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Ideal Projective Measurements Have Infinite Resource Costs

Yelena Guryanova, Nicolai Friis, Marcus Huber

arXiv:1805.11899v3quant-ph

TL;DR

Ideal projective measurements promise pure post-measurement states, while the third law rules out reaching pure ground states with finite resources. This paper analyzes measurement costs through system-pointer correlations and shows that perfect correlation is infinitely costly, though unbiased approximations are possible at finite cost.

  • Problem

    Ideal projective measurements produce pure post-measurement states, raising the question of how this can accord with the third law’s restriction on finite-resource cooling to pure ground states.

    Source: §1 · p. 1

  • Method

    The paper analyzes measurements through system-pointer correlations, identifies three model-independent ideal-measurement properties, and derives cost results for a qubit measured by an N-qubit pointer.

    Source: §1 · p. 1, §7 · p. 6

  • Results

    Faithful measurements require pure states unavailable at finite resources, while unbiased measurements can approximate ideal ones to arbitrary precision at finite energy cost.

    Source: §7 · p. 5

  • Takeaways & Limitations

    For a qubit measured by an N-qubit pointer, the paper gives analytic minimal energy costs for unitarily achieving maximal correlation across initial temperatures and pointer sizes.

    Source: §7 · p. 6

    The three properties of ideal measurement do not suffice to characterize the quantum-to-classical transition, which also involves robustness to pointer perturbations.

    Source: §7 · p. 6

Abstract

from arXiv · show

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

Ideal projective measurements leave a quantum system in a pure state correlated with the outcome, raising a tension with the third law’s finite-resource limit on reaching pure ground states. The paper resolves this tension by showing that ideal measurements have infinite resource cost, while finite-resource measurements incur quantifiable costs.

1 Introduction

  • The third law forbids cooling a quantum system to its ground state in finite time with finite resources, apparently conflicting with the pure post-measurement state of an ideal projection.The projection postulate assigns the measured system a pure eigenstate with Born-rule probability.Source: §1 · p. 1
  • Ideal projective measurements have infinite cost with finite resources, and the framework identifies unbiasedness, faithfulness, and non-invasiveness as properties that cannot all hold together under finite resources.The analysis gives quantitative results for an example case and avoids claims about the measurement problem.Source: §1 · p. 1
  • Past approaches obtained finite costs by assuming unlimited supplies of pure states, whereas measurements using thermal environments produce errors that can be reduced by cooling or larger devices.Both strategies have thermodynamic costs, for which the paper provides exact analytic results; even the simplest qubit measurements are not free.Source: §1 · pp. 1–2

2 Ideal measurements

An ideal measurement correlates each system eigenstate with a corresponding pointer outcome, yielding a post-interaction state that can be entangled or classically correlated. The paper characterizes ideality through unbiasedness, faithfulness, and non-invasiveness.

2 Ideal measurements

  • The ideal post-interaction state associates each system basis state with a corresponding pointer state, with diagonal weights matching the system’s initial populations.The resulting correlation may be classical or entangled; off-diagonal terms are not fixed by the stated form.Source: §2 · p. 2
  • Unbiasedness means the pointer reproduces the system’s measurement statistics, faithfulness means each pointer outcome identifies one post-measurement system state, and non-invasiveness preserves basis-state probabilities.Non-invasiveness does not require preservation of off-diagonal coherences.Source: §2 · p. 2
  • The paper’s faithfulness measure is the average probability of correctly inferring the post-measurement state from the pointer, reaching one exactly for the specified ideally correlated state when the measurement is unbiased.Classical correlation suffices for perfect correlation under this measure; quantum correlations are not necessary.Source: §2 · p. 2
  • A controlled-NOT interaction between a qubit system and a ground-state qubit pointer realizes an ideal measurement in the stated example.The construction is also extended to equal-dimensional systems and pointers with an initially pure pointer state.Source: §2 · p. 2

3 Non-ideal measurements

Finite-resource measurements cannot satisfy all three ideal properties simultaneously, and faithfulness requires pointer states that cannot be prepared with finite resources. The paper therefore considers unbiased measurements and how closely they can approach faithfulness.

3 Non-ideal measurements

  • For a fixed input state, faithfulness plus unbiasedness implies non-invasiveness, and faithfulness plus non-invasiveness implies unbiasedness; unbiasedness plus non-invasiveness implies faithfulness only across all input states.A measurement is non-ideal if any of the three properties fails.Source: §3 · p. 3, Figure 1 · p. 3
  • A faithful measurement requires a pointer state of rank at most dP/dS; because non-full-rank states require infinite resources under the third law, faithful and ideal measurements are not physically feasible with finite resources.For equal system and pointer dimensions, the pointer must start pure; larger pointers may be mixed but not full rank.Source: §3 · p. 3
  • The analysis uses a thermal pointer, which has full rank at every finite temperature, and any departure from its thermal state requires work.The paper prioritizes unbiasedness because repeated measurements then yield a mean matching the underlying system’s measured-observable mean.Source: §3 · p. 3

4 Maximally faithful measurements

For unbiased unitary measurements, maximal correlation is bounded by the thermal pointer’s populations, and perfect correlation requires sufficiently low pointer rank. The section characterizes the optimal correlation structure and the conditions for reaching it.

4 Maximally faithful measurements

  • For unbiased unitary measurements, the largest pointer populations are assigned to the correlated subspace, yielding a dimension- and temperature-dependent maximum correlation Cmax.The maximum is independent of the system’s diagonal input-state populations.Source: §4 · p. 4
  • For any unbiased measurement, C cannot exceed Cmax, which can be interpreted as the maximum probability that the post-measurement system matches the pointer’s state.An appropriate pointer basis can be selected for a measurement achieving Cmax.Source: §4 · p. 4
  • Perfect correlation is possible only at absolute zero or when the pointer rank is at most dP/dS; the unitary measurement’s remaining choices can still affect its energy cost.The system’s state change is not completely fixed by the measurement form described in the passage.Source: §4 · p. 4
  • The exact lowest-energy unbiased, maximally correlated state depends on both the system state and the pointer Hamiltonian, which the paper diagonalizes for an example case.The passage refers to an appendix for step-by-step instructions.Source: §4 · p. 4

5 Energy cost of measurements

The measurement protocol first cools a pointer and then correlates it with the system, with total energy cost divided between those operations. Pointer size and temperature determine the maximum achievable correlation.

5 Energy cost of measurements

  • Increasing pointer size at fixed temperature or lowering its initial temperature at fixed size increases the maximum correlation Cmax.The example uses a single-qubit system and an N-qubit pointer with identical, initially non-interacting qubits.Source: §5 · p. 4
  • The protocol models refrigeration with one fridge qubit per pointer qubit, so the fridge and pointer have the same size.The pointer is cooled before the unitary that correlates it with the system.Source: §5 · p. 4, Figure 2 · p. 5
  • A larger fridge could reduce cooling cost by a constant factor, but the cost still diverges near zero temperature unless the fridge is infinite.Source: §5 · p. 5
  • The total energy cost is the sum of pointer cooling and the subsequent system-pointer correlation costs, ∆E = ∆EI + ∆EII.The protocol uses consecutive unitary operations for cooling and correlating.Source: §5 · p. 5

6 Minimal energy cost

The section considers minimum energy costs for cooling and correlating a pointer to its maximal achievable correlation. The optimal correlation unitary has an analytic cost, while perfect correlation requires limiting resources.

6 Minimal energy cost

  • For odd N, the analysis seeks the minimum correlation-stage energy cost while achieving Cmax(β).Source: §6 · p. 5
  • An optimal unitary achieves algebraically maximal correlation at minimum energy cost for arbitrary N and β, with an analytic ∆EII expression in terms of β, N, and ρS.Source: §6 · p. 5
  • For finite N, correlations above Cmax(β) require cooling the pointer; reaching the ground state in finite time costs infinite energy in this cooling paradigm.Source: §6 · p. 5
  • As N approaches infinity, achieving Cmax(β) costs finite energy but requires infinite time or full control over N-body interactions.Source: §6 · p. 5

7 Discussion

The discussion concludes that ideal finite-time projective measurements have infinite cost, while finite-energy measurements can approximate them. It reports analytic costs for a qubit-pointer model and notes that ideal-measurement properties do not capture robustness in the quantum-to-classical transition.

7 Discussion

  • Ideal finite-time projective measurements have infinite cost when their existence is not assumed.Source: §7 · p. 5
  • In the six-qubit example, the plotted refrigeration cost significantly exceeds the cost of correlating the system and pointer.The figure considers cooling from room temperature and tracks maximal correlation as the cooling setting changes.Source: §7 · p. 6, Figure 3 · p. 6
  • For a single-qubit system and an N-qubit pointer, the paper gives analytic minimum energy costs for maximal correlation at any initial temperature and pointer size.Correlation can be increased by enlarging or cooling the pointer.Source: §7 · p. 6
  • Unbiased, faithful, and non-invasive properties do not suffice to characterize the quantum-to-classical transition, which also involves robustness to pointer perturbations.Source: §7 · p. 6
  • The authors identify measurement imperfection and its work cost as relevant to work-fluctuation studies and quantum information-based engines.They also point to bounding syndrome-measurement power consumption in quantum computers.Source: §7 · p. 6

Appendix: Mathematical Model for Measurement Procedures

The appendix motivates quantifying projective-measurement costs and focuses on projective measurements because general POVMs can be realized with an auxiliary system.

Appendix: Mathematical Model for Measurement Procedures

A.I Framework

The framework models a measurement as a transformation of an unknown system state and a thermal pointer into a joint post-measurement state. The system is measured in its energy eigenbasis, and pointer states are assigned to the system outcomes.

A.I Framework

  • The system is described by an arbitrary unknown density operator and measured in the eigenbasis of its Hamiltonian.For the qubit example, the ground state has zero energy and the gap is E_S = ℏΩ.Source: A.I Framework · p. 8
  • The measurement procedure is a CPTP map on the joint system–pointer state, initially formed from the system state and the pointer’s thermal state.A measurement in another orthonormal basis can be included by an additional unitary transformation, with its energy cost.Source: A.I Framework · p. 8
  • Pointer outcomes correspond to orthogonal projectors that form a complete set, with pointer states assigned to system eigenstates.In an ideal measurement, observing outcome i leaves the system in |i⟩_S.Source: A.I Framework · p. 9

A.II Example: 2-Qubit Measurements

The two-qubit example uses a pointer qubit and a controlled-NOT interaction to illustrate ideal correlations and how finite-temperature pointer preparation degrades them. It also shows that correlation, unbiasedness, and faithfulness need not coincide.

A.II Example: 2-Qubit Measurements

  • With the pointer prepared in its ground state, the controlled-NOT interaction leaves system and pointer perfectly correlated, and the measurement is faithful, unbiased, and non-invasive.The pointer probabilities match the initial system statistics, while the system’s diagonal elements are preserved.Source: A.II Example: 2-Qubit Measurements · p. 9
  • Unbiasedness and strong correlation do not imply one another: alternative interactions can retain one while failing the other.One example exchanges the pointer outcome probabilities relative to the system statistics; another is unbiased but generally uncorrelated.Source: A.II Example: 2-Qubit Measurements · p. 9
  • At finite temperature, the controlled-NOT measurement has correlation C = p = Z^-1 < 1 and is generally biased.The pointer state is a thermal mixture with 0 < p < 1.Source: A.II Example: 2-Qubit Measurements · p. 9
  • Replacing the controlled-NOT with an alternative interaction preserves unbiasedness at the same imperfect correlation C = p = Z^-1.Finite-resource procedures cannot prepare the pure pointer states required for ideal measurements.Source: A.II Example: 2-Qubit Measurements · p. 10

A.III Non-Ideal Measurement Procedures

The section examines how unbiasedness, faithfulness, and non-invasiveness relate for non-ideal measurements, distinguishing properties that hold for a particular input state from those required for all states. Under finite-resource constraints, faithfulness cannot be retained, so the analysis focuses on unbiased measurements with the largest possible correlations.

A.III Non-Ideal Measurement Procedures

  • No single property alone distinguishes ideal from non-ideal measurements, and examples show that each can hold without the other two.The examples include a faithful procedure with neither unbiasedness nor non-invasiveness, a trivial non-invasive procedure, and an unbiased state exchange.Source: A.III Non-Ideal Measurement Procedures · p. 10
  • For a fixed input state, faithful and unbiased measurements imply non-invasiveness, and faithful and non-invasive measurements imply unbiasedness.The section proves these two implications while showing that the remaining pair does not generally imply faithfulness for a fixed state.Source: A.III Non-Ideal Measurement Procedures · p. 10
  • For a particular input state, unbiasedness and non-invasiveness need not imply faithfulness: the example has correlation C = 3/8.Its reduced system and pointer states equal the initial system state, despite the imperfect correlation.Source: A.III Non-Ideal Measurement Procedures · p. 10
  • The section cautions that properties satisfied for one input state may fail for others, motivating definitions that require them to hold for all system states.For example, an equal mixture can satisfy all three properties even though the procedure is biased and imperfectly correlated for almost all other states.Source: A.III Non-Ideal Measurement Procedures · p. 11
  • For measurements required to satisfy properties for every input state, unbiasedness and non-invasiveness together imply faithfulness, making the measurement ideal.The argument uses the structure of maps satisfying each property for arbitrary system states.Source: A.III Non-Ideal Measurement Procedures · p. 11
  • Finite resources rule out faithful realistic measurements, leaving a choice between unbiasedness and non-invasiveness; the section pursues unbiased measurements with maximal correlations and minimal energy costs.The text argues that biased measurements lacking faithfulness are of little use because their outcomes provide no certainty about either system state.Source: A.III Non-Ideal Measurement Procedures · p. 11

A.IV General Unbiased Measurements

The measurement model separates the process into pointer preparation and system–pointer correlation.

A.IV General Unbiased Measurements

A.IV.1 Step I: Preparation

Preparing the pointer in a chosen state costs at least its free-energy difference from the initial thermal state, while the exact work cost depends on available control and auxiliary degrees of freedom. The paper therefore assumes refrigeration to a lower-temperature thermal state, a practical restriction that does not explore the pointer’s full Hilbert space.

A.IV.1 Step I: Preparation

  • The energy cost of preparing a pointer state is bounded below by its free-energy difference from the initial thermal state.The exact work cost depends on control and available auxiliary degrees of freedom and may exceed this bound.Source: A.IV.1 Step I: Preparation · p. 11
  • Preparing arbitrary states is difficult to quantify precisely because the work cost depends on control and auxiliary resources.A CPTP preparation can be represented using an auxiliary pure state, whose own preparation cost must then be accounted for.Source: A.IV.1 Step I: Preparation · p. 11
  • The paper restricts preparation to refrigeration, lowering the pointer temperature from T to T0 ≤ T to reach a thermal state.This practical assumption does not explore the entire pointer Hilbert space; the resulting thermal state is energetically optimal at fixed entropy and maximizes entropy at fixed energy.Source: A.IV.1 Step I: Preparation · p. 12

A.IV.2 Step II: Correlating

The section characterizes correlating maps and structural constraints on unbiased measurements, including how maximal correlation is independent of the system state and assigns the largest pointer-state eigenvalues to correlated subspaces. It develops an energy-minimizing construction for a qubit and an N-qubit pointer, generalizes it to arbitrary systems with maximally mixed states and thermal pointers, and notes that the optimal state need not be unique.

A.IV.2 Step II: Correlating

  • The correlating step is a CPTP interaction between system and pointer; classical correlations in the measurement basis and a chosen pointer basis suffice, though genuine quantum correlations may also arise.When the correlating map is unitary, the joint system can be treated as closed for that step, allowing its energy cost to be calculated.Source: A.IV.2 Step II: Correlating · p. 12

A.IV.3 Unbiased measurements

  • Finite-resource unbiased measurements split into pointer preparation and a correlating map that sends each system-basis subspace to its corresponding pointer-outcome subspace.At nonzero temperature, finite resources require the overall measurement map to be rank non-increasing; unbiasedness also requires dP ≥ dS.Source: A.IV.3 Unbiased measurements · pp. 12–14

A.V Extremal Measurements

  • For unitary unbiased measurements, the pointer dimension must be dP = λdS for an integer λ ≥ 1, and the correlating unitary factors into rearrangements and swaps.The rearrangements act within fixed system-basis subspaces; the swaps place the resulting sub-blocks in the corresponding system and pointer-outcome subspaces.Source: A.V Extremal Measurements · pp. 15–16

A.VI Unbiased and Non-invasive Measurements are Faithful

  • A measurement that is both unbiased and non-invasive for every system state is faithful: outcome i occurs with probability ρii.The two requirements constrain the map to the joint subspace associated with system state i and pointer outcome i.Source: A.VI Unbiased and Non-invasive… · p. 16

A.VII Maximally Correlating Unbiased Measurements

  • For an unbiased unitary correlator acting on a thermal pointer, an algebraic maximum correlation is attainable regardless of the initial system state, and the maximizing unitary need not be unique.The correlation is invariant under operations acting separately within the correlated and non-correlated subspaces.Source: A.VII Maximally Correlating Unbiased… · p. 16
  • For unbiased maximal correlation, each system outcome receives the dP/dS largest pointer eigenvalues, making Cmax independent of ρS; remaining weights occupy the non-correlated subspace.The maximizing state is block diagonal across the correlated and non-correlated subspaces.Source: A.VII Maximally Correlating Unbiased… · p. 16

A.VIII Optimally Correlating Unitary for a Single-Qubit System and N-Qubit Pointer

  • For a qubit and N-qubit pointer, minimizing energy first in the correlated and then in the non-correlated subspaces yields a global minimum among maximally correlating unitaries.The example assumes a maximally mixed initial qubit and pointer qubits with energy gaps matching the system gap.Source: A.VIII Optimally Correlating Unitary… · pp. 17–19
  • The minimum-energy unbiased state with maximal correlation is diagonal in the joint Hamiltonian eigenbasis, reducing the remaining basis choices to permutations.Unbiasedness and maximal correlation impose block structure but do not require off-diagonal elements.Source: A.VIII Optimally Correlating Unitary… · p. 19
  • In the correlated subspace, the energy minimum pairs nearest-neighbor pointer levels, assigning the smallest thermal weights to the largest pair-energy sums.This pairing constructs the vector that majorizes all other allowed pairings.Source: A.VIII Optimally Correlating Unitary… · p. 19
  • After fixing the pointer energy basis through the correlated-subspace optimum, minimizing non-correlated-subspace energy reduces to choosing permutations of the remaining populations.Source: A.VIII Optimally Correlating Unitary… · p. 20
  • The correlation cost is finite and no longer depends on the system energy gap.Source: A.VIII Optimally Correlating Unitary… · p. 20
  • For a pure pointer, optimal correlation reaches C = 1, and with ground-state pointer qubits the cost is independent of N and equals the cost of exciting one qubit, modulated by ρ00 = ρ11 = 1/2.The passage states that perfect correlation is the maximal correlation in the pure-pointer limit.Source: A.VIII Optimally Correlating Unitary… · p. 20

A.IX Construction of the optimal unitary for arbitrary systems

  • The qubit-pointer construction is an energy minimum and generalizes to unknown maximally mixed systems and thermal pointers with arbitrary Hamiltonians.Source: A.IX Construction of the optimal… · p. 21
  • The general construction assigns the pointer’s largest dP/dS eigenvalues to the correlated subspace, then pairs remaining largest weights with smallest energies in the non-correlated subspace.Source: A.IX Construction of the optimal… · p. 21
  • The optimal final state is generally degenerate because it depends on how the energy basis and excitations are represented.The authors leave investigation of any preferred state within this class open.Source: A.IX Construction of the optimal… · p. 21
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