Minimal sufficient statistical experiments on von Neumann algebras

Abstract

A statistical experiment on a von Neumann algebra is a parametrized family of normal states on the algebra. This paper introduces the concept of minimal sufficiency for statistical experiments in such operator algebraic situations. We define equivalence relations of statistical experiments indexed by a common parameter set by completely positive or Schwarz coarse-graining and show that any statistical experiment is equivalent to a minimal sufficient statistical experiment unique up to normal isomorphism of outcome algebras. We also establish the relationship between the minimal sufficiency condition for statistical experiment in this paper and those for subalgebra. These concepts and results are applied to the concatenation relation for completely positive channels with general input and outcome von Neumann algebras.In the case of the quantum-classical channel corresponding to the positive-operator valued measure (POVM), we prove the equivalence of the minimal sufficient condition previously proposed by the author and that in this paper. We also give a characterization of the discreteness of a POVM up to postprocessing equivalence in terms of the corresponding quantum-classical channel.

Entanglement-breaking channels with general outcome operator algebras

“…Equation(6) implies that the Fourier transform of the integrable function f (β)e − |β| is zero. Hence, by the injectivity of the Fourier transform, we obtain f (α) = 0 for almost all α ∈ C, proving the injectivity of Γ B .By using the concept of the minimal sufficiency of channels,27 we can show that Λ M is not randomization-equivalent to any QC channel if M is not commutative. A normalchannel Γ ∈ Ch σ (N → M in ) is called minimal sufficient if Γ • Φ = Γ implies Φ = id N for any Φ ∈ Ch σ (N → N ).…”

Quantum incompatibility of channels with general outcome operator algebras

“…By following the construction given in Ref. 13 (Lemma 2), there exist a von Neumann algebra M 0 and a normal channel Λ 0 ∈ Ch σ (M 0 → L(H in )) such that Λ ∼ CPσ Λ 0 and Λ 0 is faithful. We take a minimal Stinespring representation (K 0 , π 0 , V 0 ) of Λ 0 and define conjugate channels Λ c 0 ∈ Ch σ (π 0 (M 0 ) ′ → L(H in )) and Λ cc 0 ∈ Ch σ (π 0 (M 0 ) → L(H in )) in the same way as Λ c and Λ cc .…”

Accessible Information Without Disturbing Partially Known Quantum States on a von Neumann Algebra

“…A statistical experiment E = (M ,Θ , (ϕ θ ) θ ∈Θ ) is called minimal sufficient if E satisfies either of the following equivalent conditions ( [14], Theorem 2):…”

“…Such a minimal sufficient statistical experiment E 0 is unique up to normal isomorphism and, in this sense, we may say that E 0 is the minimal sufficient statistical experiment normally CP equivalent to E . If E is faithful, E 0 can be constructed as follows [14,17]. Define…”

“…First, we fix arbitrary ξ ∈ Ξ and write s ξ := s(ψ ξ ) ands ξ := 1 M ⊗ s(ψ ξ ). Consider the following faithful statistical experiment: is normally isomorphic to E , the assumption of the minimal sufficiency of E implies that M ⊗Cs ξ is the minimal sufficient subalgebra of M ⊗(s ξ N s ξ ) with respect to the family (ϕ θ ⊗ψ ξ ) θ ∈Θ ( [14], Theorem 3). Therefore for a normal channel Γ ∈…”