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

Scientia Sinica Mathematica

Convex combination preserving maps and geometric features for selective quantum measurements

Xiaofei Qi, Hou Jinchuan

Published Apr 3, 2023• 0 citations
Publisher

Abstract

Let H and K be the complex Hilbert spaces and mathcal SH) theconvex set of all quantum states (i.e., the positive operators withtrace one) on H. In this paper, the convex combination preservingmaps from mathcal SH) into mathcal SK)∪0and theirconnection to quantum measurements are discussed. It is shown that amap Φ:mathcal SH )→ mathcal SK) with 2≤ dimH≤∞ preserves pure states and strict convex combinations ifand only if one of the following holds: (1) there exists a purestate σ0 on K such that Φ(ρ)= σ0 for allρ∈ mathcal SH); (2) Φ is an injective selective quantummeasurement map up to the transpose, i.e., there exists an injective bounded linear operator M: H→ K such thatΦ(ρ)=fracMρ M^*Tr(Mρ M^*)

DOI
10.1360/ssm-2022-0133

Related papers

Quantum measurement and maps preserving convex combinations of separable states

Jinchuan Hou, Liang Liu

A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.

Journal of Physics A Mathematical and Theoretical•May 4, 2012•5 citations

Quantum measurements and maps preserving strict convex combinations and pure states

Lihua Yang, Jinchuan Hou

In this paper, a characterization of maps between quantum states that preserve pure states and strict convex combinations is obtained. Based on this characterization, a structure theorem for maps between multipartite quantum states that preserve separable pure states and strict convex combinations is established. Then, these results are applied to characterize injective (local) quantum measurements and answer some conjectures proposed in Hou and Liu (2012 J. Phys. A: Math. Theor. 45 205305).

arXiv:1301.7551•Journal of Physics A Mathematical and Theoretical•Apr 29, 2013•1 citation

Hilbert's projective metric in quantum information theory

David Reeb, Michael J. Kastoryano, Michael M. Wolf

We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.

arXiv:1102.5170•Journal of Mathematical Physics•Aug 1, 2011•61 citations

Maps on Quantum States in -algebras Preserving von Neumann Entropy or Schatten -norm of Convex Combinations

Marcell Gaál

Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten $p$ -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a $C^{\ast }$ -algebra.

Canadian Mathematical Bulletin•Mar 22, 2018•3 citations

Preservers for the Tsallis Entropy of Convex Combinations of Density Operators

Xiaochun Fang, Yihui Lao

Let H be a complex separable Hilbert space; we first characterize the unitary equivalence of two density operators by use of Tsallis entropy and then obtain the form of a surjective map on density operators preserving Tsallis entropy of convex combinations.

Advances in Mathematical Physics•Jun 4, 2018•3 citations

Nonlinear Maps Preserving Convex Combination of Tensor Products of Operators

Liu Lian

In this paper,we give a structure feature of convex combination preserving bijective maps which send separable states to separable states of bipartite system.It is shown that such a map Φ has the form of Φ(ρσ)=(ST)Ψ(ρσ)(ST)*Tr((ST)Ψ(ρσ)(ST)*),where Ψ is one of the identity,transpose,partial transpose,or a combination of above mentioned map with the swap.

Journal of Taiyuan University of Technology•Jan 1, 2015•0 citations
Let H and K be the complex Hilbert spaces and mathcal SH) theconvex set of all quantum states (i.e., the positive operators withtrace one) on H. In this paper, the convex combination preservingmaps from mathcal SH) into mathcal SK)∪0and theirconnection to quantum measurements are discussed. It is shown that amap Φ:mathcal SH )→ mathcal SK) with 2≤ dimH≤∞ preserves pure states and strict convex combinations ifand only if one of the following holds: (1) there exists a purestate σ0 on K such that Φ(ρ)= σ0 for allρ∈ mathcal SH); (2) Φ is an injective selective quantummeasurement map up to the transpose, i.e., there exists an injective bounded linear operator M: H→ K such thatΦ(ρ)=fracMρ M^Tr(Mρ M^)
A map ϕ between convex sets is said to be convex combination preserving if, for any elements x , y in its domain and any number 0 < t < 1, there is some s with 0 < s < 1, such that ϕ( tx + (1 − t ) y ) = s ϕ( x ) + (1 − s )ϕ( y ). Every quantum measurement described by a measurement operator M introduces a map (i.e. if ) that is convex combination preserving. In this paper, we give a characterization of convex combination preserving bijective maps on the set of all separable states in a bipartite quantum system H 1 ⊗ H 2 and show that, under a mild additional condition, such a map is a composition of an invertible local quantum measurement (i.e. the map of the form with S , T invertible) and some of the following maps: the transpose, the partial transpose and the swap.
We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.