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:1203.3329

On quantum information

Adam Paszkiewicz, Tomasz Sobieszek

Published Mar 15, 2012• 1 citation
arXivHTMLPDF

Abstract

We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator ρ, tr ρ=1. Let I(P)\in R be defined for any partition P = (P_1,...,P_m), P_1+ ... +P_m=1_H, P_i \in proj H$ and let I(P_i Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q_1,..., Q_n), \sum Q_j = 1_H and P_iQ_j = Q_j P_i, tr ρ P_iQ_j = tr ρ P_i tr ρ Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I.

DOI
10.48550/arxiv.1203.3329

Related papers

Quantum Information Theory

Yorick Hardy, Willi-Hans Steeb

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as $$tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } $$ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states $$\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|$$ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and $$pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1$$

Birkhäuser Basel eBooks•Jan 1, 2001•8 citations

QUANTUM INFORMATION, ENTANGLEMENT AND ENTROPY

Siddharth Sharma

In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix.

•Dec 5, 2020•1 citation

Information-entropy and the space of decoherence functions in generalised quantum theory

Isham, C. J., Linden, N.

In standard quantum theory, the ideas of information-entropy and of pure states are closely linked. States are represented by density matrices $ρ$ on a Hilbert space and the information-entropy $-tr(ρ\logρ)$ is minimised on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalised quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions we demonstrate that there are no `pure' decoherence functions. The main content of the paper is a new notion of information-entropy in generalised quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information-entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information-entropy suggests an intrinsic way of giving a consistent set selection criterion.

arXiv:quant-ph/9612035•Dec 12, 1996•1 citation

Some More Results of Generalized Parametric R-Norm Entropy in Quantum Logic

Amandeep Singh

In the present communication, I deal with the modelling of Generalized Parametric R-Norm Information measure in quantum logic. This information measure is extended to Joint and Conditional measure in quantum logic.

International Journal of Science and Research (IJSR)•Aug 5, 2022•0 citations

Information gain in quantum continual measurements

Alberto Barchielli, G. Lupieri

Inspired by works on information transmission through quantum channels, we propose the use of a couple of mutual entropies to quantify the efficiency of continual measurement schemes in extracting information on the measured quantum system. Properties of these measures of information are studied and bounds on them are derived.

arXiv:quant-ph/0612010•Dec 1, 2006•0 citations

Bargmann invariants of Gaussian states

Jianwei Xu

Given a set of ordered quantum states, described by density operators {ρj}j=1n, the Bargmann invariant of {ρj}j=1n is defined as tr(ρ1ρ2, …, ρn). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr(ρ1ρ2, …, ρn) for any m-mode bosonic Gaussian states {ρj}j=1n in terms of the means and covariance matrices of {ρj}j=1n. We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.

arXiv:2508.07155•Journal of Mathematical Physics•May 1, 2026•0 citations
We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator ρ, tr ρ=1. Let I(P)\in R be defined for any partition P = (P_1,...,P_m), P_1+ ... +P_m=1_H, P_i \in proj H$ and let I(P_i Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q_1,..., Q_n), \sum Q_j = 1_H and P_iQ_j = Q_j P_i, tr ρ P_iQ_j = tr ρ P_i tr ρ Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I.
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=∑j=1n⟨βj∣A∣βj⟩tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } tr(A):=j=1∑n​⟨βj​∣A∣βj​⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=∑j=1nPj∣⟨aj⟨aj∣\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|ρ=j=1∑n​Pj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,∑j=1npj=1pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1pj∈R,pj​0,j=1∑n​pj​=1
In standard quantum theory, the ideas of information-entropy and of pure states are closely linked. States are represented by density matrices ρρρ on a Hilbert space and the information-entropy −tr(ρlog⁡ρ)-tr(ρ\logρ)−tr(ρlogρ) is minimised on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalised quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions we demonstrate that there are no `pure' decoherence functions. The main content of the paper is a new notion of information-entropy in generalised quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information-entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information-entropy suggests an intrinsic way of giving a consistent set selection criterion.
Given a set of ordered quantum states, described by density operators {ρj}j=1n, the Bargmann invariant of {ρj}j=1n is defined as tr(ρ1ρ2, …, ρn). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr(ρ1ρ2, …, ρn) for any m-mode bosonic Gaussian states {ρj}j=1n in terms of the means and covariance matrices of {ρj}j=1n. We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.