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数理解析研究所講究録

Upper bound on our knowledge about noncommuting observables for a qubit system (Non-Commutative Analysis and Micro-Macro Duality)

Takahiro Sagawa, Yuji Kurotani, Masahito Ueda

Published Jul 1, 2008• 0 citations
OpenAlex

Abstract

We formulate the accuracy of quantum measurement for a qubit system in terms of a 3 by 3 matrix.This matrix, which we refer to as the accuracy matrix, can be calculated from a POVM corresponding to the quantum measurement.Based on the accuracy matrix, we derive new $trad\triangleright 0ff$ relations between the measurement accuracy of two or three noncommuting observables of a qubit system.These trade-off relations offer a quantitative information-theoretic representation of Bohr's principle of complementarity.We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by quantum measurement.

Related papers

Accuracy matrix in a generalized simultaneous measurement of a qubit system

Takahiro Sagawa, Masahito Ueda

We formulate the accuracy of a quantum measurement for a qubit (spin-1/2) system in terms of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. This matrix, which we refer to as the accuracy matrix, can be calculated from a positive operator-valued measure (POVM) corresponding to the quantum measurement. Based on the accuracy matrix, we derive trade-off relations between the measurement accuracy of two or three noncommuting observables of a qubit system. These trade-off relations offer a quantitative information-theoretic representation of Bohr's principle of complementarity. They can be interpreted as the uncertainty relations between measurement errors in simultaneous measurements and also as the trade-off relations between the measurement error and back-action of the measurement. A no-cloning inequality is derived from the trade-off relations. Furthermore, our formulation and the results obtained can be applied to analyze quantum-state tomography. We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by a finite number of quantum measurements.

arXiv:0707.3872•Physical Review A•Jan 14, 2008•9 citations

Upper bound on our knowledge about noncommuting observables for a qubit system

Yuji Kurotani, Takahiro Sagawa, Masahito Ueda

A trade-off relation on our knowledge about two noncommuting observables of a qubit system in simultaneous measurement is formulated. The obtained inequality offers a quantitative information-theoretic representation of Bohr's principle of complementarity, and can be interpreted as a trade-off relation on the asymptotic accuracy of the maximum-likelihood estimation of the probability distributions of observables.

arXiv:quant-ph/0703130•Physical Review A•Aug 22, 2007•9 citations

Unsharp Measurements and Joint Measurability

H. S. Karthik, A. R. Usha Devi, A. K. Rajagopal

We give an overview of joint unsharp measurements \nof non-commuting observables using positive operator \nvalued measures (POVMs). We exemplify the role \nplayed by joint measurability of POVMs in entropic \nuncertainty relation for Alice’s pair of non-commuting \nobservables in the presence of Bob’s entangled quantum \nmemory. We show that Bob should record the \noutcomes of incompatible (non-jointly measurable) \nPOVMs in his quantum memory so as to beat the entropic \nuncertainty bound. In other words, in addition \nto the presence of entangled Alice–Bob state, implementing \nincompatible POVMs at Bob’s end is necessary \nto beat the uncertainty bound and hence predict \nthe outcomes of non-commuting observables with \nimproved precision. We also explore the implications \nof joint measurability to validate a moment matrix \nconstructed from average pairwise correlations of \nthree dichotomic non-commuting qubit observables. \nWe prove that a classically acceptable moment matrix \n– which ascertains the existence of a legitimate \njoint probability distribution for the outcomes of all \nthe three dichotomic observables – could be realized if \nand only if compatible POVMs are employed.

Current Science•Dec 1, 2015•0 citations

Quantum estimation for quantum technology

Matteo G. A. Paris

Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology.

arXiv:0804.2981•Apr 18, 2008•0 citations

Quantum estimation for quantum technology

Matteo G. A. Paris

Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology. 1.

•Apr 18, 2008•0 citations

Universal Sample Complexity Bounds in Quantum Learning Theory via Fisher Information Matrix

Hyukgun Kwon, Seok Hyung Lie, Liang Jiang

We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.

quant-pharXiv:2602.21510•PRX Quantum•Feb 26, 2026•0 citations
We formulate the accuracy of quantum measurement for a qubit system in terms of a 3 by 3 matrix.This matrix, which we refer to as the accuracy matrix, can be calculated from a POVM corresponding to the quantum measurement.Based on the accuracy matrix, we derive new trad▹0fftrad\triangleright 0fftrad▹0ff relations between the measurement accuracy of two or three noncommuting observables of a qubit system.These trade-off relations offer a quantitative information-theoretic representation of Bohr's principle of complementarity.We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by quantum measurement.
We formulate the accuracy of a quantum measurement for a qubit (spin-1/2) system in terms of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. This matrix, which we refer to as the accuracy matrix, can be calculated from a positive operator-valued measure (POVM) corresponding to the quantum measurement. Based on the accuracy matrix, we derive trade-off relations between the measurement accuracy of two or three noncommuting observables of a qubit system. These trade-off relations offer a quantitative information-theoretic representation of Bohr's principle of complementarity. They can be interpreted as the uncertainty relations between measurement errors in simultaneous measurements and also as the trade-off relations between the measurement error and back-action of the measurement. A no-cloning inequality is derived from the trade-off relations. Furthermore, our formulation and the results obtained can be applied to analyze quantum-state tomography. We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by a finite number of quantum measurements.
We give an overview of joint unsharp measurements \nof non-commuting observables using positive operator \nvalued measures (POVMs). We exemplify the role \nplayed by joint measurability of POVMs in entropic \nuncertainty relation for Alice’s pair of non-commuting \nobservables in the presence of Bob’s entangled quantum \nmemory. We show that Bob should record the \noutcomes of incompatible (non-jointly measurable) \nPOVMs in his quantum memory so as to beat the entropic \nuncertainty bound. In other words, in addition \nto the presence of entangled Alice–Bob state, implementing \nincompatible POVMs at Bob’s end is necessary \nto beat the uncertainty bound and hence predict \nthe outcomes of non-commuting observables with \nimproved precision. We also explore the implications \nof joint measurability to validate a moment matrix \nconstructed from average pairwise correlations of \nthree dichotomic non-commuting qubit observables. \nWe prove that a classically acceptable moment matrix \n– which ascertains the existence of a legitimate \njoint probability distribution for the outcomes of all \nthe three dichotomic observables – could be realized if \nand only if compatible POVMs are employed.
Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology.
Several quantities of interest in quantum information, including entanglement and purity, are nonlinear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e the determination of the most precise estimator, unavoidably involves an optimization procedure. We review local quantum estimation theory and present explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states. Estimability of a parameter is defined in terms of the quantum signal-to-noise ratio and the number of measurements needed to achieve a given relative error. The connections between the optmization procedure and the geometry of quantum statistical models are discussed. Our analysis allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices in quantum technology. 1.
We show that the sample complexity required in quantum learning theory within a general parametric framework is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error, with high success probability under maximum-likelihood estimation. Notably, both the upper and lower bounds are determined by the supremum of the maximum diagonal entry of the inverse Fisher information matrix. We then apply the general bounds to Pauli channel learning and Pauli expectation value learning, which serve as representative tasks in quantum channel and state learning, respectively, in the asymptotic small-error regime. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value learning without quantum memory by comparing the quantum Fisher information matrix and the classical Fisher information matrix. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators, deriving the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli channel learning with entangled probes. We highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, the learning sample complexity is governed by the inverse Fisher information matrix, which is the central quantity in quantum metrology that determines the ultimate achievable mean squared error.