Conditional expectations in von Neumann algebras and a theorem of Takesaki

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. Among the applications the equality case for the strong subadditivity of the von Neumann entropy, the Imoto-Koashi theorem and exponential families are treated. The setting of the paper allows the underlying Hilbert space to be infinite dimensional.MSC: 46L53, 81R15, 62B05.

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“…8.3 in [13]). The dual of the embedding of a subalgebra into an algebra is called generalized conditional expectation [1].…”

Section: Appendix Dual Mappingmentioning

confidence: 99%

Sufficiency in Quantum Statistical Inference

Jenčová

Petz

2006

Commun. Math. Phys.

129

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“…We use the term coarse-graining, because the statistical aspects get emphasis. Let D 1 be a density of a quantum state on H. 1 The work was supported by the Hungarian OTKA T032662. 2 E-mail: mosonyi@math.bme.hu 3 E-mail: petz@math.bme.hu Algebraicly β in (1) is the left inverse of T as far as the densities D 1 and D 2 are concerned.…”

Section: Introductionmentioning

confidence: 99%

Structure of Sufficient Quantum Coarse-Grainings

Mosonyi

Petz

2004

Letters in Mathematical Physics

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Let H and K be finite dimensional Hilbert spaces, T : B(H) → B(K) be a coarse-graining and D 1 , D 2 be density matrices on H. In this paper the consequences of the existence of a coarse-graining β :H (p) (s = 1, 2) should hold with pairwise orthogonal summands and with commuting factors and with some probability distributions λ s (p) for 1 ≤ p ≤ r (s = 1, 2). This decomposition allows to deduce the exact condition for equality in the strong subaddivity of the von Neumann entropy.Mathematics Subject Classification: 81R15, 62B05, 94A15.

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“…Lemma 2.5 (Proposition 6.1, [1]). Let Q be a completely positive normal unital map from (A, ϕ) into (B, ψ).…”

Section: Definition 22 ([24]mentioning

confidence: 99%

“…for every σ ϕ -analytic element a ∈ A and every σ ψ -analytic element b ∈ B (see [1], [8]). There exists Q ⋆ such that…”

Section: Definition 22 ([24]mentioning

confidence: 99%

On ergodic theorems for free group actions on noncommutative spaces

Anantharaman-Delaroche

2006

Probab. Theory Relat. Fields

120

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Abstract. We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s 2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s 2n (x)) for x in noncommutative spaces L p (A). For the Cesàro means 1 n n−1 k=0 s k and p = +∞, this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota "Alternierende Verfahren" theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.

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Conditional expectations in von Neumann algebras and a theorem of Takesaki

Cited by 229 publications

References 28 publications

Sufficiency in Quantum Statistical Inference

Sufficiency in Quantum Statistical Inference

Structure of Sufficient Quantum Coarse-Grainings

On ergodic theorems for free group actions on noncommutative spaces

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