# Ideal Projective Measurements Have Infinite Resource Costs

Yelena Guryanova<EMAIL_ADDRESS>Institute for Quantum Optics and
Quantum Information – IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Nicolai Friis
<EMAIL_ADDRESS>Institute for Quantum Optics and Quantum
Information – IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Marcus Huber
<EMAIL_ADDRESS>Institute for Quantum Optics and Quantum Information
– IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria

(December 18, 2019)

###### Abstract

We show that it is impossible to perform ideal projective measurements on
quantum systems using finite resources. We identify three fundamental features
of ideal projective measurements and show that when limited by finite
resources only one of these features can be salvaged. Our framework is general
enough to accommodate any system and measuring device (pointer) models, but
for illustration we use an explicit model of an $N$-particle pointer. For a
pointer that perfectly reproduces the statistics of the system, we provide
tight analytic expressions for the energy cost of performing the measurement.
This cost may be broken down into two parts. First, the cost of preparing the
pointer in a suitable state, and second, the cost of a global interaction
between the system and pointer in order to correlate them. Our results show
that, even under the assumption that the interaction can be controlled
perfectly, achieving perfect correlation is infinitely expensive. We provide
protocols for achieving optimal correlation given finite resources for the
most general system and pointer Hamiltonians, phrasing our results as
fundamental bounds in terms of the dimensions of these systems.

##  1 Introduction

The foundations of any physical theory are laid by its axioms, postulates and
laws. In quantum theory, the projection postulate presents one of these
central pillars. It says that upon measuring a quantum system, its post-
measurement state is given by an eigenstate of the measured observable and the
corresponding probability for obtaining this state is given by the Born rule.
In this way, an ideal projective measurement leaves the system in a pure state
that is perfectly correlated with the measurement outcome.

Similarly, the key tenets of thermodynamics are formed by its three
fundamental laws. Intense efforts in quantum thermodynamics [1, 2, 3] have
placed these laws on rigorous mathematical footing [4, 5, 6, 7, 8, 9, 10, 11,
12, 13]. Of particular interest is the third law of thermodynamics in the
quantum regime, which tells us that no quantum system can be cooled to the
ground state (which, in non-degenerate cases, is a pure state) in finite time
and with finite resources [14, 15, 12, 16, 17, 11]. This is in apparent
contradiction with the projection postulate [18] — how is it that an ideal,
error-free, measurement leaves the system in a state forbidden by the laws of
thermodynamics?

In reality, we know that measurements in the lab are performed in finite time
and with finite resources. These measurements are prone to small errors,
implying that the post-measurement state of the system is never truly pure.
However, with technological advances making errors ever smaller, one would
assume rising thermodynamic costs as the post-measurement state of the system
approaches purity.

Here, we resolve this apparent contradiction. We show that the resource cost
of an ideal quantum measurement in a finite temperature environment is indeed
infinite. Our operational approach is based on correlations between a system
and a pointer, allowing us to make quantifiable statements about the cost.
Within this framework we identify that an ideal projective measurement has
three model-independent properties; it is: unbiased, faithful, and non-
invasive — properties that cannot hold simultaneously for measurements with
finite resources (energy and time). Our framework is general enough to
accommodate any measurement model for which we provide quantitative results
for an example case. In doing so, we refrain from making statements about what
is commonly perceived as the ‘measurement problem’ (how or why the system is
left in a particular state and what it means to obtain a ‘result’ [19, 20]).

Past approaches to quantifying the cost of a quantum measurement have
typically assumed that projective measurements can be carried out perfectly
and that their cost can be attributed to the work value of the measurement
outcome [21, 5, 22, 23, 24]. Others adopt the stance that Landauer’s erasure
bound represents the cost of resetting devices to pure states [4, 25, 26],
without providing conclusive evidence that the bound is achievable. These
works assume an unlimited supply of pure states, circumventing the third law
of thermodynamics and resulting in finite energy costs. However, when limited
to thermal environments, measurements produce errors, which can be mitigated
by either reducing the temperature of the environment, or by using larger
measuring devices. Both strategies can be quantified in terms of their
thermodynamic cost, for which we provide exact analytic results. Our results
demonstrate that even the simplest quantum measurements on qubits are never
for free.

##  2 Ideal measurements

Consider an unknown quantum system $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ and a
measuring device (pointer) $\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$. To measure
the system, one must couple it to the pointer and effect a joint
transformation that correlates them:
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}\longrightarrow\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$.
In an _ideal_ measurement, the system and pointer become perfectly correlated,
such that upon “observing” the pointer, one infers which pure state the system
is in with probability $1$. More precisely, each eigenstate
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ of
the measured observable of the system is assigned a set
$\\{\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}\\}_{n}$
of orthogonal states of the pointer corresponding to a projector
$\Pi_{i}=\sum_{n}\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right|$.
The projectors are orthogonal, forming a complete set,
$\Pi_{i}\Pi_{j}=\delta_{ij}\Pi_{i}$ and
$\sum_{i}\Pi_{i}=\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$. Upon finding the
pointer in a state
$\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$
(chosen to reflect
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$),
one concludes that the measurement outcome is “$i$”, and that the system is
left in the state
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$. Up
to arbitrary off-diagonal elements w.r.t. the basis
$\\{\left|\right.\\!{i}\\!\left.\right\rangle\otimes\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(j)$}}}}\\!\left.\right\rangle\\}_{i,j,n}$,
the ideal post-interaction state with perfect correlation has the form

$\displaystyle\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$
$\displaystyle=\,\sum_{i}\rho_{ii}\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\rho^{\raisebox{0.0pt}{\tiny{$(i)$}}}+\,\text{off-
diag.}\,,$ (1)

where
$\rho_{ii}=\left\langle\right.\\!{i}\\!\left.\right|\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\left|\right.\\!{i}\\!\left.\right\rangle$
are diagonal elements w.r.t. the basis
$\\{\left|\right.\\!{i}\\!\left.\right\rangle\\}_{\raisebox{0.0pt}{\tiny{$S$}}}$
and $\rho^{\raisebox{0.0pt}{\tiny{$(i)$}}}$ is a pointer state, associated to
one and only one of the outcomes $i$, i.e.,
$\Pi_{i}\rho^{\raisebox{0.0pt}{\tiny{$(j)$}}}=\delta_{ij}\rho^{\raisebox{0.0pt}{\tiny{$(j)$}}}$.
The form of $\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ in Eq. (1) is the
result of an ideal measurement and can be entangled or simply classically
correlated. This ideal measurement satisfies three fundamental properties:

  1. (i)

Unbiased. The probability of finding the pointer in a state associated with
outcome $i$ after the interaction is the same as the probability of finding
the system in the state
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$
before the interaction,

$\displaystyle\mathrm{tr}\bigl{[}\mathbb{I}\otimes\,\Pi_{i}\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,\mathrm{tr}\bigl{[}\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\rho_{\raisebox{0.0pt}{\tiny{$S$}}}]\,=\,\rho_{ii}\quad\forall
i\ \forall\rho_{\raisebox{0.0pt}{\tiny{$S$}}}.$ (2)

A measurement is unbiased if the pointer reproduces the measurement statistics
of the system.

  2. (ii)

Faithful. There is a one-to-one correspondence between the pointer outcome and
the post-measurement system state

$\displaystyle C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
$\displaystyle:=\sum_{i}\mathrm{tr}\bigl{[}\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\Pi_{i}\;\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,1\quad\forall\rho_{\raisebox{0.0pt}{\tiny{$S$}}},$
(3)

i.e., $\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ has perfect correlation:
on observing the pointer outcome $i$ (associated to $\Pi_{i}$), one concludes
that the system is left in the state
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$
with certainty.

  3. (iii)

Non-invasive. The probability of finding the system in the state
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ is
the same before and after the interaction with the pointer,

$\displaystyle\mathrm{tr}\bigl{[}\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,\mathrm{tr}\bigl{[}\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\bigr{]}\,=\,\rho_{ii}\
\ \forall\;i\ \forall\rho_{\raisebox{0.0pt}{\tiny{$S$}}}.$ (4)

This property only holds for the basis
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
coherences appearing on the off-diagonal can, in general, be destroyed.

These three properties, stated here without particular hierarchy, capture the
pairwise relation between (i) the pre-measurement system state and the
measurement outcome, (3) between the measurement outcome and the post-
measurement system state, and (4) between the pre- and post-measurement system
states, respectively.

All quantitative statements we make about the faithfulness of a measurement
[property (3)] depend on the function
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ in Eq. (3). This function’s
value represents the average probability of correctly inferring the post-
measurement state upon observing the pointer, which is $1$ for any unbiased
measurement if and only if the post-interaction state is of the form of
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ in Eq. (1). One could choose
more complicated functions or even measures of correlation. However, in our
paradigm it is sufficient to be classically correlated to have perfect
‘correlation’ in the sense that $C=1$. Note that quantum correlations are not
strictly necessary, since
$C(\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}})=1$.
The advantage of the expression in Eq. (3) is that it quantifies the
probability that the pointer indicates an outcome which is correct and yields
the maximal value $1$ if and only if the post-interaction state is of the form
of Eq. (1).

Example. Consider a measurement of a qubit system using a single qubit pointer
in the ground state. We model the measurement with a controlled NOT operation
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}+\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes
X_{\raisebox{0.0pt}{\tiny{$P$}}}$, where
$X=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|+\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|$.
The post-measurement state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}\;(\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}})\;U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}^{\dagger}$
is of the form of Eq. (1), meaning the measurement is unbiased, faithful, and
non-invasive. Indeed, whenever both system and pointer have dimension
$d_{\raisebox{0.0pt}{\tiny{$S$}}}=d_{\raisebox{0.0pt}{\tiny{$P$}}}=d$, and the
pointer is initially in a pure state (w.l.o.g. the ground state), we can
define a unitary
$U_{d}:=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}+\sum_{i\neq
0}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes
X^{\raisebox{0.0pt}{\tiny{$(i)$}}}_{\raisebox{0.0pt}{\tiny{$P$}}}$, where
$X^{\raisebox{0.0pt}{\tiny{$(i)$}}}_{\raisebox{0.0pt}{\tiny{$P$}}}=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|+\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|+\sum_{j\neq
0,i}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{j}\\!\left.\right|$,
realizing an ideal measurement. A more detailed discussion of this example is
presented in Appendix A.II.

##  3 Non-ideal measurements

We call a measurement in which any of the properties (i) – (iii) fails to hold
non-ideal. This is due to the fact that, in general, the properties do not
imply one another, i.e., satisfying a single property does not imply any of
the other two, as illustrated in Fig. 1. Things become more subtle when a pair
of properties is satisfied. In two cases, satisfying a pair of properties
implies the third. As we show in detail in Appendix A.III, a measurement which
is faithful and unbiased is also non-invasive, and a measurement that is
faithful and non-invasive is unbiased. However, in general, a measurement
being unbiased and non-invasive for a particular input state does not imply it
is faithful, unless it is unbiased and non-invasive _for all_ input states
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$, as illustrated in Fig. 1.

In what follows we prove that faithful measurements (perfect correlations) are
possible if and only if one can prepare states with sufficiently many
vanishing eigenvalues. Since, by the third law of thermodynamics, one cannot
prepare states of non-full rank with finite resources, property (ii) fails to
hold and therefore ideal measurements are not physically feasible. To see
this, consider the most general interaction between a system and pointer — a
completely positive and trace-preserving (CPTP) map, which can be understood
as a unitary on the system and an extended pointer. In order for such a
unitary to realize a faithful measurement according to Eq. (3), the rank of
the final state $\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ must be bounded
from above by the dimension of the pointer $d_{\raisebox{0.0pt}{\tiny{$P$}}}$
(with $d_{\raisebox{0.0pt}{\tiny{$P$}}}\geq
d_{\raisebox{0.0pt}{\tiny{$S$}}}$), since
$d_{\raisebox{0.0pt}{\tiny{$S$}}}\leq\sum_{i}\operatorname{rank}(\rho^{\raisebox{0.0pt}{\tiny{$(i)$}}})\leq
d_{\raisebox{0.0pt}{\tiny{$P$}}}$. When
$d_{\raisebox{0.0pt}{\tiny{$S$}}}=d_{\raisebox{0.0pt}{\tiny{$P$}}}=d$, this
implies that the initial rank of the pointer
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$ must be $1$, i.e., a pure state. For
larger pointers, their initial state need not be pure, but it cannot have full
rank — one must have
$\operatorname{rank}(\rho_{\raisebox{0.0pt}{\tiny{$P$}}})\leq
d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$.
Practically, this requires pure state preparation for some non-trivial pointer
subspaces. Thus, faithful and therefore ideal measurements are not possible
without a supply of pure states (states at absolute zero temperature). States
with non-full rank require infinite time, energy or complexity (interaction
range) to be prepared and are prohibited by the third law of thermodynamics
[14, 15, 12, 16, 17, 11]. Conversely, whenever the pointer state does not
start with full rank, operations such as $U_{d}$ allow one to achieve perfect
correlation.

Since faithful (and hence also ideal) measurements are not possible, we want
to determine how closely they can be approximated. Since laboratory
experiments take place at non-zero temperature, the natural state of a pointer
is in thermodynamic equilibrium with its environment, i.e., the state
$\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)$, with inverse temperature
$\beta=1/k_{\mathrm{B}}T$. At any finite temperature, a thermal state has full
rank, and any deviation from it requires an input of work.

Figure 1: The properties attributed to an ideal measurement. In a non-ideal
measurement these three properties do not hold simultaneously, and satisfying
one of them does not imply any of the other two. When
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ is fixed, in two out of three cases,
satisfying a pair of properties implies that the third property also holds. A
measurement which is faithful and unbiased implies that it is also non-
invasive, and a measurement that is faithful and non-invasive implies that it
is unbiased. When $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ is relaxed to all
initial system states, then satisfying any pair of properties implies the
third, and one recovers the last relation, namely that a measurement which is
unbiased and non-invasive (for all $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$)
implies that it is also faithful.

While faithfulness is not necessarily the most important property, it is the
one that certainly cannot be upheld in practice, whereas one of the other two
properties can in principle be maintained also for practical measurements.
Here, we take the point of view that the crucial property to demand of _any_
measurement is to be _unbiased_. This guarantees that with sufficient
repetitions, one obtains a mean value for the measured observable that
accurately reflects the mean value of the underlying system
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ and the degree of trust in this outcome
can be quantified using standard statistical methods. Conversely, if one
imposed that the measurement were non-invasive, one would be able to perform
repeated measurements on the system without changing the statistics of the
measured observable. However, without properties (i) or (ii) it would not be
possible to reliably relate the measurement data to statements about
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$. We therefore consider non-ideal
measurements between a system $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ and a
thermal pointer $\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)$ with the property
that the measurements are unbiased. In Appendix A.IV, we derive the general
structure of maps realizing unbiased measurements in full detail.

##  4 Maximally faithful measurements

After imposing the measurements to be unbiased, we are then interested in the
ones that produce the best correlations, in other words those which are as
close to faithful as possible and thus approximate ideal measurements in a
meaningful way. For such unbiased maximally correlating measurements, one may
then determine and minimize the finite energy cost. To provide a self-
contained description of this cost, we consider the joint system of $S$ and
$P$ to be closed, implying that we restrict to unitary maps. In Appendix A.V
we further discuss this restriction and show that such unitarily correlating
unbiased measurements are indeed possible at any temperature. Moreover, we
find that unitarily correlating unbiased measurements have a maximal
achievable correlation $C_{\text{max}}$ that can only be reached if sufficient
energy is supplied. To see this, let us denote the system and pointer
Hamiltonians by $H_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
$H_{\raisebox{0.0pt}{\tiny{$P$}}}$ and the corresponding dimensions by
$d_{\raisebox{0.0pt}{\tiny{$S$}}}$ and $d_{\raisebox{0.0pt}{\tiny{$P$}}}$. For
unbiased measurements, the restriction to unitary maps
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tau_{\raisebox{0.0pt}{\tiny{$P$}}}\rightarrow\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$
implies a truncation of $d_{\raisebox{0.0pt}{\tiny{$P$}}}$ at an integer
multiple of $d_{\raisebox{0.0pt}{\tiny{$S$}}}$. We then order the spectrum of
the pointer Hamiltonian and divide it into $d_{\raisebox{0.0pt}{\tiny{$S$}}}$
disjoint sets
$\\{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\\}_{i=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}$
for $k=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$S$}}}-1$, such that
$H_{\raisebox{0.0pt}{\tiny{$P$}}}=\sum_{k=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\left|\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right|$
and $E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\geq
E^{\raisebox{0.0pt}{\tiny{$(k^{\prime})$}}}_{j}$ whenever $k\geq k^{\prime}$
or when $k=k^{\prime}$ and $i\geq j$. Up to swaps between degenerate energies,
the set $\\{E^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{i}\\}_{i}$ contains the
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$ smallest
energies and consequently $\\{\tfrac{1}{\mathcal{Z}}e^{-\beta
E^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{i}\\}}\\}_{i}$ are the largest
populations. These populations are assigned to the ‘correlated subspace’. For
unbiased unitaries, there is an algebraic maximum to the achievable
correlations between any system and thermal pointer, given by

$\displaystyle
C_{\mathrm{max}}(\beta)=\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}e^{-\beta
E^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{i}}/\mathcal{Z}\,,$ (5)

which is independent of $\rho_{ii}$ precisely due to unbiasedness. A more
detailed derivation can be found in Appendix A.VII.

Note that, because we assigned the largest populations to the ‘correlated
subspace’, $C_{\mathrm{max}}$ can be interpreted as the maximum probability of
the post-measurement system being in the same state as the pointer. For an
arbitrary unbiased measurement generally
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})\leq C_{\mathrm{max}}(\beta)$.
For an arbitrary unbiased measurement achieving $C_{\mathrm{max}}$ one can
select a pointer basis
$\\{\left|\right.\\!{\tilde{\psi}^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right\rangle\\}_{i,k}$
such that the resulting state can be written

$\displaystyle\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\sum_{k=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\frac{\rho_{kk}}{\mathcal{Z}}\Bigl{(}\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}e^{-\beta
E^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{i}}\left|\right.\\!{k}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{k}\\!\left.\right|\otimes\left|\right.\\!{\tilde{\psi}_{i}^{\raisebox{0.0pt}{\tiny{$(k)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\tilde{\psi}_{i}^{\raisebox{0.0pt}{\tiny{$(k)$}}}}\\!\left.\right|$
$\displaystyle+\\!\sum_{m\neq
k}\\!\\!\\!\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\\!\\!\\!\\!e^{-\beta\tilde{E}_{i,m}}\,U^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{\mathrm{nc}}\left|\right.\\!{m}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{m}\\!\left.\right|\otimes\left|\right.\\!{\tilde{\psi}_{i}^{\raisebox{0.0pt}{\tiny{$(k)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\tilde{\psi}_{i}^{\raisebox{0.0pt}{\tiny{$(k)$}}}}\\!\left.\right|U^{\raisebox{0.0pt}{\tiny{$(k)\dagger$}}}_{\mathrm{nc}}\Bigr{)},$
(6)

where the $U^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{\mathrm{nc}}$ for
$k=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$S$}}}$ are unitaries on the non-
correlated subspaces spanned by the vectors
$\left|\right.\\!{m}\\!\left.\right\rangle\left|\right.\\!{\tilde{\psi}_{i}^{\raisebox{0.0pt}{\tiny{$(k)$}}}}\\!\left.\right\rangle$
for
$i=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1$
and $m\neq k$, $\\{\tilde{E}_{i,m\neq
k}\\}_{i,m}=\\{E^{\raisebox{0.0pt}{\tiny{$(n>0)$}}}_{i}\\}_{n,i}$. From this
form, we see that perfect correlation $C=1$ is only possible if the pointer
temperature reaches absolute zero, or, more generally, if the rank of the
pointer is bounded by $\text{rank}(\rho_{\raisebox{0.0pt}{\tiny{$P$}}})\leq
d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$. Note that
the way the system state is altered through measurement is not completely
fixed by Eq. (6). The relation between the bases
$\\{\tilde{\psi}_{i}^{\raisebox{0.0pt}{\tiny{$(k)$}}}\\}_{i,k}$ and
$\\{\left|\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right\rangle\\}_{i,k}$,
as well as the choices of $U^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{\mathrm{nc}}$
and the ordering of the energies $\tilde{E}_{i,m}$ leave room for adjusting
the final energy cost.

The exact form of the unbiased and maximally (but not perfectly) correlated
lowest energy states depends both on $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
requires diagonalization of $H_{\raisebox{0.0pt}{\tiny{$P$}}}$. We now present
a solution for an example case and refer to Appendix A.IX for a detailed step-
by-step instruction.

##  5 Energy cost of measurements

We now investigate the relation between the energy cost $\Delta E$ of an
unbiased measurement achieving the maximum correlation $C_{\mathrm{max}}$
between a system and the pointer. Here, we wish to showcase different ways of
increasing $C_{\mathrm{max}}$, which depends on temperature and dimension. It
is readily seen from Eq. (5) that $C_{\mathrm{max}}$ increases when the
pointer size is increased at fixed temperature or when the initial temperature
is lowered at fixed pointer size. While Eq. (6) provides a general form for
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$, quantitative insight about
$\Delta E$ cannot be gained without fixing the pointer Hamiltonian. An
exception is when the pointer dimension is infinite. There, the third law of
thermodynamics can be circumvented by using (a part of) the pointer as a
‘fridge’ and creating asymptotically pure subspaces (see, e.g., [27]), a
scenario which we include as a limiting case in our analysis. For the general
case we refer to Appendix A.IX, but here, as a concrete example (described in
more detail in Appendix A.VIII), we will consider a single-qubit system and a
pointer consisting of $N$ initially non-interacting qubits with identical
Hamiltonians $H_{\raisebox{0.0pt}{\tiny{$P$}}}$.

We have
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\,U_{\text{corr}}\,(\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)^{\otimes
N})\,U_{\text{corr}}^{\dagger}$, where from now on we take
$\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)={1}/{\mathcal{Z}_{\raisebox{0.0pt}{\tiny{$P$}}}}(\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|+e^{-\beta
E_{\raisebox{0.0pt}{\tiny{$P$}}}}\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|)$
with the partition function
$\mathcal{Z}_{\raisebox{0.0pt}{\tiny{$P$}}}=\mathrm{tr}[e^{-\beta
H_{\raisebox{0.0pt}{\tiny{$P$}}}}]$. Since we would like to increase
$C_{\mathrm{max}}$ as much as possible and we know that $C_{\mathrm{max}}$
depends on the initial temperature of the pointer, we also consider cooling
the pointer prior to the correlating interaction in order to get closer to a
faithful result.

In principle, there exist many models of refrigeration, e.g., [16]. To achieve
ground state cooling, however, some form of resource has to diverge. For
instance, infinite time is required in adiabatic Landauer erasure [1],
infinite energy in finite size fridges [16], or infinite complexity (or time)
in fridges of infinite size [27]. In short, quantum measurements inherit the
limitations imposed by the third law of thermodynamics [12, 15]. Since our
example also aims to quantify the energy cost of correlations with increasing
pointer size $N$, including fridges of arbitrary size may compromise
statements about correlations costs at fixed $N$. We therefore consider a
fridge model of the same type and size as the pointer, i.e., for each pointer
qubit we add one fridge qubit, see Fig. 2. For larger refrigeration systems
the cost of cooling could be decreased by a constant factor (see [16]), but
would still diverge as one approaches zero temperature unless the fridge size
is itself infinite.

Figure 2: The measurement procedure. In step ${\mathrm{I}}$ an $N$-qubit
pointer is coupled to an $N$-qubit fridge and cooled from $\beta$ to
$\beta_{0}$. In step ${\mathrm{II}}$, a unitary correlates the pointer with
the unknown qubit system.

Within this framework, we describe the measurement by two consecutive unitary
operations, which we call cooling and correlating. The total transformation on
the system, pointer, and fridge is
$U_{\text{tot}}\,(\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)^{\otimes
N}\otimes\tau_{\raisebox{0.0pt}{\tiny{$F$}}}(\beta)^{\otimes
N})\,U_{\text{tot}}^{\dagger}$, where
$\tau_{\raisebox{0.0pt}{\tiny{$F$}}}(\beta)={1}/{\mathcal{Z}_{\raisebox{0.0pt}{\tiny{$F$}}}}(\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|+e^{-\beta
E_{\raisebox{0.0pt}{\tiny{$F$}}}}\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|)$
and
$U_{\text{tot}}=(U_{corr}\otimes\mathbb{I}_{\raisebox{0.0pt}{\tiny{$F$}}})\cdot(\mathbb{I}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes
U_{\text{cool}})$, see Fig. 2. Both unitaries drive the respective systems out
of equilibrium and come at a thermodynamic cost. Neglecting the price for
perfect control over these operations, the work cost of implementing them is
lower-bounded by the total energy change of the system, pointer, and fridge,
$W\geq\Delta E$. The total cost in energy can be split into the sum of the two
parts: cooling and correlating, which we write $\Delta E=\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}+\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$. Our objective is to minimise
$\Delta E$ when performing a non-ideal measurement for a fixed value of the
correlation function
$C(\tilde{\rho}_{{}_{\raisebox{0.0pt}{\tiny{$SP$}}}})=C_{\text{max}}<1$.

##  6 Minimal energy cost

To minimise $\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$ we use Ref.
[16], which details the optimal cost for the single-qubit fridge. Cooling the
pointer from $T=1/\beta$ to $E_{\raisebox{0.0pt}{\tiny{$P$}}}/(\beta
E_{\raisebox{0.0pt}{\tiny{$F$}}})$ such that
$\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)^{\otimes
N}\mapsto\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta\tfrac{E_{\raisebox{0.0pt}{\tiny{$F$}}}}{E_{\raisebox{0.0pt}{\tiny{$P$}}}})^{\otimes
N}$ requires at least

$\displaystyle\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$
$\displaystyle=N(E_{\raisebox{0.0pt}{\tiny{$F$}}}-1)\left(\frac{1}{e^{-\beta
E_{\raisebox{0.0pt}{\tiny{$F$}}}}+1}-\frac{1}{e^{-\beta
E_{\raisebox{0.0pt}{\tiny{$P$}}}}+1}\right)\,.$ (7)

To minimise $\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ we are
interested in determining $\min_{U_{\text{corr}}}\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ such that
$C(\tilde{\rho}_{{}_{\raisebox{0.0pt}{\tiny{$SP$}}}})=C_{\text{max}}(\beta)$.
For the case of a single-qubit system and $N$-qubit pointer (with $N$ odd), we
have

$\displaystyle
C_{\text{max}}(\beta)=\;\frac{1}{\mathcal{Z}^{N}}\sum\limits_{k=0}^{N/2}{{N}\choose{k}}e^{-kE_{\raisebox{0.0pt}{\tiny{$P$}}}\beta}\,.$
(8)

For even $N$, the formula is slightly different with the same qualitative
behaviour. As expected, in the limit of infinite pointer size
($N\rightarrow\infty$) for fixed $\beta$, or in the limit of zero temperature
($\beta\rightarrow\infty$) for any $N$, the correlations become perfect,
$\lim_{N\rightarrow\infty}C_{\text{max}}(\beta)=\lim_{\beta\rightarrow\infty}C_{\text{max}}(\beta)=1$.
In Appendix A.VIII, we construct the optimal unitary $U_{\text{opt}}$ that
solves the optimisation problem for $\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ for arbitrary $N$ and $\beta$,
i.e., the unitary that achieves the algebraic maximum correlation for minimal
energy cost. In particular, this construction allows us to specify an analytic
expression for $\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ in terms of
$\beta$, $N$, and $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$, which implies an
achievable lower bound on the minimal energy cost of non-ideal measurements
that approximate ideal ones as well as possible.

Note that in the limit $N\rightarrow\infty$ the energy cost of achieving
$C_{\text{max}}(\beta)$ is finite but infinite time (or full control over
$N$-body interactions with $N\rightarrow\infty$) is required (see Appendix
A.VIII). For any finite $N$, the only way to achieve correlations higher than
$C_{\text{max}}(\beta)$ is to cool the pointer. Thus, we consider the scenario
where starting at some finite $\beta$, we cool the pointer
($\beta\rightarrow\beta_{0}>\beta$) and then correlate it with the system to
the algebraic maximum for the new temperature $C_{\text{max}}(\beta_{0})$.
Results for exemplary temperatures are shown in Fig. 3 for $N=6$. Within our
cooling paradigm, the energy cost for reaching the ground state in finite time
is infinite. Other paradigms allow cooling to the ground state using finite
energy, but require infinite time [27]. Thus, without access to pure states, a
measurement satisfying properties (i) – (iii) has an infinite resource cost.
The cost for the maximally correlating unitary $U_{\text{corr}}$ is always
finite and given by $\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\,(C=1)}=\tfrac{1}{2}E_{\raisebox{0.0pt}{\tiny{$P$}}}$.

Figure 3: Cost of a non-ideal projective measurement of a qubit system using a
$6$-qubit pointer. Each pointer qubit is in the state
$\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)$. We start from room temperature
($\approx 300$ K) and choose an energy gap in the microwave regime such that
$\beta E_{\raisebox{0.0pt}{\tiny{$P$}}}=1/30$. Each point on the horizontal
indicates the maximal algebraic correlation
$C_{\text{max}}(\beta^{\prime},N=6)$ achievable for a fixed value
$\beta^{\prime}=\frac{\beta
E_{\raisebox{0.0pt}{\tiny{$F$}}}}{E_{\raisebox{0.0pt}{\tiny{$P$}}}}$ (or
equivalently, fixed $E_{\raisebox{0.0pt}{\tiny{$F$}}}$), which is the result
of cooling the $6$-qubit pointer from $\beta$ to $\beta^{\prime}$ using
refrigeration qubits with gaps $E_{\raisebox{0.0pt}{\tiny{$F$}}}$. For each
correlation value, the refrigeration cost $\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$ and the cost $\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ of maximally correlating the
thermal state at inverse temperature $\beta^{\prime}$ are shown. The inset
shows the relevant energy scale for correlating the system and pointer since
the cooling cost significantly dominates the correlating cost.

##  7 Discussion

The projection postulate is a central concept within the foundations of
quantum mechanics, asserting that ideal projective measurements leave the
system in a pure state corresponding to the observed outcome. All
interpretations of quantum mechanics must be compatible with this statement
together with the Born rule assigning the probabilities. However, the
existence of such ‘true’ projections is usually taken for granted. Here, we
have discussed a self-contained description of measurements from a
thermodynamic point of view. We have shown that, when their existence is not
assumed, ideal finite-time projective measurements have an infinite cost.

We argued that a necessary condition for ideal measurements is to be faithful,
i.e., have perfect correlation between the system and pointer. However, this
requirement incurs infinite costs unless pure states are freely available.
Nonetheless, ideal measurements can be approximated by non-ideal, unbiased
ones to arbitrary precision at finite energy cost. We find that the
correlation achieved by the best unbiased measurement is universally bounded
by the largest
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$
eigenvalues of the pointer. To gain quantitative insight into this cost, we
considered the measurement of a single qubit by an $N$-qubit pointer. We
provided analytic expressions for the minimal energy cost for unitarily
achieving maximal correlation for any initial temperature and any $N$. We find
that correlations can be increased by increasing the pointer size and by
cooling the pointer.

While the three mentioned properties capture the basic features of ideal
measurements, they are not sufficient to characterise the ‘quantum to
classical’ transition. Classical outcomes additionally feature ‘robustness’,
where small perturbations of the pointer do not significantly alter the
observed outcomes. This is an important consideration for broadcasting
measurement outcomes. In our qubit model, this is taken into account by the
size of the subspaces the system is correlated with, i.e., the number of
pointer particles, $N$.

The insight that ideal measurements carry a diverging cost also sheds light on
thermodynamics because it implies that, in practice, all measurements that can
be performed are intrinsically non-ideal. Consequently, a central question is
how well one can approximate ideal measurements and which consequences these
approximate realizations have on tasks that arise in quantum thermodynamics.
For instance, to interpret work as a random variable in the quantum regime,
two projective measurements are commonly assumed to characterize work [28, 29,
30]. The impossibility of these measurements with finite work, prompts two
questions (recently studied in [31]): (a) what is the impact of measurement
imperfection on the observed fluctuations and (b) what is the total work cost
of observing work fluctuations imperfectly. Furthermore, in quantum
information-based engines [32, 24, 33], it would be highly relevant to
incorporate measurement imperfection and work cost into the efficiency in
addition to other constraints [34]. These insights could also be useful for
quantum information processing, e.g., bounding the minimal power consumption
of quantum computers employing syndrome measurements for error correction.

We are grateful to Tiago Debarba and Karen Hovhannisyan for valuable comments
and insights. We acknowledge support by the EU COST Action MP1209
“Thermodynamics in the quantum regime” and from the Austrian Science Fund
(FWF) through the START project Y879-N27, the project P 31339-N27, and the
joint Czech-Austrian project MultiQUEST (I 3053-N27 and GF17-33780L). YG
acknowledges support by the Austrian Science Fund (FWF) through the
Zukunftskolleg ZK03.

## References

  * Vinjanampathy and Anders [2016] Sai Vinjanampathy and Janet Anders,  _Quantum thermodynamics,_Contemp. Phys. 57, 545 (2016), arXiv:1508.06099.
  * Millen and Xuereb [2016] James Millen and André Xuereb,  _Perspective on quantum thermodynamics,_New J. Phys. 18, 011002 (2016), arXiv:1509.01086.
  * Goold  _et al._ [2016] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk,  _The role of quantum information in thermodynamics — a topical review,_J. Phys. A: Math. Theor. 49, 143001 (2016), arXiv:1505.07835.
  * Esposito and Van den Broeck [2011] Massimiliano Esposito and Christian Van den Broeck,  _Second law and landauer principle far from equilibrium,_Europhys. Lett. 95, 40004 (2011), arXiv:1104.5165.
  * Jacobs [2012] Kurt Jacobs,  _Quantum measurement and the first law of thermodynamics: the energy cost of measurement is the work value of the acquired information,_Phys. Rev. E 86, 040106(R) (2012), arXiv:1208.1561.
  * Brandão  _et al._ [2015] Fernando G. S. L. Brandão, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, and Stephanie Wehner,  _The second laws of quantum thermodynamics,_Proc. Natl. Acad. Sci. U.S.A. 11, 3275 (2015), arXiv:1305.5278.
  * Lostaglio  _et al._ [2015] Matteo Lostaglio, David Jennings, and Terry Rudolph,  _Description of quantum coherence in thermodynamic processes requires constraints beyond free energy,_Nat. Commun. 6, 6383 (2015), arXiv:1405.2188.
  * Ćwikliński  _et al._ [2015] Piotr Ćwikliński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim,  _Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics,_Phys. Rev. Lett. 115, 210403 (2015), arXiv:1405.5029.
  * Alhambra  _et al._ [2016] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry,  _Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality,_Phys. Rev. X 6, 041017 (2016), arXiv:1601.05799.
  * Wilming  _et al._ [2016] Henrik Wilming, Rodrigo Gallego, and Jens Eisert,  _Second law of thermodynamics under control restrictions,_Phys. Rev. E 93, 042126 (2016), arXiv:1411.3754.
  * Scharlau and Müller [2018] Jakob Scharlau and Markus P. Müller,  _Quantum Horn’s lemma, finite heat baths, and the third law of thermodynamics,_Quantum 2, 54 (2018), arXiv:1605.06092.
  * Masanes and Oppenheim [2017] Lluis Masanes and Jonathan Oppenheim,  _A general derivation and quantification of the third law of thermodynamics,_Nat. Commun. 8, 14538 (2017), arXiv:1412.3828.
  * Bera  _et al._ [2017] Manabendra Nath Bera, Arnau Riera, Maciej Lewenstein, and Andreas Winter,  _Generalized Laws of Thermodynamics in the Presence of Correlations,_Nat. Commun. 8, 2180 (2017), arXiv:1612.04779.
  * Schulman  _et al._ [2005] Leonard J. Schulman, Tal Mor, and Yossi Weinstein,  _Physical Limits of Heat-Bath Algorithmic Cooling,_Phys. Rev. Lett. 94, 120501 (2005).
  * Wilming and Gallego [2017] Henrik Wilming and Rodrigo Gallego,  _Third Law of Thermodynamics as a Single Inequality,_Phys. Rev. X 7, 041033 (2017), arXiv:1701.07478.
  * Clivaz  _et al._ [2019a] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber,  _Unifying paradigms of quantum refrigeration: fundamental limits of cooling and associated work costs,_Phys. Rev. E 100, 042130 (2019a), arXiv:1710.11624.
  * Clivaz  _et al._ [2019b] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber,  _Unifying Paradigms of Quantum Refrigeration: A Universal and Attainable Bound on Cooling,_Phys. Rev. Lett. 123, 170605 (2019b), arXiv:1903.04970.
  * Kieu [2019] Tien D. Kieu,  _Principle of Unattainability of absolute zero temperature, the Third Law of Thermodynamics, and projective quantum measurements,_Phys Lett A 383, 125848 (2019), arXiv:1804.04182.
  * Korbicz  _et al._ [2017] Jarosław K. Korbicz, Edgar A. Aguilar, Piotr Ćwikliński, and Paweł Horodecki,  _Generic appearance of objective results in quantum measurements,_Phys. Rev. A 96, 032124 (2017), arXiv:1604.02011.
  * Zurek [2009] Wojciech Hubert Zurek,  _Quantum Darwinism,_Nat. Phys. 5, 181 (2009), arXiv:0903.5082.
  * Sagawa and Ueda [2009] Takahiro Sagawa and Masahito Ueda,  _Minimal Energy Cost for Thermodynamic Information Processing: Measurement and Information Erasure,_Phys. Rev. Lett. 102, 250602 (2009), arXiv:0809.4098.
  * Elouard  _et al._ [2017] Cyril Elouard, David Herrera-Martí, Benjamin Huard, and Alexia Auffèves,  _Extracting work from quantum measurement in Maxwell demon engines,_Phys. Rev. Lett. 118, 260603 (2017), arXiv:1702.01917.
  * Lipka-Bartosik and Demkowicz-Dobrzanski [2018] Patryk Lipka-Bartosik and Rafal Demkowicz-Dobrzanski,  _Thermodynamic work cost of quantum estimation protocols,_J. Phys. A: Math. Theor. 51, 474001 (2018), arXiv:1805.01477.
  * Elouard and Jordan [2018] Cyril Elouard and Andrew N. Jordan,  _Efficient Quantum Measurement Engine,_Phys. Rev. Lett. 120, 260601 (2018), arXiv:1801.03979.
  * Reeb and Wolf [2014] David Reeb and Michael M. Wolf,  _An improved Landauer Principle with finite-size corrections,_New J. Phys. 16, 103011 (2014), arXiv:1306.4352.
  * Abdelkhalek  _et al._ [2016] Kais Abdelkhalek, Yoshifumi Nakata, and David Reeb,  _Fundamental energy cost for quantum measurement,_ (2016), arXiv:1609.06981.
  * Allahverdyan  _et al._ [2011] Armen E. Allahverdyan, Karen V. Hovhannisyan, Dominik Janzing, and Guenter Mahler,  _Thermodynamic limits of dynamic cooling,_Phys. Rev. E 84, 041109 (2011), arXiv:1107.1044.
  * Crooks [1999] Gavin E. Crooks,  _The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences,_Phys. Rev. E 60, 2721 (1999), arXiv:cond-mat/9901352.
  * Tasaki [2000] Hal Tasaki,  _Jarzynski Relations for Quantum Systems and Some Applications,_ (2000), arXiv:cond-mat/0009244.
  * Talkner  _et al._ [2007] Peter Talkner, Eric Lutz, and Peter Hänggi,  _Fluctuation theorems: Work is not an observable,_Phys. Rev. E 75, 050102(R) (2007), arXiv:cond-mat/0703189.
  * Debarba  _et al._ [2019] Tiago Debarba, Gonzalo Manzano, Yelena Guryanova, Marcus Huber, and Nicolai Friis,  _Work estimation and work fluctuations in the presence of non-ideal measurements,_New J. Phys. 21, 113002 (2019), arXiv:1902.08568.
  * Vidrighin  _et al._ [2016] Mihai D. Vidrighin, Oscar Dahlsten, Marco Barbieri, M. S. Kim, Vlatko Vedral, and Ian A. Walmsley,  _Photonic Maxwell’s Demon,_Phys. Rev. Lett. 116, 050401 (2016), arXiv:1510.02164.
  * Aydin  _et al._ [2019] Alhun Aydin, Altug Sisman, and Ronnie Kosloff,  _Landauer’s Principle in a Quantum Szilard Engine Without Maxwell’s Demon,_ (2019), arXiv:1908.04400.
  * Mohammady and Anders [2017] M. Hamed Mohammady and Janet Anders,  _A quantum Szilard engine without heat from a thermal reservoir,_New J. Phys. 19, 113026 (2017), arXiv:1706.00938.
  * Oszmaniec  _et al._ [2017] Michał Oszmaniec, Leonardo Guerini, Peter Wittek, and Antonio Acín,  _Simulating Positive-Operator-Valued Measures with Projective Measurements,_Phys. Rev. Lett. 119, 190501 (2017), arXiv:1609.06139.
  * Bruschi  _et al._ [2015] David E. Bruschi, Martí Perarnau-Llobet, Nicolai Friis, Karen V. Hovhannisyan, and Marcus Huber,  _The thermodynamics of creating correlations: Limitations and optimal protocols,_Phys. Rev. E 91, 032118 (2015), arXiv:1409.4647.
  * Huber  _et al._ [2015] Marcus Huber, Martí Perarnau-Llobet, Karen V. Hovhannisyan, Paul Skrzypczyk, Claude Klöckl, Nicolas Brunner, and Antonio Ac$\acute{\i}$n,  _Thermodynamic cost of creating correlations,_New J. Phys. 17, 065008 (2015), arXiv:1404.2169.
  * Vitagliano  _et al._ [2019] Giuseppe Vitagliano, Claude Klöckl, Marcus Huber, and Nicolai Friis,  _Trade-off between work and correlations in quantum thermodynamics,_ in  _Thermodynamics in the Quantum Regime_, edited by Felix Binder, Luis A. Correa, Christian Gogolin, Janet Anders, and Gerardo Adesso (Springer, 2019) Chap. 30, pp. 731–750, arXiv:1803.06884.
  * Bakhshinezhad  _et al._ [2019] Faraj Bakhshinezhad, Fabien Clivaz, Giuseppe Vitagliano, Paul Erker, Ali T. Rezakhani, Marcus Huber, and Nicolai Friis,  _Thermodynamically optimal creation of correlations,_J. Phys. A: Math. Theor. 52, 465303 (2019), arXiv:1904.07942.
  * Paris [2012] Matteo G. A. Paris,  _The modern tools of quantum mechanics,_Eur. Phys. J. S. T. 203, 61 (2012), arXiv:1110.6815.

## Appendix: Mathematical Model for Measurement Procedures

In this Appendix, we give a detailed description of _unbiased_ measurement
procedures introduced in the main text. As we have argued, ideal measurements
(unbiased, faithful, and non-invasive) are not generally implementable in
finite time or with finite energy. In practice, real measurements may
nonetheless approximate ideal measurements by investing energy; loosely
speaking, the approximation becomes better, the more energy that is invested.
To make this more precise, we will explicitly determine the fundamental energy
cost of projective111Arbitrary quantum measurements represented by positive-
operator-valued measures (POVMs) can be realized by projective measurements on
a Hilbert space obtained by appending an auxiliary system of, at most [35],
the same dimension as the original system. We therefore concentrate on
projective measurements. measurements.

###  A.I Framework

System. We consider a quantum system $S$ with Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}$ of dimension
$d_{\raisebox{0.0pt}{\tiny{$S$}}}=\dim(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}})$
initially in an arbitrary unknown quantum state represented by a density
operator
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\in\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}})$,
i.e., a Hermitian operator with
$\mathrm{tr}(\rho_{\raisebox{0.0pt}{\tiny{$S$}}})=1$ in the space of linear
operators $\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}})$ over the
system Hilbert space $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}$. We are then
interested in describing (projective) measurements of the system w.r.t. a
basis
$\\{\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}\\}_{i}$
of $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}$, which we take to be the
eigenbasis of the system Hamiltonian $H_{\raisebox{0.0pt}{\tiny{$S$}}}$, i.e.,
we can write
$H_{\raisebox{0.0pt}{\tiny{$S$}}}=\hbar\sum_{i}\Omega_{i}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}$,
where $\hbar(\Omega_{j}-\Omega_{i})$ is the energy gap between the $i$-th and
$j$-th levels. For instance, an example that we will focus on later is that of
the simplest quantum system — a qubit — with vanishing ground state energy and
energy gap $E_{\raisebox{0.0pt}{\tiny{$S$}}}=\hbar\Omega$. That is,
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}=\mathbb{C}^{2}$, and the system
Hamiltonian $H_{\raisebox{0.0pt}{\tiny{$S$}}}$ has eigenstates
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
$\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
spectral decomposition
$H_{\raisebox{0.0pt}{\tiny{$S$}}}=E_{\raisebox{0.0pt}{\tiny{$S$}}}\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}$.

Pointer. Similarly, we consider a pointer system $P$ with Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$ of dimension
$d_{\raisebox{0.0pt}{\tiny{$P$}}}=\dim(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})$
and Hamiltonian $H_{\raisebox{0.0pt}{\tiny{$P$}}}$. We then take the resource-
theoretic point of view that the pointer is initially in a state that is
freely available, i.e., a thermal state
$\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)\in\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})$
at ambient temperature $T=1/\beta$. We order the spectrum of the pointer
Hamiltonian in terms of its excitations into
$d_{\raisebox{0.0pt}{\tiny{$S$}}}$ sectors of size
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$, i.e.,
$H_{\raisebox{0.0pt}{\tiny{$P$}}}=\sum_{k=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\left|\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right|$
with $E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\leq
E^{\raisebox{0.0pt}{\tiny{$(k^{\prime})$}}}_{j}\ \forall i,j$ for
$k^{\prime}>k$. The thermal (Gibbs) state is given by

$\displaystyle\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)$
$\displaystyle=\sum_{k=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}p^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\left|\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}\\!\left.\right|$
(A.1)

where $p^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}=\exp\bigl{(}-\beta
E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}\bigr{)}/\mathcal{Z}$ and $\mathcal{Z}$
is the pointer’s partition function $\mathcal{Z}=\mathrm{tr}(e^{-\beta
H_{\raisebox{0.0pt}{\tiny{$P$}}}})=\sum_{i,k}e^{-\beta
E^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{i}}$.

Measurement procedure. We now wish to consider a measurement of the system’s
energy, i.e., of the observable $H_{\raisebox{0.0pt}{\tiny{$S$}}}$, or, in
other words, a projective measurement of the system in the energy
eigenbasis222Note that a projective measurement of the system in any other
(orthonormal) basis can be subsumed into this discussion by including an
additional unitary transformation (and its energy cost) on the initial state
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ to switch between the energy eigenbasis
and the desired measurement basis.. The corresponding _measurement procedure_
may be defined via a completely positive and trace-preserving (CPTP) map
$\mathcal{E}:\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})\rightarrow\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})$,
where
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$,
that maps
$\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}=\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tau_{\raisebox{0.0pt}{\tiny{$P$}}}$
to a post-measurement state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\in\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})$.
This may be understood as a generalized interaction between the system, the
pointer, and some auxiliary system. Here, we do not wish to address the
question of which measurement outcome is ultimately realized (which pure state
the system is left in), or how and why this may be the case. That is, we do
not attempt to make statements about what is often perceived as the
“measurement problem”, but rather take the point of view that system and
pointer are left in a joint state in which the internal states of the system
are correlated with the internal states of the pointer. Each of the latter is
designated to correspond to one of the system states
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$,
such that, upon finding the pointer in a state chosen to reflect
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$,
one concludes that the measurement outcome is “$i$”. More precisely, each
eigenstate
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ of
the measured observable of the system is assigned a set
$\\{\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}\\}_{n}$
of orthogonal states of the pointer corresponding to a projector
$\Pi_{i}=\sum_{n}\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right|$.
The projectors are chosen to be orthogonal and to form a complete set, i.e.,
$\Pi_{i}\Pi_{j}=\delta_{ij}\Pi_{i}$ and
$\sum_{i}\Pi_{i}=\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$. In an ideal
measurement, upon obtaining the outcome “$i$”, one may further conclude that
the post-measurement _system_ is left in the state
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$.
This is one of three features that can be identified for ideal projective
measurements. As explained in the main text, ideal measurements are

unbiased:
$\displaystyle\mathrm{tr}\bigl{[}\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\rho_{\raisebox{0.0pt}{\tiny{$S$}}}]=\rho_{ii}\,\forall\,i,$
(A.2) faithful: $\displaystyle
C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
$\displaystyle:=\sum_{i}\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\;\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}=1,$
(A.3) non-invasive:
$\displaystyle\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}\
\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}=\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\bigr{]}=\rho_{ii}\,\forall\,i.$
(A.4)

###  A.II Example: 2-Qubit Measurements

To illustrate the properties above and to understand why these conditions are
not met by general non-ideal measurement procedures with finite energy input,
we consider a simple example. Consider a measurement procedure using a single
pointer qubit and assume that by some means it has been prepared in the ground
state, i.e.,
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}$.
We model the interaction with the pointer by applying a controlled NOT
operation
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}+\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes
X_{\raisebox{0.0pt}{\tiny{$P$}}}$, with the usual Pauli operator
$X=\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|+\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|$.
Denoting the matrix elements of the initial state as
$\rho_{ij}=\left\langle\right.\\!{i}\\!\left.\right|\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\left|\right.\\!{j}\\!\left.\right\rangle$,
we can then write the post-measurement state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ as

$\displaystyle\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$
$\displaystyle=\,U_{\raisebox{-1.0pt}{\tiny{$\mathrm{CNOT}$}}}\,\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}^{\dagger}\,=\,\sum_{i,j=0,1}\rho_{ij}\left|\right.\\!{ii}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{jj}\\!\left.\right|\,.$
(A.5)

The system and pointer are now perfectly (classically) correlated in the sense
that whenever the pointer is found in the state
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$
($\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$),
the system is left in the corresponding state
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$
($\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$).
In other words, for the choice
$\Pi_{i}=\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}$,
we find that the measurement is faithful,

$\displaystyle C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
$\displaystyle=\,\sum\limits_{i=0,1}\mathrm{tr}\bigl{[}\left|\right.\\!{ii}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{ii}\\!\left.\right|\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,\sum\limits_{i=0,1}\rho_{ii}\,=\,\mathrm{tr}[\rho_{\raisebox{0.0pt}{\tiny{$S$}}}]\,=\,1\,.$
(A.6)

The post-measurement system state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$S$}}}=\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$P$}}}[\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}]=\sum_{i}\rho_{ii}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}$
is in general different from the initial system state since it not longer has
any off-diagonal elements w.r.t. the measurement basis, but the measurement is
nonetheless non-invasive since the diagonal elements match those of the
initial system state $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$. At the same time,
the chosen unitary $U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ guarantees
that the probabilities for finding the pointer in the states
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$ and
$\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$,
match those of the original system state, i.e., for $i=0,1$ we have

$\displaystyle\mathrm{tr}\bigl{(}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$S$}}}(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})\bigr{)}$
$\displaystyle=\,\mathrm{tr}(\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\rho_{\raisebox{0.0pt}{\tiny{$S$}}})\,=\,\rho_{ii}\,.\vspace*{-2mm}$
(A.7)

Consequently, the measurement is not biased towards one of the outcomes and
reproduces the statistics of the original system state, while being perfectly
correlated (i.e., faithful).

However, in general strong correlation and unbiasedness of the measurement do
not imply one another. For instance, one can construct an unbiased but also
generally uncorrelated measurement by replacing
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ with
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{SWAP}$}}}=\left|\right.\\!{00}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{00}\\!\left.\right|+\left|\right.\\!{01}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{10}\\!\left.\right|+\left|\right.\\!{10}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{01}\\!\left.\right|+\left|\right.\\!{11}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{11}\\!\left.\right|$,
leaving the system in the state
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ no
matter which state the pointer is in. Although all available information about
the pre-measurement system is thus stored in the pointer, measuring the latter
reveals no (additional) information about the post-measurement system.
Alternatively, consider the unitary
$\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}+\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes
X_{\raisebox{0.0pt}{\tiny{$P$}}}$ instead of
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$, both of which lead to the same
correlation $C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$, but the
probabilities for observing the two outcomes are now exchanged w.r.t. to
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$, i.e., the pointer is found in the state
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$
($\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$)
with probability $\rho_{11}$ ($\rho_{00}$) after the interaction.

For the purpose of examining real measurements, these examples are of course
pathological due to the assumption of reliably preparing the pointer in a pure
state (without having to have performed a projective measurement in order to
model a projective measurement or having to cool to the ground state using
finite resources [12]). Let us therefore relax this assumption and assume
instead that the pointer has been prepared at some finite non-vanishing
temperature $T=1/\beta$, such that
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}=p\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}+(1-p)\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}$
for some $p=(1+e^{-\omega\beta})^{-1}=\mathcal{Z}^{-1}$ with $0<p<1$. A quick
calculation then reveals that the previously perfect correlations are reduced
to $C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=p=\mathcal{Z}^{-1}<1$ and
that the measurement procedure using
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ is in general biased, i.e.,

$\displaystyle\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$S$}}}[\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}]\bigr{]}$
$\displaystyle=\,\rho_{ii}(2p-1)+1-p\,.$ (A.8)

However, while we generally have to give up the notion of a perfect projective
measurement in the sense that the outcomes are perfectly correlated with the
post-measurement states (as we have shown in the main text), one may retain
the unbiasedness of the measurement. That is, if we replace
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ by
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{unb.}$}}}=\left|\right.\\!{00}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{00}\\!\left.\right|+\left|\right.\\!{01}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{11}\\!\left.\right|+\left|\right.\\!{11}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{10}\\!\left.\right|+\left|\right.\\!{10}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{01}\\!\left.\right|$,
we obtain the same imperfect correlation value
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=p=\mathcal{Z}^{-1}$, but the
unbiasedness condition of Eq. (A.7) is satisfied. To reiterate, measurement
procedures using finite resources (finite time, finite energy, and finite
complexity, i.e., operations with finite interaction range), cannot be ideal,
since finite resources are not sufficient to prepare pointers in the required
pure states. Realistic measurement procedures hence are non-ideal.

###  A.III Non-Ideal Measurement Procedures

When any one of the three properties (A.2), (A.3) or (A.4) fails to hold, we
call the corresponding measurement procedure _non-ideal_. For non-ideal
measurements, the relation between the remaining properties is more
complicated. In particular, none of the three properties alone implies any of
the other two. For instance, consider an ideal post-interaction state

$\displaystyle\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$
$\displaystyle=\,\sum_{i}\rho_{ii}\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\rho^{\raisebox{0.0pt}{\tiny{$(i)$}}}+\,\text{off-
diag.}\,,\vspace*{-2mm}$ (A.9)

where we have not explicitly written the off-diagonal elements w.r.t. the
basis
$\\{\left|\right.\\!{i}\\!\left.\right\rangle\otimes\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(j)$}}}}\\!\left.\right\rangle\\}_{i,j,n}$
and $\rho^{\raisebox{0.0pt}{\tiny{$(i)$}}}$ is a pointer state that is
associated to one and only one of the outcomes $i$, that is,
$\Pi_{i}\rho^{\raisebox{0.0pt}{\tiny{$(j)$}}}=\delta_{ij}\rho^{\raisebox{0.0pt}{\tiny{$(j)$}}}$.
Any measurement procedure based on a map $\mathcal{E}$ for which the values
$\rho_{ii}$ in (A.9) are replaced with arbitrary probabilities
$p_{i}\neq\rho_{ii}$, results in a joint post-interaction state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ satisfying (A.3), but not (A.2)
or (A.4), resulting in a non-ideal measurement that is faithful, but neither
unbiased or non-invasive. Similarly, the state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$
obtained from a trivial interaction
$\mathcal{E}[\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}]=\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}$
complies with (A.4), but not with (A.3) or (A.2). Finally, the map
$\mathcal{E}$ realizing a complete exchange of the initial system and pointer
states (assuming, for the purpose of this example that
$d_{\raisebox{0.0pt}{\tiny{$S$}}}=d_{\raisebox{0.0pt}{\tiny{$P$}}}$), results
in an unbiased (A.2) measurement procedure that does not satisfy either (A.3)
or (A.4).

Satisfying any single one of the three properties is hence not sufficient for
distinguishing ideal from non-ideal measurements. When two out of the three
properties hold, things become more subtle. In two cases, a joint final state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ satisfying a pair of properties
implies the third property, and hence that the measurement is ideal for the
particular given initial system state $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$.
First, a measurement that is both faithful (A.3) and unbiased (A.2) implies
that it is also non-invasive (A.4). To show this, we start with the property
of unbiasedness and, summing the right-hand side of Eq. (A.2) over all $i$, we
have $\sum_{i}\rho_{ii}=1$. The left-hand side of Eq. (A.2) thus gives

$\displaystyle\sum_{i}\mathrm{tr}\bigl{[}\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,\sum_{i,j}\mathrm{tr}\bigl{[}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{j}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,1.$
(A.10)

At the same time, property (A.3) demands that the sum in the second step of
(A.10) yields $1$ already just for the terms where $i=j$, implying

$\displaystyle\sum_{i\neq
j}\mathrm{tr}\bigl{[}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{j}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\;\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,0.$ (A.11)

Since all diagonal matrix elements of a density operator are non-negative,
this further implies
$\mathrm{tr}\bigl{[}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{j}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\;\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}=0$
$\forall i\neq j$, which we can insert back into (A.2) to see that
$\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,\rho_{ii}$.
Inserting all of this into the left-hand side of Eq. (A.4) together with
$\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}=\sum_{j}\Pi_{j}$, we obtain

$\displaystyle\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\sum_{j}\Pi_{j}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,\rho_{ii},$
(A.12)

which concludes the proof that unbiased and faithful measurements are also
non-invasive.

Second, a measurement that is both faithful (A.3) and non-invasive (A.4) is
also unbiased (A.2). Now starting with (A.4), we again sum the left-hand side
over all $i$ and resolve the identity
$\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}=\sum_{j}\Pi_{j}$, to obtain

$\displaystyle\sum_{i,j}\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{j}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,\sum_{i}\rho_{ii}\,=\,1.$ (A.13)

This time, unbiasedness (A.2) implies

$\displaystyle\sum_{i\neq
j}\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{j}\;\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\,0,$ (A.14)

and in turn
$\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{j}\;\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}=0$
$\forall i\neq j$, as before. Inserting this into (A.4) then implies
$\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,\rho_{ii}$.
Finally inserting into the left-hand side of (A.2) yields

$\displaystyle\mathrm{tr}\bigl{[}\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\sum_{j}\mathrm{tr}\bigl{[}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{j}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$
$\displaystyle=\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,=\,\rho_{ii},$
(A.15)

confirming the unbiasedness condition.

For the remaining combination this is not the case. A measurement procedure
that is unbiased (A.2) and non-invasive (A.4) for a fixed system state
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ is not necessarily faithful (A.3).
Consider, e.g., the initial single-qubit system state
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}=\tfrac{3}{4}\left|\right.\\!{0}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{0}\\!\left.\right|+\tfrac{1}{4}\left|\right.\\!{1}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{1}\\!\left.\right|$,
i.e., where $\rho_{00}=\tfrac{3}{4}$ and $\rho_{11}=\tfrac{1}{4}$, and the
two-qubit final state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\sum_{m,n=0,1}\left|\right.\\!{m,n}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{m,n}\\!\left.\right|$
with $p_{01}=p_{10}=p_{11}=\tfrac{1}{8}$ and $p_{00}=\tfrac{5}{8}$. For
$\Pi_{i}=\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}}$,
one has the reduced states
$\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$S$}}}(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$P$}}}(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$,
so we have unbiasedness and non-invasiveness, but
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=\tfrac{3}{4}<1$.

Moreover, the measurement procedure (corresponding to the transformation above
(whose details are not given in the example) may not be unbiased or non-
invasive for other initial system states
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$. This singles out the property of
faithfulness when one is interested in checking the properties of a
measurement procedure for any given initial system state. Nonetheless, caution
is advisable here. For a particular initial state
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ of the system, all three properties may
be satisfied, yet, this may not be so for other initial states. Simply
consider the example in Eq. (A.8). For an initial state with
$\rho_{00}=\rho_{11}=\tfrac{1}{2}$, the measurement satisfies all three
properties, but for $p<1$ the measurement procedure is biased and has non-
perfect correlation for almost all (other) states
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$.

Indeed, demanding that any of the properties hold only for particular initial
system states $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ is somewhat contradictory
to the notion of performing a measurement that reveals previously unknown
information about a system. In other words, measurements should not require
detailed knowledge about $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ to ensure that
one may trust the measurement outcomes, or inferences made from them. The
definitions of the properties (A.2), (A.3) and (A.4) must hence be extended to
demand that measurement procedures are only called unbiased, faithful, or non-
invasive, if the respective properties (A.2), (A.3) or (A.4) hold _for all
initial systems states_ $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$.

With such an extended definition, one then indeed finds that any two
properties imply the third. In particular, it is then the case that
measurement procedures that are unbiased _and_ non-invasive, are also
faithful, and thus ideal. The proof of this statement, relies on insights into
the general structure of all maps representing unbiased measurement
procedures, and as such appears later in Appendix A.VI. In purely qualitative
terms, maps that are either unbiased or non-invasive need to transfer the
diagonal elements of the system state $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ to
particular (different) subspaces of the joint Hilbert space of system and
pointer. The only way to simultaneously satisfy both the requirements for
unbiasedness and non-invasiveness for arbitrary
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ forces all information to be
concentrated in the subspaces corresponding to the images of the projectors
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\Pi_{i}$,
such that the resulting state $\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$
satisfies (A.3) independently of the details of
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$.

Ultimately, this means that only one of the three properties can be satisfied
exactly for all initial system states in any realistic measurement procedure.
Given that the constraint of finite resources rules out that realistic non-
ideal measurements are faithful, we have a choice between the measurement
being unbiased or non-invasive. Arguably, biased measurements that are not
even faithful are of little use, since the outcomes would not provide any
level of certainty about either the pre- or post-measurement system state. In
the following, we are therefore interested in unbiased measurement procedures
for which the correlations between the system and the pointer are as large as
possible. Given such non-ideal measurement, we then wish to minimize the
associated energy costs.

###  A.IV General Unbiased Measurements

Here, we will identify the basic structure and important properties of a
general model of non-ideal measurement procedures. To do so, we separate what
we believe to accurately model such a measurement procedure into two steps.
These are:

I Preparation: Some energy is invested to prepare the pointer system in a
suitable quantum state. II Correlating: The pointer interacts with the system
to be measured, creating correlations between them.

####  A.IV.1 Step I: Preparation

Before interacting with the system, the pointer can be prepared in a suitable
quantum state $\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$ at the expense of some
initial energy investment $\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$
accounting for the (CPTP) transformation
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}:\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})\rightarrow\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})$
mapping $\tau(\beta)$ to
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}=\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}[\tau(\beta)]$.
In particular it may be desirable to lower the entropy of the initial pointer
state. In principle, one may use any given amount of energy to prepare an
arbitrary pointer state that is compatible with the specified energy and whose
entropy is lower than that of $\tau(\beta)$. The energy cost for reaching a
particular state $\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$ is bounded from below
by the free energy difference, i.e.,

$\displaystyle\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$
$\displaystyle\geq\,\Delta
F\bigl{(}\tau(\beta)\rightarrow\rho_{\raisebox{0.0pt}{\tiny{$P$}}}\bigr{)}\,=\,\Delta
E_{\raisebox{0.0pt}{\tiny{$P$}}}-T\Delta S_{\raisebox{0.0pt}{\tiny{$P$}}}\,,$
(A.16)

with $\Delta
E_{\raisebox{0.0pt}{\tiny{$P$}}}=\mathrm{tr}\bigl{[}H_{\raisebox{0.0pt}{\tiny{$P$}}}(\rho_{\raisebox{0.0pt}{\tiny{$P$}}}-\tau(\beta))\bigr{]}$
and $\Delta
S_{\raisebox{0.0pt}{\tiny{$P$}}}=S\bigl{(}\rho_{\raisebox{0.0pt}{\tiny{$P$}}}\bigr{)}-S\bigl{(}\tau(\beta)\bigr{)}$,
and where $S(\rho)=-\mathrm{tr}\bigl{[}\rho\log(\rho)\bigr{]}$ is the von
Neumann entropy. However, the exact work cost of the preparation depends on
the control over the system and the available auxiliary degrees of freedom,
and may exceed this bound. In particular, the free energy difference to the
ground state is finite although this state cannot be reached with a finite
work investment in finite time [12]. The precise resource requirements in
terms of energy, control, and time for preparing arbitrary quantum states are
hence difficult to capture333Certainly, any such preparation can be described
by a CPTP map, which in turn can be seen as a unitary acting on
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}\otimes\left|\right.\\!{\Phi}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\Phi}\\!\left.\right|$
in a Hilbert space enlarged by an auxiliary system with Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$A$}}}\ni\left|\right.\\!{\Phi}\\!\left.\right\rangle$.
However, this brings one back to the question of quantifying the cost for
preparing the pure state $\left|\right.\\!{\Phi}\\!\left.\right\rangle$ of the
auxiliary system., whereas the refrigeration of quantum systems is a well-
understood task, whose energy cost has been quantified for various levels of
control one assumes about the quantum systems involved in the cooling
procedure [16].

It is therefore practically useful (and reasonable) to assume that the
preparation only involves refrigeration. That is, we assume in the following
that the temperature of the pointer is lowered from $T$ to $T_{0}\leq T$,
reaching a thermal state $\tau(\beta_{0})$ with $\beta_{0}=1/T_{0}$. On the
one hand, step I thus becomes less general than it could potentially be since
one does not explore the entire Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$. On the other hand, the thermal
state can be considered to be energetically optimal, since it minimizes the
energy at fixed entropy. Moreover, at fixed energy the thermal state also
maximizes the entropy and hence minimizes the free energy, which in turn
bounds the work cost from below.

####  A.IV.2 Step II: Correlating

During the second step of the measurement procedure, the system interacts with
the pointer in such a way that correlations between the two are established
via a CPTP map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}:\,\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})\rightarrow\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
that maps
$\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}=\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$
to
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}\bigl{[}\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}$.
A particularly important special case is the case of unitary correlating maps
$U$, i.e., such that
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=U\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}U^{\dagger}$,
representing measurement procedures where the joint system of $S$ and $P$ can
be considered to be closed for the purpose of the correlating step. Then, the
energy cost for the second step can be calculated via

$\displaystyle\Delta E_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$
$\displaystyle=\,\mathrm{tr}\bigl{[}(H_{\raisebox{0.0pt}{\tiny{$S$}}}+H_{\raisebox{0.0pt}{\tiny{$P$}}})(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}-\rho_{\raisebox{0.0pt}{\tiny{$SP$}}})\bigr{]}.$
(A.17)

In any case the generated correlations can in principle be (but need not be)
genuine quantum correlations. For (non-ideal) projective measurements as
defined here, it nonetheless suffices that classical correlations are
established with respect to the measurement basis (here the eigenbasis of
$H_{\raisebox{0.0pt}{\tiny{$S$}}}$) and a chosen basis of the pointer system.
More specifically, we assign a set of orthogonal projectors

$\displaystyle\Pi_{i}$
$\displaystyle:=\,\sum\limits_{n}\left|\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right|,$
(A.18)

with $\Pi_{i}\Pi_{j}=\delta_{ij}\Pi_{i}$ (in particular,
$\left\langle\right.\tilde{\psi}_{m}^{\raisebox{0.0pt}{\tiny{$(i)$}}}\left|\right.\tilde{\psi}_{n}^{\raisebox{0.0pt}{\tiny{$(j)$}}}\left.\right\rangle=\delta_{ij}\delta_{mn}$)
and $\sum_{i}\Pi_{i}=\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$. The
orthogonality and completeness of the projectors ensure that every pointer
state is associated with a state of the measured system, i.e., every outcome
provides a definitive measurement result “i”. We further amend the
correlations function defined in Eq. (A.6) for a single-qubit pointer to
reflect the use of the more general projectors, i.e., we redefine the
quantifier $C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ as

$\displaystyle C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
$\displaystyle:=\,\sum\limits_{i}\mathrm{tr}\bigl{[}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\,\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,.$
(A.19)

####  A.IV.3 Unbiased measurements

We are now in a position to give a formal definition of what we consider as an
abstract measurement procedure.

Definition 1:  Measurement procedure A measurement procedure
$\mathcal{M}(\beta)$ that realizes a (non-ideal) projective measurement at
ambient temperature $T=1/\beta$ of an (unknown) quantum state
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\in\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}})$
w.r.t. to an orthonormal basis
$\\{\left|\right.\\!{n}\\!\left.\right\rangle\\}_{n}$ of
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}$ is given by the tuple
$(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}},H_{\raisebox{0.0pt}{\tiny{$P$}}},\Pi,\mathcal{E})$,
consisting of a pointer Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$, a pointer Hamiltonian
$H_{\raisebox{0.0pt}{\tiny{$P$}}}\in\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})$,
a complete set $\Pi=\\{\Pi_{i}\\}_{i}$ of orthogonal projectors on
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$, and a CPTP map
$\mathcal{E}:\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})\rightarrow\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
with
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$,
together with the induced CPTP map
$\mathcal{E}_{\mathcal{M}}:\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}})\rightarrow\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
given by $\displaystyle\mathcal{E}_{\mathcal{M}}:\
\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\mapsto\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\mathcal{E}[\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tau(\beta)].$
(A.20)

Note that any definition of a thermal state $\tau(\beta)$ implies that the
state has full rank. This definition includes, in particular, the case that we
consider here, where the map
$\mathcal{E}=\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}\circ(\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}})$
is split into two separate steps. As we have already motivated in our earlier
example, we are interested in considering measurement procedures that
represent the measured quantum state without bias. While perfect correlations
cannot be guaranteed in this way, one may however ask that averages of the
measured quantity match for the pointer and the system. Moreover, it is
desirable that this is so independently of the specific initial states of the
system and the pointer. All of this is captured by the following definition.

Definition 2:  Unbiased measurement A measurement procedure
$\mathcal{M}(\beta)$ is called _unbiased_ , iff
$\mathrm{tr}\bigl{[}\Pi_{i}\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$S$}}}(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})\bigr{]}=\mathrm{tr}[\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\rho_{\raisebox{0.0pt}{\tiny{$S$}}}]$
$\forall i$ and $\forall\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$.

Since we wish to restrict our further considerations to unbiased measurements,
it will be useful to know more about the structure of these measurement
procedures, in particular, about the involved CPTP map $\mathcal{E}$ and
projectors $\Pi_{i}$, given that one has selected a suitable pointer system
with Hilbert space $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$ and
Hamiltonian $H_{\raisebox{0.0pt}{\tiny{$P$}}}$. To this end, note that our
previous example using $U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ was
unbiased only for pointers that can be prepared in the ground state (or any
pure state for that matter). This can only be the case if the initial
temperature vanishes or if infinite resources are available in step I, whereas
we are interested in describing more realistic conditions. To capture this, we
formalise the following:

Definition 3:  Finite-resource measurement A measurement procedure
$\mathcal{M}(\beta)$ at a nonzero ambient temperature $T=1/\beta$ uses finite
resources if the map $\mathcal{E}_{\mathcal{M}}$ is rank non-increasing.

On the other hand, measurement procedures which reduce the rank use either
infinite energy $E$, take infinite time $t$ (an infinite sequence of finite
interaction range operations) or are infinitely complex (infinite interaction
range operations) [12, 14, 15, 16, 17, 27]. Now, in order to take a first step
towards unraveling the structure of unbiased measurements we formulate the
following lemma.

Lemma 1 All unbiased finite-resource measurement procedures
$\mathcal{M}(\beta)$ with (thermal, full-rank) pointer system with Hilbert
space $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$, Hamiltonian
$H_{\raisebox{0.0pt}{\tiny{$P$}}}$, and orthogonal projectors $\Pi_{i}$ can be
realized by CPTP maps $\mathcal{E}$ of the form
$\mathcal{E}=\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}\circ(\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}})$,
where $\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$ is a CPTP map from
$\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})$ to itself
(achievable in finite time $t$ and satisfying $\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}<\infty$), and the CPTP map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ from
$\mathcal{L}(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ to itself has Kraus
operators $K_{l}=\sum\limits_{i}K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$ with
$\displaystyle K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
$\displaystyle=\,\sum\limits_{j=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\sum\limits_{n=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}-1}\sum\limits_{m=0}^{d_{i}-1}k_{jmn}^{\raisebox{0.0pt}{\tiny{$(i,l)$}}}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\left|\right.\\!{\tilde{\psi}_{m}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\psi_{n}}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}},$
(A.21) with
$d_{\raisebox{0.0pt}{\tiny{$S$}}}=\dim(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}})$,
$d_{\raisebox{0.0pt}{\tiny{$P$}}}=\dim(\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}})\geq
d_{\raisebox{0.0pt}{\tiny{$S$}}}$, and coefficients
$k_{jmn}^{\raisebox{0.0pt}{\tiny{$(i,l)$}}}$ such that
$\displaystyle\sum_{l}(K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}})^{\dagger}K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
$\displaystyle=\,\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}.$
(A.22)

Proof of Lemma A.IV.3. Before we get into the technical details of the proof,
let us phrase the Lemma A.IV.3 more informally. It states that the map
$\mathcal{E}$ consists of an arbitrary (finite energy, $\Delta
E_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}<\infty$, finite time $t<\infty$)
preparation of the pointer
($\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$), followed by a map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ that maps the subspaces
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ to
those corresponding to $\Pi_{i}$, respectively. Moreover, note that
unbiasedness of course implies that the pointer system must be large enough
($d_{\raisebox{0.0pt}{\tiny{$P$}}}\geq d_{\raisebox{0.0pt}{\tiny{$S$}}}$) to
accommodate all the possible measurement outcomes. Let us then prove the
lemma. As mentioned before, the CPTP map $\mathcal{E}$ may be separated into a
map $\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$ acting nontrivially
only on the pointer Hilbert space, and a CPTP map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ acting on the resulting
state
$\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}=\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$,
which we can write with respect to the basis
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ as

$\displaystyle\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}$
$\displaystyle=\,\left(\begin{array}[]{c|c|c|c}\tikzmarkin[ver=stylegreen,linewidth=0mm,]{a1}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$}}\tikzmarkend{a1}&\cdot&\cdot&\cdots\\\
\hline\cr\cdot&\tikzmarkin[ver=styleorange,linewidth=0mm,]{a2}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$}}\tikzmarkend{a2}&\cdot&\cdots\\\
\hline\cr\cdot&\cdot&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{a33}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$}}\tikzmarkend{a33}&\cdots\\\
\hline\cr\vdots&\vdots&\vdots&\ddots\end{array}\right),$ (A.27)
$\displaystyle\phantom{=\,(}\begin{array}[]{cccc}&&&\\\\[-5.69054pt] \hskip
2.84526pt$\upbracefill$&$\upbracefill$&$\upbracefill$&\\\\[-2.84526pt] \hskip
6.82864pt\text{\scriptsize{$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$}}&\hskip
0.28453pt\text{\scriptsize{$\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$}}&\hskip
0.28453pt\text{\scriptsize{$\left|\right.\\!{2}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$}}&\text{\scriptsize{$\cdots$}}\end{array}$
(A.31)

Without loss of generality, we can then write the final state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}=\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}[\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}]$
with respect to the product basis
$\\{\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\left|\right.\\!{\tilde{\psi}_{m}^{\raisebox{0.0pt}{\tiny{$(j)$}}}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}\\}_{i,j,m}$
in the form

where we have indicated the columns corresponding to the subspaces of fixed
vectors
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
projectors $\Pi_{i}$ for $i=1,2,3$, and dots indicate matrix elements that may
be nonzero but are not explicitly shown. In particular, the latter can include
subspaces for $i>3$, and the case for $d_{\raisebox{0.0pt}{\tiny{$S$}}}\leq 3$
can be obtained by truncating the shown matrix by removing the corresponding
rows and columns. The colored sub-blocks $A_{0i}$, $A_{1i}$, $A_{2i}$ and so
forth are $d_{i}\times d_{i}$ block matrices in terms of which the
unbiasedness condition of Def. A.IV.3 can be written as

$\displaystyle\sum\limits_{j=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\mathrm{tr}[A_{ji}]$
$\displaystyle=\,\rho_{ii}\ \ \forall i.$ (A.33)

Crucially, the unbiasedness condition in Eq. (A.33) is to hold for all
possible system states $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$, and hence for
all possible values of $\rho_{ii}$. This, in turn, implies that all sub-blocks
with second subscript $i$ must be proportional to $\rho_{ii}$. That is
$A_{ji}=\rho_{ii}\tilde{A}_{ji}$ $\;\forall\,j$ with
$\sum_{j=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\mathrm{tr}[\tilde{A}_{ji}]=1$
$\forall i$. Since the unbiasedness condition is not sensitive to terms
appearing in the off-diagonal blocks, a convenient representation of the
relevant terms of the post-interaction state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ under the map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ is :

$\displaystyle\Gamma_{\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}}=\,\left[\begin{array}[]{c|c|c|c}\tikzmarkin[ver=stylegreen,linewidth=0mm,]{d1}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\tilde{A}_{00}$}}\tikzmarkend{d1}&\tikzmarkin[ver=styleorange,linewidth=0mm,]{dd2}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\tilde{A}_{01}$}}\tikzmarkend{dd2}&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{w1}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\tilde{A}_{02}$}}\tikzmarkend{w1}&\cdots\\\
\hline\cr\tikzmarkin[ver=stylegreen,linewidth=0mm,]{dd1}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\tilde{A}_{10}$}}\tikzmarkend{dd1}&\tikzmarkin[ver=styleorange,linewidth=0mm,]{d2}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\tilde{A}_{11}$}}\tikzmarkend{d2}&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{w3}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\tilde{A}_{12}$}}\tikzmarkend{w3}&\cdots\\\
\hline\cr\tikzmarkin[ver=stylegreen,linewidth=0mm,]{d33}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\tilde{A}_{20}$}}\tikzmarkend{d33}&\tikzmarkin[ver=styleorange,linewidth=0mm,]{d44}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\tilde{A}_{21}$}}\tikzmarkend{d44}&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{d5}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\tilde{A}_{22}$}}\tikzmarkend{d5}&\ldots\\\
\hline\cr\vdots&\vdots&\vdots&\ddots\end{array}\right]\,,$ (A.38)

which we call the correlation matrix. Here it can immediately be seen that the
unbiasedness condition, which implies
$\sum_{j=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\mathrm{tr}[\tilde{A}_{ji}]=1$
says that the sum of the trace of the blocks in column $i$ of
$\Gamma_{\mathcal{E}}$ must be equal to $\rho_{ii}$ for unbiasedness to
hold444Note that this representation is not square, since in principle the
dimension of the matrix-valued entries of each column are different. In the
case that the map $\mathcal{E}$ representing the measurement is unitary, the
representation becomes square and the dimension of all blocks across all
columns is equal..

From the initial state (A.31) and the final state (LABEL:eq:appendrow) it
becomes apparent that unbiasedness can be guaranteed for maps that connect the
subspaces corresponding to
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$
only with those corresponding to $\Pi_{i}$. More precisely, each of these
connecting maps can be viewed as an arbitrary CPTP map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
from the $d_{\raisebox{0.0pt}{\tiny{$P$}}}$-dimensional space spanned by the
vectors in the set
$\\{\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\left|\right.\\!{\psi_{n}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}\\}_{n=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$P$}}}-1}$,
where
$\\{\left|\right.\\!{\psi_{n}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}\\}_{n}$
is an arbitrary basis of $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$, to the
$(d_{\raisebox{0.0pt}{\tiny{$S$}}}\times d_{i})$-dimensional space spanned by
the vectors in the set
$\\{\left|\right.\\!{j}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\left|\right.\\!{\tilde{\psi}_{m}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}\\}_{\raisebox{0.0pt}{\tiny{$\begin{subarray}{c}j=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$S$}}}-1\\\
m=0,\ldots,d_{i}-1\end{subarray}$}}}$. The Kraus operators for these CPTP maps
are precisely the $\\{K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}}\\}_{l}$ of Eq.
(A.21) and in matrix notation we may denote these maps as

$\displaystyle\rho_{ii}\rho_{P}$
$\displaystyle\,\begin{subarray}{c}\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\raisebox{0.0pt}{\tiny{$(i)$}}}\\\
\longmapsto\end{subarray}\,\left(\begin{array}[]{c|c|c|c}\\!\\!A_{0i}\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\cdots\\!\\!\\\
\hline\cr\\!\\!\cdot\\!\\!&\\!\\!A_{1i}\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\cdots\\!\\!\\\
\hline\cr\\!\\!\cdot\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!A_{2i}\\!\\!&\\!\\!\cdots\\!\\!\\\
\hline\cr\\!\\!\vdots\\!\\!&\\!\\!\vdots\\!\\!&\\!\\!\vdots\\!\\!&\\!\\!\ddots\\!\\!\end{array}\right)=\,\rho_{ii}\left(\begin{array}[]{c|c|c|c}\\!\\!\tilde{A}_{0i}\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\cdots\\!\\!\\\
\hline\cr\\!\\!\cdot\\!\\!&\\!\\!\tilde{A}_{1i}\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\cdots\\!\\!\\\
\hline\cr\\!\\!\cdot\\!\\!&\\!\\!\cdot\\!\\!&\\!\\!\tilde{A}_{2i}\\!\\!&\\!\\!\cdots\\!\\!\\\
\hline\cr\\!\\!\vdots\\!\\!&\\!\\!\vdots\\!\\!&\\!\\!\vdots\\!\\!&\\!\\!\ddots\\!\\!\end{array}\right).$
(A.47)

Since the domains as well as the images for different $i$ lie in orthogonal
subspaces of $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}}$, the maps
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
can be combined to the map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ with Kraus
operators555Note that the number of nonzero Kraus operators may be different
for each $i$, but one may always add trivial (vanishing) Kraus operators to
each set $\\{K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}}\\}_{l}$ with fixed $i$.
$\\{K_{l}=\sum_{i}K_{l}^{\raisebox{0.0pt}{\tiny{$(i)$}}}\\}_{l}$. Once can
check that the unbiasedness condition is satisfied for these Kraus operators
by a simple calculation, which we will not repeat here. If the initial ambient
temperature is nonzero and the measurement procedure uses finite resources
(time and energy), the pointer state $\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$ has
full rank and unbiasedness can only be achieved via maps of the form
mentioned, as claimed in Lemma A.IV.3, which concludes the proof. ∎

Inspecting again the example from Appendix A.II, one notes that the controlled
NOT operation $U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ is not of the
form required for a finite-resource unbiased measurement, as expected.
However, when the pointer can be prepared in a pure state (w.l.o.g. the ground
state
$\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$P$}}}$)
one observes that the measurement procedure using
$U_{\raisebox{0.0pt}{\tiny{$\mathrm{CNOT}$}}}$ in the correlating step becomes
unbiased because some of the sub-blocks $A_{ji}$ are only trivially
proportional to $\rho_{ii}$. In particular,
$A_{01}=\tilde{A}_{01}=A_{10}=\tilde{A}_{10}=0$.

Having understood the general structure of all unbiased measurements, we now
want to turn to some specific instances of such measurement procedures.

###  A.V Extremal Measurements

With the help of Lemma A.IV.3 we can now describe the set of _all_ unbiased
measurements for a given quantum system $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$
and pointer. The measurement within this set may further be categorized
according to their specific properties, in particular, their energy cost, the
amount of correlations created between the system and the pointer (how
faithful the measurement is), and the level of control required for their
implementation (e.g., what type of auxiliary systems are available and which
operations can be performed on the system, pointer, and auxiliaries). Given an
(unknown) quantum system $S$ it would ideally be desirable to answer the
question: What is the maximal correlation achievable between the system and
_any suitable_ pointer given a fixed work input $\Delta E$? A more restricted
version of this question is: What is the maximal correlation achievable
between the system and _a particular_ pointer given a fixed work input $\Delta
E$?

Since we assume that the system state $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ is
unknown before the measurement, the correlation measure $\bar{C}$ that we are
interested in optimizing is obtained from averaging
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ from Eq. (A.3) over all
system states. We observe that for any particular systems state
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$, the correlation measure
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ does not depend on any of
the matrix elements of $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ except for those
on the diagonal, i.e.,

$\displaystyle C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
$\displaystyle=\,\rho_{00}\mathrm{tr}[\tilde{A}_{00}]+\rho_{11}\mathrm{tr}[\tilde{A}_{11}]+\rho_{22}\mathrm{tr}[\tilde{A}_{22}]+\dots\,.$
(A.48)

Averaging over all states $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ is hence
equivalent to an average over all probability distributions corresponding to
the diagonal of $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$. Moreover, for each of
these values $\rho_{ii}$ $(i=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$S$}}}-1)$,
the average over all probability distributions results in the value
$1/d_{\raisebox{0.0pt}{\tiny{$S$}}}$, such that the average of
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ is given by

$\displaystyle\bar{C}$
$\displaystyle=\,\tfrac{1}{d_{\raisebox{0.0pt}{\tiny{$S$}}}}\mathrm{tr}\bigl{[}\tilde{A}_{00}+\tilde{A}_{11}+\tilde{A}_{22}+\dots\bigr{]},$
(A.49)

which, in terms of the representation presented in Eq. (A.38), corresponds to
taking the trace of the blocks appearing along the diagonal.
Despite this simple form of $\bar{C}$, the optimization over all pointer
systems and operations thereon is a daunting task. Indeed, even for a fixed
pointer at initial temperature $T=1/\beta$, identifying the optimal
measurement procedure in terms of the best ratio of (average) correlation
increase per unit energy cost (averaged over the input system states) is
highly nontrivial. To illustrate the difficulty, first note that an
(attainable) bound exists for correlating (quantified by the mutual
information) two arbitrary systems that are initially thermal at the same
temperature at optimal energy expenditure [36, 37]. While the known protocol
for attaining this bound is in general not unitary (it involves lowering the
temperature), in some cases the bound is tight already when one correlates the
systems unitarily. However, it was recently shown [38, 39] that the optimal
(in the sense of the mentioned bound being tight) trade-off between
correlations and energy cost cannot always be achieved unitarily.

Of course, in our case, the initial state of the system is not known, and
cannot be expected to be thermal in general. Moreover, the mutual information
is not a suitable figure of merit for quantifying the desired correlations
between system and pointer because the latter don’t distinguish between
classical and genuinely quantum correlations. For instance, for a single-qubit
system and pointer, the states
$\left|\right.\\!{\Phi^{+}}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$SP$}}}=\bigl{(}\left|\right.\\!{00}\\!\left.\right\rangle+\left|\right.\\!{11}\\!\left.\right\rangle\bigr{)}/\sqrt{2}$
and
$\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}=\tfrac{1}{2}\bigl{(}\left|\right.\\!{00}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{00}\\!\left.\right|+\left|\right.\\!{11}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{11}\\!\left.\right|\bigr{)}$
have different values of mutual information but are equally well (i.e.,
perfectly) correlated w.r.t. to the desired measurement basis. The above
arguments on optimally correlating protocols hence do not apply directly, but
with the added complication of the unknown system state and the unbiasedness
condition, we cannot rule out the possibility that the optimal unbiased
measurement procedures are not realized by a unitary correlating step.

Nonetheless, it can be argued that any nonunitary realization of the second
part $\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ of the CPTP map
$\mathcal{E}$ must require higher levels of control than a corresponding
unitary realization due to the requirement of realizing nonunitary maps
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ as unitaries on a
larger Hilbert space. Specifically, any CPTP map
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ can be thought of as a
unitary on a larger Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$E$}}}$
(with a factoring initial condition) [40], that is, one may write any
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ as

$\displaystyle\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}[\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}]$
$\displaystyle=\,\mathrm{tr}_{\raisebox{0.0pt}{\tiny{$E$}}}\bigl{[}U_{\raisebox{0.0pt}{\tiny{$SPE$}}}\bigl{[}\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}\otimes\left|\right.\\!{\chi}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\chi}\\!\left.\right|\bigr{]}U_{\raisebox{0.0pt}{\tiny{$SPE$}}}^{\dagger}\bigr{]}$
(A.50)

for some unitary $U_{\raisebox{0.0pt}{\tiny{$SPE$}}}$ on
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$SP$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$E$}}}$
and for some pure state
$\left|\right.\\!{\chi}\\!\left.\right\rangle\in\mathcal{H}_{\raisebox{0.0pt}{\tiny{$E$}}}$.
At the same time, employing a unitary to correlate pointer and system enables
us to unambiguously quantify the work cost of the correlating step without
assumptions about the Hamiltonian of potential auxiliary systems.

We are therefore particularly interested in describing all unbiased
measurement procedures, where
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ is realized unitarily,
such that

$\displaystyle\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$
$\displaystyle=\,\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}[\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}]\,=\,U\rho_{\raisebox{0.0pt}{\tiny{$SP$}}}U^{\dagger}$
(A.51)

with $UU^{\dagger}=U^{\dagger}U=\mathds{1}_{\raisebox{0.0pt}{\tiny{$SP$}}}$.
In this sense, our focus lies on unbiased measurement procedures where all
control that one may have over external systems (beyond $S$ and $P$) is used
in the initial step represented by
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$ to prepare the pointer
in a suitable state, e.g., by lowering its temperature. Here we make use of
the fact that the work cost of refrigeration with various levels of control
has been extensively studied [16]. This leaves us with the task of analyzing
the structure of the representations $U$ of the unitary maps
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$.

A first step towards the completion of this task is to note that the
unbiasedness condition for measurement procedures with a _unitary_ correlating
step $\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ means that it is
inefficient (in terms of energy cost) to use a pointer Hilbert space
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$ whose dimension is not an integer
multiple of the system dimension. This can be explained in the following way.
By inspection of the maps in Eq. (A.47), one notes that
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
maps the $d_{\raisebox{0.0pt}{\tiny{$P$}}}\times
d_{\raisebox{0.0pt}{\tiny{$P$}}}$ density matrix
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}$ to a
$d_{\raisebox{0.0pt}{\tiny{$S$}}}\operatorname{rank}(\Pi_{i})\times
d_{\raisebox{0.0pt}{\tiny{$S$}}}\operatorname{rank}(\Pi_{i})$ density matrix.
That is, the size of each of the $d_{\raisebox{0.0pt}{\tiny{$S$}}}$ blocks
$\tilde{A}_{ji}$ $\;\forall\,j$ is determined by the rank of $\Pi_{i}$. If the
map $\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ is unitary, this
implies that all
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
are unitary, and hence

$\displaystyle d_{\raisebox{0.0pt}{\tiny{$P$}}}$
$\displaystyle=\,d_{\raisebox{0.0pt}{\tiny{$S$}}}\,\operatorname{rank}(\Pi_{i})\quad\forall\,i.$
(A.52)

Conversely, this implies that all projectors $\Pi_{i}$ have the same rank
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$, which
must be an integer larger or equal to 1,
$d_{\raisebox{0.0pt}{\tiny{$P$}}}=\lambda d_{\raisebox{0.0pt}{\tiny{$S$}}}$
for $\lambda\in\mathbb{N}$. The implication of this for the correlation matrix
in (A.38) is that it is now square

$\displaystyle\Gamma_{U_{\text{unb}}}=\,\left[\begin{array}[]{c|c|c|c}\tikzmarkin[ver=stylegreen,linewidth=0mm,]{D1}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\tilde{A}_{00}$}}\tikzmarkend{D1}&\tikzmarkin[ver=styleorange,linewidth=0mm,]{DD2}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\tilde{A}_{01}$}}\tikzmarkend{DD2}&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{W1}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\tilde{A}_{02}$}}\tikzmarkend{W1}&\cdots\\\
\hline\cr\tikzmarkin[ver=stylegreen,linewidth=0mm,]{DD1}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\tilde{A}_{10}$}}\tikzmarkend{DD1}&\tikzmarkin[ver=styleorange,linewidth=0mm,]{D2}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\tilde{A}_{11}$}}\tikzmarkend{D2}&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{W3}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\tilde{A}_{12}$}}\tikzmarkend{W3}&\cdots\\\
\hline\cr\tikzmarkin[ver=stylegreen,linewidth=0mm,]{D33}\raisebox{2.0pt}{\footnotesize{$\rho_{00}\tilde{A}_{20}$}}\tikzmarkend{D33}&\tikzmarkin[ver=styleorange,linewidth=0mm,]{D44}\raisebox{2.0pt}{\footnotesize{$\rho_{11}\tilde{A}_{21}$}}\tikzmarkend{D44}&\tikzmarkin[ver=stylecyan,linewidth=0mm,]{D5}\raisebox{2.0pt}{\footnotesize{$\rho_{22}\tilde{A}_{22}$}}\tikzmarkend{D5}&\ldots\\\
\hline\cr\vdots&\vdots&\vdots&\ddots\end{array}\right]\,.$ (A.57)

In principle, one could initially consider a pointer with a Hilbert space
dimension larger than required for the desired $\lambda$. However, the energy
levels exceeding $\lambda d_{\raisebox{0.0pt}{\tiny{$S$}}}$ would have to be
truncated before the preparation step to avoid unnecessary additional energy
costs. The general form of the unitaries realising such unbiased measurement
procedures is summarised below.

Lemma 2 Let $\mathcal{M}_{U}(\beta)$ be an unbiased finite-resource ($\Delta
E<\infty,T=1/\beta>0$) measurement procedure with _unitary_ correlating step
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}$ using a pointer system
Hilbert space $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$ and Hamiltonian
$H_{\raisebox{0.0pt}{\tiny{$P$}}}$ with
$d_{\raisebox{0.0pt}{\tiny{$P$}}}=\lambda d_{\raisebox{0.0pt}{\tiny{$S$}}}$
for $\lambda\in\mathbb{N}$. The unitary map $U$ realizing the correlating
step, i.e.,
$\displaystyle\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}[\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}]$
$\displaystyle=\,U\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}U^{\dagger},$
(A.58) can be split into two consecutive unitary operations, $U=V\tilde{U}$,
where $\tilde{U}$ and $V$ are of the form $\displaystyle\tilde{U}$
$\displaystyle=\,\sum\limits_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tilde{U}^{\raisebox{0.0pt}{\tiny{$(i)$}}},$
(A.59) $\displaystyle V$
$\displaystyle=\,\sum\limits_{i,j=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\sum\limits_{m=1}^{\lambda}\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\left|\right.\\!{\tilde{\psi}_{m}^{\raisebox{0.0pt}{\tiny{$(i)$}}}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\tilde{\psi}_{m}^{\raisebox{0.0pt}{\tiny{$(j)$}}}}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$P$}}},$
(A.60) and $\tilde{U}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$ are arbitrary
unitaries on $\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$.

Proof of Lemma A.V. The structure of the unitaries in the correlating step can
be understood by noting that unitaries have only a single non-trivial Kraus
operator. The operators $\tilde{U}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$ in the
first unitary $\tilde{U}$ then simply correspond to the single Kraus operators
of the maps
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$
from Eq. (A.47), rearranging the joint density matrix only in the subspaces of
fixed
$\left|\right.\\!{i}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$,
creating the distinction between the sub-blocks $\tilde{A}_{ji}$ for different
$j$. The second part, realized by the unitary $V$ then just swaps these sub-
blocks, such that all $\tilde{A}_{0i}$ are left in the subspace corresponding
to $\left|\right.\\!{0}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$
and $\Pi_{i}$, all $\tilde{A}_{1i}$ are left in the subspace corresponding to
$\left|\right.\\!{1}\\!\left.\right\rangle_{\raisebox{0.0pt}{\tiny{$S$}}}$ and
$\Pi_{i}$, and so forth. The only freedom in choosing unitary correlation
steps for unbiased measurements hence lies in the choice of the
$\tilde{U}^{\raisebox{0.0pt}{\tiny{$(i)$}}}$. ∎

###  A.VI Unbiased and Non-invasive Measurements are Faithful

With the compact result of Lemmas A.IV.3 and A.V at hand, let us now briefly
return to the relationship between the properties unbiasedness and non-
invasiveness. As we have now seen, measurement procedures that are unbiased
for all initial system states $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ are
required to map the subspace of the joint system-pointer Hilbert space
corresponding to the image of the operator
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$
to the subspace corresponding to the image of
$\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}$ for all $i$. At this
point it becomes clear that, formulating the analogous statements to Lemmas
A.IV.3 and A.V for measurement procedures that are non-invasive instead of
unbiased for all $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$, results in maps from
the subspace corresponding to the image of the operator
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$
to itself. Since the image of
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$
is spanned by the set of non-trivial joint eigenvectors of the set of
projectors
$\\{\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{j}\\}_{j}$,
and the image of $\mathds{1}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}$ is
spanned by the set of non-trivial joint eigenvectors of the set of projectors
$\\{\left|\right.\\!{j}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{j}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\\}_{j}$,
a map that is supposed to satisfy both unbiasedness and non-invasiveness for
all $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$ must be a map from the image of
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathds{1}_{\raisebox{0.0pt}{\tiny{$P$}}}$
to the span of the set of non-trivial eigenvectors of
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}$,
such that
$\mathrm{tr}(\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=\rho_{ii}\
\forall i$. By construction, one thus has
$\sum_{i}\mathrm{tr}(\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\Pi_{i}\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=\sum_{i}\rho_{ii}=1$,
and the measurement is faithful.

###  A.VII Maximally Correlating Unbiased Measurements

To gain further insight into the fundamental limitations of non-ideal
measurements, we now wish to focus on a special case where Lemma A.V applies,
that is, an unbiased measurement procedure with unitary correlating step, such
that — at least for the purpose of controlling their interaction — the joint
system of pointer and measured system can be considered closed. That is,
procedures where
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{II}$}}}[\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}]=U\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\rho_{\raisebox{0.0pt}{\tiny{$P$}}}U^{\dagger}$.
Apart from this restriction, we will only consider preparation steps that
modify the temperature of the initial pointer system, i.e.
$\mathcal{E}_{\raisebox{0.0pt}{\tiny{$\mathrm{I}$}}}$ is a refrigeration step.
In such a scenario, $T=1/\beta$ might be the initial temperature of the
pointer, or, e.g., one below the ambient temperature, reached by investing
energy for cooling the pointer. For such measurement procedures, we now wish
to find the maximum attainable correlation between the system and pointer. As
we show, there is an algebraic maximum $C_{\text{max}}$ for the correlations
that can be unitarily created between the system and the thermal pointer,
regardless of the initial system state $\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$.
Recall the definition of the correlation function in Eq. (A.19), which we
rewrite as

$\displaystyle C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$
$\displaystyle:=\,\sum\limits_{i}\mathrm{tr}\bigl{[}\tilde{\Pi}_{ii}\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\bigr{]}\,,$
(A.61)

by making the association
$\tilde{\Pi}_{ij}=\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\Pi_{j}\;\forall\,i,j$.
By definition, the correlation function is only sensitive to terms appearing
along the diagonal w.r.t. any chosen common eigenbasis (with nontrivial
eigenvalues) of the set of operators $\tilde{\Pi}_{ii}$. We refer to the
subspace of
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}$
spanned by these eigenvectors as $\mathcal{H}_{\mathrm{corr}}$, and to its
complement as $\mathrm{H}_{\mathrm{nc}}$, such that
$\mathcal{H}_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\mathcal{H}_{\raisebox{0.0pt}{\tiny{$P$}}}=\mathcal{H}_{\mathrm{corr}}\oplus\mathcal{H}_{\mathrm{nc}}$.
In particular, this implies that the unitary transformation achieving the
algebraic maximum (over all unitaries $U_{\mathrm{unb}}$ realizing unbiased
measurement procedures in the sense of Lemma A.V)

$\displaystyle\max_{U_{\mathrm{unb}}}C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})=C_{\text{max}}$
(A.62)

is not unique since $C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ is
invariant under operations of the form $U_{\mathrm{corr}}\oplus
U_{\mathrm{nc}}$, where $U_{\mathrm{corr}}$ and $U_{\mathrm{nc}}$ act
nontrivially only on the subspaces $\mathcal{H}_{\mathrm{corr}}$ and
$\mathcal{H}_{\mathrm{nc}}$, respectively.

Within the orbit of all unitaries that one may perform (including those
corresponding to biased measurements) on
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}\otimes\tau_{\raisebox{0.0pt}{\tiny{$P$}}}$,
the global maximum value of the function
$C(\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}})$ is achieved when the state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ is block-diagonal w.r.t. to the
subspace partition into
$\mathcal{H}_{\mathrm{corr}}\oplus\mathcal{H}_{\mathrm{nc}}$, and the
eigenvalues of the joint final state restricted to the
$d_{\raisebox{0.0pt}{\tiny{$P$}}}$-dimensional correlated subspace
$\mathcal{H}_{\mathrm{corr}}$, given by
$\tilde{\rho}_{\mathrm{corr}}=\Pi_{\mathrm{corr}}\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}\Pi_{\mathrm{corr}}$
with
$\Pi_{\mathrm{corr}}=\sum_{i}\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\Pi_{i}$,
are the $d_{\raisebox{0.0pt}{\tiny{$P$}}}$ largest eigenvalues of
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$. These eigenvalues depend on
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$. However, we have to take into account
unbiasedness and the fact that we are looking for a unitary. In particular,
from Eq. (A.52) we know that the $\Pi_{i}$ must all have the same rank, namely
$\operatorname{rank}(\Pi_{i})=d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}=2^{N-1}\
\forall\,i$. It then becomes apparent that one is restricted to selecting the
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$ largest
eigenvalues of
$\rho_{\raisebox{0.0pt}{\tiny{$P$}}}=\tau_{\raisebox{0.0pt}{\tiny{$P$}}}(\beta)$
for each of the $d_{\raisebox{0.0pt}{\tiny{$S$}}}$ subspaces corresponding to
the image of a projector
$\left|\right.\\!{i}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{i}\\!\left.\right|\otimes\Pi_{i}$.
Since the assignment of eigenvalues to each subspace labelled by $i$ is the
same, and the corresponding matrix elements of the initial system state sum to
1, the maximal achievable correlation is independent of
$\rho_{\raisebox{0.0pt}{\tiny{$S$}}}$. In the notation of Eq. (A.1), this
corresponds to all probability populations that belong to the sector where
$(k=0)$, i.e.,

$\displaystyle C_{\text{max}}(\beta)$
$\displaystyle=\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\sum_{j=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}\rho_{ii}\,p^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{j}\,=\,\tfrac{1}{\mathcal{Z}}\sum_{i=0}^{d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1}e^{-\beta
E^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{i}}\;.$ (A.63)

As mentioned in Eq. (A.49), this function is independent of the system. The
remaining probability weights (i.e., the
$p^{\raisebox{0.0pt}{\tiny{$(k)$}}}_{j}$ for $k\neq 0$ and
$j=0,\ldots,d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}-1$)
are distributed in the non-correlated subspace.

The corresponding general form of the joint final state
$\tilde{\rho}_{\raisebox{0.0pt}{\tiny{$SP$}}}$ of any unitarily maximally
correlating unbiased measurement procedure (starting from an initially thermal
pointer state) can then be compactly specified in terms of its correlation
matrix $\Gamma_{U_{C_{\mathrm{max}}}}$ as defined in Eq. (A.38). To write
$\Gamma_{U_{C_{\mathrm{max}}}}$ in a simple form, let
$a^{\raisebox{0.0pt}{\tiny{$(0)$}}}_{i}$ for
$i=0,1,\ldots,d_{\raisebox{0.0pt}{\tiny{$S$}}}$ be
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}\times
d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$ Hermitian
matrices whose eigenvalues are the
$d_{\raisebox{0.0pt}{\tiny{$P$}}}/d_{\raisebox{0.0pt}{\tiny{$S$}}}$ largest
eigenvalues of the initial pointer state