Abstract
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.
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1 Introduction
Let \({\mathscr {M}}\) be a von Neumann algebra, let \({\mathscr {N}}\) be its von Neumann subalgebra, and let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a family of normal states on \({\mathscr {M}}\). In [4] the notion of strong sufficiency of the subalgebra \({\mathscr {N}}\) for the family \(\{\rho _{\theta }:\theta \in \Theta \}\) 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 \({\mathscr {N}}\) means the existence of a normal two-positive map \(\alpha :{\mathscr {M}}\rightarrow {\mathscr {N}}\) such that
If the map \(\alpha \) is a normal conditional expectation onto \({\mathscr {N}}\), 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 \({\mathscr {N}}\) 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 \(\sigma \)-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 \({\mathbb {B}}({\mathcal {H}})\) 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.
2 Preliminaries and notation
Let \({\mathcal {H}}\) be a Hilbert space with a scalar product \(\langle \cdot |\cdot \rangle \) and norm \(\Vert \cdot \Vert \), and let \({\mathscr {M}}\) be an arbitrary von Neumann algebra of operators acting on \({\mathcal {H}}\), with identity \(\mathbb {1}\), and predual \({\mathscr {M}}_*\). \({\mathscr {M}}\) is said to be \(\sigma \)-finite (or countably decomposable) if any family of pairwise orthogonal projections in \({\mathscr {M}}\) is countable. In our considerations, we restrict attention to such algebras.
A state on \({\mathscr {M}}\) is a bounded positive linear functional \(\rho :{\mathscr {M}}\rightarrow {\mathbb {C}}\) of norm one. A state \(\rho \) is said to be normal if it is continuous in the \(\sigma \)-weak topology on \({\mathscr {M}}\), in other words, \(\rho \in {\mathscr {M}}_*\). For a normal state \(\rho \), its support, denoted by \({\text {s}}(\rho )\), is defined as the smallest projection e in \({\mathscr {M}}\) such that \(\rho (e)=\rho (\mathbb {1})\). In particular, we have
and if \(\rho ({\text {s}}(\rho )A{\text {s}}(\rho ))=0\) for \({\text {s}}(\rho )A{\text {s}}(\rho )\geqslant 0\) then \({\text {s}}(\rho )A{\text {s}}(\rho )=0\).
A normal state \(\rho \) is said to be faithful if for each positive element \(A\in {\mathscr {M}}\) from the equality \(\rho (A)=0\) it follows that \(A=0\). It is easily seen that the faithfulness of \(\rho \) is equivalent to the relation \({\text {s}}(\rho )=\mathbb {1}\). The existence of a normal faithful state on \({\mathscr {M}}\) is equivalent to \({\mathscr {M}}\) being \(\sigma \)-finite.
To each unit vector \(\psi \in {\mathcal {H}}\) 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 \(\psi \) itself as a pure state. By \(P_{[\psi ]}\) shall be denoted the projection onto the space spanned by the vector \(\psi \) (in Dirac notation, \(P_{[\psi ]}=|\psi \rangle \langle \psi |\)).
Let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a family of normal states on a von Neumann algebra \({\mathscr {M}}\). This family is said to be faithful if for each positive element \(A\in {\mathscr {M}}\) from the equality \(\rho _{\theta }(A)=0\) for all \(\theta \in \Theta \) 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
In particular, for a family of pure states \(\{\psi _{\theta }:\theta \in \Theta \}\), its faithfulness amounts to the equality \(\overline{{{\,\mathrm{Lin}\,}}\{\psi _{\theta }:\theta \in \Theta \}}={\mathcal {H}}\).
Let P be a projection in a von Neumann algebra \({\mathscr {M}}\). A reduced von Neumann algebra \({\mathscr {M}}_P\) is defined as an algebra of operators acting on the Hilbert space \(P({\mathcal {H}})\) by the formula
Let \(\{{\mathscr {M}}_i:i\in I\}\) be a family of von Neumann algebras acting on Hilbert spaces \({\mathcal {H}}_i\). The direct sum von Neumann algebra is defined as
where the operators \(A=(A_i)_{i\in I}\) act on the Hilbert space
as
Let \({\mathscr {M}}\) and \({\mathscr {N}}\) be von Neumann algebras. A linear map \(\alpha :{\mathscr {M}}\rightarrow {\mathscr {N}}\) is said to be normal if it is continuous in the \(\sigma \)-weak topologies on \({\mathscr {M}}\) and \({\mathscr {N}}\), respectively. It is called unital if \(\alpha (\mathbb {1})=\mathbb {1}\).
The algebra of all bounded linear operators on \({\mathcal {H}}\) will be denoted by \({\mathbb {B}}({\mathcal {H}})\). For arbitrary \({\mathscr {T}}\subset {\mathbb {B}}({\mathcal {H}})\) by \({\mathscr {T}}_h\) will be denoted the set of hermitian elements of \({\mathscr {T}}\).
The Jordan product of two operators \(A,B\in {\mathbb {B}}({\mathcal {H}})\) will be denoted by the symbol ‘\(\circ \)’:
and by a slight abuse of notation, the same symbol will be used for superpositions of a functional \(\varphi \) on \({\mathscr {M}}\) and a map \({\mathbb {E}}\) on \({\mathscr {M}}\): \(\varphi \circ {\mathbb {E}}\), as well as of two maps \({\mathbb {E}},{\mathbb {F}}\) on \({\mathscr {M}}\): \({\mathbb {E}}\circ {\mathbb {F}}\).
For arbitrary \({\mathscr {T}}\subset {\mathbb {B}}({\mathcal {H}})\), by \(W^*({\mathscr {T}})\), we shall denote the von Neumann algebra generated by \({\mathscr {T}}\), i.e., the smallest von Neumann algebra containing \({\mathscr {T}}\).
Let \({\mathscr {M}}\) be a von Neumann algebra, let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a family of normal states on \({\mathscr {M}}\), and let \({\mathscr {N}}\) be a von Neumann subalgebra of \({\mathscr {M}}\). \({\mathscr {N}}\) is said to be sufficient for the family of states \(\{\rho _{\theta }:\theta \in \Theta \}\) if there exists a normal positive linear unital map \(\alpha :{\mathscr {M}}\rightarrow {\mathscr {N}}\) such that
If the map \(\alpha \) above is two-positive, then \({\mathscr {N}}\) is said to be strongly sufficient, and if it is a conditional expectation onto \({\mathscr {N}}\), then \({\mathscr {N}}\) 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 [4]. If the algebra \({\mathscr {N}}\) is sufficient and contained in any other sufficient (sufficient in strong or Umegaki’s sense) algebra, then \({\mathscr {N}}\) is said to be minimal. It is clear that a minimal sufficient (in any sense) subalgebra is unique (if it exists). On account of [6, Theorem 1], for a faithful family of normal states on \({\mathscr {M}}\) there exists the minimal sufficient von Neumann subalgebra \({\mathscr {M}}_{\text {min}}\) of \({\mathscr {M}}\).
3 Sufficiency in finite von Neumann algebras
In this section, we show that for a \(\sigma \)-finite finite von Neumann algebra \({\mathscr {M}}\) and a faithful family of normal states on \({\mathscr {M}}\) the minimal sufficient von Neumann subalgebra of \({\mathscr {M}}\) 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 \(\varphi \) be a normal faithful state on \({\mathscr {M}}\). There exists a group of automorphisms of \({\mathscr {M}}\), \(\{\sigma _t^\varphi :t\in {\mathbb {R}}\}\), called the modular automorphism group, associated with \(\varphi \) in a canonical way. The algebra of fixed points of the modular automorphism group, denoted by \({\mathscr {M}}^\varphi \), is called the centralizer of \(\varphi \), and the following relation holds
Let \({\mathscr {N}}\) be a von Neumann subalgebra of \({\mathscr {M}}\). If \(\sigma _t^\varphi ({\mathscr {N}})={\mathscr {N}}\) for all \(t\in {\mathbb {R}}\), then there is a normal faithful conditional expectation \({\mathbb {F}}\) from \({\mathscr {M}}\) onto \({\mathscr {N}}\) such that
Let \({\mathscr {A}}\) be a subspace of \({\mathscr {M}}\) containing the identity operator \(\mathbb {\mathbb {1}}\). \({\mathscr {A}}\) is said to be a JW*-subalgebra of \({\mathscr {M}}\) if it is \(\sigma \)-weakly closed and closed with respect to the operation of taking adjoints and the Jordan product. Then the self-adjoint part of \({\mathscr {A}}\) is a JW-algebra, in particular, it is generated by projections (cf. [3]). Let \({\mathbb {E}}\) be a normal positive faithful unital projection from the JW-algebra \({\mathscr {M}}_h\) onto a JW-algebra \({\mathscr {A}}\). On account of [3, Lemma 4.4.13], we have
for all \(A\in {\mathscr {A}}_h\), \(B\in {\mathscr {M}}_h\). Let now \({\mathbb {E}}\) be a normal positive faithful unital projection from \({\mathscr {M}}\) onto a JW*-algebra \({\mathscr {A}}\). Then \({\mathbb {E}}|{\mathscr {M}}_h\) is as above , and hence it satisfies Eq. (2). Decomposing arbitrary \(A\in {\mathscr {A}}\) and \(B\in {\mathscr {M}}\) as \(A=A_1+iA_2\), \(B=B_1+iB_2\) where \(A_1,A_2\in {\mathscr {A}}_h\), \(B_1,B_2\in {\mathscr {M}}_h\), we obtain the equality
for all \(A\in {\mathscr {A}}\), \(B\in {\mathscr {M}}\). Using the formula
we get
for each \(B\in {\mathscr {M}}\) and each symmetry \(S\in {\mathscr {A}}_h\) (symmetry means that \(S^2=\mathbb {1}\)).
The lemma below is a JW*-version of a known result on uniqueness of conditional expectation for von Neumann algebras.
Lemma 1
Let \(\varphi \) be a normal faithful state on \({\mathscr {M}}\), let \({\mathscr {A}}\) be a JW*-subalgebra of \({\mathscr {M}}\), and let \({\mathbb {E}}_i:{\mathscr {M}}\rightarrow {\mathscr {A}}\), \(i=1,2\), be normal positive unital projections onto \({\mathscr {A}}\) such that
Then \({\mathbb {E}}_1={\mathbb {E}}_2\).
Proof
Observe first that the family \(\{\varphi _H:H\in {\mathscr {A}}\}\) of normal linear functionals on \({\mathscr {M}}\) defined as
separates the points of \({\mathscr {A}}\). Indeed, if we have \(\varphi _H(A)=0\) for some \(A\in {\mathscr {A}}\) and all \(H\in {\mathscr {A}}\), then taking \(H=A^*\) we get
and the faithfulness of \(\varphi \) yields \(A^*A=0\), i.e., \(A=0\). Now for arbitrary \(B\in {\mathscr {M}}\), we have by virtue of the equality (3)
and analogously
showing that \(\varphi _H({\mathbb {E}}_1(B))=\varphi _H({\mathbb {E}}_2(B))\), hence \({\mathbb {E}}_1(B)={\mathbb {E}}_2(B)\). \(\square \)
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.
Theorem 2
Let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a faithful family of normal states on a \(\sigma \)-finite finite von Neumann algebra \({\mathscr {M}}\) with a normal faithful tracial state \(\tau \). The minimal sufficient von Neumann subalgebra \({\mathscr {M}}_{\text {min}}\) of \({\mathscr {M}}\) is sufficient in Umegaki’s sense.
Proof
As in the proof of Theorem 1 in [6], we let \({\mathfrak {S}}\) be the family of all normal positive linear unital maps on \({\mathscr {M}}\) such that the states \(\rho _{\theta }\) are invariant with respect to the maps from \({\mathfrak {S}}\). It is seen that \({\mathfrak {S}}\) is a non-empty (because it contains the identity map) semigroup. Let \({\mathscr {A}}\) be the set of fixed points of the maps from \({\mathfrak {S}}\), i.e.,
Then \({\mathscr {A}}\) is a JW*-algebra, and on account of [6, Theorem 1] we have
moreover, there is a normal positive faithful unital projection \({\mathbb {E}}\) from \({\mathscr {M}}\) onto \({\mathscr {A}}\) such that for all \(\theta \in \Theta \)
Put
Then \(\varphi \) is a normal state on \({\mathscr {M}}\) such that
Let S be an arbitrary symmetry in \({\mathscr {A}}_h\). Taking BS instead of B in the formula (4), we get
hence
An arbitrary projection P in \({\mathscr {A}}_h\) has the form \(P=\frac{1}{2}(S+\mathbb {1})\) where \(S=2P-\mathbb {1}\) is a symmetry, thus
for every such projection. Since \({\mathscr {A}}_h\) is a JW-algebra, linear combinations of its projections are \(\sigma \)-weakly dense in \({\mathscr {A}}_h\) (cf. [3]) and the \(\sigma \)-weak continuity of \(\varphi \) yields
for each \(A\in {\mathscr {A}}_h\). Since \(B\in {\mathscr {M}}\) was arbitrary, this means that \(A\in {\mathscr {M}}^\varphi \), thus
Since \({\mathscr {M}}^\varphi \) is a von Neumann algebra, this yields
(The reasoning above follows the proof of Haagerup and Størmer [2, Lemma 2.2].) Since for the modular automorphism group we have
we get \(\sigma _t^\varphi ({\mathscr {M}}_{\text {min}})={\mathscr {M}}_{\text {min}}\) for all \(t\in {\mathbb {R}}\), thus there is a normal faithful conditional expectation \({\mathbb {F}}\) from \({\mathscr {M}}\) onto \({\mathscr {M}}_{\text {min}}\) such that
We have
showing that \({\mathbb {E}}\circ {\mathbb {F}}\) is a normal positive unital projection onto \({\mathscr {A}}\). Moreover,
and Lemma 1 yields
Thus for each \(\theta \in \Theta \), we have
which means that \({\mathscr {M}}_{\text {min}}\) is sufficient in Umegaki’s sense. \(\square \)
An important factorization theorem was formulated in [4, Theorem 4] in the case when the algebra \({\mathscr {M}}\) 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 a linear operator A on a Hilbert space \({\mathcal {H}}\), by \({\mathcal {D}}(A)\), we denote the domain of A, and by \({\bar{A}}\)—the closure of A.
Let \({\mathscr {M}}\) be a finite von Neumann algebra with a normal faithful finite trace \(\tau \), let \({\mathscr {M}}_0\) be a von Neumann subalgebra of \({\mathscr {M}}\), and let \({\mathscr {M}}_1={\mathscr {M}}'_0\cap {\mathscr {M}}\)be the relative commutant of \({\mathscr {M}}_0\). By \({\widetilde{{\mathscr {M}}}}\) we denote the algebra of measurable operators, and by \(L^1({\mathscr {M}},\tau ),\,L^1({\mathscr {M}}_0,\tau )\), \(L^1({\mathscr {M}}_1,\tau )\) the corresponding \(L^1\)-spaces. (Here we make use of the assumption that \({\mathscr {M}}\) is finite, otherwise there is no guarantee that \(\tau |{\mathscr {M}}_0\) or \(\tau |{\mathscr {M}}_1\) is semifinite.) Let \(\{\rho _{\theta }:\theta \in \Theta \}\) be a faithful family of normal states on \({\mathscr {M}}\), and let, as before, for some convex combination
\(\omega \) be a faithful state on \({\mathscr {M}}\). Denote
and let \(D_{\omega }\in L^1({\mathscr {M}},\tau ),\,D_{\omega _0}\in L^1({\mathscr {M}}_0,\tau _0),\, D_{\omega _1}\in L^1({\mathscr {M}}_1,\tau _1)\) be the densities of \(\omega ,\,\omega _0\) and \(\omega _1\), respectively. Now the reasoning of [4, p. 269] can be repeated word for word upon observing that the modular automorphism groups \(\{\sigma _t^{\omega }\}\), \(\{\sigma _t^{\omega _0}\}\) and \(\{\sigma _t^{\omega _1}\}\) 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 \({\mathscr {M}}_1\).
Denote by \(D_{\theta },\,D_{\theta ,0}\) the densities of the states \(\rho _{\theta }\) and \(\rho _{\theta }|{\mathscr {M}}_0\), respectively. Now the factorization theorem reads
Theorem 3
The subalgebra \({\mathscr {M}}_0\) is sufficient for a faithful family of normal states \(\{\rho _{\theta }:\theta \in \Theta \}\) if and only if
where z is a positive self-adjoint operator with trivial null-space, affiliated with the centre of \({\mathscr {M}}_1\).
Proof
Fix \(\theta \), and let
be the spectral decompositions of \(D_{\theta ,0},\,D_{\omega _1}\) and z. Since \(D_{\theta ,0}\) is affiliated with \({\mathscr {M}}_0\), \(D_{\omega _1}\) is affiliated with \({\mathscr {M}}_1\), and z is affiliated with the centre of \({\mathscr {M}}_1\), it follows that the spectral measures \(E_0,\,E_1\) and \(E_z\) take values in \({\mathscr {M}}_0,\,{\mathscr {M}}_1\) and the centre of \({\mathscr {M}}_1\), respectively, thus they mutually commute. Consequently, there exists a spectral measure E in \({\mathscr {M}}\), 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 \({\mathscr {M}}\), i.e., it belongs to the algebra \({\widetilde{{\mathscr {M}}}}\) of measurable operators, while on the left hand side we have a product of operators from \({\widetilde{{\mathscr {M}}}}\). From the properties of \({\widetilde{{\mathscr {M}}}}\), 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 \(D_{\theta },\,D_{\omega },\,D_{\theta ,0}\) and \(D_{\omega _0}\) the following equalities for the Connes cocycles hold true
(cf. Araki and Masuda [1, Theorem C.1] for the chain rule in the first equality above, Strătilă [10, Corollary 3.4] for the second equality, and Strătilă [10, Corollary 4.8] for the third). Similarly,
Assume that \({\mathscr {M}}_0\) is sufficient. Then on account of [4, Theorem 1], we have \([D\rho _{\theta }:D_{\omega }]_t=[D\rho _{\theta ,0}:D_{\omega _0}]_t\), thus from (9), (10) and the fact that \(D_{\omega }^{it}\) are unitary, we get
Using (6) together with the fact that \(D_{\omega _0}^{it}\) are also unitary, we obtain
From this easily follows that \(D_{\theta }=B\). Indeed, denoting by \({\mathcal {N}}(T)\) the null-space of the operator T, we have from the spectral theorem
so
Denote by P the projection onto the common null-space above, and let
be the spectral decomposition of \(D_{\theta }\). Then
where \(\chi _{\{0\}}\) denotes the indicator function of the set \(\{0\}\). Hence
and similarly
which on account of the equality (11) yields
But the operators \(P+D_{\theta }\) 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 \({\widetilde{{\mathscr {M}}}}\), 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 \({\mathscr {M}}_*\) and \(L^1({\mathscr {M}},\tau )\), we have
where \(s -\sum \) denotes the strong sum. The same holds of course for \(\omega _{n,0}=\omega _n|{\mathscr {M}}_0\), i.e.,
so from the formula (7) and the fact that \({\widetilde{{\mathscr {M}}}}\) is an algebra, we obtain
Since \(\omega _n\rightarrow \omega \) in norm in \({\mathscr {M}}_*\), we infer that \(D_{\omega _n}\rightarrow D_{\omega }\) in the norm \(\Vert \cdot \Vert _1\), and by the same token \(D_{\omega _{n,0}}\rightarrow D_{\omega _0}\). According to Lemma 5 (see the Appendix), we have \(D_{\omega _n}\rightarrow D_{\omega }\) and \(D_{\omega _{n,0}}\rightarrow D_{\omega _0}\) in measure. Since \({\widetilde{{\mathscr {M}}}}\) 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 \(D_{\omega _0},\,D_{\omega _1}\) and z commute, from the equalities (7) and (13), we obtain
Thus on account of commutation properties and the fact that \(\{D_{\omega _1}^{it}\}\) and \(\{z^{it}\}\) are unitary groups, we get
which by virtue of [4, Theorem 1] yields the sufficiency of \({\mathscr {M}}_0\). \(\square \)
4 Sufficiency for pure states on \({\mathbb {B}}({\mathcal {H}})\)
In this section, we shall be dealing with the full algebra \({\mathscr {M}}={\mathbb {B}}({\mathcal {H}})\) and pure (\(\equiv \) vector) states. Let \(\{\psi _{\theta }:\theta \in \Theta \}\) be a faithful family of pure states. According to [6, Theorem 1], there exists the minimal sufficient for this family subalgebra of \({\mathbb {B}}({\mathcal {H}})\). It turns out that this algebra admits a straightforward description.
Theorem 4
Let \(\{\psi _{\theta }:\theta \in \Theta \}\) be a faithful family of pure states on \({\mathbb {B}}({\mathcal {H}})\), and let \({\mathscr {M}}_{\text {min }}\) be the minimal sufficient for this family subalgebra of \({\mathbb {B}}({\mathcal {H}})\). Then
-
(i)
\({\mathscr {M}}_{\text {min }}\) is generated by \(P_{[\psi _{\theta }]}\), i.e., \({\mathscr {M}}_{\text {min }}=W^*(\{P_{[\psi _{\theta }]}:\theta \in \Theta \})\).
-
(ii)
There exists a decomposition
$$\begin{aligned} {\mathcal {H}}=\bigoplus _i{\mathcal {H}}_i \end{aligned}$$such that
$$\begin{aligned} {\mathscr {M}}_{\text {min }}=\sum _i{}^{\oplus }{\mathbb {B}}({\mathcal {H}}_i). \end{aligned}$$(14)
Proof
First observe that \({\text {s}}(\psi _{\theta })=P_{[\psi _{\theta }]}\). As in Theorem 2, denote by \({\mathfrak {S}}\) the family of all normal positive unital maps on \({\mathscr {M}}\) such that the states \(\psi _{\theta }\) are invariant with respect to the maps from \({\mathfrak {S}}\). On account of [12, Theorem 4], for every \(\theta \in \Theta \), we have \(\alpha (P_{[\psi _{\theta }]})=P_{[\psi _{\theta }]}\) for each \(\alpha \in {\mathfrak {S}}\), which by virtue of [6, Theorem 1] yields the relation \(P_{[\psi _{\theta }]}\in {\mathscr {M}}_{\text {min}}\).
For the sake of brevity, set \({\mathcal {R}}=\{\psi _{\theta }:\theta \in \Theta \}\), and define on \({\mathcal {R}}\) an equivalence relation as follows: \(\psi _{\theta '}\equiv \psi _{\theta ''}\) if there exists a finite string \(\psi _{\theta _1},\dots ,\psi _{\theta _m}\) such that \(\psi _{\theta _1}=\psi _{\theta '}\), \(\psi _{\theta _m}=\psi _{\theta ''}\) and
Let
be the partition of \({\mathcal {R}}\) determined by the relation \(\equiv \). Denote \({\mathcal {R}}_{i}=\{\psi _{\theta }:{\theta } \in {\Theta _{i}} \}\). It is obvious that for all \(i',\,i''\) such that \(i'\ne i''\), we have \({\mathcal {R}}_{i'}\perp {\mathcal {R}}_{i''}\). Put
It follows that \(P_i\) are pairwise orthogonal, and the faithfulness of \(\{\psi _{\theta }:\theta \in \Theta \}\) yields
Denote \({\mathcal {H}}_i=P_i({\mathcal {H}})\). Then
Fix arbitrary i and consider the family of pure states \({\mathcal {R}}_i=\{\psi _{\theta }:\theta \in \Theta _i\}\) on the Hilbert space \({\mathcal {H}}_i\). Take arbitrary \(\psi _{\theta '},\psi _{\theta ''}\in {\mathcal {R}}_i\), and let \(\psi _{\theta _1},\dots ,\psi _{\theta _m}\), with \(\psi _{\theta _1}=\psi _{\theta '}\) and \(\psi _{\theta _m}=\psi _{\theta ''}\) be the string as in the definition of the equivalence relation \(\equiv \). From this definition, it follows that \(\psi _{\theta _j}\equiv \psi _{\theta _k}\) for all \(j,k=1,\dots ,m\), thus \(\psi _{\theta _j}\in {\mathcal {R}}_i\) for all \(j=1,\dots ,m\). Consequently, \(P_{[\psi _{\theta _j}]}\leqslant P_i\), so we can consider the projections \(P_{[\psi _{\theta _j}]}\) on the Hilbert space \({\mathcal {H}}_i\). We have
where
and for \(\xi ,\eta \in {\mathcal {H}}\) the operator \(E_{\xi ,\eta }\) 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 \(E_{\xi ,\eta }=|\xi \rangle \langle \eta |\).)
Denote \({\mathscr {N}}=W^*(\{P_{[\psi _{\theta }]}:\theta \in \Theta \})\). On account of the above considerations, we infer that \(E_{\psi _{\theta '},\psi _{\theta ''}}\in {\mathscr {N}}\) for all \(\psi _{\theta '},\psi _{\theta ''}\in {\mathcal {R}}_i\).
Let
be an arbitrary finite linear combination of vectors from \({\mathcal {R}}_i\). We have
(This is again most easily seen by using Dirac notation since then
It follows that \(P_{[\xi ]}\in {\mathscr {N}}\). Vectors \(\xi \) as above lie densely in \({\mathcal {H}}_i\), so arbitrary \(\eta \in {\mathcal {H}}_i\) is a limit of some \(\xi _n\), consequently,
hence \(P_{[\eta ]}\in {\mathscr {N}}\) for each \(\eta \in {\mathcal {H}}_i\). Linear combinations of these projections restricted to the subspace \({\mathcal {H}}_i\) lie densely in \({\mathbb {B}}({\mathcal {H}}_i)\), so for the reduced von Neumann algebra \({\mathscr {N}}_{P_i}\), we get the inclusion
and thus
From the definition of \(P_i\), it follows that \(P_i\) commutes with all \(P_{[\psi _{\theta }]}\), \(\theta \in \Theta \), because
Since the algebra \({\mathscr {N}}\) is generated by the projections \(P_{[\psi _{\theta }]}\), we obtain that \(P_i\in {\mathscr {N}}'\). For each \(A\in {\mathscr {N}}\), we have
showing that
and the formula (17) gives the representation
In the first part of the proof, we obtained the relation \(P_{[\psi _{\theta }]}\in {\mathscr {M}}_{\text {min}}\) for all \(\theta \in \Theta \) which yields the inclusion
Consider a map \({\mathbb {E}}:{\mathbb {B}}({\mathcal {H}})\rightarrow {\mathscr {N}}\) defined as
It is easily seen that \({\mathbb {E}}\) is a normal conditional expectation such that the states \(\psi _{\theta }\) are \({\mathbb {E}}\)-invariant, thus \({\mathscr {N}}\) is sufficient in Umegaki’s sense. Since \({\mathscr {M}}_{\text {min}}\) is minimal sufficient, we obtain
showing that \({\mathscr {M}}_{\text {min}}={\mathscr {N}}\), and the formula (18) together with the definition of \({\mathscr {N}}\) as \(W^*(\{P_{[\psi _{\theta }]}:\theta \in \Theta \})\) finish the proof. \(\square \)
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The author is indebted to the referee for valuable comments on the previous version of the paper, in particular, for pointing out an inaccuracy in the proof of Theorem 2.
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Appendix
Appendix
In this section, we collect some facts from the theory of measurable operators and noncommutative \(L^1\)-space over a von Neumann algebra with a normal faithful finite trace. A fuller account of this theory, also for a more general case of a semifinite trace and spaces \(L^p\), can be found, e.g., in [8, 9, 11, 15].
Thus let \({\mathscr {M}}\) be a von Neumann algebra with a normal faithful finite trace \(\tau \) acting on a Hilbert space \({\mathcal {H}}\) . The algebra of measurable operators \({\widetilde{{\mathscr {M}}}}\) is defined as a topological \({}^*\)-algebra of densely defined closed operators on \({\mathcal {H}}\) affiliated with \({\mathscr {M}}\) with strong addition \(\dotplus \) and strong multiplication \(\cdot \), i.e.,
and the translation-invariant measure topology defined by a fundamental system of neighbourhoods of 0, \(\{N(\varepsilon ,\delta ): \varepsilon ,\delta >0\}\), given by
Thus for operators \(A_n, A\) in \({\widetilde{{\mathscr {M}}}}\), the sequence \((A_n)\) converges to A in measure if for any \(\varepsilon ,\delta >0\) there exists \(n_0\) such that for each \(n\geqslant n_0\) there exists a projection \(P\in {\mathscr {M}}\) such that
The following ‘technical’ form of convergence in measure proved in [15, Proposition 2.7] is useful. Let
be the spectral decomposition of \(|A_n-A|\) with spectral measures \(E_n\) taking values in \({\mathscr {M}}\) since \(A_n-A\), and thus \(|A_n-A|\), are affiliated with \({\mathscr {M}}\). Then \(A_n\rightarrow A\) in measure if and only if for each \(\varepsilon >0\)
Introduce on \({\mathscr {M}}\) the norm \(\Vert \cdot \Vert _1\) as follows
The space \(L^1({\mathscr {M}},\tau )\) is defined as the completion of \({\mathscr {M}}\) with respect to this norm. It turns out that \(L^1({\mathscr {M}},\tau )\) consists of measurable operators, \(\tau \) can be extended to a linear bounded positive functional on \(L^1({\mathscr {M}},\tau )\) such that \(\Vert A\Vert _1=\tau (|A|)\) for \(A\in L^1({\mathscr {M}},\tau )\), and \(L^1({\mathscr {M}},\tau )\) is isometrically isomorphic to \({\mathscr {M}}_*\) with the isomorphism given by the formula
The operator D is called the density of \(\varphi \). Moreover, if \(\varphi \) is hermitian then \(D=D^*\), and \(\varphi \geqslant 0\) if and only if \(D\geqslant 0\).
For an arbitrary \(A\in L^1({\mathscr {M}},\tau )\) we have the spectral decomposition
Thus for any \(\varepsilon >0\), 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 \(L^1({\mathscr {M}},\tau )\) converges in \(\Vert \cdot \Vert _1\)-norm, then it converges in measure.
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Łuczak, A. Some aspects of quantum sufficiency for finite and full von Neumann algebras. Lett Math Phys 111, 95 (2021). https://doi.org/10.1007/s11005-021-01428-8
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DOI: https://doi.org/10.1007/s11005-021-01428-8