Sufficiency in Quantum Statistical Inference

Abstract

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.

Markov property and strong additivity of von Neumann entropy for graded quantum systems

“…4 It has been shown that the Markov property is tightly related to the sufficiency of conditional expectations through the strong subadditivity of von Neumann entropy: A state of a threecomposed tensor-product system is Markovian if and only if it takes the equality for the strong subadditivity inequality of entropy, which will be referred to as "the strong additivity of entropy." 16,6,12,7 We show that a similar equivalence relation of the Markov property and the strong additivity of entropy is valid for graded quantum systems. Its proof proceeds in much the same way as that for the tensor-product case following Ref.…”

Lüders Instruments, Generalised Lüders Theorem, and Some Aspects of Sufficiency

“…Obviously, sufficiency in Umegaki's sense implies sufficiency in Petz's sense which in turn implies sufficiency. Some aspects of sufficiency, Petz's sufficiency, and Umegaki's sufficiency were investigated in [13], [6,7], and [17,18], respectively, while in [8,9] …”

Weakly sufficient quantum statistics

“…The most general notion of sufficiency was introduced in [6,7] (and further studied in [2]) as follows. Let M be a von Neumann algebra and let N be its von Neumann subalgebra.…”

“…[6,7] and [2]) adapted to the present setup, and to compare it with weak sufficiency. Since the algebra N is abelian, the two-positivity of the map α : B(H) → N defining sufficiency is equivalent to its positivity.…”

“…of T , where e : B(R) → P(H) is the spectral (≡ projection-valued) measure of T , the quantity e(E)ϕ, ϕ = e(E)ϕ 2 …”