Ideal Projective Measurements Have Infinite Resource Costs

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.

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 tells us that upon measuring a quantum system, its post-measurement state is given by one of the eigenstates of the measured observable and that 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.
On the other hand, the backbone of the theory of thermodynamics is formed by its three fundamental laws. Intense recent efforts in quantum thermodynamics (for reviews see Refs. [1][2][3]) have been able to place these laws on rigorous mathematical footing, see for example Refs. [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 [12,[14][15][16][17]. This is in apparent contradiction to 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, of course, prone to small errors (e.g., dark counts in a photon detector), which implies that the post-measurement state of the system is never truly pure. However, with technological advances making these errors ever smaller, one would assume an increasingly large thermodynamic cost as the post-measurement state of the system approaches a pure state.
Here, we resolve this apparent contradiction. We show that the cost of an ideal quantum measurement in a finite temperature environment is indeed infinite but that it may be approximated by non-ideal measurements at finite cost. We pursue an operational approach based on correlations between a system and a pointer, which allows us to make quantifiable statements about the corresponding energy cost. Within this framework, we identify three model-independent properties of ideal projec-tive measurements, which are unbiased, faithful, and noninvasive, as we explain in the following. We find that these properties cannot hold simultaneously for measurements using finite resources. Such non-ideal measurements may satisfy one of these properties exactly, while the others are approximated-the better, the more energy is provided. The framework is general enough to accommodate any reasonable measurement model, but we focus on the case where one measures a single qubit, for which we provide quantitative results. In particular, we discuss the trade-off between the energy cost of non-ideal measurements and the amount of information gained about the post-measurement state. 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 post-measurement state and what it means to obtain a 'result' [19,20]).
Past approaches to quantifying the relationship between thermodynamics and the cost of a quantum measurement almost always assumed that projective measurements can be carried out perfectly and that their cost can be attributed to the work value of the measurement outcome [5, [21][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 for whether the bound is achievable or not. These works inherently assume an infinite supply of pure states, circumventing the third law of thermodynamics and resulting in finite energy costs even for ideal measurements. Indeed, access to pure states implies the ability to perform ideal measurements, in turn allowing the creation of the required pure states. In other words, ideal measurements produce pure states, i.e., states at temperature zero.
However, when limited to thermal environments, measurements produce errors. That is, the pointer states are not perfectly correlated with those of the system. Such errors can be mitigated by either reducing the temperature of the environment, or by using larger and larger measurement devices. Both of these strategies can be quantified in terms of their thermodynamic cost for which we provide exact analytic arXiv:1805.11899v1 [quant-ph] 30 May 2018 results. Our results thus demonstrate that even the simplest quantum measurements on qubits are never for free.
Ideal measurements. Consider an initially unknown quantum system in a state ρ S and a measuring device, which we call a pointer, represented by an initial state ρ P . In order to measure the system, one must couple it to the pointer such that the two undergo a joint transformation which correlates them In an ideal measurement, the system and pointer are left perfectly correlated. That is, upon "observing" the pointer, one infers which pure state the system is left in with probability 1. More precisely, each eigenstate i ⟩ S of the measured observable of the system is assigned a set { ψ (i) n ⟩ P } n of orthogonal states of the pointer corresponding to a projector Π i = ∑ n ψ (i) n ⟩⟨ψ (i) n . The projectors are chosen to be orthogonal and to form a complete set, i.e., Π i Π j = δ ij Π i and ∑ i Π i = 1 P , such that each possible measurement outcome is represented and can be unambiguously identified. Upon finding the pointer in a state ψ (i) n ⟩ P (chosen to reflect i ⟩ S ), one then concludes that the measurement outcome is "i", and that the system is left in the state i ⟩ S . Up to arbitrary off-diagonal elements w.r.t. the basis { i ⟩ ⊗ ψ (j) n ⟩} i,j,n , such an ideal post-interaction state with perfect correlation is of the formρ where ρ ii = ⟨ i ρ S i ⟩ are the diagonal elements of the matrix ρ S w.r.t. the measured basis { i ⟩} S and ρ (i) is a pointer state that can be associated to one and only one of the outcomes i, that is, such that Π i ρ (j) = δ ij ρ (j) . The stateρ SP can be a pure entangled state, or simply a classically correlated state (if all off-diagonal elements vanish). The form ofρ SP in (2) is the result of an ideal measurement. Such an ideal measurement satisfies three fundamental properties, it is: (i) Unbiased. The probability of finding the pointer in a state associated with the outcome i after the interaction is the same as the probability of finding the system in the state i ⟩ S before the interaction, that is, A measurement is unbiased if the pointer reproduces the measurement statistics of the (undisturbed) system.
There is a one-to-one correspondence between the pointer outcome and the postmeasurement system state That is, the post-measurement state is perfectly correlated: On observing the pointer outcome i (associated to Π i ), one concludes that the system is left in the state i ⟩ S with certainty.
(iii) Non-invasive. The probability of finding the system in the state i ⟩ S is the same before and after the interaction with the pointer, i.e., This property only holds for the basis i ⟩ S and coherences appearing on the off-diagonal can, in general, be destroyed even for perfect measurements.
Note that any statements we make about the faithfulness of a measurement are dependent on the function we have chosen in (4). In principle one could choose more complicated functions or even measures for correlation. However, in our paradigm it is sufficient to be classically correlated in order to have perfect correlation. The advantage of this function is that it quantifies the probability that the pointer indicates an outcome which is correct (in the sense that the system is left in the corresponding state) and yields the maximal value 1 if and only if the post-interaction state is of the form (2).
Example. Consider a measurement of a single-qubit system using a pointer, itself composed of a single qubit, that was (somehow) prepared in the ground state. We may model the measurement procedure with a controlled NOT operation U CNOT = 0 ⟩⟨ 0 S ⊗1 P + 1 ⟩⟨ 1 S ⊗X P , where X = 0 ⟩⟨ 1 + 1 ⟩⟨ 0 . The post-measurement statẽ is of the form (2), meaning the measurement is unbiased, faithful, and non-invasive. Indeed, whenever both system and pointer have dimension d S = d P = d, and the pointer is initially in a pure state (w.l.o.g. the ground state), we can define a unitary Non-ideal measurements. Conversely, a measurement in which any of the properties (i) -(iii) fails to hold is nonideal. This is due to the fact that in general, the properties do not imply one another (see Fig. 1). In what follows we argue that faithful measurements (i.e., those which produce perfect correlations) are possible if and only if one can prepare pure states. Since, by the third law of thermodynamics, one cannot prepare pure states, 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 measured system and an extended pointer. In order for such a unitary to realize a faithful measurement, i.e., to create perfect correlations as per faithful unbiased noninvasive FIG. 1. Properties attributed to an ideal measurement. In a non-ideal measurement these three properties do not all hold simultaneously, and satisfying one of them does not imply any of the other two. However, in two cases, satisfying a pair of properties implies the third: As discussed in Appendix A.1, a measurement which is faithful and unbiased implies that it is also non-invasive, and a measurement that is faithful and noninvasive implies that it is unbiased. In general, a measurement which is unbiased and non-invasive does not imply that it is faithful.
Eq. (4), the rank of the final stateρ SP must be bounded from above by the dimension of the pointer d P (with this implies that the initial rank of the pointer ρ P must be 1, i.e., a pure state. For larger pointers, the pointer state need not be initially pure, but it cannot have full rank, i.e., one must have rank(ρ P ) ≤ d P d S . Practically, this corresponds to the requirement of preparing pure states at least for some non-trivial subspaces of the pointer. Thus, perfectly faithful (including ideal) measurements are not possible without a supply of pure states, i.e., states at absolute zero temperature (or unitarily equivalent states). In turn, such states with non-full rank require infinite resources or infinite time to be prepared and are prohibited by the third law of thermodynamics [12,[14][15][16][17]. Conversely, whenever the pointer state is not of full rank initially, operations such as U d in the example above allow one to achieve perfect correlation.
Given the above-mentioned problems with ideal measurements, we are thus interested in describing non-ideal measurements and in quantifying their energy cost. Since all real laboratory experiments take place in non-zero temperature environments, we argue that the natural state of a pointer is in thermodynamic equilibrium with its environment. At any finite temperature, the corresponding thermal state is of full rank (the set of nonfull rank states being of measure zero), thereby avoiding self-referential descriptions of the measurement process relying on a supply of pure states. We therefore take the pointer to be in the state τ P (β), with β = 1 T the inverse temperature. Any deviation from this state requires an input of work.
Having established that faithful measurements are not possible in such a case, we are interested in determining how closely they can be approximated. Here, we take the point of view that the crucial property to demand of any useful measurement is to be unbiased. This property guarantees that, with sufficiently many repetitions, one can obtain a mean value for the measured observable that accurately reflects the mean value of the underlying system state ρ S , and the degree of trust in this outcome can be quantified using standard statistical methods. Conversely, if one were to insist on the measurement being non-invasive, one would be able to perform repeated measurements on the same physical 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 ρ S . We therefore consider non-ideal measurements between a system ρ S and a thermal pointer τ P (β) with the property that the measurements are unbiased.
Energy cost of unbiased measurements. We wish to investigate the relation between the energy cost ∆E of realizing an unbiased measurement and the amount of correlation (as measured by C(ρ SP ) in Eq. (4)) created between the system and the pointer. Loosely speaking, we are interested in determining how faithful an unbiased non-ideal measurement can be and how much energy is required. The general structure of the maps realizing such measurements is discussed in detail in the Appendix. Without specifying the dimensions and Hamiltonians of the system and pointer, precise statements about the relation are not possible. We therefore focus on the simple but important special case where the system is a single qubit and the pointer consists of N non-interacting qubits. We focus on the case where the correlations between the system and pointer are generated unitarily such that the corresponding energy cost can be expressed without reference to external auxiliary systems. In this setup, the post-measurement state is thus where from now on we will take τ P (β) = 1 Z P ( 0 ⟩ ⟨ 0 + e −βE P 1 ⟩ ⟨ 1 ) with the partition function Z P = tr[e −βH P ]. Since we would like to bringρ SP as close as possible to the form in (2), we also consider cooling the pointer prior to this unitary correlating interaction in order to produce a better measurement. In principle, various models of refrigeration are available to chose from in [16], and although their specific energy costs, reachable temperatures, and assumed levels of control differ, the one feature they have in common is that reductions in temperature have non-zero energy costs that diverge as one approaches the ground state. For the purpose of illustrating the costs for our chosen special case, we choose a paradigm [16], where each qubit in the pointer is cooled by exactly one qubit from the refrigerator, i.e. the single-qubit fridge. We can thus describe the entire process by two consecutive unitary operations, which we call cooling and correlating. The total transformation on the system, pointer, and fridge is thus The measurement procedure. In the first step a pointer composed of N qubits is coupled to an N -qubit fridge and cooled from β to β0. In the second step a unitary correlates the pointer with the unknown qubit system. Fig. 2. It is clear that both unitaries drive the respective systems out of equilibrium and thus 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 i.e., W ≥ ∆E. The total cost in energy can be split into the sum of the two parts: cooling and correlating Our objective is to minimise the energy cost ∆E of performing a non-ideal measurement for a fixed value of the correlation function in (12), i.e., C(ρ SP ) = c < 1.
Minimal energy cost. We proceed to minimise each term in Eq. (9). In order to minimise ∆E cool , we turn to Ref. [16], where the optimal cooling cost is obtained for the single-qubit refrigeration paradigm chosen here. That is, cooling the pointer from The minimisation of ∆E corr , on the other hand, requires some additional effort. For this we are interested in determining where C max (β, N ) is the algebraic maximum that an unbiased measurement can achieve for the correlation function in (4). For the particular qubit case that we study Cost of a non-ideal projective measurement of a qubit system using 6 pointer qubits, each with energy gap EP and starting from inverse temperature β. For the purpose of illustration we choose room temperature (≈ 300 K) and an energy gap in the microwave regime such that βEP = 1 30. Each point on the horizontal axis of the shown parametric plot indicates the maximal algebraic correlation Cmax(β ′ , N = 6) achievable for a fixed value β ′ = βE F E P (or equivalently, fixed EF ), which is the result of cooling the 6-qubit pointer from β to β ′ using refrigeration qubits with gaps EF . For each of these values, the refrigeration cost ∆E cool and the cost ∆Ecorr of maximally correlating the thermal state at inverse temperature β ′ are shown on the vertical axis. The cost of cooling significantly dominates the cost of correlating. The inset shows the relevant energy scale for correlating the system and pointer. this maximum has a closed form expression 1 In the limit of infinite pointer size (N → ∞) for fixed temperature, or in the limit of zero temperature (β → ∞) for any pointer size the correlations become perfect 2 , i.e., lim N →∞ C max (β, N ) = lim β→∞ C max (β, N ) = 1. Note that since we assume unbiased measurements and in the limit we recover faithfulness then Fig. 1 implies that such a measurement is also non-invasive. Thus a measurement satisfying all three properties has an infinite cost without access to pure states. For arbitrary N and β, one may analytically construct the optimal unitary U opt that solves the optimisation problem in (11), i.e., find the unitary that achieves the maximum algebraic correlation at minimal energy cost. In addition, an analytic expression for ∆E corr can be specified in terms of β, N , and the system state 3 ρ S . We present the explicit details of these constructions and the derivation of (12) in Appendix A.5. From (12), we see explicitly that for fixed N the only way to achieve correlations higher than C max (β, N ) is to cool the pointer. We therefore now consider the scenario where starting at some finite inverse temperature β, we cool the pointer (β → β 0 > β) and then correlate it with the system to the algebraic maximum for that temperature C max (β 0 , N ). For any fixed N and finite temperature, achieving perfect correlations (C = 1) is hence only possible if infinite resource are invested to cool the pointer to the ground state, i.e., β 0 → ∞. At the same time, the additional cost for the perfectly correlating unitary U corr is always finite and given by ∆E Discussion. The projection postulate is a central concept within the foundation of quantum mechanics, asserting that ideal projective measurements leave the system in a pure state corresponding to the observed measurement outcome. All interpretations of quantum mechanics are (and indeed must be) compatible with this statement as well as with the Born rule assigning the associated probabilities. However, the existence of such "true" projections is usually assumed. Here, we discuss a self-contained (unitary) description of measurement processes from a thermodynamic point of view. We show that, when their existence is not assumed a priori, ideal projective measurements have an infinite energetic cost. This can also be interpreted as a challenge: If it is possible to provably perform a measurement for costs below the ones required by a self-contained description, it would imply the existence of true projections.
In this work, we take the stance that the basic premise of a projective measurement is to correlate the measured system with the measurement apparatus in such a way that the observation of the apparatus provides full information about the system. This correlation is brought about by an interaction between the system and appara-tus. We cast these features into three properties, arguing that ideal projective measurements are faithful, unbiased, and non-invasive, and show that a measurement simultaneously satisfying all of these properties has an infinite energy cost, unless pure states are freely available.
Nonetheless, non-ideal projective measurements may approximate the ideal case to arbitrary precision (correlations arbitrarily close to 1) at a finite energy cost. To quantify this energy cost, we consider the example of a measurement of a single qubit by an N -qubit apparatus. For this system, we provide analytic expressions of the minimal energy cost for unitarily achieving maximal correlation for any initial temperature and for any N . We find that the correlations can be increased with increasing energy input, and show that perfect correlations (and hence ideal measurements) necessarily require this energy cost to diverge.
Apart from the relevance for the foundations of quantum mechanics, the insight that ideal measurements carry an intrinsically diverging cost also sheds new light on different thermodynamic considerations. To interpret work as a random variable in the quantum regime, two projective measurements (TPM) are commonly assumed to characterize work [27][28][29]. If these measurements themselves carry an infinite work cost, an observation of microscopic work may not be strictly possible. Furthermore, in quantum engines driven by classical information, such as information-based engines [24,30], it would be highly challenging to approximate a positive work output, since the act of obtaining information not only costs the work value of the obtained bit string, but more generally the thermodynamic cost of the quantum to classical transition itself. In this Appendix, we give a detailed mathematical description of unbiased measurement procedures introduced in the main text. As we have argued there, ideal measurements that are unbiased, faithful, and noninvasive, are not generally implementable in finite time or with finite energy. In practice, measurements may nonetheless approximate such ideal measurements by investing energy and, loosely speaking, the approximation becomes better, if more energy is invested. To make this statement more precise, we are interested in explicitly determining the fundamental energy cost of projective 4 measurements. This requires a specific microscopic model for the measurement procedure, during which a quantum system S to be measured interacts with a pointer system P (the measurement device). In Sec. A.1, we formally define the mathematical models used for the system, pointer, and measurement procedure.

A.1. Framework
System. We consider a quantum system S with Hilbert space H S of dimension d S = dim(H S ). The system is initially in an arbitrary unknown quantum state that is represented by a density operator ρ S ∈ L(H S ), i.e., a Hermitean operator with tr(ρ S ) = 1 in the space of linear operators L(H S ) over the system Hilbert space H S . We are then interested in describing (projective) measurements of the system w.r.t. a basis { i ⟩ S } i of H S , which we take to be the eigenbasis of the system Hamiltonian H S , i.e., we can write 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 S = ̵ hΩ. That is, H S = C 2 , and the system Hamiltonian H S has eigenstates 0 ⟩ S and 1 ⟩ S and spectral decomposition Pointer. Similarly, we consider a pointer system P with Hilbert space H P of dimension d P = dim(H P ) and Hamiltonian H 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 τ (β) ∈ L(H P ) at ambient temperature T = 1 β, where for arbitrary inverse temperatures β, and Hamiltonians H P with eigenbasis { ψ n ⟩} n and corresponding eigenvalues E n such that H P ψ n ⟩ P = E n ψ n ⟩ P , the thermal (Gibbs) state is given by Here, Z P denotes the pointer's partition function Z P = tr(e −βH P ) = ∑ n e −Enβ . As an example that will also be of relevance later on, one could take the pointer system to be a collection of N qubits with , where H P ≡ (C 2 ) ⊗N , and the subscript i indicates an operator acting nontrivially only on the i-th (pointer) qubit, i.e., while 0 ⟩ i and 1 ⟩ i denote the ground and excited states of the i-th qubit, respectively. Here, we have further assumed that all of these pointer qubits have the same energy gap ω for simplicity.
Measurement procedure. We now wish to consider a measurement of the system's energy, i.e., of the observable H S , or, in other words, a projective measurement of the system in the energy eigenbasis 5 . The corresponding measurement procedure may be defined via a completely positive and trace-preserving (CPTP) Note 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 ρS to switch between the energy eigenbasis and the desired measurement basis.
that maps ρ SP = ρ S ⊗ τ P to a post-measurement statẽ ρ SP ∈ L(H 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 i ⟩ S , such that, upon finding the pointer in a state chosen to reflect i ⟩ S , one concludes that the measurement outcome is "i", and, in an ideal measurement, that the system is left in the state i ⟩ S . More specifically, three features of ideal projective measurements can be identified. As explained in the main text, a perfect measurement is In a non-ideal measurement procedure, the three properties of unbiasedness, faithfulness, and non-invasiveness can therefore not all be maintained simultaneously. In particular, none of the three alone implies any of the other two. For instance, replacing ρ ii in Eq. (2) with arbitrary probabilities p i ≠ ρ ii , one obtains a state satisfying (ii), but not (i) or (iii). Similarly, the state ρ S ⊗ ρ P obtained from a trivial interaction complies with (iii), but not with (ii) or (i), and, as we will discuss in more detail later, unbiased measurements are generally neither faithful nor non-invasive. However, faithful measurements that are also unbiased (non-invasive) imply non-invasiveness (unbiasedness), while the combination of (i) and (iii) does not imply (ii) 6 .

A.2. Example: 2-Qubit Measurement Procedures
Let us consider a measurement using a single pointer qubit, assuming that by some means it has been prepared in the ground state, i.e., ρ P = 0 ⟩⟨ 0 P . Let us further model the interaction with the pointer by applying a controlled NOT operation U CNOT = 0 ⟩⟨ 0 S ⊗1 P + 1 ⟩⟨ 1 S ⊗X P , with the usual Pauli operator X = 0 ⟩⟨ 1 + 1 ⟩⟨ 0 . Denoting the matrix elements of the initial state as ρ ij = ⟨ i ρ S j ⟩, we can then write the post-measurement statẽ ρ SP as As one can observe, system and pointer are now perfectly (classically) correlated in the sense that whenever the pointer is found in the state 0 ⟩ P ( 1 ⟩ P ), the system is left in the corresponding state 0 ⟩ S ( 1 ⟩ S ). In other words, for the choice Π i = i ⟩⟨ i P , we find that the measurement is faithful, i.e., The post-measurement system stateρ S = tr P [ρ SP ] = ∑ i ρ ii i ⟩⟨ i 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 ρ S . At the same time, the chosen unitary U CNOT guarantees that the probabilities for finding the pointer in the states 0 ⟩ P and 1 ⟩ P , match those of the original system state, i.e., for i = 0, 1 we have 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 CNOT with U SWAP = 00 ⟩⟨ 00 + 01 ⟩⟨ 10 + 10 ⟩⟨ 01 + 11 ⟩⟨ 11 , leaving the system in the state 0 ⟩ 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. In some sense this case is hence pathological, since the question of the energy cost of the projective measurement is just relayed from the original system to the pointer. Alternatively, consider the unitary 1 ⟩⟨ 1 S ⊗ 1 P + 0 ⟩⟨ 0 S ⊗ X P instead of U CNOT , both of which lead to the same correlation C(ρ SP ), but the probabilities for observing the two outcomes are now exchanged w.r.t. to ρ S , i.e., the pointer is found in the state 0 ⟩ P ( 1 ⟩ P ) with probability ρ 11 (ρ 00 ) after the interaction.
These examples are of course problematic 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 try to relax this assumption and assume instead that the pointer has been prepared at some finite non-vanishing temperature T = 1 β, such that ρ P = p 0 ⟩⟨ 0 P +(1−p) 1 ⟩⟨ 1 P for some p = (1 + e −ωβ ) −1 = Z −1 with 0 < p < 1. A quick calculation then reveals that the previously perfect correlations are reduced to C(ρ SP ) = p = Z −1 < 1 and that the measurement procedure using U CNOT is in general biased, i.e., 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 postmeasurement states (as we have shown in the main text), one may retain the unbiasedness of the measurement. That is, if we replace U CNOT by U unb. = 00 ⟩⟨ 00 + 01 ⟩⟨ 11 + 11 ⟩⟨ 10 + 10 ⟩⟨ 01 , we obtain the same imperfect correlation value C(ρ SP ) = p = Z −1 , but the unbiasedness condition of Eq. (A.4) is satisfied. 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, and the associated energy costs.

A.3. Unbiased Measurement Procedures
Despite the simplicity of the previous examples, these can help us to identify the basic structure and important properties of a general model of imperfect measurement procedures that we will attempt to construct now. 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.
In the following Sections A.3.I and A.3.II we will motivate and describe these steps in more detail.

A.3.I. Step I: Preparation
Before interacting with the system, the pointer can be prepared in a suitable quantum state ρ P at the expense of some initial energy investment ∆E I accounting for the (CPTP) transformation E I ∶ L(H P ) → L(H P ) mapping τ (β) to ρ P = E I [τ (β)]. 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 τ (β). The energy cost for reaching a particular state ρ P is bounded from below by the free energy difference, i.e., with ∆E P = tr H P (ρ P − τ (β)) and ∆S P = S ρ P − S τ (β) , and where S(ρ) = −tr ρ log(ρ) 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 capture 7 , 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 ≤ T , reaching a thermal state τ (β 0 ) with β 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 H 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.3.II. 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 E II ∶ L(H SP ) → L(H SP ) that maps ρ SP = ρ S ⊗ ρ P toρ SP = E II ρ SP . A particularly important special case is the case of unitary correlating maps U , i.e., such thatρ SP = U ρ SP U † , 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 In any case the generated correlations in principle can 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 S ) and a chosen basis of the pointer system. More specifically, we assign a set of orthogonal projectors n ⟩ = δ ij δ mn ) and ∑ i Π i = 1 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.3) for a single-qubit pointer to reflect the use of the more general projectors, i.e., we redefine the quantifier C(ρ SP ) as (A.9)

A.3.III. Unbiased measurements
We are now in a position to give a formal definition of what we consider as an abstract measurement procedure.
This definition includes, in particular, the case that we consider here, where the map E = E II ○ (1 S ⊗ E 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
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 E and projectors Π i , given that one has selected a suitable pointer system with Hilbert space H P and Hamiltonian H P . To this end, note that our previous example using U 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 take a first step towards unraveling the structure of unbiased measurements we formulate the following lemma.

Lemma 1
All unbiased finite-resource (∆E < ∞, T = 1 β > 0) measurement procedures M(β) with pointer system Hilbert space H P , Hamiltonian H P , and orthogonal projectors Π i can be realized by CPTP maps E of the form E = E II ○ (1 S ⊗ E I ), where E I is a CPTP map from L(H P ) to itself (achievable in finite time and satisfying ∆E I < ∞), and the CPTP map E II from L(H SP ) to itself has Kraus operators K l = ∑ i K (i) l with Proof of Lemma 1. Before we get into the technical details of the proof, let us phrase the Lemma 1 more informally. It states that the map E consists of an arbitrary (finite energy, ∆E < ∞) preparation of the pointer (E I ), followed by a map E II that maps the subspaces i ⟩ S to those corresponding to Π i , respectively. Moreover, note that unbiasedness of course implies that the pointer system must be large enough (d P ≥ d S ) to accommodate all the possible measurement outcomes. Let us then prove the lemma. As mentioned before, the CPTP map E may be separated into a map E I acting nontrivially only on the pointer Hilbert space, and a CPTP map E II acting on the resulting state ρ SP = ρ S ⊗ ρ P . Without loss of generality, we can then write the final stateρ SP = E II [ρ SP ] with respect to the product basis where we have indicated the columns corresponding to the subspaces of fixed vectors i ⟩ S and projectors Π 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 S ≤ 3 can be obtained by truncating the shown matrix by removing the corresponding rows and columns. The colored sub-blocks A i , B i , C i and so forth are d i ×d i block matrices in terms of which the unbiasedness condition of Def. 2 can be written as Crucially, the unbiasedness condition in Eq. (A.14) is to hold for all possible system states ρ S , and hence for all possible values of ρ ii . This, in turn, implies that all sub-blocks with subscript i must be proportional to ρ ii . That is, If one compares this structure with the original state ρ SP , which we can write with respect to the basis i ⟩ S as it becomes apparent that unbiasedness can be guaranteed for maps that connect the subspaces corresponding to i ⟩ S only with those corresponding to Π i . More precisely, each of these connecting maps can be viewed as an arbitrary CPTP map E (i) II from the d P -dimensional space by the vectors in the set { i ⟩ S ⊗ ψ n ⟩ P } n=0,...,d P −1 , where { ψ n ⟩ P } n is an arbitrary basis of H P , to the (d S × d i )dimensional space spanned by the vectors in the set . The Kraus operators for these CPTP maps are precisely the {K (i) l } l of Eq. (A.11) and in matrix notation we may denote these maps as Since the domains as well as the images for different i lie in orthogonal subspaces of H SP , the maps E (i)

II
can be combined to the map E II with Kraus operators 8 {K l = ∑ i K (i) l } l . That the unbiasedness condition is satisfied for these Kraus operators can be checked 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 ρ P has full rank and unbiasedness can only be achieved via maps of the form mentioned, as claimed in Lemma 1, which concludes the proof.
Inspecting again the example from Sec. A.2, one notes that the controlled NOT operation U 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 0 ⟩ P ) one observes that the measurement procedure using U CNOT in the correlating step becomes unbiased because some of the sub-blocks A i and B i are only trivially proportional to ρ ii . In particular, Having understood the general structure of all unbiased measurements, we now want to turn to some specific instances of such measurement procedures.

A.4. Extremal measurement procedures
With the help of Lemma 1 we can now describe the set of all unbiased measurement procedures for a given quantum system ρ S and pointer. The measurement procedures 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 procedure 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 ∆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 ∆E?
Since we assume that the system state ρ S is unknown before the measurement, the correlation measureC that we are interested in optimizing is obtained from averaging C(ρ SP ) from Eq. (4) over all system states. We observe that for any particular systems state ρ S , the correlation measure C(ρ SP ) does not depend on any of the matrix elements of ρ S except for those on the diagonal, i.e., Averaging over all states ρ S is here hence equivalent to an average over all probability distributions corresponding to the diagonal of ρ S . Moreover, for each of these values ρ ii (i = 0, . . . , d S − 1), the average over all probability distributions results in the value 1 d S , such that the average of C(ρ SP ) is given bȳ Despite this simple form ofC, the optimization over all pointer systems and operations thereon is a daunting task. Indeed, even for a fixed pointer at initial temperature T = 1 β, 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 [32,33]. 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 [34] 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 Φ + ⟩ SP = 00 ⟩ + 11 ⟩ √ 2 and ρ SP = 1 2 00 ⟩⟨ 00 + 11 ⟩⟨ 11 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 E II of the CPTP map E must require higher levels of control than a corresponding unitary realization due to the requirement of realizing nonunitary maps E II as unitaries on a larger Hilbert space. Specifically, any CPTP map E II can be thought of as a unitary on a larger Hilbert space H SP ⊗ H E (with a factoring initial condition) [35], that is, one may write any E II as for some unitary U SP E on H SP ⊗ H E and for some pure state χ ⟩ ∈ H 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 E II is realized unitarily, such that with U U † = U † U = 1 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 E 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 E 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 E II means that it is inefficient (in terms of energy cost) to use a pointer Hilbert space H P whose dimension is not an integer multiple of the system dimension. This can be explained in the following way. Inspection of the maps in Eq. (A.16), one notes that E (i) Conversely, this implies that all projectors Π i have the same rank d P d S , which must be an integer larger or equal to 1, d P = λd S for λ ∈ N. In principle, one could initially consider a pointer with a Hilbert space dimension larger than required for the desired λ. However, the energy levels exceeding λd S would have to be truncated before the preparation step to avoid unnecessary additional energy costs. We can then give the general form of the unitaries involved in realizing such unbiased measurement procedures.

Lemma 2
Let M U (β) be an unbiased finite-resource (∆E < ∞, T = 1 β > 0) measurement procedure with unitary correlating step E II using a pointer system Hilbert space H P and Hamiltonian H P with d P = λd S for λ ∈ N. The unitary map U realizing the correlating step, i.e., can be split into two consecutive unitary operations, U = VŨ , whereŨ and V are of the form andŨ (i) are arbitrary unitaries on H P .
Proof of Lemma 2. 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 operatorsŨ (i) in the first unitaryŨ then simply correspond to the single Kraus operators of the maps E (i) II from Eq. (A.16), rearranging the joint density matrix only in the subspaces of fixed i ⟩ S , creating the distinction between the sub-blocksÃ i ,B i ,C i , and so forth. The second part, realized by the unitary V then just swaps these sub-blocks, such that allÃ i are left in the subspace corresponding to 0 ⟩ S and Π i , allB i are left in the subspace corresponding to 1 ⟩ S and Π i , and so forth. The only freedom in choosing unitary correlation steps for unbiased measurements hence lies in the choice of theŨ (i) .

A.5. Construction of the optimal unitary
In the previous section we have identified the structure of all unbiased non-ideal (i.e., finite-resource) measurement procedures that create correlations unitarily. We are now interested in quantifying the energy cost of such measurement procedures. Since the energy cost crucially depends on the chosen Hamiltonians for system and pointer, we focus on the interesting case where the measured system is a single qubit and the measurement apparatus is modelled as an N -qubit pointer. That is, we assume the system Hamiltonian to have en-ergy gap E S , such that H S = E S 1 ⟩⟨ 1 S , and for each of the pointer qubits we have the same local Hamiltonian H Pn ≡ E P 1 ⟩⟨ 1 Pn . The total pointer Hamiltonian is then H P = ∑ n H Pn with eigenvalues E i for i ∈ {0, 1, . . . , 2 N − 1}, which we label such that they are ordered by non-decreasing values such that E i ≤ E j for i < j. In terms of the energy gap E P we have E i = tE P , where t = 0 if i = 0, and for i ≥ 1 the prefactor t is implicitly given by The initial thermal state of the N -qubit pointer is thus We now construct the unitary that correlates the system and pointer as much as possible for the least energy, i.e., the optimal unitary U opt , given bỹ that solves the optimisation problem in Eq. (11), that is, Here, C max (β, N ) is the maximum algebraic correlation for an N −qubit thermal pointer and is given by N k e −kE P β , (A. 29) and the cost of correlating is given by the energy difference of the initial and final states From (4) we have that the correlation function is which sums the populations in the correlated subspaces. It will be useful to decompose the energy in (A.30) in terms of the correlated and anti-correlated subspaces where the combined system-pointer Hamiltonian is H SP = H S ⊗ I P + I S ⊗ H P . In our model, we take From Eq. (A.21) we know that since we are looking for a unitary, then the Π i must all have the same rank, namely rank The class of unitaries that achieves C max (β, N ) rearranges the elements of ρ SP to place the heaviest (in terms of probabilities) populations into these subspaces. The unitaries in this class are therefore permutation matrices that can be understood as the product of SWAP operations.
From this constraint we already know which elements of the post-interaction stateρ SP are assigned to which subspaces. In order to minimise the energy, one must therefore find the optimal assignment of the energy eigenbasis to these subspaces. The unitary which minimises ∆E corr assigns the lowest energies to heaviest populations, which are in the correlated subspaces.
Since the initial energy is fixed, i.e., where the p i are given in (A.26), the term to minimise is the post-interaction energy E(ρ SP ). To begin with consider minimising the post-interaction energy in the correlated subspace E ↑↑ (ρ SP ). Since the interaction we implement is unbiased, from Eq. (A.16) it must be that the correlated subspaces are proportional to the system weights ρ ii . More precisely, i ⟩⟨ i ⊗ Π iρSP ∝ ρ ii I ∀ i.
When the system ρ S is a qubit and ρ P is composed of N qubits, the post-interaction state is given by Since C max (β, N ) is achieved by placing the heaviest populations in the correlated subspaces, the matricesÃ 0 andB 1 are permutations of precisely the same elements, which we denote by the vector p = (p 0 , p 1 , ⋯p 2 N −1 −1 ), where the p i are given in (A.26). We fix the elements to be in non-increasing order p 0 ≥ p 1 ≥ ⋯. Let S N be the set of all energies associated to each level i ⟩ of the pointer in (A.26) also ordered in non-decreasing order. For example a 3-qubit pointer with gap E P = 1 and 0 1 1 1 2 2 2 3 0 1 1 1 2 2 2 3 x 0 x 1 x 2 x 3 vanishing ground state would be associated with the set S 3 = {0, 1, 1, 1, 2, 2, 2, 3}. We now rewrite the energy in the correlated subspace as where x = (x i ) i ∈ R 2 N −1 is the vector with elements x i (with 0 ≤ i ≤ 2 N −1 − 1) which themselves are the sum of two energies from the subspaces Π 0 and Π 1 , respectively, i.e., from the matricesÃ 0 andB 1 There are several statements we can make immediately about the set S N . First, it has 2 N elements which are distributed binomially such that the energy kE P appears N k times. Second, the sum of the elements is constant, which in turn implies that the sum of all energy pairs is constant, namely Since x i ≥ 0 ∀i, this means we can treat the set {x i c} i as a probability distribution, a fact that will come in useful later. Let X denote the set of all possible vectors x. This set can also be understood as the set of all possible ways of choosing pairs from S N without replacement (modulated by the weights ρ 00 and ρ 11 ). The size of X (denoted X ) grows factorially with N , so enumerating all possible x and testing for the optimal vector is not a feasible option.
However, as we will now argue, there is a unique vector x * ∈ X that presents the optimal solution. To find this vector, let us first slightly reformulate the problem and define a vector p ↑ = (p 2 N −1 −1 , ⋯, p 1 , p 0 ) which corresponds to the reversed order of the components of p, i.e., p 2 N −1 −1 ≤ ⋯ ≤ p 1 ≤ p 0 . Now, for any given vector x = (x 0 , . . . , x 2 N −1 −1 ), we can also define the reversely ordered vector x ↓ = (x 2 N −1 −1 , . . . , x 0 ) such that E ↑↑ (ρ SP ) = x ⋅ p = x ↓ ⋅ p ↑ . The solution x * ↓ for the minimization problem min x∈X (x ↓ ⋅ p ↑ ) = x * ↓ ⋅ p ↑ is then given by the vector x * ↓ that pairs the smallest weights p i with the largest values x i . To be more specific, let v, w ∈ R 2 N −1 be two normalised vectors with their components ordered in non-increasing order such that v 0 ≥ v 1 ⋯ and w 0 ≥ w 1 ⋯. We say that v majorises w, written v ≻ w, when ∑ k i v i ≥ ∑ k i w i ∀ k. In other words the cumulative sum of the components of the vector v grows faster than for w. The vector x * ↓ that presents the solution to the optimization problem is hence the vector that majorises all other vectors x ∈ X, i.e., x * ↓ ≻ x ∀ x ∈ X. (A.40) The corresponding reversed vector x * is then the solution of min x∈X (x ⋅ p) = x * ⋅ p. The vector x * is constructed by maximising each x * i term by term. In doing so, we populate the components of x * in reverse order such that x 2 N −1 −1 ≥ ⋯ ≥ x 0 . This construction is equivalent to picking nearest neighbour pairs from the set S N , starting with the largest pair, as illustrated in Fig. (A.1). For the case that ρ 00 ≥ ρ 11 (see (A.37)) we must choose E l ≥ E j in the pair (when ρ 00 < ρ 11 we choose E l ≤ E j ). The general prescription for each component is given by By construction, the majorisation of Eq. (A.40) is satisfied. Finally, it remains to find the optimal vector y * that achieves the global minimum in the anti-correlated subspace. By constructing x * from nearest neighbour pairs in S N , we have fixed the energy eigenbasis in the subspaces such that We now apply the same argument to the anti-correlated subspace to find the vector y * , which again pairs the highest energies with the lowest populations.
Thus from the above and (A.34) we have In the case that ρ S is unknown and thus maximally mixed this reduces to where E S (the gap of the system) no longer plays any role. Finally, observe that the cost of correlating is always finite. If one substitutes for the p i as they are in (A.26) and takes the limit in which the pointer is composed of N -pure states, i.e. β → ∞, then the cost of correlating is given by