Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2205.11411

Comment on "Thermodynamic Principle for Quantum Metrology"

Shane Dooley, Michael J. Kewming, Mark T. Mitchison, John Goold

Published May 23, 2022• 1 citation
arXivHTMLPDF

Abstract

In Phys. Rev. Lett. 128, 200501 (2022) the authors consider the thermodynamic cost of quantum metrology. One of the main results is $\mathcal{S} \geq \log(2) \| h_λ\|^{-2} F_Q [ψ_λ]$, which purports to relate the Shannon entropy $\mathcal{S}$ of an optimal measurement (i.e., in the basis of the symmetric logarithmic derivative) to the quantum Fisher information $F_Q$ of the pure state $|ψ_λ\rangle$. However, we show that in the setting considered by the authors we have $\mathcal{S} = \log(2)$ and $\| h_λ\|^{2} = \max_{ψ_λ} F_Q[ψ_λ]$, so that their inequality reduces to the trivial inequality $\max_{ψ_λ} F_Q[ψ_λ] \geq F_Q[ψ_λ]$, and does not in fact relate the entropy $\mathcal{S}$ to the quantum Fisher information. Moreover, for pure state quantum metrology, there exist optimal measurements (though not in the basis of the symmetric logarithmic derivative) for which $0 \leq \mathcal{S} \leq \log(2)$, leading to violations of the inequality for some states $|ψ_λ\rangle$.

DOI
10.48550/arxiv.2205.11411

Related papers

Minimum-cost quantum measurements for quantum information

Petros Wallden, Vedran Dunjko, Erika Andersson

Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.

arXiv:1312.5205•Journal of Physics A Mathematical and Theoretical•Mar 10, 2014•18 citations

Quantum Fisher Information for Entropy of Gibbs States

Francis J. Headley

We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.

quant-pharXiv:2603.16456•Mar 18, 2026•0 citations

Extended convexity of quantum Fisher information in quantum metrology

S. Alipour, A. T. Rezakhani

We prove an extended convexity for quantum Fisher information of a mixed state with a given convex decomposition. This convexity introduces a bound which has two parts: (i) The classical part associated with the Fisher information of the probability distribution of the states contributing to the decomposition, and (ii) the quantum part given by the average quantum Fisher information of the states in this decomposition. Next we use a non-Hermitian extension of a symmetric logarithmic derivative in order to obtain another upper bound on quantum Fisher information, which helps to derive a closed form for the bound in evolutions having the semigroup property. We enhance the extended convexity with this concept of a non-Hermitian symmetric logarithmic derivative (which we show is computable) to lay out a general metrology framework where the dynamics is described by a quantum channel and derive the ultimate precision limit for open-system quantum metrology. We illustrate our results and their applications through two examples where we also demonstrate how the extended convexity allows identifying a crossover between quantum and classical behaviors for metrology.

arXiv:1403.8033•Physical Review A•Apr 7, 2015•55 citations

Maximum Fisher information in mixed state quantum systems

Alessandra Luati

We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.

arXiv:math/0410095•The Annals of Statistics•Aug 1, 2004•26 citations

MAXIMUM FISHER INFORMATION IN MIXED STATE QUANTUM SYSTEMS

Alessandra Luati

We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states. 1. Introduction. Quantum

•Jan 1, 2004•28 citations

Fisher information in quantum statistics

Ole Eiler Barndorff-Nielsen, Richard David Gill

Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.

arXiv:quant-ph/9808009•Journal of Physics A Mathematical and General•Jun 9, 2000•208 citations
In Phys. Rev. Lett. 128, 200501 (2022) the authors consider the thermodynamic cost of quantum metrology. One of the main results is S≥log⁡(2)∥hλ∥−2FQ[ψλ]\mathcal{S} \geq \log(2) \| h_λ\|^{-2} F_Q [ψ_λ]S≥log(2)∥hλ​∥−2FQ​[ψλ​], which purports to relate the Shannon entropy S\mathcal{S}S of an optimal measurement (i.e., in the basis of the symmetric logarithmic derivative) to the quantum Fisher information FQF_QFQ​ of the pure state ∣ψλ⟩|ψ_λ\rangle∣ψλ​⟩. However, we show that in the setting considered by the authors we have S=log⁡(2)\mathcal{S} = \log(2)S=log(2) and ∥hλ∥2=max⁡ψλFQ[ψλ]\| h_λ\|^{2} = \max_{ψ_λ} F_Q[ψ_λ]∥hλ​∥2=maxψλ​​FQ​[ψλ​], so that their inequality reduces to the trivial inequality max⁡ψλFQ[ψλ]≥FQ[ψλ]\max_{ψ_λ} F_Q[ψ_λ] \geq F_Q[ψ_λ]maxψλ​​FQ​[ψλ​]≥FQ​[ψλ​], and does not in fact relate the entropy S\mathcal{S}S to the quantum Fisher information. Moreover, for pure state quantum metrology, there exist optimal measurements (though not in the basis of the symmetric logarithmic derivative) for which 0≤S≤log⁡(2)0 \leq \mathcal{S} \leq \log(2)0≤S≤log(2), leading to violations of the inequality for some states ∣ψλ⟩|ψ_λ\rangle∣ψλ​⟩.
Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.
We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.
We prove an extended convexity for quantum Fisher information of a mixed state with a given convex decomposition. This convexity introduces a bound which has two parts: (i) The classical part associated with the Fisher information of the probability distribution of the states contributing to the decomposition, and (ii) the quantum part given by the average quantum Fisher information of the states in this decomposition. Next we use a non-Hermitian extension of a symmetric logarithmic derivative in order to obtain another upper bound on quantum Fisher information, which helps to derive a closed form for the bound in evolutions having the semigroup property. We enhance the extended convexity with this concept of a non-Hermitian symmetric logarithmic derivative (which we show is computable) to lay out a general metrology framework where the dynamics is described by a quantum channel and derive the ultimate precision limit for open-system quantum metrology. We illustrate our results and their applications through two examples where we also demonstrate how the extended convexity allows identifying a crossover between quantum and classical behaviors for metrology.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states.
We deal with the maximization of classical Fisher information in a quantum system depending on an unknown parameter. This problem has been raised by physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101–102] a quantum counterpart of classical Fisher information, which has been found to constitute an upper bound for classical information itself [Braunstein and Caves (1994) Phys. Rev. Lett. 72 3439–3443]. It has then become of relevant interest among statisticians, who investigated the relations between classical and quantum information and derived a condition for equality in the particular case of two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys. A 33 4481–4490]. In this paper we show that this condition holds even in the more general setting of two-dimensional mixed state systems. We also derive the expression of the maximum Fisher information achievable and its relation with that attainable in pure states. 1. Introduction. Quantum
Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett. 72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.