Some aspects of quantum sufficiency for finite and full von Neumann algebras

Some features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.


Introduction
Let M be a von Neumann algebra, let N be its von Neumann subalgebra, and let {ρ θ : θ ∈ } be a family of normal states on M. In [4] the notion of strong sufficiency of the subalgebra N for the family {ρ θ : θ ∈ } was introduced (and further investigated in [5]) as a generalisation of sufficiency in Umegaki's sense considered earlier in [13,14]. In this operator algebra setup, strong sufficiency of N means the existence of a normal two-positive map α : M → N such that If the map α is a normal conditional expectation onto N, then we get sufficiency in Umegaki's sense. An obvious and natural generalisation of the notion of strong sufficiency would consist in giving up the, rather technical, requirement of two-positivity and replacing it by positivity. Some work in this setup was done in [6], and in the present paper, we continue this line of investigation. As was pointed out in [6], this approach is additionally motivated by considerations from quantum hypothesis testing theory, where a different notion of sufficiency is employed, namely, the sufficiency of N means that we can find there an optimal measurement minimising the Bayes risk (cf. [7]).
In the first part of the paper, we show that for a σ -finite finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki's sense, and obtain a proper version of the factorization theorem due to Jenčová and Petz. In the second part, we are dealing with the full algebra B(H) and a family of pure states. There we give a description of the minimal sufficient subalgebra together with an explicit form of the conditional expectation defining this sufficiency.

Preliminaries and notation
Let H be a Hilbert space with a scalar product ·|· and norm · , and let M be an arbitrary von Neumann algebra of operators acting on H, with identity 1, and predual M * . M is said to be σ -finite (or countably decomposable) if any family of pairwise orthogonal projections in M is countable. In our considerations, we restrict attention to such algebras.
A state on M is a bounded positive linear functional ρ : M → C of norm one. A state ρ is said to be normal if it is continuous in the σ -weak topology on M, in other words, ρ ∈ M * . For a normal state ρ, its support, denoted by s(ρ), is defined as the smallest projection e in M such that ρ(e) = ρ (1). In particular, we have and if ρ(s(ρ)A s(ρ)) = 0 for s(ρ)A s(ρ) ≥ 0 then s(ρ)A s(ρ) = 0. A normal state ρ is said to be faithful if for each positive element A ∈ M from the equality ρ(A) = 0 it follows that A = 0. It is easily seen that the faithfulness of ρ is equivalent to the relation s(ρ) = 1. The existence of a normal faithful state on M is equivalent to M being σ -finite.
To each unit vector ψ ∈ H there corresponds a normal state, denoted by the same symbol and called a vector state, defined as By a slight abuse of language, we shall speak of the vector ψ itself as a pure state. By P [ψ] shall be denoted the projection onto the space spanned by the vector ψ (in Dirac notation, P [ψ] = |ψ ψ|).
Let {ρ θ : θ ∈ } be a family of normal states on a von Neumann algebra M. This family is said to be faithful if for each positive element A ∈ M from the equality ρ θ (A) = 0 for all θ ∈ it follows that A = 0. Similarly to the case of one state, it is seen that the faithfulness of the family is equivalent to the relation θ∈ s(ρ θ ) = 1.
Let P be a projection in a von Neumann algebra M. A reduced von Neumann algebra M P is defined as an algebra of operators acting on the Hilbert space P(H) by the formula Let {M i : i ∈ I } be a family of von Neumann algebras acting on Hilbert spaces H i . The direct sum von Neumann algebra is defined as where the operators A = (A i ) i∈I act on the Hilbert space Let M and N be von Neumann algebras. A linear map α : M → N is said to be normal if it is continuous in the σ -weak topologies on M and N, respectively. It is called unital if α(1) = 1.
The algebra of all bounded linear operators on H will be denoted by B(H). For arbitrary T ⊂ B(H) by T h will be denoted the set of hermitian elements of T.
The Jordan product of two operators A, B ∈ B(H) will be denoted by the symbol '•': and by a slight abuse of notation, the same symbol will be used for superpositions of a functional ϕ on M and a map E on M: ϕ • E, as well as of two maps E, F on M: For arbitrary T ⊂ B(H), by W * (T), we shall denote the von Neumann algebra generated by T, i.e., the smallest von Neumann algebra containing T.

A. Łuczak
Let M be a von Neumann algebra, let {ρ θ : θ ∈ } be a family of normal states on M, and let N be a von Neumann subalgebra of M. N is said to be sufficient for the family of states {ρ θ : θ ∈ } if there exists a normal positive linear unital map α : M → N such that If the map α above is two-positive, then N is said to be strongly sufficient, and if it is a conditional expectation onto N, then N is said to be sufficient in Umegaki's sense. It can be shown that for finite-dimensional von Neumann algebras and families of faithful states sufficiency and strong sufficiency are equivalent. The notion of sufficiency in Umegaki's sense appeared in [13,14], while strong sufficiency was introduced in [

Sufficiency in finite von Neumann algebras
In this section, we show that for a σ -finite finite von Neumann algebra M and a faithful family of normal states on M the minimal sufficient von Neumann subalgebra of M is sufficient in Umegaki's sense. Moreover, we also obtain a version of the factorization theorem.
In what follows, we shall need a few basic facts from modular theory of von Neumann algebras; however, this theory will be employed only to a limited extent. For its full account, the reader is referred to [10]. Let ϕ be a normal faithful state on M. There exists a group of automorphisms of M, {σ ϕ t : t ∈ R}, called the modular automorphism group, associated with ϕ in a canonical way. The algebra of fixed points of the modular automorphism group, denoted by M ϕ , is called the centralizer of ϕ, and the following relation holds Let A be a subspace of M containing the identity operator 1. A is said to be a JW*subalgebra of M if it is σ -weakly closed and closed with respect to the operation of taking adjoints and the Jordan product. Then the self-adjoint part of A is a JW-algebra, in particular, it is generated by projections (cf. [3]). Let E be a normal positive faithful unital projection from the JW-algebra M h onto a JW-algebra A. On account of [3, Let now E be a normal positive faithful unital projection from M onto a JW*-algebra A. Then E|M h is as above , and hence it satisfies Eq. (2).
for all A ∈ A, B ∈ M. Using the formula we get for each B ∈ M and each symmetry S ∈ A h (symmetry means that S 2 = 1).
The lemma below is a JW*-version of a known result on uniqueness of conditional expectation for von Neumann algebras.
Proof Observe first that the family {ϕ H : H ∈ A} of normal linear functionals on M defined as and the faithfulness of ϕ yields A * A = 0, i.e., A = 0. Now for arbitrary B ∈ M, we have by virtue of the equality (3) and analogously Now we are ready to prove that for a faithful family of normal states the minimal sufficient subalgebra is also sufficient in Umegaki's sense. Proof As in the proof of Theorem 1 in [6], we let S be the family of all normal positive linear unital maps on M such that the states ρ θ are invariant with respect to the maps from S. It is seen that S is a non-empty (because it contains the identity map) semigroup. Let A be the set of fixed points of the maps from S, i.e., Then A is a JW*-algebra, and on account of [6, Theorem 1] we have moreover, there is a normal positive faithful unital projection E from M onto A such that for all θ ∈ Then ϕ is a normal state on M such that An arbitrary projection P in A h has the form P = 1 2 (S + 1) where S = 2P − 1 is a symmetry, thus for every such projection. Since A h is a JW-algebra, linear combinations of its projections are σ -weakly dense in A h (cf. [3]) and the σ -weak continuity of ϕ yields for each A ∈ A h . Since B ∈ M was arbitrary, this means that A ∈ M ϕ , thus Since M ϕ is a von Neumann algebra, this yields We have showing that E • F is a normal positive unital projection onto A. Moreover, and Lemma 1 yields Thus for each θ ∈ , we have which means that M min is sufficient in Umegaki's sense.

A. Łuczak
An important factorization theorem was formulated in [4,Theorem 4] in the case when the algebra M is a countable direct sum of type I factors. However, as it was shown in [6, Examples 2 and 3], its setup needs substantial corrections, and we shall see that this theorem holds in full generality in finite von Neumann algebras. For this purpose, let us introduce some necessary notions. Since the proof of this theorem is in its main part a repetition of the proof in [4,Section 4] with some additional reasoning, we shall follow closely the notation employed there. However, we shall use some parts of the theory of measurable operators and noncommutative L p -spaces (actually only L 1 -space) for a brief account of which we refer the reader to the "Appendix".
For and let D ω ∈ L 1 (M, τ ), D ω 0 ∈ L 1 (M 0 , τ 0 ), D ω 1 ∈ L 1 (M 1 , τ 1 ) be the densities of ω, ω 0 and ω 1 , respectively. Now the reasoning of [4, p. 269] can be repeated word for word upon observing that the modular automorphism groups {σ ω t }, {σ ω 0 t } and {σ ω 1 t } have the same form as in [4], i.e., for instance and similarly for the remaining two (cf. [10,Proposition 4.7]). This leads to the equality for some positive self-adjoint z with trivial null-space, affiliated with the centre of M 1 . Denote by D θ , D θ,0 the densities of the states ρ θ and ρ θ |M 0 , respectively. Now the factorization theorem reads where z is a positive self-adjoint operator with trivial null-space, affiliated with the centre of M 1 .

Proof Fix θ , and let
be the spectral decompositions of D θ,0 , D ω 1 and z. Since D θ,0 is affiliated with M 0 , D ω 1 is affiliated with M 1 , and z is affiliated with the centre of M 1 , it follows that the spectral measures E 0 , E 1 and E z take values in M 0 , M 1 and the centre of M 1 , respectively, thus they mutually commute. Consequently, there exists a spectral measure E in M, and nonnegative Borel functions g 0 , g 1 and h such that From these equalities, we obtain Operator B on the right hand side of the above relation is a self-adjoint operator affiliated with M, i.e., it belongs to the algebra M of measurable operators, while on the left hand side we have a product of operators from M. From the properties of M, it follows that Moreover, we have Now the proof follows the lines of the proof of Theorem 4 in [4]. The crucial observation is that also in the case of, in general unbounded, operators D θ , D ω , D θ,0 and D ω 0 the following equalities for the Connes cocycles hold true Assume that M 0 is sufficient. Then on account of [4, Theorem 1], we have [Dρ θ : (9), (10) and the fact that D it ω are unitary, we get Using (6) together with the fact that D it ω 0 are also unitary, we obtain From this easily follows that D θ = B. Indeed, denoting by N (T ) the null-space of the operator T , we have from the spectral theorem Denote by P the projection onto the common null-space above, and let be the spectral decomposition of D θ . Then where χ {0} denotes the indicator function of the set {0}. Hence and similarly which on account of the equality (11) yields But the operators P + D θ and P + B have trivial null-spaces, which means that the equality above is the identity of two unitary groups. It follows that their generators must be the same, so i.e., and the equality (8) proves the claim. Assume now that the equality (7) holds. In terms of the algebra M, it can be rewritten as (where the central dot stands for the strong multiplication-see the "Appendix"). Denote Then by virtue of the isomorphism between M * and L 1 (M, τ ), we have where s − denotes the strong sum. The same holds of course for ω n,0 = ω n |M 0 , i.e., so from the formula (7) and the fact that M is an algebra, we obtain Since ω n → ω in norm in M * , we infer that D ω n → D ω in the norm · 1 , and by the same token D ω n,0 → D ω 0 . According to Lemma 5 (see the Appendix), we have D ω n → D ω and D ω n,0 → D ω 0 in measure. Since M is a topological * -algebra with respect to the measure topology, the formula (12) yields in other words Now we proceed as in the first part of the proof. Since the operators D ω 0 , D ω 1 and z commute, from the equalities (7) and (13), we obtain Thus on account of commutation properties and the fact that {D it ω 1 } and {z it } are unitary groups, we get which by virtue of [4, Theorem 1] yields the sufficiency of M 0 .

Sufficiency for pure states on B(H)
In this section, we shall be dealing with the full algebra M = B(H) and pure (≡ vector) states. Let {ψ θ : θ ∈ } be a faithful family of pure states. According to [6,Theorem 1], there exists the minimal sufficient for this family subalgebra of B(H). It turns out that this algebra admits a straightforward description. For the sake of brevity, set R = {ψ θ : θ ∈ }, and define on R an equivalence relation as follows: ψ θ ≡ ψ θ if there exists a finite string ψ θ 1 , . . . , ψ θ m such that ψ θ 1 = ψ θ , ψ θ m = ψ θ and It follows that P i are pairwise orthogonal, and the faithfulness of {ψ θ : θ ∈ } yields i P i = 1.
Fix arbitrary i and consider the family of pure states R i = {ψ θ : θ ∈ i } on the Hilbert space H i . Take arbitrary ψ θ , ψ θ ∈ R i , and let ψ θ 1 , . . . , ψ θ m , with ψ θ 1 = ψ θ and ψ θ m = ψ θ be the string as in the definition of the equivalence relation ≡. From this definition, it follows that ψ θ j ≡ ψ θ k for all j, k = 1, . . . , m, thus ψ θ j ∈ R i for all j = 1, . . . , m. Consequently, P [ψ θ j ] ≤ P i , so we can consider the projections P [ψ θ j ] on the Hilbert space H i . We have where a = ψ θ 1 |ψ θ 2 · · · ψ θ m−1 |ψ θ m = 0, and for ξ, η ∈ H the operator E ξ,η is defined as (The formula (15) is obvious when using Dirac notation since then on the left hand side we have simply which is just the right-hand side. In this notation, we have . On account of the above considerations, we infer that E ψ θ ,ψ θ ∈ N for all ψ θ , ψ θ ∈ R i . Let be an arbitrary finite linear combination of vectors from R i . We have (This is again most easily seen by using Dirac notation since then It follows that P [ξ ] ∈ N. Vectors ξ as above lie densely in H i , so arbitrary η ∈ H i is a limit of some ξ n , consequently, From the definition of P i , it follows that P i commutes with all P [ψ θ ] , θ ∈ , because otherwise.
Since the algebra N is generated by the projections P [ψ θ ] , we obtain that P i ∈ N . For each A ∈ N, we have showing that In the first part of the proof, we obtained the relation P [ψ θ ] ∈ M min for all θ ∈ which yields the inclusion Consider a map E : B(H) → N defined as It is easily seen that E is a normal conditional expectation such that the states ψ θ are E-invariant, thus N is sufficient in Umegaki's sense. Since M min is minimal sufficient, we obtain Thus for any ε > 0, we get Consequently, we obtain the Chebyschev inequality Taking into account the above-mentioned 'technical' form of convergence in measure, we have Lemma 5 If a sequence (A n ) of operators in L 1 (M, τ ) converges in · 1 -norm, then it converges in measure.