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:1501.05271 · Physical Review A

Geometric lower bound for a quantum coherence measure

Diego Paiva Pires, Lucas C. Céleri, Diogo O. Soares-Pinto

Published Apr 23, 2015• 106 citations
arXivHTMLPDF

Abstract

Nowadays, geometric tools are being used to treat a huge class of problems of quantum information science. By understanding the interplay between the geometry of the state space and information-theoretic quantities, it is possible to obtain less trivial and more robust physical constraints on quantum systems. Here we establish a geometric lower bound for the Wigner-Yanase skew information (WYSI), a well-known information-theoretic quantity recently recognized as a proper quantum coherence measure. In the case of a mixed state evolving under unitary dynamics generated by a given observable, the WYSI between the state and the observable is bounded from below by the rate of change of the state's statistical distinguishability from its initial value. Our result shows that, since WYSI fits in the class of Petz's metrics, this lower bound is the change rate of its respective geodesic distance on quantum state space. The geometric approach is advantageous because it raises several physical interpretations of this inequality under the same theoretical umbrella.

DOI
10.1103/physreva.91.042330

Related papers

Metric adjusted skew information

Frank Hansen

We extend the concept of Wigner-Yanase-Dyson skew information to something we call "metric adjusted skew information" (of a state with respect to a conserved observable). This "skew information" is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the "lambda-skew information," parametrized by a lambda in (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.

arXiv:math-ph/0607049•Proceedings of the National Academy of Sciences•Jul 17, 2008•145 citations

Generalized geometric speed limits for quantum observables

Jacob Bringewatt, Zach Steffen, Martin A. Ritter +6 more

Leveraging quantum information geometry, we derive generalized quantum speed limits on the rate of change of the expectation values of observables. These bounds subsume and, for Hilbert space dimension ≥ 3 , tighten existing bounds—in some cases by an arbitrarily large multiplicative constant. Our theoretical results are supported by illustrative examples and an experimental demonstration using a superconducting qutrit. We also derive two upper bounds on the generalized quantum Fisher information in terms of the condition number of the density matrix. One of these bounds applies only to coherent dynamics and depends also on the variance of the Hamiltonian. The other bound depends also on the so-called Wigner-Yanase skew information. These bounds generalize well-known bounds on the symmetric logarithmic derivative quantum Fisher information and are tighter than the existing bounds for sufficiently mixed states (e.g., for sufficiently high temperature thermal states).

arXiv:2409.04544•Physical Review Research•Aug 20, 2025•1 citation

Wigner–Yanase information on quantum state space: The geometric approach

Paolo Gibilisco, Tommaso Isola

In the search of appropriate Riemannian metrics on quantum state space, the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner–Yanase information. Using a well-known approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner–Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible.

arXiv:math/0304170•Journal of Mathematical Physics•Aug 21, 2003•105 citations

Geometric uncertainty relations on Wigner–Yanase skew information

Bin Chen, Pan Lian

Abstract We formulate uncertainty relations based on Wigner–Yanase skew information. By using the Kirillov–Kostant–Souriau Kähler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kähler structure plays a vital role in the proof.

Journal of Physics A Mathematical and Theoretical•Jun 7, 2023•4 citations

Wigner-Yanase Skew Information and Uncertainty Relations

Shunlong Luo

The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.

Physical Review Letters•Oct 29, 2003•380 citations

Generalized Wigner-Yanase skew information and the affiliated inequality

Ma-Cheng Yang, Cong‐Feng Qiao

A family of skew information quantities is obtained, in which the well-known Wigner-Yanase skew information and quantum Fisher information stand as special cases. A transparent proof of convexity of the generalized skew information is given, implying a simple proof of the Wigner-Yanase-Dyson conjecture. We find in this work an exact skew information inequality for a qubit system, which may be regarded as the information counterpart of the uncertainty relation. A lower bound for the generalized skew information of a pair of incompatible observables in arbitrary dimension and also the upper bound for a qubit system are achieved.

arXiv:2205.06988•Physical Review A•Nov 1, 2022•11 citations
Nowadays, geometric tools are being used to treat a huge class of problems of quantum information science. By understanding the interplay between the geometry of the state space and information-theoretic quantities, it is possible to obtain less trivial and more robust physical constraints on quantum systems. Here we establish a geometric lower bound for the Wigner-Yanase skew information (WYSI), a well-known information-theoretic quantity recently recognized as a proper quantum coherence measure. In the case of a mixed state evolving under unitary dynamics generated by a given observable, the WYSI between the state and the observable is bounded from below by the rate of change of the state's statistical distinguishability from its initial value. Our result shows that, since WYSI fits in the class of Petz's metrics, this lower bound is the change rate of its respective geodesic distance on quantum state space. The geometric approach is advantageous because it raises several physical interpretations of this inequality under the same theoretical umbrella.
We extend the concept of Wigner-Yanase-Dyson skew information to something we call "metric adjusted skew information" (of a state with respect to a conserved observable). This "skew information" is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the "lambda-skew information," parametrized by a lambda in (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.
Leveraging quantum information geometry, we derive generalized quantum speed limits on the rate of change of the expectation values of observables. These bounds subsume and, for Hilbert space dimension ≥ 3 , tighten existing bounds—in some cases by an arbitrarily large multiplicative constant. Our theoretical results are supported by illustrative examples and an experimental demonstration using a superconducting qutrit. We also derive two upper bounds on the generalized quantum Fisher information in terms of the condition number of the density matrix. One of these bounds applies only to coherent dynamics and depends also on the variance of the Hamiltonian. The other bound depends also on the so-called Wigner-Yanase skew information. These bounds generalize well-known bounds on the symmetric logarithmic derivative quantum Fisher information and are tighter than the existing bounds for sufficiently mixed states (e.g., for sufficiently high temperature thermal states).
In the search of appropriate Riemannian metrics on quantum state space, the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner–Yanase information. Using a well-known approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner–Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible.
Abstract We formulate uncertainty relations based on Wigner–Yanase skew information. By using the Kirillov–Kostant–Souriau Kähler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kähler structure plays a vital role in the proof.
The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.
A family of skew information quantities is obtained, in which the well-known Wigner-Yanase skew information and quantum Fisher information stand as special cases. A transparent proof of convexity of the generalized skew information is given, implying a simple proof of the Wigner-Yanase-Dyson conjecture. We find in this work an exact skew information inequality for a qubit system, which may be regarded as the information counterpart of the uncertainty relation. A lower bound for the generalized skew information of a pair of incompatible observables in arbitrary dimension and also the upper bound for a qubit system are achieved.