Toeplitz Covariance Matrices and the von Neumann Relative Entropy

2003 ◽  
pp. 23-29 ◽  
Author(s):  
Tryphon T. Georgiou
Keyword(s):  
Relative Entropy ◽  
Covariance Matrices ◽  
Von Neumann

scholarly journals Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

2021 ◽  
Author(s):  
Ivan Bardet ◽  
Ángela Capel ◽  
Cambyse Rouzé
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Spin Systems ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Finite Dimensional ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

scholarly journals Symmetry-like Relation of Relative Entropy Measure of Quantum Coherence

2020 ◽  
Author(s):  
Cheng-yang Zhang ◽  
Zhi-hua Guo ◽  
H.X. Cao
Keyword(s):  
Relative Entropy ◽  
Quantum Coherence ◽  
Quantum Physics ◽  
Information Science ◽  
Upper Bounds ◽  
Von Neumann Entropy ◽  
Entropy Measure ◽  
Lower And Upper Bounds ◽  
Von Neumann ◽  
Natural Logarithm

Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.

scholarly journals MONOTONICITY OF QUANTUM RELATIVE ENTROPY REVISITED

2003 ◽  
Vol 15 (01) ◽  
pp. 79-91 ◽  
Author(s):  
DÉNES PETZ
Keyword(s):  
Relative Entropy ◽  
Coarse Graining ◽  
Von Neumann Entropy ◽  
Trace Inequality ◽  
Von Neumann ◽  
Quantum Relative Entropy ◽  
Crucial Property

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden–Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.

scholarly journals Relating Entropies of Quantum Channels

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1028
Author(s):  
Dariusz Kurzyk ◽  
Łukasz Pawela ◽  
Zbigniew Puchała
Keyword(s):  
Relative Entropy ◽  
Upper Bound ◽  
Quantum Channel ◽  
Von Neumann Entropy ◽  
Quantum Channels ◽  
Von Neumann ◽  
Channel One ◽  
Depolarizing Channel

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

Further results on the relative entropy

1987 ◽  
Vol 101 (2) ◽  
pp. 363-373 ◽  
Author(s):  
Matthew J. Donald
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebra ◽  
Measurement Problem ◽  
Bounded Operators ◽  
Many Worlds ◽  
Von Neumann ◽  
Normal States ◽  
Injective Von Neumann Algebra ◽  
Made In

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.

RELATIVE ENTROPY FOR ABELIAN SUBALGEBRAS

2010 ◽  
Vol 21 (04) ◽  
pp. 537-550 ◽  
Author(s):  
RUI OKAYASU
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Abelian Subalgebras ◽  
Finite Dimensional ◽  
Finite Von Neumann Algebra

For finite dimensional abelian subalgebras of a finite von Neumann algebra, we obtain the value of conditional relative entropy defined by Choda. We also consider the modified invariant defined by Pimsner and Popa.

On Energy Differences in Many Body Systems

1983 ◽  
Vol 38 (12) ◽  
pp. 1276-1284
Author(s):  
Eberhard. E. Müller ◽  
Wolfgang Feist
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Standard Form ◽  
Energy Operator ◽  
Many Body ◽  
Small Energy ◽  
Von Neumann ◽  
Many Body Systems

Abstract Number-and operator-valued energy differences in von Neumann algebras are discussed by means of the Tomita-Takesaki theory. Small energy differences can be evaluated by the anti-commutator of a sort of relative entropy operator and the energy operator. The New-Tamm-Dancoff procedure is studied in terms of the standard form.

A New Generalization of von Neumann Relative Entropy

2017 ◽  
Vol 56 (11) ◽  
pp. 3405-3424
Author(s):  
Jing Li ◽  
Huaixin Cao
Keyword(s):  
Relative Entropy ◽  
Von Neumann

The Θ-KMS adjoint and time reversed quantum Markov semigroups

2015 ◽  
Vol 18 (03) ◽  
pp. 1550016 ◽  
Author(s):  
Jorge R. Bolaños-Servin ◽  
Roberto Quezada
Keyword(s):  
Relative Entropy ◽  
Detailed Balance ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Standard Quantum ◽  
Von Neumann ◽  
Quantum Markov Semigroup

We introduce the notion of Θ-KMS adjoint of a quantum Markov semigroup, which is identified with the time reversed semigroup. The break of Θ-KMS symmetry, or Θ-standard quantum detailed balance in the sense of Fagnola–Umanità,11 is measured by means of the von Neumann relative entropy of states associated with the semigroup and its Θ-KMS adjoint.

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