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:2210.15738 · Entropy

Entropy of Quantum Measurements

Stanley Gudder

Published Nov 18, 2022• 0 citations
arXivHTMLPDF

Abstract

If a is a quantum effect and ρ is a state, we define the ρ-entropy Sa(ρ) which gives the amount of uncertainty that a measurement of a provides about ρ. The smaller Sa(ρ) is, the more information a measurement of a gives about ρ. In Entropy for Effects, we provide bounds on Sa(ρ) and show that if a+b is an effect, then Sa+b(ρ)≥Sa(ρ)+Sb(ρ). We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their ρ-entropies. In Entropy of Observables and Instruments, we employ Sa(ρ) to define the ρ-entropy SA(ρ) for an observable A. We show that SA(ρ) directly provides the ρ-entropy SI(ρ) for an instrument I. We establish bounds for SA(ρ) and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider ρ-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.

DOI
10.3390/e24111686

Related papers

Sequential Products of Quantum Measurements

Stanley Gudder

Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.

arXiv:2108.07925•Jun 6, 2007•2 citations

Properties of Sequential Products

Stanley Gudder

Our basic concept is the set $\mathcal{E}(H)$ of effects on a finite dimensional complex Hilbert space $H$. If $a,b\in\mathcal{E}(H)$, we define the sequential product $a[\mathcal{I}]b$ of $a$ then $b$. The sequential product depends on the operation $\mathcal{I}$ used to measure $a$. We begin by studying the properties of this sequential product. It is observed that $b\mapsto a[\mathcal{I}]b$ is an additive, convex morphism and we show by examples that $a\mapsto a[\mathcal{I}]b$ enjoys very few conditions. This is because a measurement of $a$ can interfere with a later measurement of $b$. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on $a[\mathcal{I}]b$ that imply commutativity. We introduce the concept of an effect $b$ given an effect $a$ and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.

arXiv:2307.16327•Jul 30, 2023•0 citations

Further results on the relative entropy

Matthew J. Donald

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

Mathematical Proceedings of the Cambridge Philosophical Society•Mar 1, 1987•48 citations

The information entropy of quantum-mechanical states

A Stotland, A. A Pomeransky, E Bachmat +1 more

It is well known that a Shannon-based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the von Neumann entropy as the corresponding quantum-mechanical definition. But the latter is problematic from the quantum information point of view. Consequently, we introduce a new definition of entropy that reflects the inherent uncertainty of quantum-mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.

arXiv:quant-ph/0401021•Europhysics Letters (EPL)•Aug 25, 2004•11 citations

Dual Quantum Instruments and Sub-observables

Stanley Gudder

We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Holero and constant state instruments. Various conjectures for future research are presented.

arXiv:2302.12243•Feb 23, 2023•0 citations

Coarse-Graining of Observables

Stanley Gudder

We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have 0–1 stochastic kernels. We next consider finite observables and instruments and show that in these cases, stochastic kernels are replaced by stochastic matrices. We also show that coarse-graining is the same as post-processing in this finite case. We then consider sequential products of observables and discuss the sequential product of a post-processed observable with another observable. We briefly discuss SIC observables and the example of qubit observables.

arXiv:2109.07019•Quantum Reports•Oct 3, 2022•1 citation
If a is a quantum effect and ρ is a state, we define the ρ-entropy Sa(ρ) which gives the amount of uncertainty that a measurement of a provides about ρ. The smaller Sa(ρ) is, the more information a measurement of a gives about ρ. In Entropy for Effects, we provide bounds on Sa(ρ) and show that if a+b is an effect, then Sa+b(ρ)≥Sa(ρ)+Sb(ρ). We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their ρ-entropies. In Entropy of Observables and Instruments, we employ Sa(ρ) to define the ρ-entropy SA(ρ) for an observable A. We show that SA(ρ) directly provides the ρ-entropy SI(ρ) for an instrument I. We establish bounds for SA(ρ) and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider ρ-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.
Our basic structure is a finite-dimensional complex Hilbert space HHH. We point out that the set of effects on HHH form a convex effect algebra. Although the set of operators on HHH also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.
Our basic concept is the set E(H)\mathcal{E}(H)E(H) of effects on a finite dimensional complex Hilbert space HHH. If a,b∈E(H)a,b\in\mathcal{E}(H)a,b∈E(H), we define the sequential product a[I]ba[\mathcal{I}]ba[I]b of aaa then bbb. The sequential product depends on the operation I\mathcal{I}I used to measure aaa. We begin by studying the properties of this sequential product. It is observed that b↦a[I]bb\mapsto a[\mathcal{I}]bb↦a[I]b is an additive, convex morphism and we show by examples that a↦a[I]ba\mapsto a[\mathcal{I}]ba↦a[I]b enjoys very few conditions. This is because a measurement of aaa can interfere with a later measurement of bbb. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on a[I]ba[\mathcal{I}]ba[I]b that imply commutativity. We introduce the concept of an effect bbb given an effect aaa and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.
Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.
It is well known that a Shannon-based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the von Neumann entropy as the corresponding quantum-mechanical definition. But the latter is problematic from the quantum information point of view. Consequently, we introduce a new definition of entropy that reflects the inherent uncertainty of quantum-mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.
We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Holero and constant state instruments. Various conjectures for future research are presented.
We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have 0–1 stochastic kernels. We next consider finite observables and instruments and show that in these cases, stochastic kernels are replaced by stochastic matrices. We also show that coarse-graining is the same as post-processing in this finite case. We then consider sequential products of observables and discuss the sequential product of a post-processed observable with another observable. We briefly discuss SIC observables and the example of qubit observables.