maximisation

Emanuel Schwarzhans<EMAIL_ADDRESS>Institute for Quantum Optics and
Quantum Information - IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Atominstitut, Technische Universität
Wien, 1020 Vienna, Austria  Felix C. Binder<EMAIL_ADDRESS>School
of Physics, Trinity College Dublin, Dublin 2, Ireland  Marcus Huber
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien, 1020
Vienna, Austria  Institute for Quantum Optics and Quantum Information - IQOQI
Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
Maximilian P. E. Lock<EMAIL_ADDRESS>Atominstitut,
Technische Universität Wien, 1020 Vienna, Austria  Institute for Quantum
Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria

(August 30, 2024)

###### Abstract

Textbook quantum physics features two types of dynamics, reversible unitary
dynamics and irreversible measurements. The latter stands in conflict with the
laws of thermodynamics and has evoked debate on what actually constitutes a
measurement. With the help of modern quantum statistical mechanics, we take
the first step in formalising the hypothesis that quantum measurements are
instead driven by the natural tendency of closed systems to maximize entropy,
a notion that we call the Measurement-Equilibration Hypothesis. In this
paradigm, we investigate how objective measurement outcomes can emerge within
an purely unitary framework, and find that: (i) the interactions used in
standard measurement models fail to spontaneously feature emergent objectivity
and (ii) while ideal projective measurements are impossible, we can (for a
given form of Hamiltonian) approximate them exponentially well as we collect
more physical systems together into an “observer” system. We thus lay the
groundwork for self-contained models of quantum measurement, proposing
improvements to our simple scheme.

##  I Introduction

The mathematical formulation of quantum theory includes two types of dynamics
for a closed quantum system: unitary evolution, and measurement. The latter is
often referred to as a collapse of the wavefunction, instantaneously updating
the state of the system, and is at the heart of many interpretational
discussions of quantum mechanics (e.g. the “measurement problem”). For
projective measurements, this is an apparent contradiction of the laws of
thermodynamics, as it does not conserve energy, allows for an arbitrary
decrease in entropy [1], and allows for cooling to zero-temperature with
finite resources, in violation of Nernst’s formulation of the $3^{\text{rd}}$
law [2, 3]. In this work, we investigate a potential resolution to this
apparent contradiction by modelling the process of measurement as the system
of interest and its environment undergoing an equilibration process, thus
being driven by an _increase_ in entropy.

In the widely-used von Neumann measurement scheme [4], subsequently developed
into what is sometimes referred to as the standard model of measurement [5], a
carefully-timed interaction correlates the system of interest with some probe
or pointer system. This allows, for example, an investigation of the
restrictions placed by quantum theory on incompatible observables. Quantum
Darwinism [6, 7, 8] broadens this model to include the environment, and thus
the observer itself, highlighting the role of the redundant encoding of
information about the system state in the measurement basis (the “pointer
basis”) into the environment degrees of freedom, arguably removing the need
for an explicit “Heisenberg cut”. Demanding that this redundancy be such that
multiple observers (i.e. parts of the environment) independently agree on the
system state, and that the only correlations between them be their shared
information about the system, implies that the total state of the system and
observers necessarily exhibits Spectrum Broadcast Structure [9, 10, 11]. The
appearance of this structure has been demonstrated theoretically for a
dielectric sphere in a photonic environment [12], in a spin-spin model [13,
14], and for Quantum Brownian Motion [15].

The process of measurement, specifically the transformation from quantum
indeterminacy to an objective fact, is irreversible, and occurs without
precise control of all microscopic degrees of freedom of the system’s
environment. A dynamical model of measurement must therefore explain how the
system and its environment evolve spontaneously and irreversibly towards a
state structure exhibiting objectivity. Since spontaneous and irreversibile
processes in nature are those associated with the process of equilibration, we
hypothesise that any measurement is an entropy-increasing transition towards
an equilibrium state, and refer to this as the Measurement-Equilibration
Hypothesis. The notion that measurements correspond to an increase in entropy
has been discussed at various points since the early years of quantum theory,
e.g. [16, 4, 17, 18, 19], but is absent from dynamical models of measurement.
Recent advances in quantum statistical mechanics, specifically in the
equilibration of closed quantum systems [20] allow us to address this absence,
and as a first step in this direction, we seek to determine the circumstances
under which a general system and environment equilibrate to states exhibiting
objective properties, including the Spectrum Broadcast Structure mentioned
above.

In this work, we constrain the form of the interactions capable of producing
such equilibrium states, and show that while it is impossible to reach this
state structure exactly without violating the laws of thermodynamics, one can
approach it exponentially as we increase the marcroscopicity of “observers”
for whom the measurement outcome should be objective. We find that the usual
form of interactions employed in the standard model of measurement in fact
does not result in an equilibrium state exhibiting objectivity, or even any
correlations whatsoever between the system and its environment.

We begin by revisiting the concepts of objectivity and Spectrum Broadcast
Structure and how they connect, as well as the role of the time-averaged state
in equilibration phenomena. We then derive our main results and discuss the
consequences of our findings as well as future directions and open questions.

##  II Equilibration of quantum systems

To investigate the conditions under which objectivity emerges as a result of
equilibration, we will employ the notion of equilibration on average [20],
whereby the equilibrium state is given by the infinite-time average. That is,
if a system evolving as $\rho(t)$ equilibrates on average, then the
equilibrium state is given by

$\displaystyle\rho_{\infty}:=\lim_{T\rightarrow\infty}\dfrac{1}{T}\int_{0}^{T}\rho(t)dt.$
(1)

This is the unique state which maximises the von Neumann entropy given all
conserved quantities [21, 20]. To illustrate the sense in which states
equilibrate to their time average, we now review two examples of equilibration
on average.

First, consider a system evolving unitarily via some time-independent
Hamiltonian $H$, so that $\rho(t)=U(t)\rho(0)U^{\dagger}(t)$ with
$U(t)=e^{iHt}$, and denote by $\\{p_{j}\\}_{j}$ the diagonal elements of
$\rho(0)$ in the eigenbasis of $H$. The effective dimension – sometimes called
the inverse participation ratio – is defined as
$d_{\mathrm{eff}}[\rho(0)]:=(\sum_{j}p_{j}^{2})^{-1}$. Assuming that no two
energy gaps of $H$ are equal in magnitude (which also implies that $H$ is non-
degenerate), then an arbitrary bounded operator $O$ satisfies [22, 23]

$\langle|\textnormal{Tr}[O\rho(t)-O\rho_{\infty}]|^{2}\rangle\leq\frac{||O||^{2}}{d_{\mathrm{eff}}[\rho(0)]},$
(2)

where $\langle\cdot\rangle$ denotes the average over time and $||\cdot||$ is
the standard operator norm. As a second example, consider $\rho(t)$ to be some
subsystem of a larger, unitarily evolving system, i.e.
$\rho(t)=\textnormal{Tr}_{X}[U(t)\tilde{\rho}(0)U^{\dagger}(t)]$ for some
$\tilde{\rho}(0)$ and tensor factor $X$, and again assume that no two energy
gaps of $H$ are equal. In this case, the average trace distance between
$\rho(t)$ and $\textnormal{Tr}_{X}[\tilde{\rho}_{\infty}]$ satisfies [24]

$\langle
D\left[\rho(t),\textnormal{Tr}_{X}[\tilde{\rho}_{\infty}]\right]\rangle\leq\frac{1}{2}\sqrt{\frac{d^{2}}{d_{\mathrm{eff}}[\tilde{\rho}(0)]}},$
(3)

where $d$ is the dimensionality of the Hilbert space to which the subsystem
$\rho(t)$ belongs.

We note three issues for both of these examples to be considered
equilibration. First, both rely on an effective dimension
$d_{\mathrm{eff}}[\rho(0)]$ being in some sense large. This quantity measures
the number of significantly occupied energy levels participating in the
evolution of $\rho(t)$, and its typically large size for large systems can be
argued for on both physical and mathematical grounds. Second, there is the
necessity for some finite equilibration time, and third, the possibility of
Poincaré recurrences. All three of these issues, as well as generalisations
beyond the condition of no equal energy gaps, are addressed in detail in [20].

##  III Objectivity and quantum states

Let $\mathcal{H}_{\mathrm{S}}$ denote the Hilbert space corresponding to some
$d_{\mathrm{S}}$-dimensional system of interest, and let
$\\{\mathcal{H}_{k}\\}$ with $k=1,\ldots,N$ denote the Hilbert spaces
corresponding to $N$ systems which form the environment of the system of
interest, and which we will associate with observers, for whom we seek
objectivity. Note that we do not _a priory_ impose any criteria for what is to
be considered as an observer. In order for a fact about the system to be
considered objective (and thus classical), multiple observers must be able to
discover it in such a way that they agree with each other and that none of
them changes the fact through their observation. As noted above, formalising
this notion in the context of quantum theory [25] and combining it with the
requirement that observers be uncorrelated except via the system (“strong
independence”), is equivalent to demanding that the state of the system and
the observers exhibits Spectrum Broadcast Structure [9]. A state has Spectrum
Broadcast Structure with respect to a pointer basis
$\\{|{i}\rangle\\!\langle{i}|\\}$, where $i=1,\ldots,d_{\mathrm{S}}$, if it is
of the form

$\displaystyle\rho_{\mathrm{SBS}}=\sum_{i=1}^{d_{\mathrm{S}}}p_{i}\,|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\bigotimes_{k=1}^{N}\rho_{k}^{(i)},$
(4)

where $p_{i}$ is the probability associated with outcome
$\left|\right.\\!{i}\\!\left.\right\rangle$, and $\rho_{k}^{(i)}$ is the state
on $\mathcal{H}_{k}$ conditioned on that outcome, and where
$\rho_{k}^{(i)}\rho_{k}^{(j)}=0$ for $i\neq j$. As in [15], we will find it
more convenient to express this orthogonality in terms of the fidelity, namely
$F(\rho_{k}^{(i)},\rho_{k}^{(j)})=0$ for $i\neq j$, where
$F(\sigma,\rho):=\textnormal{Tr}\left(\sqrt{\sqrt{\sigma}\rho\sqrt{\sigma}}\right)^{2}$.

The structure in Eq. (4) thus represents a state where quantum information has
become redundantly encoded, and through this can be considered classical [26].
Note, however, that this is only true in the pointer basis, in accordance with
no-cloning theorem [27]. This state structure holds if and only if the
conditions of Quantum Darwinism and Strong Independence are met [10]. A
projective measurement can then be understood as a post-selection performed on
the process by which an initially uncorrelated state between system and
observers tends to a state with Spectrum Broadcast Structure in the
measurement basis. Consequently, to say that a projective measurement has been
performed, one must first arrive at a state structure exhibiting objectivity.

##  IV Measurement as an equilibration process

With this framework, we can now state precisely what it means to demand that
objectivity emerge as a consequence of equilibration, and thus the
maximization of entropy. To wit, we assume that system and observers are
initially uncorrelated and seek the conditions under which they equilibrate to
a state of objectivity, specifically when $\rho_{\infty}=\rho_{SBS}$. Note
that this demand does not restrict the observer states $\rho_{k}^{(i)}$ in Eq.
(4), beyond the orthogonality requirement described above. As above, we denote
the measurement basis on $\mathcal{H}_{\mathrm{S}}$ by
$\\{\left|\right.\\!{i}\\!\left.\right\rangle\\}_{i}$, and the outcome
probabilities by $\\{p_{i}\\}$, i.e.
$p_{i}=\left\langle\right.\\!{i}\\!\left.\right|\rho_{\mathrm{S},0}\left|\right.\\!{i}\\!\left.\right\rangle$,
where $\rho_{\mathrm{S},0}$ is the initial state of the system.

Note that the infinite-time average is a pinching map with respect to the
Hamiltonian generating the evolution [21], i.e.

$\rho_{\infty}=\sum_{E}\Pi_{E}\rho(0)\Pi_{E},$ (5)

where $H=\sum_{E}E\,\Pi_{E}$ is the spectral decomposition of $H$, and we
therefore have $[H,\rho_{\infty}]=0$. Consequently, to equilibrate to a state
with a given structure, the Hamiltonian must commute with all states of that
structure. In particular, for Spectrum Broadcast Structure, we must have
$[H,\rho_{\mathrm{SBS}}]=0$, which can be interpreted as an extension of the
commutation relation defining the pointer basis [28] to the observers’
systems. As we discuss below, no such Hamiltonian exists, which precludes
exact equilibration to states with this structure.

##  V Exact projective measurements are impossible

Let us first examine a weaker requirement than that of Spectrum Broadcast
Structure. We write the initial state
${\rho(0)=\rho_{\mathrm{S},0}\otimes\rho_{E,0}}$, where $\rho_{E,0}$ is the
state on $\bigotimes_{k=1}^{N}\mathcal{H}_{k}$, and we demand a “classical-
quantum” [29] equilibrium state between the system and the observers, i.e.
$\rho_{\infty}=\sum_{i=1}^{d_{\mathrm{S}}}p_{i}\,|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes\rho_{E}^{(i)}$,
where the $\\{\rho_{E}^{(i)}\\}_{i}$ are some density matrices on
$\bigotimes_{k=1}^{N}\mathcal{H}_{k}$. Such a state may violate the Strong
Independence condition. As shown in Appendix B, this requirement implies that
$H$ takes the form

$H=\sum_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes
H_{\mathrm{E}}^{(i)},$ (6)

for some Hermitian matrices $\\{H_{E}^{(i)}\\}_{i}$ on
$\bigotimes_{k=1}^{N}\mathcal{H}_{k}$. Note that Eq. (6) can only be
considered as a sum of a free system Hamiltonian $H_{\mathrm{S}}$ and some
interaction $H_{int}$ in the case where the pointer basis consists of
eigenstates of $H_{\mathrm{S}}$. For measurement in any other basis, it
corresponds to an interaction term alone, necessitating the so-called
measurement limit, often used in dynamical models of measurement, where free
terms in the Hamiltonian are neglected i.e. $H\approx H_{int}$ (see e.g. [5]).

When we now demand, in addition to the above, that the observers’ states
unambiguously encode the system state, i.e. that
$F(\rho_{E}^{(i)},\rho_{E}^{(j)})=0$ for $i\neq j$, we find that this only
holds when $\rho_{E}^{(i)}=0$ $\forall i$ (see Appendix B for proof). In
particular, $F(\rho_{k}^{(i)},\rho_{k}^{(j)})$ is bounded below by
$1/d_{E}^{2}$, where $d_{E}$ is the dimensionality of
$\bigotimes_{k=1}^{N}\mathcal{H}_{k}$. Consequently, it is impossible for an
initially uncorrelated state to equilibriate exactly to a state with the
properties listed above, as one might expect. This precludes equilibration to
a state exhibiting objectivity, as defined in [9]. The stronger condition of
Spectrum Broadcast Structure (which combines objectivity with Strong
Independence) is then likewise impossible. In [1], the authors found that it
was possible to perform an ideal projective measurement given access to a
zero-temperature pointer state, but here we see that, adopting the
Measurement-Equilibration Hypothesis, it is impossible even given access to
such a resource. That the lower bound of $F(\rho_{k}^{(i)},\rho_{k}^{(j)})$
rapidly decreases with dimensionality however, does allow the approximation an
objective state as the size of the environment increases, as we will discuss
later.

##  VI Equilibration and the standard model of measurement

As discussed above, dynamical models of measurement usually assume a degree of
temporal control such that a particular unitary transformation can be
performed on the system and some apparatus/environment, generated the product
of observables acting on the system and the apparatus [5]. For example, one
such model was used to investigate the transition to Spectrum Broadcast
Structure in [15]. In our context, let us denote such a Hamiltonian by
$H=X_{\mathrm{S}}\otimes Y$, where $X_{\mathrm{S}}$ is any operator whose
eigenbasis is the pointer basis, and where $Y$ is an observable on
$\bigotimes_{k=1}^{N}\mathcal{H}_{k}$. In this case, as shown in Appendix C,
the equilibrium state is given by

$\rho_{\infty}=\sum_{i=1}^{d_{\mathrm{S}}}p_{i}\,|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes\rho_{E},$
(7)

for some state $\rho_{E}$ on $\bigotimes_{k=1}^{N}\mathcal{H}_{k}$. This
highlights the distinction between the standard model of measurement and the
equilibration paradigm; given temporal control, one can of course generate the
desired correlations between the system and the observers, but considering a
spontaneous and uncontrolled process, one finds that on average there are no
correlations whatsoever between the system and the observers.

##  VII Approximating projective measurements

While exact equilibration to a state with Spectrum Broadcast Structure is
impossible, there remains the question of how well an equilibrium state can
approximate this structure. We saw earlier that, with increasing dimension of
$\bigotimes_{k=1}^{N}\mathcal{H}_{k}$, one can approach the desired
orthogonality condition required for objectivity (though recall that this was
a weaker condition than Spectrum Broadcast Structure). In the same vein, we
will now see that by collecting the observers’ systems together into composite
systems, which we call macro-observers, one can approach Spectrum Broadcast
Structure exponentially as the size of these macro-observers increases.

As discussed above, equilibration to a state exhibiting the appropriate
statistics in the pointer basis, with no coherences between different pointer
states, and allowing correlations between the observers and the system in this
basis requires a conditional structure of the Hamiltonian given in Eq. (6).
With the Strong Independence condition in mind, which demands that the
observers should be uncorrelated except via their correlations with the
system, we expand this to consider an interaction of the form

$H=\sum_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes\sum_{k=1}^{N}c_{k}H_{k}^{(i)}.$
(8)

We can see this structure as coupling $k$ to $\mathrm{S}$ with coupling
strength $c_{k}$, such that for the system state $i$, the action $H_{k}^{(i)}$
is performed. Assuming $H$ in Eq. (8) to be non-degenerate and starting with
the state ${\rho(0)=\rho_{\mathrm{S},0}\bigotimes_{k=1}^{N}\rho_{k,0}}$, we
arrive at the equilibrium state

$\displaystyle\rho_{\infty}=\sum_{i=1}^{d_{\mathrm{S}}}p_{i}\,|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\bigotimes_{k=1}^{N}\rho_{k}^{(i)},$
(9)

where $\rho_{k}^{(i)}$ are obtained from $\rho_{k,0}$ by applying the pinching
map with respect to $H_{k}^{(i)}$. This state has the form required for
Spectrum Broadcast Structure, but it remains to check the orthogonality
condition. To that end, let
$\left|\right.\\!{\varepsilon_{m,k}^{(i)}}\\!\left.\right\rangle$ denote the
eigenvector of $H_{k}^{(i)}$ corresponding to the eigenvalue
$\varepsilon_{m,k}^{(i)}$. For simplicity of presentation, we assume that
$H_{k}^{(i)}$ and $H_{k}^{(j)}$ have no common eigenvectors, and in Appendix D
we show how to obtain a similar result without this assumption. We will find
it useful to define the quantity

$\eta_{k}^{(ij)}:=\sum_{m}\sqrt{\left\langle\right.\\!{\varepsilon_{m,k}^{(i)}}\\!\left.\right|\rho_{k}^{(i)}\rho_{k}^{(j)}\left|\right.\\!{\varepsilon_{m,k}^{(i)}}\\!\left.\right\rangle}.$
(10)

Now, let $M<N$ denote the number of macro-observers, and let $N_{q}$ denote
the set of $k$ labels to be collected together to form a given macro-observer
$q$ (see Fig. 1). We can then write Eq. (9) in terms of macro-observers as
$\rho_{\infty}=\sum_{i=1}^{d_{\mathrm{S}}}p_{i}\,|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\bigotimes_{q=1}^{M}\tilde{\rho}_{q}^{(i)}$,
where $\\{\tilde{\rho}_{q}^{(i)}:=\bigotimes_{k\in N_{q}}\rho_{k}^{(i)}\\}$
are states on $\bigotimes_{k\in N_{q}}\mathcal{H}_{k}$. As we show in Appendix
D, the fidelity between the conditional states of the macro-observers
satisfies

$F(\tilde{\rho}_{q}^{(i)},\tilde{\rho}_{q}^{(j)})\leq
e^{-\gamma_{q}^{(ij)}|N_{q}|},$ (11)

where

$\gamma_{q}^{(ij)}:=-2\ln\left(\max_{k\in N_{q}}\eta_{k}^{(ij)}\right)>0,$
(12)

Thus as we increase each $|N_{q}|$, i.e. the the number of degrees of freedom
collected together to form each macro-observer $q$, the equilibrium state
tends exponentially towards one with Spectrum Broadcast Structure for the
macro-observers. In other words, as we coarse-grain our view of observers
towards ever-larger collections of microscopic systems, we tend exponentially
towards a state of objectivity commensurate with the occurrence of a
measurement; macroscopic observers will see objective facts about the system.

Figure 1: A schematic representation of the conditional Hamiltonian given in
Eq. 8. It depicts the interaction of system $\mathcal{H}_{\mathrm{S}}$ (in
yellow) with individual parts of the environment $\mathcal{H}_{k}$ (in red).
The coupling strength between the system and each environment is indicated by
the dashed lines and the labels $c_{k}$. The light-red frames exemplify ways
of coarse-graining the environments into macro-observers $N_{k}$.

##  VIII Discussion and conclusions

Quantum measurement is often described as the ‘collapse of the wavefunction’
when a quantum system makes contact with an observer. One is then led to the
question, what exactly can constitute an observer? [30]. The Measurement-
Equilibration Hypothesis allows us to constrain this question, giving the
answer that an observer is whatever system interacts with the system of
interest such that the result equilibrates to a state that approximates
objective state structure asymptotically. In this paradigm, we have shown the
impossibility of exact projective measurements (in accordance with known
results [1]). However, they can be approximated exponentially as more and more
microscopic systems are incorporated into a given macroscopic observer. This
accords with the intuition that small systems should not be considered
observers [31], but that larger ones can be [32, 33].

Our results provide a physical interpretation of the long-time objectivity of
the spin-spin model studied in [13]. Furthermore, understanding the initially
uncorrelated state as an out-of-equilibrium resource, the Measurement-
Equilibration Hypothesis provides a novel perspective on engines based on the
extraction of work via quantum measurements [34, 35], and thus a potential
path to resolving the foundational questions regarding the source of this
energy [36].

In the simple version of an equilibriating measurement presented above, the
Strong Independence condition on the equilibrium state led to the demand that
observers interact via a Hamiltonian with a conditional form, and
surprisingly, that the particular conditional form given by the standard model
of measurement does not satisfy the most minimal requirement for measurement,
namely correlation of observer and system. Thus, while an average over such
Hamiltonians appears to result in objectivity at a given time [37], any given
Hamiltonian of the ensemble does not equilibrate to an objective one. More
broadly, the conditional form imposed by Strong Independence does not resemble
the usual forms of interaction found in nature, suggesting that a the model
needs to be expanded if the Measurement-Equilibration Hypothesis is to be
employed in the modelling of real-world experiments.

In this regard, two immediate generalisations of our work may suffice. First,
one might allow for vanishingly small correlations violating the Strong
Independence condition. It seems likely that the requirement that such
violations be vanishingly small will nonetheless strongly constrain the form
of the interaction. A second, and in our view more promising generalisation,
would be the inclusion of an initial “pre-measurement” step, whereby one
allows some time-controlled interaction to correlate an apparatus with the
system of interest, whose subsequent equilibration with some observers leads
to a state of objectivity between the system of interest and the observers; in
this case the requirement that the system of interest be undisturbed in the
measurement basis would be less restrictive in the equilibrium step. Indeed
many measurements modelled via projection operators do not leave the system in
an eigenstate corresponding to the measurement outcome, e.g. photon detection.

Our work provides a new thermodynamically consistent framework for tackling
questions regarding the role of the preparation/driving of measurement
apparatuses and the characterisation of the necessary out-of-equilibrium
resources in the measurement process. Since the equilibrium states considered
are not thermal, it is of interest to examine the relationship between our
model and many-body localisation [38] and in particular whether a pre-
thermalisation phenomenon [39] may be at play. This is reinforced by the work
of [40], showing that thermality and objectivity are often at odds. In
addition, one may go beyond the simple statement of the Measurement-
Equilibration Hypothesis above to investigate whether the quality of the
measurement depends on the amount of entropy dissipated in the process, and
how this may relate to the timescale of the measurement [41].

While ours is only a first step towards a thermodynamically self-contained
model for quantum measurements, we hope that this will resolve many of the
foundational issues that manifest through the apparent inconsistency of text-
book postulates of quantum measurements and thermodynamic laws.

##  IX Acknowledgements

The authors thank Sophie Engineer, Nicolai Friis, Erickson Tjoa and Faraj
Bakhshinezhad for helpful discussions, and Tom Rivlin for comments on this
manuscript. This project/ publication was made possible through the support of
Grant 62423 from the John Templeton Foundation. The opinions expressed in this
publication are those of the author(s) and do not necessarily reflect the
views of the John Templeton Foundation. E.S. acknowledges support from the
Austrian Science Fund (FWF) through the START project Y879-N27, and from the
ESQ Discovery grant “Emergence of physical laws: From mathematical foundations
to applications in many body physics”, F.C.B. acknowledges support by grant
number FQXi-RFP-IPW-1910 from the Foundational Questions Institute and Fetzer
Franklin Fund, a donor advised fund of Silicon Valley Community Foundation;
and Irish Research Council Laureate Award IRCLA/2022/3922. E.S., M.H. and
M.P.E.L. acknowledge funding from the European Research Council (Consolidator
grant “Cocoquest” 101043705).

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## Appendices

##  Appendix A Setting and notation

In the following appendices, we prove the results stated in the main text.
First, for ease of reference, we recall the mathematical setting as well as
some notation.

We assume a Hilbert space structure given by
$\mathcal{H}=\mathcal{H}_{\mathrm{S}}\otimes\mathcal{H}_{\mathrm{E}}=\mathcal{H}_{\mathrm{S}}\bigotimes_{k=1}^{N}\mathcal{H}_{k}$,
where we denote the system that is being measured by the subscript S, the
total environment by the subscript E and the observers’ systems which
constitute the environment by $k$. The Hamiltonian of the system will be
denoted by $H$, and the measurement (i.e. “pointer”) basis by
$\\{|{i}\rangle\\!\langle{i}|\\}_{i}$. The probability of measurement outcome
$i$ is then given by

$p_{i}:=\textnormal{Tr}\left[\rho_{\mathrm{S},0}|{i}\rangle\\!\langle{i}|\right],$
(13)

where $\rho_{\mathrm{S},0}$ is the initial state on
$\mathcal{H}_{\mathrm{S}}$. The time average of a quantity $A(t)$ is denoted
by

$\displaystyle\langle A(t)\rangle_{T}:=\dfrac{1}{T}\int_{0}^{T}A(t)dt.$

For equilibrating systems the equilibrium state is given by the infinite-time
average state

$\displaystyle\rho_{\infty}:=\lim_{T\rightarrow\infty}\dfrac{1}{T}\int_{0}^{T}e^{-iHt}\rho(0)e^{iHt}dt.$

Equivalently, this state is given by the pinching map with respect to the
Hamiltonian [21]:

$\displaystyle\rho_{\infty}=\sum_{n}\Pi_{n}\rho(0)\Pi_{n},$ (14)

where the $\Pi_{n}$ are projectors onto degenerate subspaces of $H$ such that
$H=\sum_{n}E_{n}\Pi_{n}$.

For simplicity, when an operator acts non-trivially only on a given tensor
factor, we will suppress tensor products with the identity operator, e.g.
writing $\bigotimes_{k=1}^{j-1}\mathds{1}_{k}\otimes
H_{j}\otimes\bigotimes_{k^{\prime}=j+1}^{N}\mathds{1}_{k^{\prime}}$ as simply
$H_{j}$.

We recall the definition of Spectrum Broadcast Structure: a state has this
structure with respect to the measurement basis
$\\{|{i}\rangle\\!\langle{i}|\\}_{i}$ if it is of the form

$\displaystyle\rho_{\mathrm{SBS}}=\sum_{i=1}^{d_{\mathrm{S}}}p_{i}\,|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\bigotimes_{k=1}^{N}\rho_{k}^{(i)},$
(15)

where $\rho_{k}^{(i)}$ is the state on $\mathcal{H}_{k}$ conditioned on the
state $\left|\right.\\!{i}\\!\left.\right\rangle$ on
$\mathcal{H}_{\mathrm{S}}$, and where $\rho_{k}^{(i)}\rho_{k}^{(j)}=0$ for
$i\neq j$. Since, for states $\sigma$ and $\rho$, $\sigma\rho=0\iff
F(\sigma,\rho)=0$, where the Uhlmann fidelity is defined as
$F(\sigma,\rho)=\textnormal{Tr}\left(\sqrt{\sqrt{\sigma}\rho\sqrt{\sigma}}\right)^{2}$,
one can equivalently use the condition $F(\rho_{k}^{(i)}\rho_{k}^{(j)})=0$ in
the definition of Spectrum Broadcast Structure.

##  Appendix B The impossibility of ideal projective measurements

In this appendix, we demonstrate the impossibility of performing an exact
projective measurement according to the Measurement Equilibration Hypothesis.
In particular, we prove that an initially uncorrelated state cannot
equilibrate to one with Spectrum Broadcast Structure; this follows from a
stronger result, described in the following. Firstly, we allow the possibility
of initial correlations between the observers’ systems, but not with the
system to be measured i.e.
$\rho(0)=\rho_{\mathrm{S},0}\otimes\rho_{\mathrm{E},0}$, where
$\rho_{\mathrm{S},0}$ is the initial state of the system and
$\rho_{\mathrm{E},0}$ the initial state of the environment. Secondly, we
demand equilibration to a classical-quantum state [29] with respect to the
measurement basis, with the appropriate probabilities, i.e. a state of the
form

$\displaystyle\rho_{\text{CQ}}=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes\rho_{\mathrm{E}}^{(i)}$
(16)

for some $\rho_{\mathrm{E}}^{(i)}$ on $\mathcal{H}_{\mathrm{E}}$. Thirdly, we
demand orthogonality of the conditional states
$\\{\rho_{\mathrm{E}}^{(i)}\\}_{i}$, i.e.
$\rho_{\mathrm{E}}^{(i)}\rho_{\mathrm{E}}^{(j)}=0$, or equivalently
$F(\rho_{\mathrm{E}}^{(i)},\rho_{\mathrm{E}}^{(j)})=0$, for $i\neq j$. These
are weaker requirements than for Spectrum Broadcast Structure, as can be seen
by comparing Eqs. (16) and (15). Thus when we prove that they cannot be
simultaneously satisfied, the impossibility of equilibration to a state with
Spectrum Broadcast Structure follows as a consequence.

The proof proceeds by first constraining the form of the Hamiltonian that
would be necessary to equilibrate to a classical-quantum state, and then
showing that the resulting fidelity between the conditional states
$\\{\rho_{\mathrm{E}}^{(i)}\\}_{i}$ is necessarily non-zero.

###  B.1 Constraining the Hamiltonian

Recalling that if a system equilibrates on average, it equilibrates to its
infinite-time average [20], our first step is to find which Hamiltonians can
give rise to an infinite-time average of the required clasical-quantum form,
i.e. Eq. (16). Since the infinite-time average is also given by the pinching
map with respect to the Hamiltonian, Eq. (14), it will commute with the
Hamiltonian: $[\rho_{\infty},H]=0$. Consequently, if
$\rho_{\infty}=\rho_{\mathrm{CQ}}$, then $[\rho_{\mathrm{CQ}},H]=0$. We now
seek the constraints that the latter equation imposes on the form of $H$.
Letting $\\{\left|\right.\\!{\lambda}\\!\left.\right\rangle\\}$ denote an
arbitrary basis of $\mathcal{H}_{\mathrm{E}}$, we can take the tensor product
with the pointer basis on $\mathcal{H}_{\mathrm{S}}$ to write any Hamiltonian
in the form

$H=\sum_{ij\lambda\lambda^{\prime}}C_{ij\lambda\lambda^{\prime}}|{i}\rangle\\!\langle{j}|\otimes|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|,$
(17)

where $C_{ij\lambda\lambda^{\prime}}$ are the matrix elements of $H$ in this
basis. The relevant commutator is then

$\displaystyle[H,\rho_{\mathrm{CQ}}]$
$\displaystyle=\sum_{l}p_{l}\sum_{ij\lambda\lambda^{\prime}}C_{ij\lambda\lambda^{\prime}}\left[|{i}\rangle\\!\langle{j}|\otimes|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\,,\,|{l}\rangle\\!\langle{l}|\otimes\rho_{\mathrm{E}}^{(l)}\right]$
$\displaystyle=\sum_{j}p_{j}\sum_{i\lambda\lambda^{\prime}}\left(C_{ij\lambda\lambda^{\prime}}|{i}\rangle\\!\langle{j}|\otimes|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\rho_{\mathrm{E}}^{(j)}-C_{ji\lambda\lambda^{\prime}}|{j}\rangle\\!\langle{i}|\otimes\rho_{\mathrm{E}}^{(j)}|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\right)$
(18) $\displaystyle\overset{!}{=}0.$

Now, since we seek a Hamiltonian which equilibrates to the SBS state for any
initial system state, i.e. any $\\{p_{i}\\}$, this implies that

$\sum_{i\lambda\lambda^{\prime}}\left(C_{ij\lambda\lambda^{\prime}}|{i}\rangle\\!\langle{j}|\otimes|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\rho_{\mathrm{E}}^{(j)}-C_{ji\lambda\lambda^{\prime}}|{j}\rangle\\!\langle{i}|\otimes\rho_{\mathrm{E}}^{(j)}|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\right)=0\quad\forall
j.$ (19)

Noting that $\forall i,j$ with $i\neq j$, the tensor factors
$|{i}\rangle\\!\langle{j}|$, $|{j}\rangle\\!\langle{i}|$ and
$|{j}\rangle\\!\langle{j}|$ are all linearly independent, we then see that Eq.
(19) implies the following

$\displaystyle\sum_{\lambda\lambda^{\prime}}\left(C_{ij\lambda\lambda^{\prime}}|{i}\rangle\\!\langle{j}|\otimes|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\rho_{\mathrm{E}}^{(j)}-C_{ji\lambda\lambda^{\prime}}|{j}\rangle\\!\langle{i}|\otimes\rho_{\mathrm{E}}^{(j)}|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\right)$
$\displaystyle=0\quad\forall i,\,j,\,i\neq j,$ (20)
$\displaystyle\sum_{\lambda\lambda^{\prime}}C_{ii\lambda\lambda^{\prime}}|{i}\rangle\\!\langle{i}|\otimes\left[|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|,\rho_{\mathrm{E}}^{(i)}\right]$
$\displaystyle=0\quad\forall i.$ (21)

Again, due to the linear independence of $|{i}\rangle\\!\langle{j}|$ and
$|{j}\rangle\\!\langle{i}|$, we must have that
$C_{ij\lambda\lambda^{\prime}}=0$ $\forall i\neq j,\lambda,\lambda^{\prime}$,
in order for Eq. (20) to be satisfied. For the infinite-time average state to
have the structure of Eq. (16) (for an arbitrary initial system state), the
Hamiltonian must then be of the form

$H=\sum_{i}|{i}\rangle\\!\langle{i}|\otimes H_{\mathrm{E}}^{(i)}\text{,
where}\;\;\;\;H_{\mathrm{E}}^{(i)}:=\sum_{\lambda\lambda^{\prime}}C_{ii\lambda\lambda^{\prime}}|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|.$
(22)

This generates unitary evolution of the form

$\displaystyle e^{-iHt}=\sum_{i}|{i}\rangle\\!\langle{i}|\otimes
U_{\mathrm{E}}^{(i)}(t),$ (23)

where
$U_{\mathrm{E}}^{(i)}(t):=\exp\left[-it\sum_{\lambda\lambda^{\prime}}C_{ii\lambda\lambda^{\prime}}|{\lambda}\rangle\\!\langle{\lambda^{\prime}}|\right]$.
We can then write

$\displaystyle\rho_{\infty}=\Big{\langle}\sum_{i}|{i}\rangle\\!\langle{i}|\otimes
U_{\mathrm{E}}^{(i)}(t)\rho_{\mathrm{S},0}\otimes\rho_{\mathrm{E},0}\sum_{j}|{j}\rangle\\!\langle{j}|\otimes
U_{\mathrm{E}}^{(j)\dagger}(t)\Big{\rangle}_{\infty}$
$\displaystyle=\sum_{i,j}\left\langle\right.\\!{i}\\!\left.\right|\rho_{\mathrm{S},0}\left|\right.\\!{j}\\!\left.\right\rangle|{i}\rangle\\!\langle{j}|\otimes\Big{\langle}U_{\mathrm{E}}^{(i)}(t)\rho_{\mathrm{E},0}U_{\mathrm{E}}^{(j)\dagger}(t)\Big{\rangle}_{\infty}\stackrel{{\scriptstyle!}}{{=}}\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|\otimes\rho_{\mathrm{E}}^{(i)}$
(24)

which implies that

$\displaystyle\Big{\langle}U_{\mathrm{E}}^{(i)}(t)\rho_{\mathrm{E},0}U_{\mathrm{E}}^{(j)\dagger}(t)\Big{\rangle}_{\infty}$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}\,\begin{cases}0&i\neq j\\\
\rho_{\mathrm{E}}^{(i)}&i=j.\end{cases}$ (25)

Now, let us denote the eigenvalues of $H_{\mathrm{E}}^{(i)}$ by
$\\{\varepsilon_{n}^{(i)}\\}_{n}$, with associated eigenvectors
$\\{\left|\right.\\!{\varepsilon_{n}^{(i)}}\\!\left.\right\rangle\\}_{n}$.
Writing $U_{\mathrm{E}}^{(i)}(t)$ and $U_{\mathrm{E}}^{(j)}(t)$ in the left
hand side of Eq. (25) in the eigenbasis of $H_{\mathrm{E}}^{(i)}$ and
$H_{\mathrm{E}}^{(j)}$ respectively, we have

$\Big{\langle}U_{\mathrm{E}}^{(i)}(t)\rho_{\mathrm{E},0}U_{\mathrm{E}}^{(j)\dagger}(t)\Big{\rangle}_{\infty}=\sum_{n,m}\Big{\langle}e^{-i(\varepsilon_{n}^{(i)}-\varepsilon_{m}^{(j)})t}\Big{\rangle}_{\infty}\left\langle\right.\\!{\varepsilon_{n}^{(i)}}\\!\left.\right|\rho_{\mathrm{E},0}\left|\right.\\!{\varepsilon_{m}^{(j)}}\\!\left.\right\rangle|{\varepsilon_{n}^{(i)}}\rangle\\!\langle{\varepsilon_{m}^{(j)}}|,$
(26)

and so in order to satisfy Eq. (25), we must have that

$\Big{\langle}e^{-i(\varepsilon_{n}^{(i)}-\varepsilon_{m}^{(j)})t}\Big{\rangle}_{\infty}\left\langle\right.\\!{\varepsilon_{n}^{(i)}}\\!\left.\right|\rho_{\mathrm{E},0}\left|\right.\\!{\varepsilon_{m}^{(j)}}\\!\left.\right\rangle\stackrel{{\scriptstyle!}}{{=}}0\quad\forall
n,m\;\mathrm{where}\;n\neq m.$ (27)

If the $\\{H_{\mathrm{E}}^{(i)}\\}_{i}$ are each nondegenerate, and have no
eigenvalues in common, i.e. $\varepsilon_{n}^{(i)}\neq\varepsilon_{m}^{(j)}$
$\forall n,m$ with $n\neq m$, then the above equation holds independently of
the initial state $\rho_{\mathrm{E},0}$. Note that both of these conditions
are true if and only if the total Hamiltonian is non-degenerate. In the case
where there are values of $\varepsilon_{n}^{(i)}$ and $\varepsilon_{m}^{(j)}$
that coincide, _i.e._ there are some degeneracies, then we must have
$\left\langle\right.\\!{\varepsilon_{n}^{(i)}}\\!\left.\right|\rho_{\mathrm{E},0}\left|\right.\\!{\varepsilon_{m}^{(j)}}\\!\left.\right\rangle=0$
for each pair of coinciding eigenvalues, requiring a initially rank-deficient
state. Assuming either non-degeneracy of the total Hamiltonian, or the
appropriate rank-deficiency of the initial environmental states in the case of
degeneracies, we arrive at a state of the form

$\rho_{\infty}=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|\otimes\rho_{\mathrm{E}}^{(i)}\quad\text{with}\quad\rho_{\mathrm{E}}^{(i)}=\sum_{n}\left\langle\right.\\!{\varepsilon_{n}^{(i)}}\\!\left.\right|\rho_{\mathrm{E},0}\left|\right.\\!{\varepsilon_{n}^{(i)}}\\!\left.\right\rangle|{\varepsilon_{n}^{(i)}}\rangle\\!\langle{\varepsilon_{n}^{(i)}}|.$
(28)

As an aside, we note that this state satisfies the faithfulness requirement in
[1] (i.e.:
$\sum_{i}\textnormal{Tr}[|{i}\rangle\\!\langle{i}|\otimes\Pi_{i}\rho_{\infty}]=1$
for orthogonal projectors $\Pi_{i}$ with $[\Pi_{i},\rho_{E}^{(i)}]=0$).

We have so far made no demands on the microscopic structure on
$\mathcal{H}_{\mathrm{E}}$, aside from the potential rank-deficient initial
states required to deal with degeneracies in $H$. A Hamiltonian of the form of
Eq. 22 is therefore necessary for the formation of classical-quantum states in
general.

###  B.2 Bounding the fidelity between conditional states

We now show that the orthogonality requirement, i.e.
$F(\rho_{\mathrm{E}}^{(i)},\rho_{\mathrm{E}}^{(j)})=0$ for $i\neq j$, cannot
be satisfied exactly. In particular, we show that

$\displaystyle
F(\rho_{\mathrm{E}}^{(i)},\rho_{\mathrm{E}}^{(j)})\geq\dfrac{1}{d_{\mathrm{E}}^{2}},$
(29)

where $d_{\mathrm{E}}:=\dim(\mathcal{H}_{\mathrm{E}})$.

First, note that the conditional time-averaged state $\rho_{\mathrm{E}}^{(i)}$
in Eq. (28) is a result of the pinching map with respect to the projectors
$\\{|{\varepsilon_{n}^{(i)}}\rangle\\!\langle{\varepsilon_{n}^{(i)}}|\\}_{n}$.
Denoting the number of these projectors (i.e. the number of distinct
eigenvalues of $H_{\mathrm{E}}^{(i)}$) by $d^{(i)}$, we can rewrite the
pinching map as a mixed-unitary channel [42, 43] as follows

$\displaystyle\rho_{\mathrm{E}}^{(i)}=\sum_{n=1}^{d^{(i)}}|{\varepsilon_{n}^{(i)}}\rangle\\!\langle{\varepsilon_{n}^{(i)}}|\rho_{\mathrm{E},0}|{\varepsilon_{n}^{(i)}}\rangle\\!\langle{\varepsilon_{n}^{(i)}}|=\dfrac{1}{d^{(i)}}\sum_{y=1}^{d^{(i)}}U_{y}^{(i)}\rho_{\mathrm{E},0}U_{y}^{(i)\dagger}$
(30)

where $U_{y}^{(i)}:=\sum_{n=1}^{d^{(i)}}e^{-i\tfrac{2\pi
ny}{d^{(i)}}}|{\varepsilon_{n}^{(i)}}\rangle\\!\langle{\varepsilon_{n}^{(i)}}|$.
Noting that $U^{(i)}_{d^{(i)}}=\mathds{1}$, we write

$\displaystyle\rho_{\mathrm{E}}^{(i)}=\dfrac{\rho_{\mathrm{E},0}}{d^{(i)}}+\dfrac{1}{d^{(i)}}\sum_{y=1}^{d^{(i)}-1}U_{y}^{(i)}\rho_{\mathrm{E},0}U_{y}^{(i)\dagger}$
(31)

and thus

$\displaystyle\textnormal{Tr}[(\rho_{\mathrm{E}}^{(i)}\rho_{\mathrm{E}}^{(j)}]=$
$\displaystyle\dfrac{1}{d^{(i)}d^{(j)}}\Bigg{\\{}\textnormal{Tr}\Big{[}\rho_{\mathrm{E},0}^{2}\Big{]}+\textnormal{Tr}\Big{[}\rho_{\mathrm{E},0}\sum_{y=1}^{d^{(i)}-1}U_{y}^{(j)}\rho_{\mathrm{E},0}U_{y}^{(j)\dagger}\Big{]}+\textnormal{Tr}\Big{[}\sum_{y=1}^{d^{(i)}-1}U_{y}^{(i)}\rho_{\mathrm{E},0}U_{y}^{(i)\dagger}\rho_{\mathrm{E},0}\Big{]}$
$\displaystyle+\textnormal{Tr}\Big{[}\sum_{y=1}^{d^{(i)}-1}U_{y}^{(i)}\rho_{\mathrm{E},0}U_{y}^{(i)\dagger}\sum_{y^{\prime}=1}^{d^{(i)}-1}U_{y^{\prime}}^{(j)}\rho_{\mathrm{E},0}U_{y^{\prime}}^{(j)\dagger}\Big{]}\Bigg{\\}}.$
(32)

Since that the trace of the product of two positive operators is itself
positive, we can remove the second, third and fourth terms to obtain the
inequality

$\displaystyle\textnormal{Tr}(\rho_{\mathrm{E}}^{(i)}\rho_{\mathrm{E}}^{(j)})\geq\dfrac{\textnormal{Tr}(\rho_{\mathrm{E},0}^{2})}{d^{(i)}d^{(j)}}.$
(33)

Now, note that the fidelity between two states satisfies
$F(\rho,\sigma)\geq\textnormal{Tr}(\rho\sigma)$ [44], allowing us to bound the
Fidelity

$\displaystyle
F(\rho_{\mathrm{E}}^{(i)},\rho_{\mathrm{E}}^{(j)})\geq\dfrac{\textnormal{Tr}(\rho_{\mathrm{E},0}^{2})}{d^{(i)}d^{(j)}}.$
(34)

Thus, the fidelity between any two time-averaged states with the same initial
state cannot be zero. It is therefore impossible to equilibrate to a
classical-quantum state, Eq. (16), satisfying the orthogonality condition. As
a consequence, it is impossible for an initially uncorrelated state to
equilibrate to a state with Spectrum Broadcast Structure, as this is a
stronger condition, as discussed above.

Finally, we note that since $d^{(i)}$ is bounded by the Hilbert space
dimension of the environment, i.e. $d_{\mathrm{E}}\geq d^{(i)}$, Eq. (29)
follows immediately from Eq. (34).

##  Appendix C The standard measurement model does not equilibrate to a
correlated state

In this section we consider the Hamiltonian used in the von-Neumann
measurement scheme (or the “standard model of measurement”) [5]:

$\displaystyle H=X_{\mathrm{S}}\otimes Y_{\mathrm{E}},$ (35)

where $X_{\mathrm{S}}$ is diagonal in the measurement basis,
$X_{\mathrm{S}}=\sum_{i}x_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}$. By
carefully controlling the time for which this Hamiltonian is applied, one can
correlate the system S with the environment E in this basis. However, without
temporal control – specifically in an equilibration paradigm – this
Hamiltonian leads to an equilibrium state that does not feature any
correlations between system and environment, as we now show.

The unitary operator generated by Eq. (35) is given by

$\displaystyle e^{-\mathrm{i}Ht}=e^{-iX_{\mathrm{S}}\otimes
Y_{\mathrm{E}}t}=e^{-i(\sum_{i}x_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes
Y_{\mathrm{E}})t}=\sum_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes
e^{-ix_{i}Y_{\mathrm{E}}t},$ (36)

As in the previous section, we assume that system and environment are
initially uncorrelated, such that the initials state is given by

$\displaystyle\rho(0)=\rho_{\mathrm{S},0}\otimes\rho_{{}_{\mathrm{E}},0},$
(37)

and we calculate the time evolved state of the total system

$\displaystyle\rho(t)=e^{-iHt}\rho(0)e^{iHt}=e^{-iX_{\mathrm{S}}\otimes
Y_{\mathrm{E}}t}\Big{(}\rho_{\mathrm{S,0}}\otimes\rho_{\mathrm{E},0}\Big{)}e^{iX_{\mathrm{S}}\otimes
Y_{\mathrm{E}}t}=\sum_{i,j}p_{ij}|{i}\rangle\\!\langle{j}|\otimes\rho_{\mathrm{E}}^{(ij)}(t)$
(38)

where $p_{ij}$ are the elements of the initial system state in the pointer
basis, _i.e._
$\rho_{\mathrm{S},0}=\sum_{ij}^{d_{s}}p_{ij}|{i}\rangle\\!\langle{j}|$, and we
have defined
$\rho_{\mathrm{E}}^{(ij)}(t):=e^{-ix_{i}Y_{\mathrm{E}}t}\rho_{\mathrm{E},0}e^{ix_{i}Y_{\mathrm{E}}t}$.
Now, writing the spectral decomposition
$Y_{\mathrm{E}}=\sum_{n}\varepsilon_{n}|{\varepsilon_{n}}\rangle\\!\langle{\varepsilon_{n}}|$,
we have

$\displaystyle\rho_{\mathrm{E}}^{(ij)}(t)=\sum_{m,n}e^{-i(x_{i}\varepsilon_{m}-x_{j}\varepsilon_{n})t}\rho_{\mathrm{E}}^{mn}|{\varepsilon_{m}}\rangle\\!\langle{\varepsilon_{n}}|$
(39)

where
$\rho_{\mathrm{E}}^{mn}:=\left\langle\right.\\!{\varepsilon_{n}}\\!\left.\right|\rho_{\mathrm{E},0}\left|\right.\\!{\varepsilon_{m}}\\!\left.\right\rangle$.
Examining this together with the total state in Eq. (38), we see that the time
average of the total state is determined by

$\displaystyle g_{mn}^{(ij)}(T):=\left\langle
e^{-i(x_{i}\varepsilon_{m}-x_{j}\varepsilon_{n})t}\right\rangle_{T}=\begin{cases}1&\mathrm{if}\;\;x_{i}\varepsilon_{m}=x_{j}\varepsilon_{n}\\\
\dfrac{ie^{[i(x_{i}\varepsilon_{m}-x_{j}\varepsilon_{n})T]}-1}{(x_{i}\varepsilon_{m}-x_{j}\varepsilon_{n})T}&\mathrm{otherwise}\end{cases}$
(40)

Assuming ${x_{i}\varepsilon_{m}\neq x_{j}\varepsilon_{n}\;\;\forall i,j,m,n}$
we find an equilibrium state of the desired diagonal form in the pointer
basis:

$\displaystyle\rho_{\infty}=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{\mathrm{S}}\otimes\sum_{m}\rho_{\mathrm{E}}^{mm}|{\varepsilon_{m}}\rangle\\!\langle{\varepsilon_{m}}|.$
(41)

However, upon examination we see that the environment is independent of $i$,
and this state features no correlations between system and environment. The
environment therefore can not reveal any information about the system, and
such von-Neumann-type Hamiltonians are incapable of (even approximately)
equilibrating to states exhibiting objectivity.

##  Appendix D Asymptotically approaching Spectrum Broadcast Structure

In this section we show an equilibrium state with Spectrum Broadcast Structure
can be obtained approximately by coarse-graining the environment. The purpose
of this is to show one example of how Spectrum Broadcast Structure can emerge.
This should be considered as a first step; further research is needed to
expand this framework into a realistic model of measurements.

As was shown in Appendix B.1, to obtain a classical-quantum equilibrium state
(of which Spectrum Broadcast Structure is an example), one requires a
conditional form of the Hamiltonian. In conjunction with the condition of
Strong Independence this prompts us to consider a “star-shaped” Hamiltonian
(see Fig. 1 in the main text) given by

$H=\sum_{i}|{i}\rangle\\!\langle{i}|_{S}\otimes\sum_{k=1}^{N}c_{k}H_{k}^{(i)}$
(42)

where $H_{k}^{(i)}\in\mathcal{L}\left(\mathcal{H}_{k}\right)$ and $c_{k}$ are
the different couplings between the system and the observer systems (e.g.
representing distance-dependent couplings to spatially distributed observers).
This generates the unitary evolution

$e^{-iHt}=\sum_{i}|{i}\rangle\\!\langle{i}|_{S}\bigotimes_{k=1}^{N}U_{k}^{(i)}(t),$
(43)

where $U_{k}^{(i)}(t):=e^{-ic_{k}H_{k}^{(i)}t}$.

Assuming a separable initial state, i.e.
$\rho(0)=\rho_{S,0}\bigotimes_{k=1}^{N}\rho_{k,0}$, the state at time $t$ can
be written

$\rho(t)=\sum_{i,j}\left\langle\right.\\!{i}\\!\left.\right|\rho_{S,0}\left|\right.\\!{j}\\!\left.\right\rangle|{i}\rangle\\!\langle{j}|_{S}\bigotimes_{k=1}^{N}\rho_{k}^{(ij)}(t)$
(44)

where

$\rho_{k}^{(ij)}(t):=U_{k}^{(i)}(t)\rho_{k,0}U_{k}^{(j)}(t).$ (45)

In order to calculate the equilibrium state $\rho_{\infty}$, we must therefore
calculate the infinite-time average of
$\bigotimes_{k=1}^{N}\rho_{k}^{(ij)}(t)$. First, letting
$\varepsilon_{n_{k}}^{(i)}$ denote the $n_{k}^{\text{th}}$ energy eigenvalue
of $H_{k}^{(i)}$, we define

$\rho_{k}^{mn(ij)}:=\left\langle\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right|\rho_{k,0}\left|\right.\\!{\varepsilon_{n_{k}}^{(j)}}\\!\left.\right\rangle.$
(46)

Then the finite-time average of $\bigotimes_{k=1}^{N}\rho_{k}^{(ij)}(t)$ over
the interval $T$ is given by

$\displaystyle\left\langle\bigotimes_{k=1}^{N}\rho_{k}^{(ij)}(t)\right\rangle_{T}=\left\langle\bigotimes_{k=1}^{N}\sum_{m_{k}n_{k}}\exp\left[-ic_{k}\left(\varepsilon_{m_{k}}^{(i)}-\varepsilon_{n_{k}}^{(j)}\right)t\right]\rho_{k}^{mn(ij)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{n_{k}}^{(j)}}|\right\rangle_{T}$
$\displaystyle=\left\langle\left(\sum_{m_{1}n_{1}}\exp\left[-ic_{1}\left(\varepsilon_{m_{1}}^{(i)}-\varepsilon_{n_{1}}^{(j)}\right)t\right]\rho_{1}^{mn(ij)}|{\varepsilon_{m_{1}}^{(i)}}\rangle\\!\langle{\varepsilon_{n_{1}}^{(j)}}|\right)\otimes\left(\sum_{m_{2}n_{2}}\exp\left[-ic_{2}\left(\varepsilon_{m_{2}}^{(i)}-\varepsilon_{n_{2}}^{(j)}\right)t\right]\rho_{2}^{mn(ij)}|{\varepsilon_{m_{2}}^{(i)}}\rangle\\!\langle{\varepsilon_{n_{2}}^{(j)}}|\right)\otimes\ldots\right\rangle_{T}$
$\displaystyle=\sum_{m_{1}n_{1}}\sum_{m_{2}n_{2}}\ldots\sum_{m_{N}n_{N}}\left\langle\exp\left[-i\sum_{k=1}^{N}c_{k}\left(\varepsilon_{m_{k}}^{(i)}-\varepsilon_{n_{k}}^{(j)}\right)t\right]\right\rangle_{T}\bigotimes_{k=1}^{N}\rho_{k}^{mn(ij)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{n_{k}}^{(j)}}|$
$\displaystyle\equiv\sum_{\\{m_{k}\\}\\{n_{k}\\}}f_{\\{m_{k}\\}\\{n_{k}\\}}^{(ij)}(T)\bigotimes_{k=1}^{N}\rho_{k}^{mn(ij)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{n_{k}}^{(j)}}|$
(47)

where we have used the notation

$\displaystyle\sum_{\\{a_{k}\\}}=\sum_{a_{1}}\sum_{a_{2}}...\sum_{a_{N}},$
(48)

and defined

$f_{\\{m_{k}\\}\\{n_{k}\\}}^{(ij)}(T):=\begin{cases}1&\text{if
}\,\sum_{k=1}^{N}c_{k}\varepsilon_{m_{k}}^{(i)}=\sum_{k=1}^{N}c_{k}\varepsilon_{n_{k}}^{(j)}\\\
i\frac{\exp\left[-i\sum_{k=1}^{N}c_{k}\left(\varepsilon_{m_{k}}^{(i)}-\varepsilon_{n_{k}}^{(j)}\right)T\right]-1}{\sum_{k=1}^{N}c_{k}\left(\varepsilon_{m_{k}}^{(i)}-\varepsilon_{n_{k}}^{(j)}\right)T}&\text{otherwise.}\end{cases}$
(49)

Noting that
$\\{\sum_{k=1}^{N}c_{k}\varepsilon_{m_{k}}^{(i)}\\}_{i,m_{1},\ldots,m_{N}}$
are the eigenvalues of $H$, which can be seen from Eq. (42), we see that the
cases where $f^{(ij)}_{\\{m_{k}\\}\\{n_{k}\\}}=1$ for $i\neq j$ and
$\\{m_{k}\\}\neq\\{n_{k}\\}$ correspond to degeneracies in $H$. Further note
that such cases are particularly “fine-tuned”, as the coupling constants and
energy eigenvalues have to conspire in order to yield this result. Thus,
assuming that the total Hamiltonian is non-degenerate, the first case in Eq.
49 never occurs, and we have

$\left\langle\bigotimes_{k=1}^{N}\rho_{k}^{(ij)}(t)\right\rangle_{\infty}=0\qquad\text{
for }i\neq j.$ (50)

Combining this with Equations 44 and 47, we obtain

$\displaystyle\left\langle\rho(t)\right\rangle_{\infty}=\sum_{i}\left\langle\right.\\!{i}\\!\left.\right|\rho_{S,0}\left|\right.\\!{i}\\!\left.\right\rangle|{i}\rangle\\!\langle{i}|_{S}\left\langle\bigotimes_{k=1}^{N}\rho_{k}^{(ii)}(t)\right\rangle_{\infty}$
$\displaystyle=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{S}\sum_{\\{m_{k}\\}\\{n_{k}\\}}f_{\\{m_{k}\\}\\{n_{k}\\}}^{(ii)}(\infty)\bigotimes_{k=1}^{N}\rho_{k,0}^{mn(ii)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{n_{k}}^{(i)}}|,$
(51)

where
$f^{(ii)}_{\\{m_{k}\\}\\{n_{k}\\}}(\infty):=\lim_{T\rightarrow\infty}f^{(ii)}_{\\{m_{k}\\}\\{n_{k}\\}}(T)$,
and again
$p_{i}:=\left\langle\right.\\!{i}\\!\left.\right|\rho_{S,0}\left|\right.\\!{i}\\!\left.\right\rangle$
is the probability of outcome $i$ for the projective measurement
$\\{|{i}\rangle\\!\langle{i}|_{S}\\}_{i}$ on the system. Due to the assumption
of non-degeneracy of the total Hamiltonian, the conditional Hamiltonians
$H_{k}^{(i)}$ have to be non-degenerate as well, and consequently
$\sum_{k=1}^{N}c_{k}\varepsilon_{m_{k}}^{(i)}\neq\sum_{k=1}^{N}c_{k}\varepsilon_{n_{k}}^{(i)}$
$\forall\,m_{k}\neq n_{k}$, and thus only the $m_{k}=n_{k}$ terms in
$f^{(ii)}_{\\{m_{k}\\}\\{n_{k}\\}}(\infty)$ are nonzero. Equation (49) then
implies that

$\displaystyle\rho_{\infty}=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{S}\sum_{m_{1}\ldots
m_{N}}\bigotimes_{k=1}^{N}\rho_{k}^{mm(ii)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{m_{k}}^{(i)}}|$
$\displaystyle=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{S}\bigotimes_{k=1}^{N}\sum_{m_{k}}\rho_{k}^{mm(ii)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{m_{k}}^{(i)}}|$
$\displaystyle\equiv\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{S}\bigotimes_{k=1}^{N}\rho_{k}^{(i)},$
(52)

where

$\rho_{k}^{(i)}:=\sum_{m_{k}}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{m_{k}}^{(i)}}|\,\rho_{k,0}\,|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{m_{k}}^{(i)}}|$
(53)

is the initial state of observer system $k$ (i.e. $\rho_{k,0}$) subject to the
pinching map with respect to $H_{k}^{(i)}$. Now let us gather together the $N$
factors of the environment into $M$ “macro-observers”, and let $N_{q}$ denote
the set of environment labels belonging to the $q^{\text{th}}$ macro-observer,
so that
${\bigotimes_{k=1}^{N}\mathcal{H}_{k}=\bigotimes_{q=1}^{M}\mathcal{E}_{q}}$
with $\mathcal{E}_{q}:=\bigotimes_{k\in N_{q}}\mathcal{H}_{k}$. Then

$\displaystyle\rho_{\infty}=\sum_{i}p_{i}|{i}\rangle\\!\langle{i}|_{S}\bigotimes_{q=1}^{M}\tilde{\rho}_{q}^{(i)},\qquad\text{
where }\qquad\tilde{\rho}_{q}^{(i)}:=\bigotimes_{k\in N_{q}}\rho_{k}^{(i)},$
(54)

We now show that the fidelity
$F\left(\tilde{\rho}_{q}^{(i)},\tilde{\rho}_{q}^{(j)}\right)$ between
conditional states $\tilde{\rho}_{q}^{(i)}$ and $\tilde{\rho}_{q}^{(j)}$ of
each macro-observer $q$ tends rapidly to 0 as the macro-observer size
increases. To do so, we use the inequality

$\displaystyle\sqrt{F(\sigma,\rho)}\leq\sum_{m}\sqrt{\textnormal{Tr}[E_{m}\rho]}\sqrt{\textnormal{Tr}[E_{m}\sigma]}$
(55)

where the $\\{E_{m}\\}$ are the elements of an arbitrary POVM, to bound
$F\left(\rho_{k}^{(i)},\rho_{k}^{(j)}\right)$. Choosing the POVM elements to
be projectors onto the eigenbasis of $H_{k}^{(i)}$, _i.e._
$\\{|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{m_{k}}^{(i)}}|\\}_{m_{k}}$.
Since
$\rho_{k}^{(i)}=\sum_{m_{k}}\rho_{k}^{mm(ii)}|{\varepsilon_{m_{k}}^{(i)}}\rangle\\!\langle{\varepsilon_{m_{k}}^{(i)}}|$
we obtain

$\displaystyle\sqrt{F(\rho_{k}^{(i)},\rho_{k}^{(j)})}\leq\sum_{m_{k}}\sqrt{\rho_{k}^{mm(ii)}}\sqrt{\textnormal{Tr}[\sum_{n_{k}}\rho_{k}^{nn(jj)}|{\varepsilon_{n_{k}}^{(j)}}\rangle\\!\langle{\varepsilon_{n_{k}}^{(j)}}|\varepsilon_{m_{k}}^{(i)}\rangle\\!\langle\varepsilon_{m_{k}}^{(i)}|]}$
$\displaystyle=\sum_{m_{k}}\sqrt{\rho_{k}^{mm(ii)}}\sqrt{\sum_{n_{k}}\rho_{k}^{nn(jj)}|\langle{\varepsilon_{m_{k}}^{(i)}}|{\varepsilon_{n_{k}}^{(j)}}\rangle|^{2}}=\sum_{m_{k}}\sqrt{\sum_{n_{k}}\rho_{k}^{mm(ii)}\rho_{k}^{nn(jj)}|\langle{\varepsilon_{m_{k}}^{(i)}}|{\varepsilon_{n_{k}}^{(j)}}\rangle|^{2}}$
$\displaystyle=\sum_{m_{k}}\sqrt{\left\langle\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right|\rho_{k}^{(i)}\rho_{k}^{(j)}\left|\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right\rangle}.$
(56)

In the cases where $H^{(i)}=H^{(j)}$ or
$\rho_{k,0}=\dfrac{1}{d_{k}}\mathds{1}$ we see that the fidelity is bounded
above by 1, as one would expect.

To see that $F\left(\tilde{\rho}_{q}^{(i)},\tilde{\rho}_{q}^{(j)}\right)$
approaches zero exponentially as the macro-observers grow, we recall that
$F(\bigotimes_{k}\sigma_{k},\bigotimes_{k}\rho_{k})=\prod_{k}F(\sigma,\rho)$.
Combining this with the inequality in (D), we have

$\displaystyle\sqrt{F(\bigotimes_{k\in N_{q}}\rho_{k}^{(i)},\bigotimes_{k\in
N_{q}}\rho_{k}^{(j)})}$ $\displaystyle=\prod_{k\in
N_{q}}\sqrt{F(\rho_{k}^{(i)},\rho_{k}^{(j)})}$ $\displaystyle\leq\prod_{k\in
N_{q}}\left(\sum_{m_{k}}\sqrt{\left\langle\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right|\rho_{k}^{(i)}\rho_{k}^{(j)}\left|\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right\rangle}\right)$
$\displaystyle\equiv\prod_{k\in N_{q}}\eta_{k}^{(ij)},$ (57)

where we have defined
$\eta_{k}^{(ij)}:=\sum_{m_{k}}\sqrt{\left\langle\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right|\rho_{k}^{(i)}\rho_{k}^{(j)}\left|\right.\\!{\varepsilon_{m_{k}}^{(i)}}\\!\left.\right\rangle}$.
Further defining $N_{q}^{(ij)}:=\\{k\in N_{q}^{(ij)}:\eta_{k}^{(ij)}<1\\}$ and
$\eta_{max}^{(ij)}:=\max_{k\in N_{q}^{(ij)}}\eta_{k}^{(ij)}$. We then have

$\displaystyle
F(\tilde{\rho}_{q}^{(i)},\tilde{\rho}_{q}^{(j)})\leq\left(\eta_{max}^{(ij)}\right)^{2|N_{q}^{(ij)}|},$
(58)

which can be simplified by defining
$\gamma^{(ij)}_{q}=-2\ln(\eta_{\mathrm{max}}^{(ij)})$ to finally obtain

$\displaystyle F(\tilde{\rho}_{q}^{(i)},\tilde{\rho}_{q}^{(j)})\leq
e^{-\gamma^{(ij)}_{q}|N_{q}^{(ij)}|}.$ (59)

Assuming that the conditional Hamiltonians $H_{k}^{(i)}$ and $H_{k}^{(j)}$
have at least one distinct eigenvector, the upper bound in (59) exponentially
tends to 0 for increasing macro-observer size. The equilibrium state therefore
tends exponentially to one with Spectrum Broadcast Structure state as the size
of the macro-observers (and thus the environment as a whole) increases.
Further assuming that $H_{k}^{(i)}$ and $H_{k}^{(j)}$ have no eigenvectors in
common, we find the simpler result stated in the main text, i.e.

$\displaystyle F(\tilde{\rho}_{q}^{(i)},\tilde{\rho}_{q}^{(j)})\leq
e^{-\gamma^{(ij)}_{q}|N_{q}|},$ (60)

where $|N_{q}|$ is the number of observer systems $k$ collected into the
macro-observer $q$.