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数理解析研究所講究録

A representation of unital completely positive maps (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability)

Marie Choda

Published Dec 1, 2012• 0 citations
OpenAlex

Abstract

Let $M_{n}(\mathbb{C})$ be the algebra of $n\cross n$ complex mairices, and let $\Phi$ be a unital completely positive map of $M_{n}(\mathbb{C})$ to $M_{k}(\mathbb{C})$ .With the notion of the von Neumann entropy for a state in mind, we give a model ofThe $r$ is uniquely determined for $\Phi$ and the $r$ -tupple is also unique up to a $r\cross r$ unitary matrix.

Related papers

Which states can be reached from a given state by unital completely positive maps?

Bojan Magajna

Abstract For a state $\omega$ on a C $^{*}$ -algebra $A$ , we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega \circ \varphi$ , where $\varphi$ is a map on $A$ of the form $\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,$ $\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1$ ( $a_i\in A$ , $n\in \mathbb {N}$ ). These are precisely the states $\rho$ that satisfy $\|\rho |J\|\leq \|\omega |J\|$ for each ideal $J$ of $A$ . The corresponding question for normal states on a von Neumann algebra $\mathcal {R}$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $\omega \circ \psi$ , where $\psi$ is a quantum channel on $\mathcal {R}$ (that is, a map of the form $\psi (x)=\sum \nolimits _ja_j^{*}xa_j$ , where $a_j\in \mathcal {R}$ are such that the sum $\sum \nolimits _ja_j^{*}a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C $^{*}$ -algebra and for properly infinite von Neumann algebras the converse also holds.

arXiv:2207.01957•Proceedings of the Edinburgh Mathematical Society•Jul 5, 2022•0 citations

QUANTIZATION OF INFORMATION THEORY (Mathematical Studies on Independence and Dependence Structure : Algebra meets Probability)

Ichiro Fujimoto, Hideo Miyata

In scope of $CP$ -convexity theory for $C^{*}$ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.

数理解析研究所講究録•Dec 1, 2012•0 citations

On Entropy Transmission for Quantum Channels

Nasir Ganikhodjaev, Farrukh Maksutovich Mukhamedov

In this paper a notion of entropy transmission of quantum channels is introduced as a natural extension of Ohya's entropy. Here by quantum channel is meant unital completely positive mappings (ucp) of $B(H)$ into itself, where $H$ is an infinite dimensional Hilbert space. Using a representation theorem of ucp mapping we associate to every ucp map a uniquely determined state, and prove that entropy of ucp map is less then Ohya's entropy of the associated state.

arXiv:quant-ph/0703245•Mar 26, 2007•1 citation

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

Characterizations of homomorphisms among unital completely positive maps

Andre Kornell

We prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C ⁎ -algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets.

arXiv:2403.07229•Linear Algebra and its Applications•Jan 15, 2025•0 citations

Unital Positive Maps and Quantum States

M. Asorey, Andrzej Kossakowski, Giuseppe Marmo +1 more

We analyze the structure of the subset of states generated by unital completely positive quantum maps. A witness certifying that a state does not belong to the subset generated by a given map is constructed. We analyse the representations of positive maps and their relation to quantum Perron-Frobenius theory.

arXiv:0808.0427•Open Systems & Information Dynamics•Jun 1, 2008•6 citations
Let Mn(C)M_{n}(\mathbb{C})Mn​(C) be the algebra of n\crossnn\cross nn\crossn complex mairices, and let Φ\PhiΦ be a unital completely positive map of Mn(C)M_{n}(\mathbb{C})Mn​(C) to Mk(C)M_{k}(\mathbb{C})Mk​(C) .With the notion of the von Neumann entropy for a state in mind, we give a model ofThe rrr is uniquely determined for Φ\PhiΦ and the rrr -tupple is also unique up to a r\crossrr\cross rr\crossr unitary matrix.
Abstract For a state ω\omegaω on a C ∗^{*}∗ -algebra AAA , we characterize all states ρ\rhoρ in the weak* closure of the set of all states of the form ω∘φ\omega \circ \varphiω∘φ , where φ\varphiφ is a map on AAA of the form φ(x)=∑i=1nai∗xai,\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,φ(x)=∑i=1n​ai∗​xai​, ∑i=1nai∗ai=1\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1∑i=1n​ai∗​ai​=1 ( ai∈Aa_i\in Aai​∈A , n∈Nn\in \mathbb {N}n∈N ). These are precisely the states ρ\rhoρ that satisfy ∥ρ∣J∥≤∥ω∣J∥\|\rho |J\|\leq \|\omega |J\|∥ρ∣J∥≤∥ω∣J∥ for each ideal JJJ of AAA . The corresponding question for normal states on a von Neumann algebra R\mathcal {R}R (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form ω∘ψ\omega \circ \psiω∘ψ , where ψ\psiψ is a quantum channel on R\mathcal {R}R (that is, a map of the form ψ(x)=∑jaj∗xaj\psi (x)=\sum \nolimits _ja_j^{*}xa_jψ(x)=∑j​aj∗​xaj​ , where aj∈Ra_j\in \mathcal {R}aj​∈R are such that the sum ∑jaj∗aj\sum \nolimits _ja_j^{*}a_j∑j​aj∗​aj​ converge to 111 in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C ∗^{*}∗ -algebra and for properly infinite von Neumann algebras the converse also holds.
In scope of CPCPCP -convexity theory for C∗C^{*}C∗ -algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational stmcture of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.
In this paper a notion of entropy transmission of quantum channels is introduced as a natural extension of Ohya's entropy. Here by quantum channel is meant unital completely positive mappings (ucp) of B(H)B(H)B(H) into itself, where HHH is an infinite dimensional Hilbert space. Using a representation theorem of ucp mapping we associate to every ucp map a uniquely determined state, and prove that entropy of ucp map is less then Ohya's entropy of the associated state.
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.
We prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C ⁎ -algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets.