Relative Entropy of States of von Neumann Algebras

Araki, Huzihiro

doi:10.2977/prims/1195191148

Cited by 329 publications

(436 citation statements)

References 4 publications

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“…The proof in Section 2.4 is based on that in [11] which (although published later) actually preceded the proof of SSA. However, without the additional ingredient of L P and R Q , which are motivated by Araki's subsequent introduction [1] of the relative modular operator, the results in [11] are not sufficient to prove SSA. The recognition that the argument in [11] could be modified to prove SSA took another 25 years [9].…”

Section: Remarks On Cauchy-schwarz Type Inequalities 41 Elementary Pmentioning

confidence: 99%

Another short and elementary proof of strong subadditivity of quantum entropy

Ruskai

2007

Reports on Mathematical Physics

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A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz inequality in elementary courses. Several consequences are proved in a way which allow an elementary proof of strong subadditivity in a few more lines. Some expository material on Schwarz inequalities for operators and the Holevo bound for partial measurements is also included.

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Section: Remarks On Cauchy-schwarz Type Inequalities 41 Elementary Pmentioning

confidence: 99%

Another short and elementary proof of strong subadditivity of quantum entropy

Ruskai

2007

Reports on Mathematical Physics

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“…for all A ∈ V. The map V + * ∋ ω → Ω ω ∈ H + is a bijection and We now recall the definition of Araki's relative modular operator [1]. For ν, ω ∈ V + * , define S ν|ω on the domain VΩ ω + (VΩ ω ) ⊥ by…”

Section: B1 Basic Definitions and Modular Structurementioning

confidence: 99%

“…In this section, we review the relative entropy introduced by Araki in [1] in the general context of normal states on von Neumann algebras. This relative entropy reduces to the classical Kullback-Leibler divergence in the case of equivalent probability measures, and to Umegaki's quantum relative entropy [43] in the case of states defined through density operators ρ, σ defined on a fixed separable Hilbert space, where supp ρ ⊂ supp σ.…”

Section: B Araki's Generalized Relative Entropymentioning

confidence: 99%

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Contractivity properties of a quantum diffusion semigroup

Datta

Pautrat

Rouzé

2017

Journal of Mathematical Physics

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We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

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“…The mathematical study of entanglement as a special type of quantum correlations from an operational point of view has been initiated in [5,6]. In these papers the entangled mutual information was introduced as the von Neumann entropy of the entangled compound state related to the product of marginal states in the sense of Lindblad, Araki and Umegaki relative entropy [7,8,9]. The corresponding quantum mutual information leads to an entropy bound for quantum capacity, the additivity of which is not obvious for non-trivial quantum channels.…”

Section: Introductionmentioning

confidence: 99%

On Entangled Information and Quantum Capacity

Belavkin

2001

Open Syst. Inf. Dyn.

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Abstract. The pure quantum entanglement is generalized to the case of mixed compound states on an operator algebra to include the classical and quantum encodings as particular cases. The true quantum entanglements are characterized by quantum couplings which are described as transpose-CP, but not Completely Positive (CP), trace-normalized linear positive maps of the algebra.The entangled (total) information is defined in this paper as a relative entropy of the conditional (the derivative of the compound state with respect to the input) and the unconditional output states. Thus defined the total information of the entangled states leads to two different types of the entropy for a given quantum state: the von Neumann entropy, or c-entropy, which is achieved as the supremum of the information over all c-entanglements and thus is semi-classical, and the true quantum entropy, or q-entropy, which is achieved at the standard entanglement. The q-capacity, defined as the supremum over all entanglements, coincides with the topological entropy. In the case of the simple algebra it doubles the c-capacity, coinciding with the rank-entropy. The conditional q-entropy based on the q-entropy, is positive, unlike the von Neumann conditional entropy, and the q-information of a quantum channel is proved to be additive.

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Relative Entropy of States of von Neumann Algebras

Cited by 329 publications

References 4 publications

Another short and elementary proof of strong subadditivity of quantum entropy

Another short and elementary proof of strong subadditivity of quantum entropy

Contractivity properties of a quantum diffusion semigroup

On Entangled Information and Quantum Capacity

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