# Energy conservation and fluctuation theorem are incompatible for quantum
work

Karen V. Hovhannisyan<EMAIL_ADDRESS>University of
Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476
Potsdam, Germany  Alberto Imparato<EMAIL_ADDRESS>Department of Physics
and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus, Denmark

###### Abstract

Characterizing fluctuations of work in coherent quantum systems is notoriously
problematic. We reveal the ultimate source of the problem by proving that
($\mathfrak{A}$) energy conservation and ($\mathfrak{B}$) the Jarzynski
fluctuation theorem cannot be observed at the same time. Condition
$\mathfrak{A}$ stipulates that, for any initial state of the system, the
measured average work must be equal to the difference of initial and final
average energies, and that untouched systems must exchange deterministically
zero work. Condition $\mathfrak{B}$ is only for thermal initial states and
encapsulates the second law of thermodynamics. We prove that $\mathfrak{A}$
and $\mathfrak{B}$ are incompatible for work measurement schemes that are
differentiable functions of the state and satisfy two mild structural
constraints. This covers all existing schemes and leaves the theoretical
possibility of jointly observing $\mathfrak{A}$ and $\mathfrak{B}$ open only
for a narrow class of exotic schemes. For the special but important case of
state-independent schemes, the situation is much more rigid: we prove that,
essentially, only the two-point measurement scheme is compatible with
$\mathfrak{B}$.

##  1 Introduction

Work is a central notion in both mechanics and thermodynamics, being
essentially the only touching point between the two theories. When information
about the full statistics of work is required, which is especially relevant
for small systems, classical mechanics answers by means of trajectories [1,
2], leaving no prescription for accessing the statistics of work in the
quantum regime [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. This uncertainty has bred
a number of approaches towards measuring work in the quantum regime [1, 3, 4,
13, 5, 14, 7, 15, 16, 17, 18, 8, 19, 20, 10, 21, 12, 22, 23, 24, 25].
Nevertheless, as amply discussed in the literature [5, 7, 8, 9, 26, 11], and
in Sec. 3.2, they all become problematic in certain regimes.

In this paper, we show that the fundamental source of all these problems is
that energy conservation and Jarzynski fluctuation theorem (or simply
Jarzynski equality) [2] for work cannot be observed at the same time in
quantum mechanics. Namely, we prove that essentially no quantum measurement
slated to measure work can produce a random variable that can simultaneously
satisfy $\mathfrak{A}$ energy conservation for all states and $\mathfrak{B}$
Jarzynski equality (JE) for all thermal states.

Since the JE necessarily holds in classical mechanics, requiring it also in
the quantum regime is a way of imposing a weak form of quantum–classical
correspondence principle on the distribution of work, without going into the
specifics of the quantum–classical transition itself. Thus, elegantly
incorporating both the second law of thermodynamics and the correspondence
principle, the JE can serve as a high-level filter through which physically
meaningful definitions of work fluctuations ought to be able to pass,
especially given the foundational role fluctuation theorems play in
statistical mechanics [27, 6].

Undeniably, energy conservation is another such filter, and that virtually no
definition of work fluctuations can pass through both, means that quantum work
cannot be thought of as a classical random variable. In our definition, a
measurement scheme obeys energy conservation if the work statistics measured
by the scheme reflect energy conservation for the unmeasured system.
Specifically, the average work performed on an unmeasured thermally isolated
system is equal to the difference between the final and initial average
energies of the system, and a proper measurement of work must output a random
variable the first moment of which coincides with that value. We call this
condition $\mathfrak{A}_{1}$. Another simple consequence of energy
conservation is that, if a system is untouched, then zero work with
probability $1$ is performed on it. Therefore, a correct measurement of work
must output exactly that statistics; this is condition $\mathfrak{A}_{2}$. One
may call $\mathfrak{A}_{1}$ the “average energy conservation” condition, and
to contrast with that, we will call $\mathfrak{A}$, which is
$\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ imposed jointly, the “detailed
energy conservation” condition.

Our analysis is divided into two parts. First, we focus on work measuring
schemes that do not depend on the initial state of the system. We prove that,
for such schemes, $\mathfrak{B}$ necessitates a conflict with
$\mathfrak{A}_{1}$. Moreover, the schemes that satisfy $\mathfrak{B}$ and
yield the correct average for initial thermal states produce the same
statistics as the two-point measurement (TPM) scheme [3, 4, 6, 14, 28]; see
Theorem 2. This establishes that JE $\Leftrightarrow$ TPM for state-
independent schemes.

Then, we ask whether $\mathfrak{A}$ and $\mathfrak{B}$ can be reconciliated by
using schemes that are allowed to depend on the system’s initial state.
Considering physically meaningful only those schemes that depend on the state
differentiably, we prove that, while $\mathfrak{A}_{1}$ and $\mathfrak{B}$ can
be made compatible, when requiring energy conservation to be satisfied to a
fuller extent—namely, imposing $\mathfrak{A}_{2}$ alongside
$\mathfrak{A}_{1}$—the compatibility with $\mathfrak{B}$ breaks down for
essentially all reasonable state-dependent schemes, leaving the theoretical
possibility open for only a narrow exotic class of schemes (Theorem 3).
Interestingly, merely requiring $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ to
hold at the same time already severely restricts the class of acceptable
schemes (Lemmas 2 and C.1).

##  2 Formal setup

Throughout this paper, we will focus on closed systems (also called
“conservative” in classical mechanics). This setting is the most general: any
open system is a part of a larger closed system consisting of the system
itself and all the external systems with which it is correlated, interacts,
and will interact during the process under study [29, 30]. Arguably, this
picture also incorporates the process of quantum measurement [31, 32, 33]. For
closed systems, a process is described by a time-dependent Hamiltonian $H(t)$
that at the beginning ($t=t_{\mathrm{in}}$) has some value $H$ and at the end
($t=t_{\mathrm{fin}}$) some value $H^{\prime}$, generating the unitary
evolution operator
$U=\overrightarrow{\exp}\big{[}-i\int_{t_{\mathrm{in}}}^{t_{\mathrm{fin}}}dtH(t)\big{]}$.

For any quantum process, the observed statistics of any quantity (e.g., work)
can be described by a positive operator-valued measure (POVM) [34]. Since the
whole physical arrangement through which the process is executed and measured
is encapsulated in the POVM, by a “work measuring scheme” we will understand a
POVM and a set of outcomes prescribed to it: $\\{M_{W},W\\}_{W}$, where
$M_{W}$ are the POVM elements and $W$ are the associated outcomes. Thus, any
random variable representing work is nothing but a (generalized) quantum
measurement with outcomes $W$ occurring with probabilities

$\displaystyle p_{W}=\operatorname{Tr}(\rho M_{W}),$ (1)

where $\rho$ is the initial state of the system.

When the measuring device, through which the statistics is observed, does not
depend on the system’s initial state, then, expectedly, the corresponding POVM
is also state-independent [9]. Surely, the scheme shall depend on the process
(i.e., $H$, $H^{\prime}$, and $U$). In general, however, both the POVM
elements and the outcomes will also depend on the system’s initial state. Such
state-dependent schemes comprise the most general set of work measurements.

In these terms, $\mathfrak{A}_{1}$ means that

$\displaystyle\begin{aligned} \sum_{W}W\operatorname{Tr}(\rho
M_{W})&=\operatorname{Tr}(U\rho\,U^{\dagger}H^{\prime})-\operatorname{Tr}(\rho
H)\\\ &=\operatorname{Tr}(\rho\,\Omega)\end{aligned}$ (2)

is required to hold for all $\rho$’s. Here, $\operatorname{Tr}(\rho H)$ and
$\operatorname{Tr}(U\rho U^{\dagger}H^{\prime})$ are, respectively, the
initial and final average energies, and

$\displaystyle\Omega=U^{\dagger}H^{\prime}U-H$ (3)

is the so-called Heisenberg operator of work (HOW) [5].

Condition $\mathfrak{B}$ imposes

$\displaystyle\sum_{W}e^{-\beta
W}\operatorname{Tr}(\tau_{\beta}M_{W})=e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])},$
(4)

for any $\beta>0$, where $\tau_{\beta}=\frac{1}{Z_{\beta}[H]}e^{-\beta H}$ is
the thermal state, with $Z_{\beta}[H]=\operatorname{Tr}e^{-\beta H}$ and
$F_{\beta}[H]=-\beta^{-1}\ln Z_{\beta}[H]$ being, respectively, the partition
function and equilibrium free energy.

Lastly, $\mathfrak{A}_{2}$ concerns systems that are untouched, i.e., when
$H(t)=H=\mathrm{const}$. In such a case, $\mathfrak{A}_{2}$ demands that
$p_{W=0}=\operatorname{Tr}(\rho M_{W=0})=1$ for any density matrix $\rho$.

###  2.1 Discussion of the setup

Table 1: Relationship of all existing schemes with conditions $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$, and $\mathfrak{B}$. Schemes satisfying only one of these conditions [16] or featuring quasiprobabilities [7, 15, 19, 20, 12] are not shown. Also not shown are Refs. [17, 18], since their framework does not produce any work statistics for states with coherences in the energy eigenbasis. We see that none of the schemes simultaneously satisfies all three conditions. Scheme |  $\mathfrak{A}_{1}$ |  $\mathfrak{A}_{2}$ |  $\mathfrak{B}$
---|---|---|---
[1, 5], [10], [22] |  ✔ |  ✔ |  ✘
[21], [23], [25], Appendix D |  ✔ |  ✘ |  ✔
[3, 4], [13], $\Upsilon_{\rho}$ [Eq. (10)] |  ✘ |  ✔ |  ✔

To provide additional context, we show in Table 1 the relationship of all
existing schemes with Conditions $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$, and
$\mathfrak{B}$; we see that none of them satisfies all three at the same time.
Additionally, for the sake of the forthcoming analysis, below we briefly
review the two paradigmatic work measurement schemes, TPM [3, 4] and HOW [1,
5], in the light of these conditions.

In the TPM scheme [3, 4], one first measures the energy at the beginning,
obtaining the outcome $E_{a}$ and post-measurement state $P_{a}\rho
P_{a}/\operatorname{Tr}(\rho P_{a})$ with probability $\operatorname{Tr}(\rho
P_{a})$, where $E_{a}$ and $P_{a}$ are, respectively, the eigenvalues and
eigenprojectors of $H$ ($H=\sum_{a}E_{a}P_{a}$). Then, the unitary process is
implemented and the energy of the system is measured again. This yields the
outcome $E^{\prime}_{k}$ with conditional probability
$\operatorname{Tr}\left[U\frac{P_{a}\rho P_{a}}{\operatorname{Tr}(\rho
P_{a})}U^{\dagger}P^{\prime}_{k}\right]$, where $E^{\prime}_{k}$ and
$P^{\prime}_{k}$ are the eigenvalues and eigenprojectors of $H^{\prime}$.
Thus, according to the scheme, the outcomes of work and their probabilities
are

$\displaystyle W_{ak}=E^{\prime}_{k}-E_{a},\qquad
p_{ak}^{\mathrm{TPM}}=\operatorname{Tr}\big{(}\rho
M_{ak}^{\mathrm{TPM}}\big{)},$ (5)

where

$\displaystyle M_{ak}^{\mathrm{TPM}}=P_{a}U^{\dagger}P^{\prime}_{k}UP_{a}.$
(6)

Noting that $\big{[}M_{ak}^{\mathrm{TPM}},H\big{]}=0$ for all $a$ and $k$, we
see that $\sum W_{ak}M_{ak}^{\mathrm{TPM}}$ commutes with $H$ for any $H$,
$H^{\prime}$, and $U$. Whereas there exist such $H$, $H^{\prime}$, and $U$ for
which $[U^{\dagger}H^{\prime}U,H]\neq 0$ and hence $\Omega$ does not commute
with $H$, and therefore $\Omega$ cannot be equal to $\sum
W_{ak}M_{ak}^{\mathrm{TPM}}$. In such a case, there will exist a $\rho$ for
which Eq. (2) is violated, which shows that the TPM scheme is incompatible
with $\mathfrak{A}_{1}$ [5, 8, 9, 11]. Nonetheless, we note that the TPM
scheme always satisfies $\mathfrak{A}_{2}$. Moreover, in the classical limit,
where all the commutators go to zero, the discrepancy with $\mathfrak{A}_{1}$
also vanishes (see Refs. [35, 36, 37, 38] for more details about the TPM
scheme’s classical limit).

In the HOW scheme [5], the statistics of work is identified with the
measurement statistics of the operator $\Omega$. Therefore, by construction,
the scheme satisfies both requirements $\mathfrak{A}_{1}$ and
$\mathfrak{A}_{2}$. However, $\mathfrak{B}$ is not satisfied because, in
general,

$\displaystyle\big{\langle}e^{-\beta
W}\big{\rangle}_{\mathrm{HOW}}:=\operatorname{Tr}\big{(}\tau_{\beta}\,e^{-\beta\Omega}\big{)}\geq
e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])},$

which straightforwardly follows from the Golden–Thomposon inequality [39].
When the system’s Hilbert space is finite-dimensional, the inequality is
strict when (and only when)
$\beta[\Omega,H]=\beta[U^{\dagger}H^{\prime}U,H]\neq 0$ [39]. However, the HOW
scheme satisfies $\mathfrak{B}$ in the classical limit. In finite-dimensional
spaces, this is a simple consequence of all the commutators vanishing in the
classical limit [5]. In continuous-variable systems, issues with continuity
may arise, so it is not mathematically guaranteed anymore. Nonetheless, for a
harmonic oscillator, by performing an explicit calculation, we show in
Supplementary Note S that the HOW scheme indeed satisfies $\mathfrak{B}$ in
the $\hbar\to 0$ limit.

That $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$, and $\mathfrak{B}$ become
compatible in the classical limit for the TPM and HOW schemes (and in fact for
some other schemes too [40, 12]), is yet another indication that the classical
approach to determining the statistics of work is unlikely to provide useful
guidance for the quantum regime.

##  3 Results

###  3.1 State-independent schemes

Let us define the JE class to be the set of all state-independent schemes that
satisfy $\mathfrak{B}$, i.e., the Jarzynski equality for all thermal states.
The following lemma, proven in Appendix A, gives a complete characterization
of the JE class.

######  Lemma 1 (Characterization of the JE class).

A state-independent scheme $\\{(M_{W},W)\\}_{W}$ with a discrete set of
outcomes $\\{W\\}$ satisfies $\mathfrak{B}$ iff

$\displaystyle\\{W\\}_{W}=\\{E^{\prime}_{k}-E_{a}\\}_{a=1}^{A}\phantom{|}{}_{k=1}^{K},$
(7)

so that the outcomes and the corresponding POVM elements can be labeled as
$W_{ak}=E^{\prime}_{k}-E_{a}$ and $M_{ak}$, along with

$\displaystyle\sum_{\begin{subarray}{c}a,\,b,\,m;\\\
E^{\prime}_{m}-E_{b}+E_{a}=E^{\prime}_{k}\end{subarray}}\operatorname{Tr}(M_{bm}P_{a})=\operatorname{Tr}P^{\prime}_{k}$
(8)

and

$\displaystyle\operatorname{Tr}(M_{bm}P_{a})=0\;\;\,\mathrm{if}\;\;\,E^{\prime}_{m}-E_{b}+E_{a}\not\in\\{E^{\prime}_{k}\\}_{k=1}^{K}.$
(9)

A key consequence of Lemma 1, also proven in Appendix A, is the following no-
go theorem.

######  Theorem 1 (No-go for state-independent schemes).

If a state-independent scheme satisfies $\mathfrak{B}$, then it is
incompatible with $\mathfrak{A}_{1}$.

Note that the JE class strictly contains the TPM scheme, which is expressed in
the fact that Eqs. (8) and (9) [and even the more specific Eqs. (A7) and (A8)]
allow for more general POVMs. However, if in addition to $\mathfrak{B}$, we
merely require the scheme to respect $\mathfrak{A}_{1}$ on thermal states, the
following theorem, proven in Appendix B, will hold.

######  Theorem 2 (JE $\Leftrightarrow$ TPM).

For the (generic) case of nondegenerate $H$ and nondegenerate set of outcomes,
the schemes in the JE class that satisfy Eq. (2) for thermal initial states
are equivalent to the TPM scheme.

We emphasize that here we do not require Eq. (2) to hold for nonthermal
states. Theorem 2 shows that the TPM scheme is essentially uniquely determined
by that, for thermal states, it reproduces the correct average work and
Jarzynski equality. This relation can be expressed as JE $\Rightarrow$ TPM.
The reverse, namely, that the TPM scheme satisfies the JE (i.e., JE
$\Leftarrow$ TPM), has been known from the very conception of the scheme [3,
4], and was in fact what motivated its experimental implementations [41, 42,
43]. Theorem 2 thus means that, basically, JE $\Leftrightarrow$ TPM for state-
independent schemes.

###  3.2 State-dependent schemes

Having excluded the possibility of simultaneously satisfying the conditions
$\mathfrak{A}_{1}$ and $\mathfrak{B}$ with work measuring schemes that are
limited to being state-independent, let us explore the opportunities when no
such limitation is posed. That is, when the work-measuring POVM,
$\\{M_{W}\\}$, and the corresponding set of outcomes, $\\{W\\}$, are allowed
to depend not only on the process (i.e., $U$, $H$, and $H^{\prime}$), but also
on the system’s initial state, $\rho$. These schemes comprise the largest set
of measurement protocols allowed by quantum mechanics.

On the one hand, this dramatically increases the set of POVMs to choose from.
On the other hand, this happens at the cost of abandoning “universality”—in
order to implement such a scheme, one has to acquire additional knowledge
about incoming states and readjust the measurement setup accordingly, which is
inconvenient from the practical standpoint, to say the least.

Expectedly, this additional freedom makes $\mathfrak{A}_{1}$ and
$\mathfrak{B}$, and even $\mathfrak{A}_{2}$, more compatible. In fact, by
allowing for completely arbitrary state-dependent schemes, we can resolve the
inconsistency problem altogether. For example, by implementing the following
protocol: when $[\rho,H]=0$, apply the TPM scheme; when $[\rho,H]\neq 0$,
apply the HOW scheme. Formally, this satisfies $\mathfrak{A}_{1}$,
$\mathfrak{B}$, and $\mathfrak{A}_{2}$ simultaneously. However, this scheme is
highly discontinuous. Indeed, if one adds an infinitesimal amount of coherence
to a state that is diagonal in the energy eigenbasis, the statistics will
experience a dramatic jump from having $AK$ outcomes $E^{\prime}_{k}-E_{a}$ to
having only $d$ outcomes $\Omega_{i}$, where $d$ is the system’s Hilbert space
dimension and $\Omega_{i}$ are the eigenvalues of $\Omega$ [see Eq. (3)].
Importantly, $\\{\Omega_{i}\\}$ is not a subset of
$\\{E^{\prime}_{k}-E_{a}\\}$ whenever $[U^{\dagger}H^{\prime}U,H]\neq 0$. In
addition to this discontinuity of the outcomes, a radical change of the
probability distribution itself will also take place.

We deem such discontinuous situations pathological, especially because any
measurement has a finite resolution (e.g., of how diagonal a state is).
Therefore, we require the measurement schemes to depend on the state
continuously, at least within a certain convex subset of all states.

More specifically, let us consider a closed $\epsilon$-ball in the state space
around the infinite-temperature ($\beta=0$) state $\tau_{0}=\mathbbm{1}/d$:
$\mathcal{B}_{\epsilon}:=\\{\sigma\;:\,\;\|\sigma-\tau_{0}\|<\epsilon\\}$,
where $\epsilon>0$ and $\|\cdot\|$ is the standard operator norm (in
$\ell_{2}$ space) [44]. Let us moreover choose $\epsilon>0$ such that all
$\rho$’s in $\mathcal{B}_{\epsilon}$ are positive-definite—such an $\epsilon$
always exists as all the eigenvalue of $\tau_{0}=1/d>0$, and the eigenvalues
of a Hermitian operator depend continuously on the operator [44]. Our main
continuity assumption is that there exists such an $\epsilon_{0}>0$ for which
all the elements $M_{W}$ of the POVM, and their corresponding outcomes $W$,
are continuous functions of $\rho$ in $\mathcal{B}_{\epsilon_{0}}$. If the
scheme is continuous on the set of all density matrices, then of course the
choice of $\epsilon_{0}$ will be arbitrary.

In Appendix C, we prove that requiring the POVM elements and their outcomes to
be differentiable with respect to $\rho$, which is a bit stronger than mere
continuity, is so restrictive that the following theorem, which is the main
result of this paper, holds.

######  Theorem 3 (No-go for state-dependent schemes).

No measurement scheme that is described by a POVM the elements and outcomes of
which are at least once differentiable in $\rho$ in some open ball
$\mathcal{B}_{\epsilon_{0}}$ centered at $\tau_{0}$, are at least twice
differentiable with respect to $U$ at $U=\mathbbm{1}$, and satisfy two mild
structural constraints 1 and 2 described in Appendix C, can simultaneously
satisfy $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$, and $\mathfrak{B}$.

The structural constraints 1 and 2 essentially boil down to requiring that the
only source of complexness of the matrices $M_{W}$ in the eigenbasis of $H$
can be the noncommutativity of $U$ and $\rho$ with $H$. All work measurement
schemes known to the authors satisfy these constraints.

While requiring continuity and differentiability can hardly be considered a
limitation in physics, Constraints 1 and 2, however mild they might seem (see
Appendix C), do leave a narrow perspective open for the existence of a state-
continuous scheme that could simultaneously satisfy $\mathfrak{A}_{1}$,
$\mathfrak{A}_{2}$, and $\mathfrak{B}$. Of course, there might simply be a
more general proof of incompatibility that would not require the Constraints 1
and 2. We defer the explicit description of these constraints to Appendix C
because there is a certain amount of notation that needs to be established for
stating them.

Importantly, while not being able to provide full compatibility of
$\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$, and $\mathfrak{B}$, continuously
state-dependent schemes do make those conditions more compatible. Indeed, with
state-independent schemes, we could simultaneously satisfy $\mathfrak{A}_{1}$
and $\mathfrak{A}_{2}$ (the HOW scheme) as well as $\mathfrak{A}_{2}$ and
$\mathfrak{B}$ (the TPM scheme), but not $\mathfrak{A}_{1}$ and
$\mathfrak{B}$. Now, with the additional freedom, in Appendix D, by combining
the ordinary (“forward”) TPM scheme with what we call “backward” TPM scheme,
we construct a state-dependent measurement that simultaneously satisfies
$\mathfrak{A}_{1}$ and $\mathfrak{B}$ (but not $\mathfrak{A}_{2}$). Thus, all
the requirements are pairwise compatible when continuous state-dependent POVMs
are allowed. While this paper was in preparation, three more state-dependent
schemes simultaneously satisfying $\mathfrak{A}_{1}$ and $\mathfrak{B}$ (but,
again, not $\mathfrak{A}_{2}$) were reported in Refs. [21, 23, 25].

Complementing Theorem 3, the following lemma shows that, strikingly, the two
manifestations of the energy conservation, $\mathfrak{A}_{1}$ and
$\mathfrak{A}_{2}$, are incompatible in a wide range of practically relevant
situations.

######  Lemma 2.

If, for some state $\rho$ with nonzero coherences ($[\rho,H]\neq 0$), the
outcomes of a work measuring scheme do not depend on the evolution operator
$U$, then the scheme cannot simultaneously satisfy conditions
$\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$, provided the POVM elements of the
scheme are differentiable at least once with respect to $U$ at
$U=\mathbbm{1}$.

Lemma 2 follows as a corollary from Lemma C.1 in Appendix C. It covers a large
class of schemes that includes the TPM scheme, the scheme we introduced in
Appendix D, and those later proposed in Refs. [21, 23]. This conflict between
the two manifestations of energy conservation, $\mathfrak{A}_{1}$ and
$\mathfrak{A}_{2}$, opens up a completely new perspective on the limited
applicability of these schemes for states with coherences in the energy
eigenbasis.

Notably, Lemma 2 does not cover the HOW scheme, as all its outcomes depend on
$U$; and indeed, it satisfies both $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$
by design, but, as discussed in Sec. 2, it does not satisfy $\mathfrak{B}$. In
fact, the HOW scheme is not covered also by the stronger Lemma C.1 (see
Appendix C).

In a sense, the strategy of our proof of Theorem 3 in Appendix C is inspired
by this example, in that we start with requiring $\mathfrak{A}_{1}$ and
$\mathfrak{A}_{2}$, and then show that the structure enforced by them cannot
be made compatible with $\mathfrak{B}$.

To give a non-trivial example to apply Theorem 3 to, let us, inspired by Ref.
[13], introduce the state-dependent operator of work

$\displaystyle\Upsilon_{\rho}=-\hat{\beta}(\rho)^{-1}\ln\\!\big{[}e^{\hat{\beta}(\rho)H/2}e^{-\hat{\beta}(\rho)U^{\dagger}H^{\prime}U}e^{\hat{\beta}(\rho)H/2}\big{]},~{}$
(10)

where $\hat{\beta}(\rho):=\arg\min\limits_{\beta}\|\rho-\tau_{\beta}\|$. This
defines an “operator scheme”—the POVM elements are the eigenprojectors of the
operator, and the outcomes are the corresponding eigenvalues. This scheme is
obviously designed to satisfy $\mathfrak{B}$ and, when
$[U^{\dagger}H^{\prime}U,H]=0$, $\Upsilon_{\rho}$ coincides with the
Heisenberg operator of work $\Omega$ [defined in Eq. (3)], which also means
that $\Upsilon_{\rho}$ has a proper classical limit. One also immediately
notes that this scheme satisfies $\mathfrak{A}_{2}$. In Appendix E, we show
that this scheme is indeed covered by Theorem 3, and hence, it cannot satisfy
$\mathfrak{A}_{1}$ because it already satisfies $\mathfrak{A}_{2}$ and
$\mathfrak{B}$. And indeed, in Appendix E, we explicitly show that
$\Upsilon_{\rho}$ violates $\mathfrak{A}_{1}$ even for thermal states. In
addition to that, to mark its debut, we prove some other curious facts about
$\Upsilon_{\rho}$, unrelated to the main point of this paper.

##  4 Discussion

In this paper, we unearthed the fundamental reason why, despite numerous
attempts in the literature, no satisfactory measurement scheme for quantum
work has been found so far. By taking an abstract approach, we asked whether
any quantum measurement, defined as broadly as possible, can produce
statistics that are consistent with energy conservation for the unmeasured
system and Jarzynski equality.

These conditions ensure that the measurement results reflect the reality of
the unmeasured system. Say, nothing happens to the system—there are no systems
with which it could exchange energy. Even without measuring the system, energy
conservation law already tells us that, with probability $1$, exactly $0$ work
is performed on the system. Therefore, any meaningful measurement of work,
when applied to the system, should output $0$, with probability $1$, even if
the measurement process disrupts the system. This was our condition
$\mathfrak{A}_{2}$.

By the same logic, when we take a system in a known state and drive it
unitarily according to a protocol we fully control, then, even without
measuring the system, energy conservation tells us that the average work
performed on the system is equal to the difference between the final and
initial average energies of the system. Thus, we imposed the condition
$\mathfrak{A}_{1}$ that a proper measurement of work should output a random
variable the first moment of which coincides with that average, no matter how
disruptive the measurement may be.

Note that $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ are not conditions about
the measurement process itself. Namely, we do not ask whether energy is
conserved during the joint evolution of the system, apparatus, and external
driving agents.

The main result of this paper, Theorem 3, is that no reasonable quantum
measurement can output such a random variable that would respect the above two
manifestations of energy conservation and, at the same time, satisfy the
Jarzynski equality whenever the system is in a thermal state (condition
$\mathfrak{B}$). This uncovers a deep connection between energy conservation
law and fluctuation theorems that is present only in the quantum regime.

Our proof merely assumes (i) differentiability of the measurement operators
and outcomes with respect to the state and (ii) that, in the eigenbasis of
$H$, complexness in $M_{W}$ can originate only from noncommutativity of $U$
and $\rho$ with $H$. While (i) can hardly constitute a loophole in any
conceivable physical situation, breaking (ii) could in principle provide a
leeway for devising a meaningful work measurement scheme. The possibility of
finding a more general proof that would not require (ii) is not excluded and
is an interesting problem for future research.

Exploring the extent to which state-dependent schemes can be helpful in the
related problems of quantum back-action evasion [45, 46], measuring
fluctuations of heat [47, 48], current [49, 50, 51], entropy production [52,
53, 54], and energetic cost of measurements [55, 33, 52, 56, 57, 58], is an
interesting research avenue.

##  Acknowledgments

We thank Raffaele Salvia for useful feedback on the manuscript. K.V.H. also
thanks Armen Allahverdyan, Antonio Acín, and Martí Perarnau-Llobet for
insightful discussions during eraly stages of this work. K. V. H. acknowledges
support by the University of Potsdam startup funds.

##  APPENDIX A Proofs of Lemma 1 and Theorem 1

To prove Lemma 1, let us rewrite Eq. (4) as

$\displaystyle\sum_{W}\operatorname{Tr}(M_{W}e^{-\beta H})e^{-\beta
W}=\operatorname{Tr}e^{-\beta H^{\prime}}\quad\forall\beta,$ (A1)

where $\\{M_{W}\\}_{W}$ is a POVM that does not depend on the state of the
system, but may depend on the process (i.e., $H$, $H^{\prime}$, and $U$).

Recalling the eigenresolutions $H=\sum_{a=1}^{A}E_{a}P_{a}$ and
$H^{\prime}=\sum_{k=1}^{K}E^{\prime}_{k}P^{\prime}_{k}$ and introducing the
degeneracies

$\displaystyle g_{a}:=\operatorname{Tr}P_{a}\qquad\mathrm{and}\qquad
g^{\prime}_{k}:=\operatorname{Tr}P^{\prime}_{k},$ (A2)

we obtain from Eq. (A1) that

$\displaystyle\sum_{W,a}\operatorname{Tr}(M_{W}P_{a})e^{-\beta(E_{a}+W)}=\sum_{k=1}^{K}g^{\prime}_{k}e^{-\beta
E^{\prime}_{k}}$ (A3)

must hold for all $\beta$’s. Whence, using the expansion
$e^{x}=\sum_{N=0}^{\infty}\frac{x^{N}}{N!}$, we find that

$\displaystyle\sum_{W,a}\operatorname{Tr}(M_{W}P_{a})(E_{a}+W)^{N}=\sum_{k=1}^{K}g^{\prime}_{k}(E^{\prime}_{k})^{N}$
(A4)

for all $N\in\mathbb{N}$.

Keeping in mind that everything above is invariant under a global constant
energy shift, we choose the ground state of $H^{\prime}$ to have zero energy.
Moreover, since the eigenresolution
$H^{\prime}=\sum_{k=1}^{K}E^{\prime}_{k}P^{\prime}_{k}$ already accounts for
possible degeneracies, all values of $E^{\prime}_{k}$ are distinct. Therefore,
the order of the eigenvectors of $H^{\prime}$ can be chosen such that
$0=E^{\prime}_{1}<E^{\prime}_{2}\cdots<E^{\prime}_{K}$. Let us now take
$N=2L$, $L\in\mathbb{N}$, and divide Eq. (A4) by $(E^{\prime}_{K})^{N}$. Then,
as $L\to\infty$, the right-hand side will converge to $g^{\prime}_{K}$. Now,
if, for all $a$ and $W$, $|E_{a}+W|<E^{\prime}_{K}$, then the left-hand side
will converge to $0$, which cannot be. Likewise, there cannot exist such $a$
and $W$ for which $|E_{a}+W|>E^{\prime}_{K}$, because otherwise the left-hand
side would diverge. Therefore, there have to exist at least one pair of $a$
and $W$ such that $|E_{a}+W|=E^{\prime}_{K}$, and
$|E_{b}+W^{\prime}|\leq|E_{a}+W|$ for all other pairs $b$ and $W^{\prime}$.
And since the equality between the left- and right-hand sides of Eq. (A4) is
maintained in the $L\to\infty$ limit, it also holds that

$\displaystyle\sum_{|E_{a}+W|=E^{\prime}_{K}}\operatorname{Tr}(M_{W}P_{a})=g^{\prime}_{K}.$
(A5)

Moreover, if there exist such $b$ and $W^{\prime}$ that
$E_{b}+W^{\prime}=-E^{\prime}_{K}$, then necessarily
$\operatorname{Tr}(M_{W^{\prime}}P_{b})=0$. Indeed, taking $N=2L+1$, dividing
Eq. (A4) by $(E^{\prime}_{K})^{N}$, and taking the $L\to\infty$ limit, we will
obtain the same $g^{\prime}_{K}$ on the right-hand side, but
$\operatorname{Tr}(M_{W^{\prime}}P_{b})$ will enter the left-hand side with a
negative sign, which means that equality between the left- and right-hand
sides is possible only if $\operatorname{Tr}(M_{W^{\prime}}P_{b})=0$. In other
words, for all $N\in\mathbb{N}$, $g^{\prime}_{K}(E^{\prime}_{K})^{N}$ is equal
to the sum of all the terms on the left-hand side with maximal $E_{a}+W$.
Thus, for any $N$, we can eliminate those terms from both sides of Eq. (A4)
and reiterate the arguments above. Doing so $K$ times, we will conclude that
the set of outcomes of work coincides with energy differences.

This procedure also shows that, if $E_{a}+W$ does not coincide with any of
$E^{\prime}_{k}$’s, then $\operatorname{Tr}(M_{W}P_{a})=0$ [check Eq. (A4)];
this proves Eq. (9). Finally, the fact that Eq. (A5) is obtained at each step
of the iteration (i.e., for all values of $k$) proves Eq. (8), which thereby
concludes the proof of Lemma 1.

Now, let us prove Theorem 1. For a state-independent scheme to satisfy
$\mathfrak{A}_{1}$, Eq. (2) must be satisfied for any state $\rho$, which
means that $\sum_{W}WM_{W}=\Omega$ must hold.

On the other hand, Lemma 1, and more specifically, Eq. (9), rather strongly
restricts the set of eigensubspaces of $H$ among which the matrices $M_{ak}$
are allowed to have coherences. So much so that, for any nontrivial $H$ and
$H^{\prime}$, there exists a $U$ such that $\Omega$ has coherences where
$\sum_{W}WM_{W}$ cannot.

For brevity of presentation, let us conduct the proof of this fact for the
case of nondegenerate set of work outcomes, namely, when

$\displaystyle
E^{\prime}_{k}-E_{a}=E^{\prime}_{m}-E_{b}\;\;\Longrightarrow\;\;k=m\;\;\mathrm{and}\;\;a=b.~{}~{}~{}$
(A6)

Note that this case is the generic one—those $H$ and $H^{\prime}$ for which
the set of energy differences has degeneracies constitute a $0$-measure subset
in the space of all $H$’s and $H^{\prime}$’s.

Now, for this configuration, Eq. (9) simply states that
$\operatorname{Tr}(M_{ak}P_{b})=0$ when $b\neq a$, which means that, for any
$k$, $M_{ak}$ belongs to the $a$’th eigensubspace of $H$; in other words,

$\displaystyle M_{ak}=P_{a}M_{ak}P_{a}.$ (A7)

Hence $[M_{ak},H]=0$ for any $a$ and $k$, and therefore $\sum_{W}WM_{W}$
commutes with $H$. Thus, $\sum_{W}WM_{W}$ cannot be equal to $\Omega$ whenever
$[U^{\dagger}H^{\prime}U,H]\neq 0$.

Lastly, note that, with Eq. (A7) taken into account, Eq. (8) simplifies to

$\displaystyle\sum_{a}\operatorname{Tr}M_{ak}=\operatorname{Tr}P^{\prime}_{k}.$
(A8)

A special case of this equation was proven in Ref. [59] for a certain class of
two-point energy measurement schemes.

##  APPENDIX B Proof of Theorem 2

To prove Theorem 2, let us, taking into account Eqs. (7) and (A7), rewrite Eq.
(2) with $\rho=\tau_{\beta}$ as

$\displaystyle\sum_{a}e^{-\beta E_{a}}\big{[}\sum_{k}E^{\prime}_{k}$
$\displaystyle\operatorname{Tr}(P_{a}U^{\dagger}P^{\prime}_{k}U)-g_{a}E_{a}\big{]}$
$\displaystyle=\sum_{a,k}e^{-\beta
E_{a}}(E^{\prime}_{k}-E_{a})\operatorname{Tr}M_{ak},$

which has to hold for any $\beta$. Keeping in mind that no two $E_{a}$’s are
the same, and setting $\beta=0,1,\dots,A-1$, we will arrive at a linear system
of equations with the coefficient matrix being a Vandermonde matrix [44] with
a non-zero determinant. Hence, for any $a$,

$\displaystyle\sum_{k}(E^{\prime}_{k}-E_{a})\operatorname{Tr}(P_{a}U^{\dagger}P^{\prime}_{k}U)=\sum_{k}(E^{\prime}_{k}-E_{a})\operatorname{Tr}M_{ak},$
(B9)

where we have noted that
$\sum_{k}\operatorname{Tr}(P_{a}U^{\dagger}P^{\prime}_{k}U)=g_{a}$.

At this point, we assume that the operators $M_{ak}$ do not depend on
$E^{\prime}_{k}$’s within a small neighborhood. This assumption is natural for
two reasons. First, the unitary evolution operator,
$U=\overrightarrow{\exp}\big{[}-i\int_{t_{\mathrm{in}}}^{t_{\mathrm{fin}}}dtH(t)\big{]}$
does not change when values of $H(s)$ are changed on a measure-zero set, and
since changing the eigenvalues of $H^{\prime}=H(t_{\mathrm{fin}})$ is such an
operation, $U$ remains intact. Second, since
$0=E^{\prime}_{1}<E^{\prime}_{2}<\cdots<E^{\prime}_{K}$ and $K$ is finite,
there exists an $\epsilon>0$ such that the $\epsilon$-neighborhoods of
$E^{\prime}_{k}$’s are non-overlapping; we will assume
$E^{\prime}_{k}$-independence in these neighborhoods. Now, before the last-
moment change in the eigenvalues of $H^{\prime}$, the process runs without
“knowing” about the change, so if $M_{ak}$’s were going to measure
probabilities of work outcomes for $E^{\prime}_{k}$’s, then a small variation
that preserves the order of $E^{\prime}_{k}$’s should not impact the
measurement probabilities and hence the operators.

With this assumption, we can differentiate Eq. (B9) with respect to
$E^{\prime}_{k}$’s, with all other parameters fixed, to find that

$\displaystyle\operatorname{Tr}M_{ak}=\operatorname{Tr}(P_{a}U^{\dagger}P^{\prime}_{k}U)$
(B10)

for all $a$’s and $k$’s.

When the initial Hamiltonian has no degeneracies, i.e.,
$P_{a}=|a\rangle\langle a|$ for all $a$’s, then Eq. (A7) implies that
$M_{ak}\propto|a\rangle\langle a|$, and Eq. (B10) shows that

$\displaystyle M_{ak}=|a\rangle\langle
a|U^{\dagger}P^{\prime}_{k}U|a\rangle\langle a|=M_{ak}^{\mathrm{TPM}},$ (B11)

where the second equality is due to Eq. (6). Hence, for nondegenerate $H$’s,
the JE class consists of the TPM scheme only. For degenerate $H$’s, the
statistics produced by the schemes in the JE class are guaranteed to coincide
with those of the TPM scheme only on diagonal initial states of the form
$\rho=\sum_{a}\rho_{a}P_{a}$ [which can be easily checked by using Eqs. (A7),
(B10), and (5)]. Whenever $\rho$ is not proportional to the identity operator
in one of the eigensubspaces of $H$, there will exist a POVM that is in the JE
class, satisfies Eq. (2) for thermal states, but is nevertheless such that
$\operatorname{Tr}(\rho M_{ak})\neq p_{ak}^{\mathrm{TPM}}$.

We emphasize that the $H$’s and $H^{\prime}$’s for which the JE class is
larger than the TPM scheme comprise a $0$-measure subset of the set of all
possible $H$’s and $H^{\prime}$’s. Which in particular means that, for any $H$
and $H^{\prime}$ yielding a degenerate set of energy difference, there exist
infinitesimal perturbations of $H$ and $H^{\prime}$ that remove the
degeneracy. Therefore, if we ask which schemes satisfy the JE and average work
condition for thermal initial states for all processes at least in an
arbitrarily small neighborhood of a given process, then the TPM scheme will be
the only such scheme. In this sense, the JE class is equivalent to the TPM
scheme.

##  APPENDIX C Proof of Theorem 3

To prove that certain requirements are incompatible, it is sufficient to prove
that there exists at least one state and one process for which those
requirements entail contradicting results.

We will look for such a state within the very same ball
$\mathcal{B}_{\epsilon_{0}}$ in which the scheme is continuous. The class of
processes among which we will look for the desired process will be cyclic
Hamiltonian processes ($H^{\prime}=H$), the evolution operator generated by
which has the form $U=e^{-ixh}$, where $x$ is a real number and $h$ is a
Hermitian operator. We will call this class of processes $\mathbf{\Pi}$:

$\displaystyle\mathbf{\Pi}=\big{\\{}(H,H^{\prime},U)\;:\,\;H^{\prime}=H,\;U=e^{-ixh}\big{\\}},$
(C12)

where $h$ spans all finite-norm Hermitian operators living in the system’s
Hilbert space. This class of processes will allow us to conveniently include
$\mathfrak{A}_{2}$ in our analysis, because, for any process in
$\mathbf{\Pi}$, we can make the system untouched by simply taking the $x\to 0$
limit. Conveniently, the system becomes untouched also in the limit $h\to H$.

Let us now explore the consequences of imposing $\mathfrak{A}_{2}$. To do so,
for each $\rho$ and $h$, we define

$\displaystyle\mathcal{O}_{\rho,h}=\big{\\{}W\;:\,\;\lim_{x\to 0}W=0\big{\\}}$
(C13)

to be the set of outcomes $W$ that are either $0$ or tend to $0$ as $x\to 0$.
The rest of the outcomes will tend to non-zero values in the same limit; we
denote the set of those outcomes by $\mathcal{N}_{\rho,h}$:

$\displaystyle\mathcal{N}_{\rho,h}=\big{\\{}W\;:\,\;\lim_{x\to 0}W\neq
0\big{\\}}.$ (C14)

Then, the requirement $\mathfrak{A}_{2}$ means that, in the $x\to 0$ limit,

$\displaystyle\sum_{W\in\mathcal{N}_{\rho,h}}p_{W}\to
0\quad\mathrm{and}\quad\sum_{W\in\mathcal{O}_{\rho,h}}p_{W}\to 1,$ (C15)

where $p_{W}=\operatorname{Tr}(\rho M_{W})$, as in Eq. (1). For positive-
definite states ($\rho>0$), since all their eigenvalues are strictly positive,
Eq. (C15) entails

$\displaystyle\sum_{W\in\mathcal{N}_{\rho,h}}M_{W}\to
0\quad\mathrm{and}\quad\sum_{W\in\mathcal{O}_{\rho,h}}M_{W}\to\mathbbm{1}.$
(C16)

We emphasize that, in general, like $M_{W}$’s, the outcomes $W$ depend on $H$,
$h$, $x$, and $\rho$.

If $M_{W}$ is differentiable at least once with respect to $x$ at $x=0$, then
$p_{W}=\operatorname{Tr}(M_{W}\rho)$ will also be differentiable in $x$ at
$x=0$. Hence, we can write the Taylor series with the remainder in the Peano
form:

$\displaystyle p_{W}(x)=p_{W}(0)+xp^{\prime}_{W}(0)+o(x),$ (C17)

where the small-o is as per the standard asymptotic notation.

Now, if $p_{W}\to 0$ as $x\to 0$, then $p_{W}(0)=0$. Moreover,
$p^{\prime}_{W}(0)$ will also have to be zero, because otherwise, for
sufficiently small $x$’s, $p_{W}$ would change its sign when $x\to-x$, whereas
$p_{W}$ must always remain non-negative. Therefore, while maintaining
nonnegativity,

$\displaystyle\lim_{x\to 0}p_{W}=0\quad\Longrightarrow\quad p_{W}=o(x)$ (C18)

With exactly the same reasoning,

$\displaystyle\lim_{x\to 0}M_{W}=0\quad\Longrightarrow\quad M_{W}=o(x).$ (C19)

As discussed above, $\lim_{x\to 0}p_{W}=0$ necessitates $\lim_{x\to 0}M_{W}=0$
whenever $\rho$ is positive-definite. Note that $o(x)$ is allowed to be
anything from $x^{2}\tilde{m}_{W}$, with some $\tilde{m}_{W}\geq 0$, like in
the TPM scheme, to something exotic, say, $|x|^{3/2}\tilde{m}_{W}$.

Let us now see what $\mathfrak{A}_{1}$ entails for processes in
$\mathbf{\Pi}$. Recalling Eqs. (2) and (3), namely that $\langle
W\rangle=\operatorname{Tr}(\Omega\rho)$, we can apply the Baker–Hausdorff
lemma [60] to expand $\Omega$ as

$\displaystyle\Omega=x\omega+i\frac{x^{2}}{2}[h,\omega]+O(x^{3}),$ (C20)

for $x\ll 1$, where we have introduced the operator

$\displaystyle\omega:=i[h,H],$ (C21)

which leads us to

$\displaystyle\langle
W\rangle=x\operatorname{Tr}(\rho\,\omega)+i\frac{x^{2}}{2}\operatorname{Tr}(\rho[h,\omega])+O(x^{3}).$
(C22)

At this point, we are ready to prove the following lemma that establishes
incompatibility between $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$—the two
aspects of energy conservation—in special, but practically relevant,
circumstances.

######  Lemma C.1.

If, for a state-dependent scheme $\\{(M_{W},W)\\}_{W}$, there exist a state
$\rho$ and a process in $\mathbf{\Pi}$, such that
$\operatorname{Tr}(\rho[h,H])\neq 0$, for which the outcomes $W$ either do not
go to zero as $x\to 0$ (but also do not diverge), or go to zero as $o(x)$,
then the scheme cannot simultaneously satisfy $\mathfrak{A}_{1}$ and
$\mathfrak{A}_{2}$.

Note that, since $\operatorname{Tr}(\rho[h,H])=\operatorname{Tr}(h[H,\rho])$,
the state must have nonzero coherences in the energy eigenbasis; namely,
$[\rho,H]\neq 0$. Obviously, for any such $\rho$, there always exist an
infinite amount of $h$’s for which the condition
$\operatorname{Tr}(\rho[h,H])\neq 0$ in Lemma C.1 is satisfied. Also note that
the case of $W$ simply being equal to zero is included in the $o(x)$ notation.

It is worth emphasizing that, for Lemma C.1 to hold, the scheme does not have
to be continuous in $\rho$.

###### Proof of Lemma C.1.

As we see from Eq. (C22), once we enforce $\mathfrak{A}_{1}$, the condition
$\operatorname{Tr}(\rho[h,H])\neq 0$ of the lemma necessitates that

$\displaystyle\langle W\rangle\propto x.$ (C23)

On the other hand, from Eqs. (C15) and (C18) we see that, enforcing
$\mathfrak{A}_{2}$ necessitates that, if $\lim_{x\to 0}W\neq 0$, then
$p_{W}=o(x)$, which means that $Wp_{W}=o(x)$. By the condition of the lemma,
if $\lim_{x\to 0}W=0$, then $W=o(x)$, which means that $Wp_{W}=o(x)$ also in
this case. Hence, $\langle W\rangle=\sum_{W}Wp_{W}=o(x)$, which is in clear
contradiction with Eq. (C23). This thus proves that, under the assumptions of
the lemma, $\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ are incompatible. ∎

Note that the HOW scheme provides an important example of a scheme that is not
covered by Lemma C.1. Indeed, from Eq. (C20) we observe that all the outcomes
of the HOW scheme go to zero $\propto x$, which is in straightforward
violation of the conditions of Lemma C.1.

We immediately see that Lemma 2 in the main text follows from Lemma C.1 as a
simple corollary. Indeed, as noted above, for any $\rho$ such that
$[\rho,H]\neq 0$, there always exists a $h$ such that
$\operatorname{Tr}(\rho[h,H])\neq 0$. And if the outcomes do not depend on
$U$, it means that they are either zero, which falls under the category of
$o(x)$, or they are not zero. Hence, all the conditions of Lemma C.1 are met.

Proof of Theorem 3. Our proof strategy here will be to impose
$\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$ and show that they lead to results
that are not compatible with $\mathfrak{B}$.

By the assumption of the theorem, all $M_{W}$’s are differentiable at least
twice with respect to $x$. Therefore, the argumentation between Eqs. (C17) and
(C19) holds, which means that, in view of Eq. (C16), $M_{W}=o(x)$ whenever
$W\in\mathcal{N}_{\rho,h}$. Due to the existence of the second derivative, we
can write the $o(x)$ term as $x^{2}u_{W}+o(x^{2})$, with some operator
$u_{W}\geq 0$.

Let us now add to the set $\mathcal{N}_{\rho,h}$ all those
$W\in\mathcal{O}_{\rho,h}$ for which $M_{W}\to 0$ as $x\to 0$ (if such $W$’s
exist, of course). By the very same logic as above, for such $W$’s, we will
also have $M_{W}=x^{2}u_{W}+o(x^{2})$. The addition of such elements will
extend $\mathcal{N}_{\rho,h}$ into

$\displaystyle\mathbf{N}_{\rho,h}=\big{\\{}W\;:\,\;\lim_{x\to
0}M_{W}=+0\big{\\}}\;\supseteq\mathcal{N}_{\rho,h},$ (C24)

and, for this set, we will have

$\displaystyle
W\in\mathbf{N}_{\rho,h}\Rightarrow\bigg{\\{}\\!\begin{array}[]{l}W=U_{W}+O(x),\vspace{1mm}\\\
M_{W}=x^{2}u_{W}+o(x^{2}).\end{array}$ (C27)

Both $u_{W}\geq 0$ and $U_{W}$ may depend on $\rho$, $H$, and $h$.

For the complementary set

$\displaystyle\mathbf{O}_{\rho,h}=\big{\\{}W\;:\,\;\lim_{x\to
0}M_{W}>0\big{\\}}\;\subseteq\mathcal{O}_{\rho,h},$ (C28)

the double-differentiability of $M_{W}$’s and $W$’s yields

$\displaystyle
W\in\mathbf{O}_{\rho,h}\Rightarrow\bigg{\\{}\\!\begin{array}[]{l}W=xV_{W}+x^{2}v_{W}+o(x^{2}),\vspace{1mm}\\\
M_{W}=m_{W}+x\mu_{W}+x^{2}\nu_{W}+o(x^{2}),\end{array}$ (C31)

where, by definition, $m_{W}>0$ for all $W\in\mathbf{O}_{\rho,h}$, provided
$\mathbf{O}_{\rho,h}$ is not empty. All quantities involved ($m_{W}$,
$\mu_{W}$, $\nu_{W}$, $V_{W}$, $v_{W}$) may depend on $\rho$, $H$, and $h$.
Recall that lack of zeroth-order terms in the expansion of $W$ in Eq. (C31) is
a consequence of $\mathfrak{A}_{2}$; see the discussion around Eqs.
(C13)–(C19) in this regard.

It is worth emphasizing at this point that

$\displaystyle\mathbf{N}_{\rho,h}\cap\mathbf{O}_{\rho,h}=\emptyset\quad\mathrm{and}\quad\mathbf{N}_{\rho,h}\cup\mathbf{O}_{\rho,h}=\\{W\\}.$
(C32)

Due to the fact that $\sum_{W}M_{W}=\mathbbm{1}$ must hold for any $x$, we
find from Eqs. (C27) and (C31) that

$\displaystyle\sum_{W\in\mathbf{O}_{\rho,h}}m_{W}=\mathbbm{1},\qquad\sum_{W\in\mathbf{O}_{\rho,h}}\mu_{W}=0,$
(C33)
$\displaystyle\sum_{W\in\mathbf{O}_{\rho,h}}\nu_{W}+\sum_{W\in\mathbf{N}_{\rho,h}}u_{W}=0.$
(C34)

Next, invoking $\mathfrak{A}_{1}$, we see that, on the one hand, $\langle
W\rangle=\sum_{W}W\operatorname{Tr}(\rho M_{W})$, while on the other hand, Eq.
(C22) must be true for all $x$. By comparing the first-order (in $x$) terms,
we find that

$\displaystyle\langle Y(\rho,h,H)\rangle_{\rho}=\langle\omega\rangle_{\rho},$
(C35)

where

$\displaystyle Y(\rho,h,H):=\\!\sum_{W\in\mathbf{O}_{\rho,h}}\\!V_{W}m_{W},$
(C36)

$\omega$ is defined in Eq. (C21), and $\langle
X\rangle_{\rho}:=\operatorname{Tr}(\rho X)$. By comparing the second-order
terms in the Taylor expansion, we further obtain

$\displaystyle\begin{aligned}
\sum_{W\in\mathbf{O}_{\rho,h}}\\!\\!\big{[}V_{W}&\langle\mu_{W}\rangle_{\rho}+v_{W}\langle
m_{W}\rangle_{\rho}\big{]}\\\
+&\sum_{W\in\mathbf{N}_{\rho,h}}\\!\\!U_{W}\langle
u_{W}\rangle_{\rho}=\frac{1}{2}\langle
i[h,\omega]\rangle_{\rho}\,.\end{aligned}$ (C37)

In order for our proof of incompatibility to “work,” we will need to slightly
constrain the set of POVMs. These are high-level assumptions about the
structure of allowed POVMs, and are going to be based solely on the fact that
all $M_{W}$’s and $W$’s double differentiable with respect to $x$ at $x=0$.

######  Constraint 1.

The only source of complexity of $m_{W}$’s in the eigenbasis of $H$ can be the
noncommutativity of $h$ with $H$ and $\rho$ with $H$. More precisely, it is
the Hermitian operators $\zeta:=i[\rho,H]$ and $\omega=i[h,H]$ that control
the complexity of $m_{W}$’s: the matrices $m_{W}$ are real whenever $\zeta$
and $\omega$ are real.

This is a natural assumption roughly saying that, in the classical regime of
all the operators commuting with each other, the measurement operators of the
scheme should be real in the common eigenbasis. In fact, what we require is
somewhat weaker: the condition is needed only on the level of the first-order
expansion in $x$. Note that a real $\omega$ means a purely imaginary $h$,
which in turn means a real $U$.

Of course, the scheme may be constructed in such a way that the POVM elements
depend on some quantity that does not depend on $\rho$ and $h$ and is
represented by a complex matrix in the eigenbasis of $H$; it is this kind of
situations that Constraint 1 aims to exclude.

The second constraint concerns the two positive-semidefinite operators defined
as

$\displaystyle Y^{+}=\\!\sum_{\begin{subarray}{c}W\in\mathbf{O}_{\tau,h}\\\
V_{W}>0\end{subarray}}\\!\\!V_{W}m_{W}\quad\mathrm{and}\quad
Y^{-}=\\!\sum_{\begin{subarray}{c}W\in\mathbf{O}_{\tau,h}\\\ |V_{W}|\leq
0\end{subarray}}\\!\\!V_{W}m_{W}.$

By definition (C36), these operators satisfy

$\displaystyle Y^{+}-Y^{-}=Y.$ (C38)

######  Constraint 2.

For all the Gibbs states $\tau_{\beta}\in\mathcal{B}_{\epsilon_{0}}$,
$\operatorname{Tr}(\tau_{\beta}\\{Y^{+},Y^{-}\\})\geq 0$ whenever $U$ is a
real matrix in the eigenbasis of $H$.

Despite the fact that $Y^{+}\geq 0$ and $Y^{-}\geq 0$ by construction, their
anticommutator will not be positive-semidefinite whenever $[Y^{+},Y^{-}]\neq
0$. Therefore, Constraint 2 is not a guaranteed in general. However, it is
easy to see that

$\displaystyle\operatorname{Tr}(\\{Y^{+},Y^{-}\\})=2\operatorname{Tr}(Y^{+}\,Y^{-})\geq
0,$ (C39)

and therefore it is natural to expect that Constraint 2 should be generically
satisfied whenever the state is close enough to $\mathbbm{1}/d$. There are two
specific situations—that are fairly general in and of themselves—where
Constraint 2 is satisfied by construction.

Situation 1: $[Y^{+},Y^{-}]=0$ when the state is thermal and $U$ is real. In
this case, $\\{Y^{+},Y^{-}\\}\geq 0$, and thus Constraint 2 is satisfied.
$Y^{+}$ and $Y^{-}$ will commute, for example, when all $m_{W}$’s commute with
each other. That trivially happens, e.g., when $m_{W}$’s are the
eigenprojectors of some observable. Of course, the scenarios where $m_{W}$’s
commute with each other are much more diverse than that.

An alternative scenario where $[Y^{+},Y^{-}]=0$ is when all $|V_{W}|$’s are
equal to each other; call that value $v>0$. Indeed, in such a case, $Y^{+}=vm$
and $Y^{-}=vn$, where

$\displaystyle m=\\!\sum_{\begin{subarray}{c}W\in\mathbf{O}_{\tau,h}\\\
V_{W}>0\end{subarray}}\\!\\!m_{W}\quad\mathrm{and}\quad
n=\\!\sum_{\begin{subarray}{c}W\in\mathbf{O}_{\tau,h}\\\ V_{W}\leq
0\end{subarray}}\\!\\!m_{W}.$

In view of Eq. (C33), $m+n=\mathbbm{1}$, and therefore

$\displaystyle[Y^{+},Y^{-}]$
$\displaystyle=v^{2}[m,n]=v^{2}[m,\mathbbm{1}-m]=0.$

Note that, in this scenario, individual $m_{W}$’s do not have to commute with
each other.

Situation 2: $\lim\limits_{\beta\to
0}\operatorname{Tr}(\tau_{\beta}\\{Y^{+},Y^{-}\\})\neq 0$; let us call this
value $\xi$. We immediately see that $\xi>0$. Indeed, $\lim_{\beta\to
0}\tau_{\beta}=\mathbbm{1}/d$, and therefore, $\xi>0$ follows from the
observation (C39). Now, by the assumption of the theorem, both $Y^{+}$ and
$Y^{-}$ are continuous functions of $\rho$ in $\mathcal{B}_{\epsilon_{0}}$,
and hence, $\operatorname{Tr}(\tau_{\beta}\\{Y^{+},Y^{-}\\})$ is continuous in
$\beta$ in a certain interval around $\beta=0$. Therefore, by the very
definition of the notion of continuity, there exist a $\beta_{0}>0$ such that
$\tau_{\beta_{0}}\in\mathcal{B}_{\epsilon_{0}}$ and
$\forall\,\beta\in[0,\beta_{0}]$,
$\big{|}\operatorname{Tr}(\tau_{\beta}\\{Y^{+},Y^{-}\\})-\xi\big{|}\leq\xi/2$.
Hence, $\operatorname{Tr}(\tau_{\beta}\\{Y^{+},Y^{-}\\})\geq\xi/2>0$. Thus, at
least for this specific $\beta_{0}>0$, Constraint 2 is satisfied.

We emphasize once again that Constraints 1 and 2 are not tied to the specific
forms (C27) and (C31) that are imposed by $\mathfrak{A}_{2}$—the possibility
of Taylor-expanding $M_{W}$’s and $W$’s is all that is needed for stating
Constraints 1 and 2.

Now, let us fix the basis to be the eigenbasis of $H$ and explore some
consequences of Eq. (C35), the differentiability of $M_{W}$’s and $W$’s with
respect to $\rho$ in $\mathcal{B}_{\epsilon_{0}}$, and Constraint 1.

Consider the family of $\omega$’s represented by real (and necessarily
symmetric due to $\omega$’s Hermiticity) matrices in the eigenbasis of $H$:

$\displaystyle\omega\;\in\;\mathbb{R}^{d\times
d},\quad\omega^{T}=\omega,\quad\mathcal{D}_{H}[\omega]=0,$ (C40)

where the operator $\mathcal{D}_{H}[R]$ acts on its argument as
$\mathcal{D}_{H}[R]:=\sum_{a}P_{a}RP_{a}$, where $P_{a}$’s are the
eigenprojectors of $H$. The last equality in (C40), saying that $\omega$ is
zero in the diagonal subspace of $H$, is an immediate consequence of the fact
that $\omega=i[h,H]$. Note that, with this choice of $\omega$’s, $U$ is a real
matrix, too.

Next, consider the family of $\rho$’s defined as

$\displaystyle\rho=\tau_{\beta_{0}}+zR,$ (C41)

where $\tau_{\beta_{0}}$ is some Gibbs state in $\mathcal{B}_{\epsilon_{0}}$
with $\beta_{0}>0$, $z$ is a real adimensional parameter that is $\ll 1$ to
ensure that $\rho\in\mathcal{B}_{\epsilon_{0}}$; henceforth, we will refer to
$\tau_{\beta_{0}}$ as simply $\tau$ to simplify notation. $R$ is an arbitrary
real and symmetric matrix with zeros in the diagonal eigenspace of $H$:

$\displaystyle R\;\in\;\mathbb{R}^{d\times d},\quad
R^{\mathrm{T}}=R,\quad\mathcal{D}_{H}[R]=0;$ (C42)

we will furthermore limit the norm of $R$ from above by some constant to
ensure that $\rho$ does not fall out of $\mathcal{B}_{\epsilon_{0}}$. Note
that, although we chose $R$ to be zero in the whole of the diagonal subspace
of $H$, Eq. (C41) necessitates only that $\operatorname{Tr}R=0$.

For this family of $\rho$’s, $\zeta$’s are purely imaginary:

$\displaystyle\zeta=izQ;\quad Q=[R,H]\;\in\;\mathbb{R}^{d\times d},\quad
Q^{\mathrm{T}}=-Q.$ (C43)

Next, by the main assumption of Theorem 3, all $W$’s and $M_{W}$’s, and hence
all $V_{W}$’s and $m_{W}$’s (and thus $Y$), are at least once differentiable
with respect to $\rho$. Taylor-expanding $Y(\rho,\omega,H)$ with respect to
$\zeta$ around $\zeta=0$, we find that

$\displaystyle Y(\rho,\omega,H)=Y(\tau,\omega,H)+izG(Q,\tau,\omega,H)+o(z),$
(C44)

where $G(Q,\tau,\omega,H)$ is order-$1$ homogeneous in $Q$.

Let us now invoke Constraint 1, which states that the complexness of $Y$ is
determined solely by $\omega$ and $\zeta$. Since all the arguments of
$Y(\tau,\omega,H)$ and $G(Q,\tau,\omega,H)$ are real matrices (in the
eigenbasis of $H$), then Constraint 1 enforces that $Y(\tau,\omega,H)$ and
$G(Q,\tau,\omega,H)$ are real matrices themselves. Lastly, due to Hermiticity
of $Y(\rho,\omega,H)$, $G(Q,\tau,\omega,H)$ must be skew-symmetric; all in
all, Constraint 1 leads us to

$\displaystyle\begin{aligned}
Y(\tau,\omega,H),\;G(Q,\tau,\omega,H)\;&\in\;\mathbb{R}^{d\times d}\,;\\\
Y(\tau,\omega,H)^{\mathrm{T}}&=Y(\tau,\omega,H),\\\
G(Q,\tau,\omega,H)^{\mathrm{T}}&=-G(Q,\tau,\omega,H).\end{aligned}$ (C45)

Substituting Eqs. (C41) and (C44) into Eq. (C35), and taking into account
that, due to the skew-symmetry of $G(\tau,F,Q,H)$,
$\operatorname{Tr}\big{(}\tau G(\tau,F,Q,H)\big{)}=0$, and that, due to
$\operatorname{diag}\omega=0$, $\operatorname{Tr}(\omega\tau)=0$, we find that

$\displaystyle\operatorname{Tr}\big{(}Y(\tau,\omega,H)\tau_{\beta_{0}}\big{)}+z\operatorname{Tr}\big{(}\big{[}Y(\tau,\omega,H)-\omega\big{]}R\big{)}=o(z),$

which must be satisfied for arbitrary $z\ll 1$. Whence,

$\displaystyle\begin{aligned}
\operatorname{Tr}\big{(}Y(\tau,\omega,H)\tau\big{)}&=0,\\\
\operatorname{Tr}\big{(}\big{[}Y(\tau,\omega,H)-\omega\big{]}R\big{)}&=0.\end{aligned}$
(C46)

Introducing

$\displaystyle
Y_{\mathrm{d}}=\mathcal{D}_{H}[Y(\tau,\omega,H)]\quad\mathrm{and}\quad
Y_{\mathrm{nd}}=Y(\tau,\omega,H)-Y_{\mathrm{d}},$

and keeping in mind that $\mathcal{D}_{H}[R]=0$, we can rewrite Eqs. (C46) as

$\displaystyle\operatorname{Tr}\big{(}Y_{\mathrm{d}}\tau\big{)}$
$\displaystyle=0,$ (C47)
$\displaystyle\operatorname{Tr}\big{(}\big{[}Y_{\mathrm{nd}}-\omega\big{]}R\big{)}$
$\displaystyle=0.$ (C48)

Now, noticing that $Y_{\mathrm{nd}}(\tau,\omega,H)-\omega$ is a real symmetric
matrix that is zero in the diagonal subspace of $H$ and independent of $R$,
and that Eq. (C48) holds for arbitrary $R$ satisfying the conditions in Eq.
(C42), we immediately conclude that $Y_{\mathrm{nd}}=\omega$, and thus,

$\displaystyle Y(\tau,\omega,H)=Y_{\mathrm{d}}+\omega,$ (C49)

where $Y_{\mathrm{d}}$ is an arbitrary real, symmetric matrix satisfying
$Y_{\mathrm{d}}=\mathcal{D}_{H}[Y_{\mathrm{d}}]$ and Eq. (C47). We emphasize
that this relation was derived exclusively for real $\omega$’s. For more
general $\omega$’s, Eq. (C49) may not apply.

Let us now estimate
$\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!\beta_{0}^{2}V_{W}^{2}\langle
m_{W}\rangle_{\tau}$; the need for this will become evident shortly. To do so,
let us invoke the operator-valued Cauchy–Schwarz inequality proven in Refs.
[61, 62], which will give us

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!\\!\\!V_{W}^{2}m_{W}\\!$
$\displaystyle=\\!\Big{\|}\\!\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!(\sqrt{m_{W}})^{2}\Big{\|}\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!\\!(|V_{W}|\sqrt{m_{W}})^{2}$
$\displaystyle\geq\\!\\!\bigg{[}\sum_{W\in\mathbf{O}_{\tau,h}}\\!|V_{W}|m_{W}\bigg{]}^{2},$

where in the first line we used the fact that
$\sum_{W\in\mathbf{O}_{\rho,h}}m_{W}=\mathbbm{1}$ [see Eq. (C33)].

Next, bringing up the quantities $Y^{+}$ and $Y^{-}$, and keeping in mind Eq.
(C38), we can rewrite Eq. (LABEL:Cauchuk) as

$\displaystyle\begin{aligned}
\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!V_{W}^{2}m_{W}&\geq(Y^{+}-Y^{-})^{2}+2\\{Y^{+},Y^{-}\\}\\\
&=(Y_{\mathrm{d}}+\omega)^{2}+2\\{Y^{+},Y^{-}\\}\\\
&\geq\omega^{2}+\\{\omega,Y_{\mathrm{d}}\\}+2\\{Y^{+},Y^{-}\\}.~{}~{}\end{aligned}$
(C51)

From here, in view of Constraint 2, we immediately obtain

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!V_{W}^{2}\langle
m_{W}\rangle_{\tau}\\!$ $\displaystyle\geq$
$\displaystyle\langle\omega^{2}\rangle_{\tau}+\langle\\{\omega,Y_{\mathrm{d}}\\}\rangle_{\tau}.~{}$
(C52)

Now,

$\displaystyle\langle\\{\omega,Y_{\mathrm{d}}\\}\rangle_{\tau}=\operatorname{Tr}(\tau\\{\omega,Y_{\mathrm{d}}\\})=\operatorname{Tr}(\omega\\{\tau,Y_{\mathrm{d}}\\})=0,$

because $\\{\tau,Y_{\mathrm{d}}\\}$ can be nonzero only in the diagonal
subspace of $H$ whereas $\omega$ is zero in exactly that subspace. Thus, for
the chosen families of $\omega$’s and $\rho$’s,

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!\beta_{0}^{2}V_{W}^{2}\langle
m_{W}\rangle_{\tau}\geq\beta_{0}^{2}\langle\omega^{2}\rangle_{\tau}.$ (C53)

We are now ready for the final stage for the proof. Let us fix the
$\beta_{0}>0$ from Eq. (C40), and invoke $\mathfrak{B}$:
$\sum_{W}e^{-\beta_{0}W}\operatorname{Tr}(\tau M_{W})=1$. As before, noting
that this must hold for any $x$, and collecting all powers of $x$, we will see
that, on the zeroth- and first-order levels, $\mathfrak{B}$ is secured by Eqs.
(C33) and (C35). Whereas the second-order level produces

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\\!\bigg{[}\\!\bigg{(}\frac{\beta_{0}^{2}V_{W}^{2}}{2}$
$\displaystyle-\beta_{0}v_{W}\bigg{)}\langle
m_{W}\rangle_{\tau}-\beta_{0}V_{W}\langle\mu_{W}\rangle_{\tau}$
$\displaystyle+$
$\displaystyle\,\langle\nu_{W}\rangle_{\tau}\bigg{]}+\\!\sum_{W\in\mathbf{N}_{\tau,h}}e^{-\beta_{0}U_{W}}\langle
u_{W}\rangle_{\tau}=0,$

which, using Eqs. (C34) and (C37), we can rewrite as

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\\!\\!\beta_{0}^{2}$
$\displaystyle V_{W}^{2}\langle m_{W}\rangle_{\tau}=\beta_{0}\langle
i[h,\omega]\rangle_{\tau}$
$\displaystyle-2\sum_{W\in\mathbf{N}_{\tau,h}}\\!\\!\big{(}e^{-\beta_{0}U_{W}}+\beta_{0}U_{W}-1\big{)}\langle
u_{W}\rangle_{\tau},$

from where, by noting that $e^{-z}+z-1\geq 0$ for $\forall z\in\mathbb{R}$
(with equality only for $z=0$), we arrive at

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\beta_{0}^{2}V_{W}^{2}\langle
m_{W}\rangle_{\tau}\leq\beta_{0}\langle i[h,\omega]\rangle_{\tau}.$ (C54)

Lastly, in Subsection C.1, we prove that, as long as $\omega\neq 0$,

$\displaystyle\beta_{0}^{2}\langle\omega^{2}\rangle_{\tau}>\beta_{0}\langle
i[h,\omega]\rangle_{\tau}.$ (C55)

Substituting this into Eq. (C53), which was obtained by enforcing
$\mathfrak{A}_{1}$ and $\mathfrak{A}_{2}$, yields

$\displaystyle\sum_{W\in\mathbf{O}_{\tau,h}}\\!\beta_{0}^{2}V_{W}^{2}\langle
m_{W}\rangle_{\tau}>\beta_{0}\langle i[h,\omega]\rangle_{\tau}.$ (C56)

We see that this contradicts Eq. (C54). Thus, we have just shown that there
exist a $\rho$ and a $h$ for which $\mathfrak{A}_{1}$, $\mathfrak{A}_{2}$, and
$\mathfrak{B}$ cannot hold all at the same time, which concludes the proof of
Theorem 3.

###  C.1 Proof of inequality (C55)

To prove inequality (C55), let us switch to the eigenbasis of $H$ ordered in
such a way that its eigenvalues $E_{k}$ are ordered increasingly. By explicit
calculation, we then see that

$\displaystyle\beta^{2}\langle\omega^{2}\rangle_{\tau}=\sum\limits_{k<j}(p_{k}+p_{j})\Delta_{kj}^{2}|h_{kj}|^{2},$
(C57)

where $p_{k}=e^{-\beta E_{k}}/Z$ are the eigenvalues of $\tau$, and
$\Delta_{kj}=\beta(E_{j}-E_{k})$.

In the same notation, another simple calculation leads to

$\displaystyle\beta\langle
i[h,\omega]\rangle_{\tau}=2\sum\limits_{k<j}(p_{k}-p_{j})\Delta_{kj}|h_{kj}|^{2}\geq
0.$ (C58)

Note that this inequality holds for any passive state $\tau$ and simply
expresses the fact that, when $\rho=\tau$ in Eq. (C22), the term $\propto x$
vanishes, and the second term has to be positive because $\langle W\rangle$ is
the work performed on a passive state and thus has to be nonnegative.

Subtracting (C58) from (C57), we obtain

$\displaystyle\beta^{2}$
$\displaystyle\langle\omega^{2}\rangle_{\tau}-\beta\langle
i[h,\omega]\rangle_{\tau}$
$\displaystyle=2\sum_{k<j}|h_{kj}|^{2}(p_{k}+p_{j})\Delta_{kj}\Big{[}\frac{\Delta_{kj}}{2}-\tanh\frac{\Delta_{kj}}{2}\Big{]}.~{}~{}$
(C59)

Now, $[h,H]\neq 0$ means that $h$ has nonzero nondiagonal elements between
some eigensubspaces of $H$. In other words, there exist some $k_{0}$ and
$j_{0}$ for which $\Delta_{k_{0}j_{0}}>0$ and $h_{k_{0}j_{0}}\neq 0$.
Moreover, since $x-\tanh x>0$ whenever $x>0$, at least one of the summands in
the right-hand side of Eq (C59) is $>0$, which thereby proves the bound in Eq.
(C55).

##  APPENDIX D A scheme simultaneously satisfying $\mathfrak{A}_{1}$ and
$\mathfrak{B}$

Let us construct a scheme that satisfies $\mathfrak{A}_{1}$ and $\mathfrak{B}$
and depends continuously on the system’s state. The example illustrates that
$\mathfrak{A}_{2}$ is an independent and essential requirement for any
meaningful measurement of work.

As a first step, we devise a measurement scheme in the specific class of
processes in which the system starts out in a possibly coherent (in the
eigenbasis of $H$) state and unitarily evolves into a diagonal state:
$(H,\rho)\longrightarrow(H^{\prime},\rho_{D})$, where $\rho_{D}=V\rho
V^{\dagger}$ and $[\rho_{D},H^{\prime}]=0$. Here, we notice that the time-
reversed process, $(H^{\prime},\rho_{D})\longrightarrow(H,\rho)$, with
$\rho=V^{\dagger}\rho_{D}V$, is describable by the TPM scheme.

Now, if the work statistics of a unitary process is
$\\{(W_{\alpha},p_{\alpha})\\}_{\alpha}$, then, to the time-reversed process,
it is reasonable to prescribe the work statistics
$\\{(-W_{\alpha},p_{\alpha})\\}_{\alpha}$.

With this prescription, thinking of the
$\mathcal{P}=\\{(H,\rho)\overset{V}{\longrightarrow}(H^{\prime},\rho_{D})\\}$
process as the reverse of
$\widetilde{\mathcal{P}}=\\{(H,\rho_{D})\overset{V^{\dagger}}{\longrightarrow}(H^{\prime},\rho)\\}$,
and describing the latter by the TPM scheme [see Eqs. (5) and (6)], for the
work statistics of $\mathcal{P}$ we obtain

$\displaystyle W_{ak}$ $\displaystyle=-(E_{a}-E^{\prime}_{k}),$ $\displaystyle
p_{ak}$
$\displaystyle=\operatorname{Tr}\left(P^{\prime}_{k}\rho_{D}\right)\operatorname{Tr}\left(P_{a}\frac{V^{\dagger}P^{\prime}_{k}\rho_{D}P^{\prime}_{k}V}{\operatorname{Tr}\left(P^{\prime}_{k}\rho_{D}\right)}\right)$
$\displaystyle=\operatorname{Tr}\left(P_{a}V^{\dagger}P^{\prime}_{k}\rho_{D}P^{\prime}_{k}V\right).$

Put in other words:

$\displaystyle W_{ak}=E^{\prime}_{k}-E_{a},\qquad
p_{ak}=\operatorname{Tr}(M_{ak}\rho),$ (D60)

with

$\displaystyle
M_{ak}=V^{\dagger}P^{\prime}_{k}VP_{a}V^{\dagger}P^{\prime}_{k}V.$ (D61)

Note that $M_{ak}$ depends on the initial state through $V$. We call this
measurement “backward” TPM scheme as opposed to the standard TPM scheme that
directly measures the “forward” process. Interestingly, the work distribution
produced by the backward TPM scheme coincides with that of the scheme proposed
in Ref. [7].

Having at our disposal this scheme, we can now describe any coherent-to-
coherent process by decomposing it into coherent-to-incoherent and incoherent-
to-coherent processes. More specifically, consider an arbitrary Hamiltonian
process $(H,\rho)\longrightarrow(H^{\prime},\rho^{\prime})$, with
$\rho^{\prime}=U\rho U^{\dagger}$, for which both $[\rho,H]\neq 0$ and
$[\rho^{\prime},H^{\prime}]\neq 0$, and decompose it into

$\displaystyle(H,\rho)\overset{R}{\xrightarrow{\hskip
17.07164pt}}(H,\widetilde{\rho})\overset{UR^{\dagger}}{\xrightarrow{\hskip
17.07164pt}}(H^{\prime},\rho^{\prime}),~{}$ (D62)

where $\widetilde{\rho}=R\rho R^{\dagger}$ is diagonal in $H$. Moreover, $R$
is chosen such that the diagonal of $\widetilde{\rho}$ is ordered in the same
way as the diagonal of $\rho$. More specifically, say, $\hat{\pi}$ is the
permutation matrix that reorders $\rho_{D}^{\downarrow}$ into $\rho_{D}$:
$\rho_{D}=\hat{\pi}\rho_{D}^{\downarrow}\hat{\pi}^{\mathrm{T}}$, where
$\rho_{D}$ is a diagonal matrix whose diagonal coincides with that of $\rho$
in the eigenbasis of $H$. Then, if $r^{\downarrow}$ is the vector composed of
the eigenvalues of $\rho$, organized in the decreasing order, then
$\widetilde{\rho}=\hat{\pi}r^{\downarrow}\hat{\pi}^{\mathrm{T}}$. Thus, $R$ is
the unitary operator that rotates $\rho$ into $\widetilde{\rho}$. The operator
$R$ is chosen in this way to guarantee that, when $\rho$ is diagonal, then
$R=\mathbbm{1}$.

Now, using Eqs. (D60) and (D61), for the first part of the process (D62), we
obtain

$\displaystyle W^{\textup{I}}_{ab}=E_{b}-E_{a},\qquad
p^{\textup{I}}_{ab}=\operatorname{Tr}(M^{\textup{I}}_{ab}\rho),$ (D63)

where the POVM is given by

$\displaystyle M^{\textup{I}}_{ab}=R^{\dagger}P_{b}RP_{a}R^{\dagger}P_{b}R.$
(D64)

Following the standard TPM scheme, for the second part of the process, we find

$\displaystyle W^{\textup{II}}_{ck}=E^{\prime}_{k}-E_{c},\qquad
p^{\textup{II}}_{ck}=\operatorname{Tr}(M^{\textup{II}}_{ck}\rho),$ (D65)

where

$\displaystyle
M^{\textup{II}}_{ck}=R^{\dagger}P_{c}RU^{\dagger}P_{k}^{\prime}UR^{\dagger}P_{c}R.$
(D66)

Finally, for the overall process (D62), let us write

$\displaystyle\begin{aligned}
W_{abck}&=W^{\textup{I}}_{ab}+W^{\textup{II}}_{ck},\\\
p_{abck}&=p^{\textup{I}}_{ab}p^{\textup{II}}_{ck}=\operatorname{Tr}\left(M^{\textup{I}}_{ab}\otimes
M^{\textup{II}}_{ck}\cdot\rho\otimes\rho\right),\end{aligned}$ (D67)

where $M_{abck}=M^{\textup{I}}_{ab}\otimes M^{\textup{II}}_{ck}$ is a POVM
that depends on the initial state through $R$.

From the perspective of a single copy of the system, this scheme is simply a
state-dependent POVM. Indeed, with the same outcomes $W_{abck}$, $p_{abck}$
can be written as

$\displaystyle
p_{abck}=\operatorname{Tr}\big{[}\widetilde{M}(\rho)_{abck}\,\rho\big{]},$
(D68)

where

$\displaystyle\widetilde{M}(\rho)_{abck}=p^{\textup{II}}_{ck}M^{\textup{I}}_{ab}.$
(D69)

Obviously, the operators $\widetilde{M}(\rho)_{abck}$ constitute a POVM.

By construction, this definition satisfies $\mathfrak{A}_{1}$ and, keeping in
mind that $R=\mathbbm{1}$ for initially diagonal states, it coincides with the
TPM scheme, and therefore satisfies $\mathfrak{B}$. However, for $[\rho,H]\neq
0$, this definition does not satisfy $\mathfrak{A}_{2}$. Indeed, in order to
measure work for some $[\rho,H]\neq 0$ in the trivial case of $H^{\prime}=H$
and $U=\mathbbm{1}$, the scheme will first apply a unitary to diagonalize
$\rho$ with $H$, thereby producing some nontrivial statistics according to
Eqs. (D63) and (D64); then, it will rotate that state back to $\rho$, again
producing nontrivial statistics as per Eqs. (D65) and (D66). In total, for the
trivial process, the scheme will produce rather complicated work statistics
described by Eq. (D67).

##  APPENDIX E The other operator of work

The operator $\Upsilon_{\rho}$ is similar to the operator for work proposed in
Ref. [13]. There, the work operator was introduced according to
$e^{-\beta\hat{W}}=e^{-\beta U^{\dagger}H^{\prime}U}e^{\beta H}$. However, the
work operator defined in this way will not be Hermitian whenever
$[U^{\dagger}H^{\prime}U,H]\neq 0$. Our operator $\Upsilon_{\rho}$ is
Hermitian by construction and is defined for all states.

First of all, let us show that this scheme is covered by Theorem 3. It is
self-evident that $\hat{\beta}(\rho)$, and therefore $\Upsilon_{\rho}$, are
differentiable functions of $\rho$, which means that both the measurement
operators and outcomes of this scheme are differentiable with respect to
$\rho$. Moreover, $\Upsilon_{\rho}$ satisfies Constraint 1, because, for
processes (C12), $U^{\dagger}HU=H+x\omega+O(x^{2})$, and therefore, except for
$\omega$, all the operators in the expansion of the relevant quantities with
respect to $x$ will be operator functions of $H$. Consequently, they will all
be real in the eigenbasis of $H$, for all values of $\rho$, which means that
the dependence of complexness on $\rho$ is trivial. As regards Constraint 2,
it is satisfied by construction: $\Upsilon_{\rho}$ is an operator scheme,
which means that all $M_{W}$, and thus all $m_{W}$ are eigenprojectors of a
Hermitian operator, and are thus mutually orthogonal, which in turn means that
$Y^{+}Y^{-}=0$ [see also the discussion under Eq. (C39)].

Since $\Upsilon_{\rho}$ satisfies $\mathfrak{A}_{2}$ and $\mathfrak{B}$ by
construction, the fact that Theorem 3 applies to this scheme means that it
necessarily has to violate $\mathfrak{A}_{1}$.

To explicitly show that this is indeed the case, let us consider the processes
$\Pi$, defined in Eq. (C12), and thermal initial states $\tau\propto e^{-\beta
H}$. First, we rewrite Eq. (10) as

$\displaystyle\Upsilon_{\tau}=-\beta^{-1}\ln\big{(}\tau^{-1/2}U^{\dagger}\tau
U\tau^{-1/2}\big{)}.$ (E70)

Through a simple calculation employing the Baker–Hausdorff lemma and the
Taylor expansion of $\ln(1+x)$ around $x=0$, Eq. (E70) will lead us to

$\displaystyle\begin{aligned}
-\beta\Upsilon_{\tau}=&\,ix\tau^{-1/2}[h,\tau]\tau^{-1/2}\\\
&-\frac{x^{2}}{2}\tau^{-1/2}[h,[h,\tau]]\tau^{-1/2}\\\
&+\frac{x^{2}}{2}\tau^{-1/2}[h,\tau]\tau^{-1}[h,\tau]\tau^{-1/2}+O(x^{3}).\end{aligned}$

Therefore,

$\displaystyle\beta\langle\Upsilon_{\tau}\rangle_{\tau}=-\frac{x^{2}}{2}\operatorname{Tr}(\tau^{-1}[h,\tau]^{2})+O(x^{3}).$
(E71)

Whereas, reading from Eq. (C22), we have

$\displaystyle\beta\langle\Omega\rangle_{\tau}=\frac{x^{2}}{2}\operatorname{Tr}([h,\tau]\,[h,\beta
H])+O(x^{3}).$ (E72)

Neglecting the $O(x^{3})$ terms and working in the eigenbasis of $H$, we can
rewrite Eqs. (E71) and (E72) as

$\displaystyle\beta\langle\Upsilon_{\tau}\rangle_{\tau}$
$\displaystyle=x^{2}\sum_{k<j}|h_{kj}|^{2}(p_{k}-p_{j})^{2}\frac{p_{k}+p_{j}}{2p_{k}p_{j}}$
(E73)
$\displaystyle=x^{2}\sum_{k<j}|h_{kj}|^{2}p_{j}(e^{\Delta_{kj}}-1)\sinh\Delta_{kj}$
(E74)

and

$\displaystyle\beta\langle\Omega\rangle_{\tau}$
$\displaystyle=x^{2}\sum_{k<j}|h_{kj}|^{2}(p_{k}-p_{j})\Delta_{kj}$ (E75)
$\displaystyle=x^{2}\sum_{k<j}|h_{kj}|^{2}p_{j}(e^{\Delta_{kj}}-1)\Delta_{kj},$
(E76)

where $p_{k}=e^{-\beta E_{k}}/Z$ are the eigenvalues of $\tau$, and
$\Delta_{kj}=\beta(E_{j}-E_{k})$; when transitioning from (E73) to (E74) and
from (E75) to (E76), we used that $p_{k}=p_{j}e^{\Delta_{kj}}$.

Now, whenever $[h,H]\neq 0$, there exist such $k_{0}$ and $j_{0}$ for which
$\Delta_{k_{0}j_{0}}>0$ and $h_{k_{0}j_{0}}\neq 0$. Since $\Delta_{kj}\geq 0$
for all $k<j$, Eqs. (E74) and (E76) thus imply that

$\displaystyle\langle\Upsilon_{\tau}\rangle_{\tau}>\langle\Omega\rangle_{\tau},$
(E77)

provided $x$ is small enough for the $O(x^{3})$ terms to be irrelevant. This
inequality simply means that $\Upsilon_{\tau}$ violates $\mathfrak{A}_{1}$ for
all thermal states $\tau$ as long as $[h,H]\neq 0$.

###  E.1 Two curious properties of $\Upsilon_{\rho}$

Below, we will prove two results about $\Upsilon_{\rho}$ that are not related
to the main goal of this appendix. However, we feel compelled to present them
to more completely characterize the yet another “definition of work” that
$\Upsilon_{\rho}$ represents.

######  Proposition E.1.

$\displaystyle\langle\Omega\rangle_{\tau}-\langle\Upsilon_{\tau}\rangle_{\tau}\leq\beta^{-1}S(\tau\,\|\,U^{\dagger}\tau^{\prime}U);$
(E78)

######  Proposition E.2.

$\displaystyle\mathrm{Spec}(\Upsilon_{\tau})^{\downarrow}\succ\mathrm{Spec}(\Omega)^{\downarrow},$
(E79)

for any $\beta$.

Here, $\tau^{\prime}=e^{-\beta H^{\prime}}/\operatorname{Tr}e^{-\beta
H^{\prime}}$ and $\mathrm{Spec}(O)$ is the spectrum of the operator $O$. The
superscript $\downarrow$ indicates that the set is ordered from large to
small, and $\succ$ is the majorization relation [44].

###### Proof of Proposition E.1.

Let us first observe that, as a trivial consequence of the operator-concavity
of the logarithm,

$\displaystyle\operatorname{Tr}(\rho\ln O)\leq\ln\operatorname{Tr}(\rho O),$
(E80)

for any density operator $\rho$ and Hermitian operator $O\geq 0$. Furthermore,
noticing that $e^{\beta H/2}e^{-\beta U^{\dagger}H^{\prime}U}e^{\beta H/2}>0$,
we obtain

$\displaystyle\langle\Upsilon_{\tau}\rangle_{\tau}=-\beta^{-1}\operatorname{Tr}\Big{(}\tau\ln
e^{-\beta\small{\Upsilon_{\tau}}}\Big{)}\geq\Delta F,$ (E81)

where we took into account that, by construction,

$\displaystyle\big{\langle}e^{-\beta\small{\Upsilon_{\tau}}}\big{\rangle}_{\tau}=e^{-\beta\Delta
F}=\frac{Z_{\beta}[H^{\prime}]}{Z_{\beta}[H]}.$ (E82)

Interestingly, inequality (E81) provides another aspect by which
$\Upsilon_{\tau}$ resembles work.

Finally, observing that

$\displaystyle\Delta
F=\langle\Omega\rangle_{\tau}-\beta^{-1}S(\tau\,\|\,U^{\dagger}\tau^{\prime}U),$
(E83)

we justify Eq. (E78). ∎

###### Proof of Proposition E.2.

For this proof, we invoke the following two theorems, proven in Refs. [63,
64], that hold for arbitrary Hermitian operators $A$ and $B$.

######  Theorem E.1 (The Theorem of Ref. [63]).

An arbitrary neighbourhood of $e^{A+B}$ contains an $X$ for which there exist
two unitary operators $\mathcal{U}$ and $\mathcal{V}$ such that

$\displaystyle\mathrm{Spec}(X)=\mathrm{Spec}\big{(}Y^{\frac{1}{2}}\mathcal{U}Y^{\frac{1}{2}}\mathcal{V}\big{)},$
(E84)

where

$\displaystyle Y=e^{A/2}e^{B}e^{A/2}.$ (E85)

######  Theorem E.2 (Theorem 1 of Ref. [64]).

If $X_{1}\geq X_{2}\geq\cdots\geq X_{d}\geq 0$ and $Y_{1}\geq
Y_{2}\geq\cdots\geq Y_{d}\geq 0$ are the eigenvalues of positive-semidefinite
Hermitian matrices $X$ and $Y$, then unitary operators $\mathcal{U}$ and
$\mathcal{V}$ exist such that
$\mathrm{Spec}(X)=\mathrm{Spec}\big{(}Y^{\frac{1}{2}}\mathcal{U}Y^{\frac{1}{2}}\mathcal{V}\big{)}$
if and only if

$\displaystyle X_{1}X_{2}\cdots X_{k}\leq Y_{1}Y_{2}\cdots Y_{k},$ (E86)

for $k=1,2,...,d-1$, and

$\displaystyle\det X=\det Y.$ (E87)

In our situation, $A=\beta H$ and $B=-\beta U^{\dagger}H^{\prime}U$, and thus
$e^{A+B}=e^{-\beta\Omega}$ and $Y=e^{-\beta\Upsilon_{\tau}}$. Therefore,
Theorems E.1 and E.2 imply that, given some matrix norm $\|\cdot\|$ (e.g., the
operator norm), for any $\epsilon>0$, there exists an $\Omega^{(\epsilon)}$
such that $\|\Omega-\Omega^{(\epsilon)}\|<\epsilon$ and, according to Eq.
(E86),

$\displaystyle\Omega^{(\epsilon)}_{d}+\cdots+\Omega^{(\epsilon)}_{d-k+1}\geq\Gamma_{d}+\cdots+\Gamma_{d-k+1},$
(E88)

for $k=1,2,...,d-1$, where
$\Omega^{(\epsilon)}_{1}\geq\cdots\geq\Omega^{(\epsilon)}_{d}$ are the ordered
eigenvalues of $\Omega^{(\epsilon)}$ and $\Gamma_{1}\geq\cdots\geq\Gamma_{d}$
are the ordered eigenvalues of $\Upsilon_{\tau}$. Eq. (E87), in turn, entails

$\displaystyle\sum_{k=1}^{d}\Omega^{(\epsilon)}_{k}=\operatorname{Tr}\Omega^{(\epsilon)}=\operatorname{Tr}\Upsilon_{\tau}=\sum_{k=1}^{d}\Gamma_{k}.$
(E89)

Taking the limit $\epsilon\to 0$ in Eqs. (E88) and (E89), it is
straightforward to see that
$\sum_{m=1}^{k}\Omega_{m}\leq\sum_{m=1}^{k}\Gamma_{m}$, for $k=1,...,d-1$ and
$\operatorname{Tr}\Omega=\operatorname{Tr}\Upsilon_{\tau}$, which concludes
the proof of Proposition E.2. ∎

## References

  * Bochkov and Kuzovlev [1977] G. N. Bochkov and Y. B. Kuzovlev,  _General theory of thermal fluctuations in nonlinear systems,_Sov. Phys. JETP 45, 125 (1977).
  * Jarzynski [1997] C. Jarzynski,  _Nonequilibrium equality for free energy differences,_Phys. Rev. Lett. 78, 2690 (1997).
  * [3] J. Kurchan,  _A quantum fluctuation theorem,_ arXiv:cond-mat/0007360.
  * [4] H. Tasaki,  _Jarzynski relations for quantum systems and some applications,_ arXiv:cond-mat/0009244.
  * Allahverdyan and Nieuwenhuizen [2005] A. E. Allahverdyan and T. M. Nieuwenhuizen,  _Fluctuations of work from quantum subensembles: The case against quantum work-fluctuation theorems,_Phys. Rev. E 71, 066102 (2005).
  * Esposito  _et al._ [2009] M. Esposito, U. Harbola, and S. Mukamel,  _Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,_Rev. Mod. Phys. 81, 1665 (2009).
  * Allahverdyan [2014] A. E. Allahverdyan,  _Nonequilibrium quantum fluctuations of work,_Phys. Rev. E 90, 032137 (2014).
  * Talkner and Hänggi [2016] P. Talkner and P. Hänggi,  _Aspects of quantum work,_Phys. Rev. E 93, 022131 (2016).
  * Perarnau-Llobet  _et al._ [2017] M. Perarnau-Llobet, E. Bäumer, K. V. Hovhannisyan, M. Huber, and A. Acín,  _No-go theorem for the characterization of work fluctuations in coherent quantum systems,_Phys. Rev. Lett. 118, 070601 (2017).
  * Sampaio  _et al._ [2018] R. Sampaio, S. Suomela, T. Ala-Nissila, J. Anders, and T. G. Philbin,  _Quantum work in the Bohmian framework,_Phys. Rev. A 97, 012131 (2018).
  * Bäumer  _et al._ [2018] E. Bäumer, M. Lostaglio, M. Perarnau-Llobet, and R. Sampaio, Fluctuating work in coherent quantum systems: Proposals and limitations, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018) pp. 275–300.
  * Brodier  _et al._ [2020] O. Brodier, K. Mallick, and A. M. Ozorio de Almeida,  _Semiclassical work and quantum work identities in Weyl representation,_J. Phys. A 53, 325001 (2020).
  * Yukawa [2000] S. Yukawa,  _A quantum analogue of the Jarzynski equality,_J. Phys. Soc. Jpn. 69, 2367 (2000).
  * Talkner  _et al._ [2007] P. Talkner, E. Lutz, and P. Hänggi,  _Fluctuation theorems: Work is not an observable,_Phys. Rev. E 75, 050102(R) (2007).
  * Solinas and Gasparinetti [2015] P. Solinas and S. Gasparinetti,  _Full distribution of work done on a quantum system for arbitrary initial states,_Phys. Rev. E 92, 042150 (2015).
  * Deffner  _et al._ [2016] S. Deffner, J. P. Paz, and W. H. Zurek,  _Quantum work and the thermodynamic cost of quantum measurements,_Phys. Rev. E 94, 010103(R) (2016).
  * Åberg [2018] J. Åberg,  _Fully quantum fluctuation theorems,_Phys. Rev. X 8, 011019 (2018).
  * Alhambra  _et al._ [2016] A. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry,  _Fluctuating work: From quantum thermodynamical identities to a second law equality,_Phys. Rev. X 6, 041017 (2016).
  * Miller and Anders [2017] H. J. D. Miller and J. Anders,  _Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework,_New J. Phys. 19, 062001 (2017).
  * Xu  _et al._ [2018] B.-M. Xu, J. Zou, L.-S. Guo, and X.-M. Kong,  _Effects of quantum coherence on work statistics,_Phys. Rev. A 97, 052122 (2018).
  * Gherardini  _et al._ [2021] S. Gherardini, A. Belenchia, M. Paternostro, and A. Trombettoni,  _End-point measurement approach to assess quantum coherence in energy fluctuations,_Phys. Rev. A 104, L050203 (2021).
  * Beyer  _et al._ [2020] K. Beyer, K. Luoma, and W. T. Strunz,  _Work as an external quantum observable and an operational quantum work fluctuation theorem,_Phys. Rev. Research 2, 033508 (2020).
  * [23] K. Micadei, G. T. Landi, and E. Lutz,  _Extracting Bayesian networks from multiple copies of a quantum system,_ arXiv:2103.14570.
  * Janovitch and Landi [2022] M. Janovitch and G. T. Landi,  _Quantum mean-square predictors and thermodynamics,_Phys. Rev. A 105, 022217 (2022).
  * Beyer  _et al._ [2022] K. Beyer, R. Uola, K. Luoma, and W. T. Strunz,  _Joint measurability in nonequilibrium quantum thermodynamics,_Phys. Rev. E 106, L022101 (2022).
  * Lostaglio [2018] M. Lostaglio,  _Quantum fluctuation theorems, contextuality, and work quasiprobabilities,_Phys. Rev. Lett. 120, 040602 (2018).
  * Seifert [2012] U. Seifert,  _Stochastic thermodynamics, fluctuation theorems and molecular machines,_Rep. Prog. Phys. 75, 126001 (2012).
  * Campisi  _et al._ [2011] M. Campisi, P. Hänggi, and P. Talkner,  _Colloquium: Quantum fluctuation relations: Foundations and applications,_Rev. Mod. Phys. 83, 771 (2011).
  * Landau and Lifshitz [1980] L. D. Landau and E. M. Lifshitz,  _Statistical Physics, Part I_ (Pergamon, New York, 1980).
  * Lindblad [1983] G. Lindblad,  _Non-Equilibrium Entropy and Irreversibility_ (Reidel, Dordrecht, 1983).
  * Allahverdyan  _et al._ [2013] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen,  _Understanding quantum measurement from the solution of dynamical models,_Phys. Reps. 525, 1 (2013).
  * Masanes  _et al._ [2019] L. Masanes, T. D. Galley, and M. P. Müller,  _The measurement postulates of quantum mechanics are operationally redundant,_Nat. Commun. 10, 1361 (2019).
  * [33] K. Abdelkhalek, Y. Nakata, and D. Reeb,  _Fundamental energy cost for quantum measurement,_ arXiv:1609.06981.
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum computation and quantum information_ (Cambridge University Press, Cambridge, England, 2010).
  * Jarzynski  _et al._ [2015] C. Jarzynski, H. T. Quan, and S. Rahav,  _Quantum-classical correspondence principle for work distributions,_Phys. Rev. X 5, 031038 (2015).
  * García-Mata  _et al._ [2017] I. García-Mata, A. J. Roncaglia, and D. A. Wisniacki,  _Semiclassical approach to the work distribution,_Europhys. Lett. 120, 30002 (2017).
  * Funo and Quan [2018] K. Funo and H. T. Quan,  _Path integral approach to quantum thermodynamics,_Phys. Rev. Lett. 121, 040602 (2018).
  * Fei  _et al._ [2018] Z. Fei, H. T. Quan, and F. Liu,  _Quantum corrections of work statistics in closed quantum systems,_Phys. Rev. E 98, 012132 (2018).
  * Petz [1994] D. Petz,  _A survey of certain trace inequalities,_Banach Cent. Publ. 30, 287 (1994).
  * [40] R. Pan, Z. Fei, T. Qiu, J.-N. Zhang, and H. T. Quan,  _Quantum-classical correspondence of work distributions for initial states with quantum coherence,_ arXiv:1904.05378.
  * Huber  _et al._ [2008] G. Huber, F. Schmidt-Kaler, S. Deffner, and E. Lutz,  _Employing trapped cold ions to verify the quantum Jarzynski equality,_Phys. Rev. Lett. 101, 070403 (2008).
  * Batalhão  _et al._ [2014] T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra,  _Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system,_Phys. Rev. Lett. 113, 140601 (2014).
  * An  _et al._ [2015] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. T. Quan, and K. Kim,  _Experimental test of the quantum Jarzynski equality with a trapped-ion system,_Nat. Phys. 11, 193 (2015).
  * Horn and Johnson [2013] R. A. Horn and C. R. Johnson,  _Matrix analysis_ , 2nd ed. (Cambridge University Press, New York, 2013).
  * Wu  _et al._ [2019] K.-D. Wu, Y. Yuan, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and M. Perarnau-Llobet,  _Experimentally reducing the quantum measurement back action in work distributions by a collective measurement,_Sci. Adv. 5, 4944 (2019).
  * Wu  _et al._ [2020] K.-D. Wu, E. Bäumer, J.-F. Tang, K. V. Hovhannisyan, M. Perarnau-Llobet, G.-Y. Xiang, C.-F. Li, and G.-C. Guo,  _Minimizing backaction through entangled measurements,_Phys. Rev. Lett. 125, 210401 (2020).
  * Levy and Lostaglio [2020] A. Levy and M. Lostaglio,  _Quasiprobability distribution for heat fluctuations in the quantum regime,_PRX Quantum 1, 010309 (2020).
  * Mohammady  _et al._ [2020] M. H. Mohammady, A. Auffèves, and J. Anders,  _Energetic footprints of irreversibility in the quantum regime,_Commun. Phys. 3, 89 (2020).
  * Nazarov and Kindermann [2003] Y. V. Nazarov and M. Kindermann,  _Full counting statistics of a general quantum mechanical variable,_Eur. Phys. J. B 35, 413 (2003).
  * Hofer [2017] P. P. Hofer,  _Quasi-probability distributions for observables in dynamic systems,_Quantum 1, 32 (2017).
  * Hovhannisyan and Imparato [2019] K. V. Hovhannisyan and A. Imparato,  _Quantum current in dissipative systems,_New J. Phys. 21, 052001 (2019).
  * Elouard  _et al._ [2017] C. Elouard, D. A. Herrera-Martí, M. Clusel, and A. Auffèves,  _The role of quantum measurement in stochastic thermodynamics,_npj Quantum Inf. 3, 9 (2017).
  * Manzano  _et al._ [2018] G. Manzano, J. M. Horowitz, and J. M. R. Parrondo,  _Quantum fluctuation theorems for arbitrary environments: Adiabatic and nonadiabatic entropy production,_Phys. Rev. X 8, 031037 (2018).
  * De Chiara and Imparato [2022] G. De Chiara and A. Imparato,  _Quantum fluctuation theorem for dissipative processes,_Phys. Rev. Res. 4, 023230 (2022).
  * Sagawa and Ueda [2009] T. Sagawa and M. Ueda,  _Minimal energy cost for thermodynamic information processing: Measurement and information erasure,_Phys. Rev. Lett. 102, 250602 (2009).
  * Mohammady and Romito [2019] M. H. Mohammady and A. Romito,  _Conditional work statistics of quantum measurements,_Quantum 3, 175 (2019).
  * Guryanova  _et al._ [2020] Y. Guryanova, N. Friis, and M. Huber,  _Ideal projective measurements have infinite resource costs,_Quantum 4, 222 (2020).
  * Mohammady [2023] M. H. Mohammady,  _Thermodynamically free quantum measurements,_J. Phys. A 55, 505304 (2023).
  * Ito  _et al._ [2019] K. Ito, P. Talkner, B. P. Venkatesh, and G. Watanabe,  _Generalized energy measurements and quantum work compatible with fluctuation theorems,_Phys. Rev. A 99, 032117 (2019).
  * Baker [1905] H. F. Baker,  _Alternants and continuous groups,_Proc. London Math. Soc. s2-3, 24 (1905).
  * Lance [1995] E. C. Lance,  _Hilbert C*-Modules: A Toolkit for Operator Algebraists_ (Cambridge University Press, Cambridge, 1995).
  * Jameson [1996] G. Jameson,  _Khinchin’s inequality for operators,_Glasgow Math. J. 38, 327 (1996).
  * Lenard [1971] A. Lenard,  _Generalization of the Golden-Thompson inequality $\mathrm{Tr}(e^{A}e^{B})\geqq\mathrm{Tr}e^{A+B}$, _Indiana Univ. Math. J. 21, 457 (1971).
  * Thompson [1971] C. J. Thompson,  _Inequalities and partial orders on matrix spaces,_Indiana Univ. Math. J. 21, 469 (1971).
  * Adesso and Illuminati [2007] G. Adesso and F. Illuminati,  _Entanglement in continuous-variable systems: recent advances and current perspectives,_J. Phys. A 40, 7821 (2007).
  * Williamson [1936] J. Williamson,  _On the algebraic problem concerning the normal forms of linear dynamical systems,_Am. J. Math. 58, 141 (1936).
  * Scutaru [1998] H. Scutaru,  _Fidelity for displaced squeezed thermal states and the oscillator semigroup,_J. Phys. A 31, 3659 (1998).

Supplementary Note

##  S Classical limit of the operator of work

Here, we will explore the question of whether the operator of work starts
obeying the Jarzynski equality in the classical limit of a quantum continuous-
variable system. We will study the simplest such example—the driven harmonic
oscillator.

From the Baker–Cambell–Hausdorff formula,

$\displaystyle\begin{aligned} \langle
e^{-\beta\Omega}\rangle_{\tau_{\beta}}=&\frac{1}{Z_{\beta}[H]}\operatorname{Tr}\big{(}e^{-\beta
H}e^{-\beta\Omega}\big{)}\\\
=&\frac{1}{Z_{\beta}[H]}\operatorname{Tr}e^{-\beta(H+\Omega)+\frac{1}{2}\beta^{2}[H,\Omega]+\cdots}\\\
=&\frac{1}{Z_{\beta}[H]}\operatorname{Tr}e^{-\beta
U^{\dagger}H^{\prime}U+\frac{1}{2}\beta^{2}[H,U^{\dagger}H^{\prime}U]+\cdots},\end{aligned}$
(S1)

one could naively expect that, since the commutator terms go to zero as
$\hbar\to 0$, one would have that $\langle
e^{-\beta\Omega}\rangle_{\tau_{\beta}}$ will tend to

$\displaystyle\frac{1}{Z_{\beta}[H]}\operatorname{Tr}e^{-\beta
U^{\dagger}H^{\prime}U}=e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])}.$

However, due to the presence of infinite-norm operators, the $\hbar\to 0$
limit may not necessarily be continuous in such systems. In order to see this,
let us take an oscillator with the Hamiltonian

$\displaystyle H=\frac{1}{2}\big{(}q^{2}+p^{2}\big{)},$ (S2)

that is, the mass and the frequency are equal to $1$, so that $\beta\hbar$ is
dimensionless.

Now, performing a process as a result of which the frequency of the oscillator
changes, $\beta^{2}[H,U^{\dagger}H^{\prime}U]$ will necessarily contain a term
proportional to $\beta^{2}[p^{2},q^{2}]=-2i\beta^{2}\hbar\\{p,q\\}$, where
$\\{\cdot,\cdot\\}$ is the anticommutator. Introducing the creation and
annihilation operators $a$ and $a^{\dagger}$, we will have
$\beta^{2}[p^{2},q^{2}]=2(\beta\hbar)^{2}\left[(a^{\dagger})^{2}-a^{2}\right]$.
On the one hand, if $\\{|n\rangle\\}_{n=0}^{\infty}$ are the eigenvectors of
$a^{\dagger}a$, then

$\displaystyle\langle
n+2|\beta^{2}[p^{2},q^{2}]|n\rangle=2(\beta\hbar)^{2}\sqrt{(n+1)(n+2)},$

meaning that the norm of $\beta^{2}[p^{2},q^{2}]$ is infinity for any, no
matter how small, non-zero value of $\hbar\beta$.

On the other hand, if $\beta\hbar=0$, $\|\beta^{2}[p^{2},q^{2}]\|=0$, which
means that the $\beta\hbar\to 0$ limit is not continuous for
$\beta^{2}[H,U^{\dagger}H^{\prime}U]$, which, in turn, implies that the naïve
$\beta\hbar\to 0$ limit in Eq. (S1) has to be taken with extra care. We do
this calculation for a specific illustrative example below and show that
indeed $\langle e^{-\beta\Omega}\rangle$ converges to
$e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])}$ as $\beta\hbar\to 0$.

Our goal will thus be to explicitly calculate

$\displaystyle\Omega=U^{\dagger}H^{\prime}U-H$ (S3)

and then calculate

$\displaystyle\big{\langle}e^{-\beta\Omega}\big{\rangle}_{\tau_{\beta}}=\sum_{n}\langle\psi_{n}|\tau_{\beta}|\psi_{n}\rangle\,e^{-\beta
W_{n}},$ (S4)

where $W_{n}$ are the eigenvalues of $\Omega$ and $|\psi_{n}\rangle$ are its
eigenvectors. Then, we are going to take the $\beta\hbar\to 0$ limit. We know
that, for any finite $\beta\hbar$, $\langle
e^{-\beta\Omega}\rangle_{\tau_{\beta}}$ will be larger than
$e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])}$ [5].

Given the infinite-dimensional Hilbert space, one should of course distinguish
between $\Omega$’s with spectra bounded and unbounded from below. In this
section, we consider only $\Omega$’s whose spectra are bounded from below.

Since during the driving the Hamiltonian of the oscillator remains quadratic
in $q$ and $p$, the generated unitary evolution operator $U$ is Gaussian.
Therefore, the transformation of $q$ and $p$ in the Heisenberg picture will be
symplectic [65], which means that, introducing the column vectors
$\mathbf{z}=(q,p)$ and $\mathbf{z}^{\prime}=(U^{\dagger}qU,\,U^{\dagger}pU)$,
we can write

$\displaystyle\mathbf{z}^{\prime}=S\mathbf{z},$ (S5)

where $S$ is a $2\times 2$ symplectic matrix that depends only on $U$.

Now, if

$\displaystyle H^{\prime}=\mathbf{z}^{\mathrm{T}}\Lambda\mathbf{z},$ (S6)

where $\Lambda>0$ is a $2\times 2$ real, symmetric matrix, then

$\displaystyle
U^{\dagger}H^{\prime}U=(\mathbf{z}^{\prime})^{\mathrm{T}}\Lambda\mathbf{z}^{\prime}=\mathbf{z}^{\mathrm{T}}S^{\mathrm{T}}\Lambda
S\mathbf{z},$ (S7)

and therefore, recalling Eq. (S2),

$\displaystyle\Omega=\mathbf{z}^{\mathrm{T}}\Xi\mathbf{z}$ (S8)

with

$\displaystyle\Xi=S^{\mathrm{T}}\Lambda S-\frac{1}{2}\mathbbm{1}_{2},$ (S9)

where $\mathbbm{1}_{2}$ is the $2\times 2$ identity matrix. As mentioned
above, we will consider the only the processes for which

$\displaystyle\Xi>0.$ (S10)

In which case, by the Williamson’s theorem [66, 65], there exists a symplectic
transformation, $\Sigma$, that takes $\mathbf{z}$ to new coordinates
$\tilde{\mathbf{z}}=(\tilde{q},\tilde{p})$, i.e.,

$\displaystyle\tilde{\mathbf{z}}=\Sigma\mathbf{z},$ (S11)

in which $\Xi$ is proportional to the identity. Namely, if $w>0$ is the
symplectic eigenvalue of $\Xi$, then

$\displaystyle\Omega=\tilde{\mathbf{z}}^{T}(\Sigma^{-1})^{T}\Xi\Sigma^{-1}\tilde{\mathbf{z}}=w(\tilde{q}^{2}+\tilde{p}^{2}).$
(S12)

Introducing the symplectic form $I=\Big{(}\\!\begin{array}[]{cc}0&1\\\
-1&0\end{array}\\!\Big{)}$, one can calculate $w$ by noting that the
eigenvalues of $\Xi I$ are $\pm iw$. Using that, it is easy to show that

$\displaystyle w=\det(\Xi^{1/2})=\sqrt{\det\Xi}.$ (S13)

Moreover, the symplectic transformation in Eq. (S12) is given by

$\displaystyle\Sigma=w^{-1/2}\Xi^{1/2}.$ (S14)

In order to calculate the trace in Eq. (S4), we will make use of some ready
formulas for Gaussian states, presented in Ref. [67]. The results Ref. [67]
are given for states, therefore we will introduce density operator

$\displaystyle\kappa_{\beta}=\frac{1}{Z_{W}}e^{-\beta\Omega},$ (S15)

where

$\displaystyle Z_{W}=\operatorname{Tr}e^{-\beta\Omega}=\left(e^{\beta\hbar
w}-e^{-\beta\hbar w}\right)^{-1},$ (S16)

so that

$\displaystyle\big{\langle}e^{-\beta\Omega}\big{\rangle}_{\tau_{\beta}}=Z_{W}\operatorname{Tr}\big{(}\kappa_{\beta}\tau_{\beta}\big{)}.$
(S17)

Now, since both $\Omega$ and $H$ are purely quadratic, we can write

$\displaystyle\langle q|\kappa_{\beta}|\bar{q}\rangle$
$\displaystyle=e^{-A_{W}q^{2}-A^{*}_{W}\bar{q}^{2}-2B_{W}q\bar{q}+C_{W}},$
(S18) $\displaystyle\langle q|\tau_{\beta}|\bar{q}\rangle$
$\displaystyle=e^{-A_{H}q^{2}-A^{*}_{H}\bar{q}^{2}-2B_{H}q\bar{q}+C_{H}},$
(S19)

and in Ref. [67] it is shown that

$\displaystyle\langle
q|\kappa_{\beta}\tau_{\beta}|\bar{q}\rangle=e^{-Aq^{2}-D\bar{q}^{2}-2Bq\bar{q}+C},$
(S20)

with

$\displaystyle\begin{aligned} A&=A_{W}-\frac{B_{W}^{2}}{A_{W}^{*}+A_{H}},\\\
D&=A_{H}^{*}-\frac{B_{H}^{2}}{A_{W}^{*}+A_{H}},\\\
B&=-\frac{B_{W}B_{H}}{A_{W}^{*}+A_{H}},\\\
C&=C_{W}+C_{H}+\frac{1}{2}\ln\frac{\pi}{A_{W}^{*}+A_{H}}.~{}~{}\end{aligned}$
(S21)

Along with Eq. (S21), Eq. (S20) immediately yields

$\displaystyle\operatorname{Tr}(\kappa_{\beta}\tau_{\beta})$
$\displaystyle=\int\limits_{-\infty}^{\infty}dq\langle
q|\kappa_{\beta}\tau_{\beta}|q\rangle$ $\displaystyle=\frac{\pi
e^{C_{W}+C_{H}}}{\sqrt{\phantom{\big{|}}\\!|A^{*}_{W}+A_{H}|^{2}-(B_{W}+B_{H})^{2}}}.$
(S22)

All that is left to do now is to determine how $A_{W}$, $B_{W}$, and $C_{W}$
are expressed in terms of $\Xi$. We again use formulas from Ref. [67]: If

$\displaystyle x^{W}=\left(\\!\begin{array}[]{cc}x^{W}_{qq}&x^{W}_{qp}\\\
\\!\phantom{\Big{|}}\\!x^{W}_{qp}&x^{W}_{pp}\end{array}\\!\right),$ (S25)

with the elements defined as

$\displaystyle x^{W}_{qq}=2\langle
q^{2}\rangle_{\kappa_{\beta}},\quad\;x^{W}_{pp}=2\langle
p^{2}\rangle_{\kappa_{\beta}},\quad\;x^{W}_{qp}=\langle\\{q,p\\}\rangle_{\kappa_{\beta}},$
(S26)

then

$\displaystyle\begin{aligned} A_{W}&=\frac{1+\hbar^{-2}\det
x^{W}}{4\hbar^{-1}x_{qq}^{W}}-i\frac{x_{qp}^{W}}{2x^{W}_{qq}},~{}~{}~{}\\\
B_{W}&=\frac{1-\hbar^{-2}\det x^{W}}{4\hbar^{-1}x_{qq}^{W}},\\\
C_{W}&=-\ln\sqrt{\pi\hbar^{-1}x_{qq}^{W}}.\end{aligned}$ (S27)

Obviously, identical formulas apply for $H$.

Let us now calculate the correlators in Eq. (S26). In view of Eq. (S11),

$\displaystyle\mathbf{z}=\Sigma^{-1}\tilde{\mathbf{z}}.$ (S28)

So, keeping in mind Eq. (S12), by introducing creation-annihilation operators,
and doing the simple algebra, we arrive at

$\displaystyle x^{W}=\hbar\coth(\hbar\beta
w)(\Sigma^{-1})^{\mathrm{T}}\Sigma^{-1},$ (S29)

When taking Eqs. (S13) and (S14) into account, this formula amounts to

$\displaystyle
x^{W}=\hbar\sqrt{\det\Xi}\coth\Big{(}\hbar\beta\sqrt{\det\Xi}\Big{)}\Xi^{-1}.$
(S30)

Analogously, reading the $x$ matrix for $H$ off Eq. (S30), we get

$\displaystyle
x^{H}=\hbar\coth\Big{(}\frac{\beta\hbar}{2}\Big{)}\mathbbm{1}_{2}.$ (S31)

It is easy to check that, in the $\beta\hbar\to 0$ limit,

$\displaystyle
x^{W}=\frac{\beta^{-1}}{\det\Xi}\bigg{(}\\!\begin{array}[]{cc}\Xi_{22}&-\Xi_{12}\\\
-\Xi_{12}&\Xi_{11}\end{array}\\!\bigg{)}+O(\beta\hbar),$ (S34)

where one should keep in mind that $\Xi$ is a symmetric matrix (because
$\Lambda$ is). Similarly,

$\displaystyle x^{H}=2\beta^{-1}\mathbbm{1}_{2}+O(\beta\hbar).$ (S35)

Using these in Eq. (S27) and substituting the obtained quantities into Eq.
(S22), through a somewhat tedious algebra, we find that

$\displaystyle\operatorname{Tr}(\kappa_{\beta}\tau_{\beta})=\beta\hbar\sqrt{\frac{\det\Xi}{\det(\Xi+\mathbbm{1}_{2}/2)}}+O[(\beta\hbar)^{2}].$
(S36)

In order to finally calculate
$\big{\langle}e^{-\beta\Omega}\big{\rangle}_{\tau_{\beta}}$, let us first
notice that, due to Eq. (S9), $\det(\Xi+\mathbbm{1}_{2}/2)=\det\Lambda$. Next,
from Eq. (S16), and keeping in mind Eq. (S13), we see that

$\displaystyle Z_{W}=\frac{(\beta\hbar)^{-1}}{2\sqrt{\det\Xi}}+O(1).$ (S37)

Substituting all these into Eq. (S17), we finally obtain

$\displaystyle\big{\langle}e^{-\beta\Omega}\big{\rangle}_{\tau_{\beta}}=\frac{1}{2\sqrt{\det\Lambda}}+O(\beta\hbar).$
(S38)

Lastly, calculating $Z_{\beta}[H^{\prime}]$ and $Z_{\beta}[H]$ in the same way
as we calculated $Z_{W}$, we find that

$\displaystyle
e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])}=\frac{Z_{\beta}[H^{\prime}]}{Z_{\beta}[H]}=\frac{1}{2\sqrt{\det\Lambda}}+O(\beta\hbar),$

which means that

$\displaystyle\big{\langle}e^{-\beta\Omega}\big{\rangle}_{\tau_{\beta}}=e^{-\beta(F_{\beta}[H^{\prime}]-F_{\beta}[H])}+O(\beta\hbar).$
(S39)

This expression shows that the HOW scheme for continuous-variable systems is
consistent in the classical limit, at least in the somewhat special case of
$\Omega>0$. The case of $\Omega<0$ remains as an interesting open problem.