Entropy reduction of quantum measurements

It is observed that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a q-c channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. This observation makes it possible to define the entropy reduction for arbitrary a priori states (not only for states with finite von Neumann entropy) and to study its analytical properties by using general properties of the quantum mutual information. By using this approach one can show that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.


Introduction
The notion of a quantum measurement plays a key role in quantum theory. One of quantitative characteristics of a quantum measurement is the entropy reduction 1 defined as a difference between the von Neumann entropy of an a priori (pre-measurement) state and the mean von Neumann entropy of the corresponding posteriori (post-measurement) states. Roughly speaking, the entropy reduction characterizes a degree of purifying ("gain in purity") of a state in a measurement process. More details about the information sense of this value can be found in [11,15,21].
An interesting question concerns the sign of the entropy reduction. Groenewold has conjectured in [6] and Lindblad has proved in [12] that the entropy reduction is nonnegative for quantum measurements of the von Neumann-Luders type. The general case has been studied by Ozawa, who has proved in [21] that the entropy reduction is nonnegative if and only if the quantum measurement is quasicomplete (also called irreducible in [19]) in the sense that for an arbitrary pure a priori state the corresponding family of posteriori states consists of pure states (for almost all outcomes). The class of quasicomplete (irreducible) quantum measurements contains the class of efficient or pure measurement (cf. [10,11,15]) described in Section 3.1. Quantum measurements belonging to the gap between these two classes are characterized in the Appendix as measurements with quite singular properties.
In this paper we show that the entropy reduction of an efficient (pure) quantum measurement can be expressed via the (generalized) quantum mutual information of the quantumclassical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. This makes it possible to define the entropy reduction for arbitrary a priori states (not only for states with finite von Neumann entropy) and to study its analytical properties by using results concerning the quantum mutual information of infinite-dimensional channels [8].
The paper is organized as follows. In the first part we restrict attention to the case of quantum measurements with a discrete set of outcomes, which is more simple mathematically. In the second part we consider the case of general quantum measurements described by completely positive instruments by generalizing the notion of the quantum mutual information for channels taking values in the space of normal states of an arbitrary W * -algebra.

The discrete case
Let H be a separable Hilbert space, B(H) -the Banach space of all bounded operators in H with the operator norm · , T(H) -the Banach space of all trace-class operators in H with the trace norm · 1 = Tr| · |, containing the cone T + (H) of all positive trace-class operators. The closed convex subset S(H) = {A ∈ T + (H) | TrA = 1} is a complete separable metric space with the metric defined by the trace norm. Operators in S(H) are denoted ρ, σ, ω, ... and called density operators or states since each density operator uniquely defines a normal state on B(H) [3].
The identity operator in a Hilbert space H and the identity transformation of the set T(H) will be denoted I H and Id H correspondingly.
For an arbitrary state ω ∈ S(H ⊗ K) the partial states Tr K ω and Tr H ω will be denoted ω H and ω K .
We will use the following natural extension of the von Neumann entropy H(ρ) = −Trρ log ρ of a quantum state ρ ∈ S(H) to the cone T + (H) of all positive trace-class operators The quantum relative entropy is defined for arbitrary operators A and B in T + (H) as follows where {|i } is the orthonormal basis of eigenvectors of A and it is assumed that H(A B) = +∞ if suppA is not contained in suppB [13]. A linear completely positive trace-preserving map Φ : T(H) → T(H ′ ) is called a quantum channel [9,17].
By the Stinespring dilation theorem there exist a separable Hilbert space H ′′ and an isometry V : The quantum channel is called complementary 3 to the channel Φ, it is uniquely defined up to unitary equivalence [7]. The quantum mutual information is an important entropic characteristic of a channel Φ : T(H) → T(H ′ ) related to the entanglement-assisted classical capacity of this channel [9,17]. In finite dimensions it is defined at arbitrary state ρ ∈ S(H) by the expression (cf. [1]) In infinite dimensions this expression may contain uncertainty "∞ − ∞", but it can be modified to avoid this problem as follows where K is a Hilbert space isomorphic to H,ρ is a purification 4 of the state ρ in the space H⊗K and ̺ = Tr Hρ is a state in S(K) isomorphic to ρ. Analytical properties of the function (ρ, Φ) → I(ρ, Φ) defined by (5) in the infinite dimensional case are studied in [8].
An efficient quantum measurement M with a countable outcome set X = {x i } i∈I is described by a set {V i } i∈I of operators in B(H) such that i∈I V * i V i = I H . Applying this measurement to an arbitrary a priori state ρ ∈ S(H) results in the posteriori probability distribution {π i (ρ)} i∈I , where π i (ρ) = TrV i ρV * i is the probability of the outcome x i , and the corresponding family of posteriori states {ρ i } i∈I , where ρ i = (π i (ρ)) −1 V i ρV * i . Thus i∈I π i (ρ)H(ρ i ) = i∈I H(V i ρV * i ) is the mean entropy of posteriori states. The entropy reduction of the quantum measurement M at an a priori state ρ with finite entropy is the following value Let H X be a Hilbert space having dimension coinciding with the cardinality of the outcome set X. Consider the quantum channel where {|ϕ i } i∈I is a particular orthonormal basis in H X . The channel Π M is a quantum modification of the quantum-classical channel mapping a state ρ to the probability distribution for any state ρ in S(H) with finite entropy. If H ({π i (ρ)} i∈I ) < +∞ this equality directly follows from (4) In general case it can be easily deduced 5 from [8, Proposition 3 and Theorem 1]. Equality (7) obtained under the condition H(ρ) < +∞ makes it possible to consider the entropy reduction of an efficient quantum measurement with a countable outcome set as a function on the whole space of a priori states. Definition 1. The entropy reduction of an efficient quantum measurement M = {V i } i∈I at an arbitrary a priori state ρ is defined as follows where Π M is the quantum channel defined by (6).
By equality (7) this definition is consistent with the conventional one. Its main advantage consists in possibility to study the function ρ → ER(ρ, M) on the whole space of a priori states by using properties of the quantum mutual information (many of them follow from the corresponding properties of the quantum relative entropy). 3) monotonicity: for arbitrary efficient measurements M = {V i } i∈I and N = {U j } j∈J in a Hilbert space H with the outcome sets X and Y the inequality holds for any ρ ∈ S(H), where N • M is the measurement in the space H with the outcome set X × Y determined by the family {U j V i } i∈I,j∈J ; 4) subadditivity: for arbitrary efficient measurements M = {V i } i∈I and N = {U j } j∈J in Hilbert spaces H and K with the outcome sets X and Y the inequality holds for any ω ∈ S(H ⊗ K), where M ⊗ N is the measurement in the space H ⊗ K with the outcome set X × Y determined by the family {V i ⊗ U j } i∈I,j∈J .
" ⇒ " Relations (11) mean that P V * i V i P = π i (ρ)P for each i, where P = k |k k| is the projector on the support of the state ρ. Thus (π i (ρ)) −1/2 V i P is a partial isometry and hence ρ i = (π i (ρ)) −1 V i P ρP V * i ∼ = ρ for each i such that π i (ρ) = 0.
3) This follows from the 1-st chain rule for the quantum mutual information (property 3 in Proposition 1 in [8]). Indeed, Consider the question of continuity of the entropy reduction with respect to "perturbation" of quantum measurements.
Let M(H) be the set of all efficient quantum measurements in the Hilbert space H with finite or countable set of outcomes identified with the set of all sequences with the topology of coordinate-wise strong operator convergence. Proposition 1 in [8] and Corollary 2 in [8] imply the following assertion.
Proposition 1. The function is lower semicontinuous on the set S(H) × M(H). Let A be an arbitrary subset of S(H) on which the von Neumann entropy is continuous. Then function (12) is continuous on the set A × M(H).
By Proposition 1 and Proposition 6.6 in [18] function (12) is continuous on the set K H,h ×M(H), where K H,h is the set of states with the mean energy TrHρ not exceeding h > 0 provided the Hamiltonian H of the quantum system satisfies the condition Tre −λH < +∞ for all λ > 0 (which holds, for example, for the Hamiltonian of the system of quantum oscillators).

On properties of efficient (pure) instruments
A general quantum measurement in a Hilbert space H with the measurable outcome set {X, F } is described by a special mathematical object called instrument, which was introduced by Davis and Lewis [5]. An instrument M (in a space of states) is a σ-additive measure on {X, F } taking values within the set of quantum operations -completely positive tracenon-increasing linear transformations of T(H) such that M(X) is a channel (see the detailed definition in [2,9,14]).
Let ρ be an arbitrary a priori state in S(H). Then the outcome of the measurement M is contained in a set F ∈ F with probability TrM(F ) [ρ]. If this probability is nonzero is the corresponding posteriori state of the system. Thus is the posteriori probability measure on the outcome set {X, F } corresponding to the a priori state ρ.
Ozawa proved in [20] existence of a family {ρ x } x∈X of posteriori states defined for µ ρ -almost all x such that the function x → TrAρ x is F -measurable for any A ∈ B(H) and By using the family {ρ x } x∈X one can consider the mean entropy of posteriori state X H(ρ x )µ ρ (dx) and assuming that H(ρ) < +∞ one can define the entropy reduction as follows This is a natural generalization of the entropy reduction considered in Section 2 for the class of measurements with a countable set of outcomes.
Ozawa proved in [21] that ER(ρ, M) is nonnegative if and only if the instrument M is irreducible in the sense of the following definition. 7 Definition 2. An instrument M is called irreducible if for an arbitrary pure a priori state ρ the posteriori states ρ x are pure for µ ρ -almost all x.
An arbitrary instrument M in a Hilbert space H can be represented as follows 8 where V is an isometry from H into H ⊗ H 0 and P (F ) is a spectral measure in H 0 [19] (see also [10]).
The following notion introduced in [10] is a natural generalization of the notion of an efficient measurement with a countable outcome set.
Definition 3. An instrument M is called efficient or pure if it has representation (13) with the spectral measure P (F ) of multiplicity one. 9 In [10] it is shown that an efficient instrument is irreducible. The converse assertion is not true (see Example 1 in the Appendix). A characterization of a quantum instrument, which is irreducible but not efficient, and a simple sufficient condition providing equivalence of efficiency and irreducibility are presented respectively in Proposition 5 and in Corollary 3 in the Appendix. This characterization shows that a quantum instrument, which is irreducible but not efficient, has quite singular properties.
It is the class of efficient instruments to which the results of Section 2 can be extended. We will use the following characterization of such instruments, where M ⊗ J K denotes the instrument M(·) ⊗ Id K in the Hilbert space H ⊗ K with the outcome set of the instrument M.
Proposition 2. Let M be an instrument in a Hilbert space H with the outcome set {X, F }. The following statements are equivalent: for any ϕ ∈ D, F ∈ F , A ∈ B(H).
(v) statement (iv) holds with D = lin{|ϕ i }, where {|ϕ i } is a given arbitrary orthonormal basis in H.
Proof. (i) ⇒ (iii). It is easy to see that the instruments M and M ⊗ J K have representation (13) with the same spectral measure. Hence efficiency of M is equivalent to efficiency of M ⊗ J K . As mentioned after the proof of Theorem 1 in [10] any efficient instrument is irreducible.
(iii) ⇒ (ii) is obvious, since the instrument in (ii) can be considered as a restriction of the instrument in (iii).
(ii) ⇒ (v). By Theorem 1 in [10] and its proof there exist a positive σ-finite measure µ on {X, F } and a countable family is an orthonormal basis of the space K. By using the polarization identity it is easy to show that Hence for the instrument M ⊗ J K and a pure a priori stateρ = |φ φ| (whereφ ∈ D) we have while the posteriori state corresponding to the outcome x ∈ X \ X ŝ ϕ iŝ is a set such that µρ(X ŝ ϕ ) = 0 [10]. By noting that the instrument M ⊗ J K is irreducible and by using (14) one can show existence of a set Xφ ∈ F such that µ(X \ Xφ) = 0 and the above stateρ x is pure for any x ∈ Xφ \ X ŝ ϕ . Let D 0 be a countable subset of D consisting of finite linear combinations of the vectors of the family {ϕ i ⊗φ j } ij with rational coefficients. Let X 0 = φ∈ D 0 Xφ ∈ F . Then µ(X \X 0 ) = 0 and the stateρ x defined by (15) is pure for allφ ∈ D 0 and all x ∈ X 0 \X ŝ ϕ . Hence the family {| V k (x)φ } k consists of collinear vectors for allφ ∈ D 0 and all x ∈ X 0 . Since the rank of the (v) ⇒ (vi) is obvious.
(iv) ⇒ (i). By the condition the linear operator is isometrical and hence it can be extended to the isometry V from H into H ⊗ L 2 (X, F , µ). A direct verification shows that is the indicator function of the set F ∈ F . Thus the instrument M is efficient.
We will also use the following simple observations.
Proof. It is easy to show that the instruments N•M and M⊗N have representation (13) with the spectral measure P M ⊗ P N , where P M and P N are spectral measures corresponding to the instruments M and N.

A representation of the entropy reduction
To extend the results of the previous section to the case of general type measurement consider the construction proposed by Barchielli and Lupieri in [2]. Choose a positive complete measure µ 0 on (X, F ) such that µ ρ is absolutely continuous with respect to µ 0 for all ρ in S(H) (this can be done by using the measure µ ρ 0 , where ρ 0 is a given full rank state in S(H)). Let L ∞ (X, F , µ 0 , B(H)) be the W*-algebra of µ 0 -essentially bounded B(H)-valued weakly* measurable functions on X with the predual Banach space L 1 (X, F , µ 0 , T(H)) of T(H)-valued Bochner µ 0 -integrable functions on X. By Theorem 2 in [2] with an arbitrary instrument M one can associate a channel Λ * M : The preadjoint channel Λ M : T(H) → L 1 (X, F , µ 0 , T(H)) produces the posteriori family as follows where σ(x) is a particular representative of the class Λ M (ρ), while the function Trσ(x) is a probability density (the Radon-Nikodym derivative) of the measure µ ρ with respect to the measure µ 0 . Consider the channel Π * M : L ∞ (X, F , µ 0 ) → B(H) defined by the relation The preadjoint channel Π M : T(H) → L 1 (X, F , µ 0 ) maps an arbitrary a priori state ρ to the probability density of the posteriori measure µ ρ with respect to the measure µ 0 and hence it can be considered as a natural generalization of the channel Π M defined by (6).
Since the channel Π M defined by (17) has no purely quantum modification, to extend the results of Section 2 we have to generalize the notion of quantum mutual information.
where K is a Hilbert space isomorphic to H,ρ is a purification of the state ρ in the space H ⊗ K, ̺ = Tr Hρ is a state in S(K) isomorphic to ρ and H(· ·) is the relative entropy for two states in (A ⊗ B(K)) * .

Remark 1.
It is natural to ask about validity for the above-defined value of the properties of the quantum mutual information of a purely quantum infinite dimensional channel presented in Propositions 1 and 4 in [8]. Since the proofs of these propositions can not be directly generalized to the case of a channel considered in Definition 4, the above question is not trivial.
Here we note only that for an arbitrary channel Φ : T(H) → A * the function ρ → I(ρ, Φ) is nonnegative and lower semicontinuous and that for an arbitrary channel Ψ : A * → B * , where B is an other W * -algebra, the inequality I(ρ, Ψ • Φ) ≤ I(ρ, Φ) holds for all ρ (the 1-st chain rule). These properties follow from nonnegativity and lower semicontinuity of the relative entropy, Lemma 2 in [8] and Uhlmann's monotonicity theorem [18].
The relative entropy for two states σ 1 and σ 2 in L 1 (X, F , µ 0 , T(H)) can be expressed as follows where µ 1 (dx) = Trσ 1 (x)µ 0 (dx) (see [2, formula (4)]). In this expression H q denotes the quantum relative entropy for two positive trace class operators defined by (1), while H c denotes the classical relative entropy for two probability measures, that is H c (µ 1 µ 2 ) = X log Trσ 1 (x) Trσ 2 (x) µ 1 (dx). For an arbitrary instrument M equality (7) does not hold, but one can prove the following estimation. Let {ρ x } be the family of posteriori states obtained via a given representative of the class Λ M (ρ) by the rule similar to (16). It is easy to see that {ρ x . = Tr Kρx } is the family of posteriori states for the instrument M corresponding to the a priori state ρ. Since the instrument M is localized in the space H we have X Tr Hρx µ ρ (dx) = Tr Hρ = ̺ ∼ = ρ. By using expression (18) we obtain By the triangle inequality the absolute value of the last term in this expression is majorized by X H(ρ x )µ ρ (dx).
where Π M is the quantum-classical channel defined by (17).
The following theorem is an extended version of Theorem 1.
holds for any ρ ∈ S(H); 10 4) subadditivity: for arbitrary efficient instruments M and N in separable Hilbert spaces H and K the inequality holds for any ω ∈ S(H ⊗ K). 11 Proof. By Definition 5 lower semicontinuity of the function ρ → ER(ρ, M) follows from the second part of Remark 1.
Concavity of the function ρ → ER(ρ, M) on the convex subset of states with finite entropy follows from inequality (56) in [2]. Concavity of this function on the set S(H) can be proved by using lower semicontinuity of this function and Lemma 2 below (see the proof of Proposition 1 in [8]).
Below we will use the notations introduced in the proof of Proposition 3. 1) Note that "⇐" Let ̺ x = Tr Hρx . Since the instrument M is irreducible (by Proposition 2), we have ρ x ∼ = ̺ x for µ ρ -almost all x and hence ̺ x ∼ = ̺ = Tr Hρ for µ ρ -almost all x. Since the instrument M is localized in the space H, we have ̺ = X ̺ x µ ρ (dx). Thus Lemma 6 in the Appendix implies (21).
2) This property follows from identity (24) in Lemma 3 below, since the both summands in the left side of this identity are lower semicontinuous functions on the set S(H) by the second part of Remark 1.
3) This follows from the 1-st chain rule for the generalized quantum mutual information mentioned in the second part of Remark 1. Indeed, let µ 0 be a measure on {X × Y, F ⊗ E} chosen in accordance with the Barchielli-Lupieri construction for the instrument N • M and ν 0 be a measure on for any F ∈ F and ρ ∈ S(H). Hence the measure ν 0 can be used in the Barchielli-Lupieri construction for the instrument M. Let Ξ be a channel from L 1 (X × Y, F ⊗ E, µ 0 ) to L 1 (X, F , ν 0 ) preadjoint to the channel .
which can be proved easily by noting that N(Y ) is a trace preserving map. 4) Let µ 0 and ν 0 be measures on {X, F } and on {Y, E} chosen in accordance with the Barchielli-Lupieri construction for the instruments M and N correspondingly. Then for the instrument M ⊗ N one can take the measure µ 0 ⊗ ν 0 . By noting that it is easy to show that Λ * M⊗N = Λ * M ⊗ Λ * N . Let ω be a state in S(H ⊗ K) such that H(ω H ) and H(ω K ) are finite. Then inequality Proof. Let K ∼ = H and P n = n i=1 |e i e i | be a projector in K, n = 1, 2... Consider the value λ i λ j |e i e j | ⊗ |e i e j |,ρ n = c −1 n I H ⊗ P n ·ρ 0 · I H ⊗ P n , ̺ 0 = Tr Hρ0 , ̺ n = Tr Hρn and Ψ n (·) = P n (·)P n is a map from T(K) to itself. We will show that lim Let σ 1 (x) and σ 2 (x) be representatives of the classes Π M ⊗ Id K (ρ 0 ) and Π M (ρ 0 ) ⊗ ̺ 0 correspondingly. Then c −1 n P n σ 1 (x)P n and c −1 n P n σ 2 (x)P n are respectively representatives of the classes c −1 n Π M ⊗ Ψ n (ρ 0 ) and c −1 n Π M (ρ 0 ) ⊗ Ψ n (̺ 0 ). Expression (18) implies Hence (23) follows from Lemma 4 in [13] and the monotone convergence theorem. By using expression (18) we obtain since c n µ ρn (F ) ≤ µ ρ 0 (F ) for all F ∈ F . Hence lim n (I n − ER(ρ n , M)) = 0. This and (23) imply the assertion of the lemma.
Proof. Since the instrument M is irreducible (by Proposition 2) and is localized in the space H we have H(ρ x ) = 0 for µ ρ -almost all x ∈ X and X Tr Hρx µ ρ (dx) = Tr Hρ = ̺ ∼ = ρ. By using expression (18) we obtain This expression and Corollary 1 imply (24).

Remark 2.
By proving concavity of the function ρ → I(ρ, Λ M ) and by using Lemma 6 in [23] one can show validity of equality (24) for any ρ in S(H). By comparing this equality with the assertion of Theorem 1 in [8] we see that the channel Λ M plays the role of the complementary channel to the channel Π M . Strictly speaking, this holds in the discrete case when the q-c channel Π M can be considered as a purely quantum channel (see Section 2).
The first assertion follows from lower semicontinuity of the relative entropy with respect to pointwise convergence of states [18,Corollary 5.12].
. By means of (27) it is easy to show that the last three terms in this expression are less than ε A for all n = 0, 1, 2, ... As proved before (25) implies that the first term tends to X TrP ε A(x)P ε σ 0 (x)µ 0 (dx), since P ε A(x)P ε ∈ B(P ε (H)). Thus we conclude that (26) holds.

A characterization of a quantum instrument which is irreducible but not efficient
In for any ϕ ∈ D, F ∈ F , A ∈ B(H). We may assume that for each x the all nonzero operators from the family V k (x) are not proportional to each other, since if V k ′ (x 0 ) = λV k (x 0 ) for some x 0 and λ ∈ C then we may replace V k (x 0 ) by 1 + |λ| 2 V k (x 0 ) and consider that V k ′ (x 0 ) = 0.
The instrument M is efficient if and only if V k (x) = 0 for k > 1 and µ-almost all x. This follows from the proof of Theorem 1 in [10] and the proof of the implication (iv) ⇒ (i) in Proposition 2.
So, if the instrument M is not efficient then there exists a subset F s of X such that M(F s ) = 0 and V 2 (x) = 0 for all x ∈ F s . 12 Proposition 5. If the instrument M is irreducible but not efficient then for µ-almost all x in F s the all nonzero operators V k (x) has the same one dimensional range (depending on x), that is V k (x)|ϕ = ω k (x)[ϕ]|ψ x for all ϕ ∈ D, where ω k (x) is a linear functional defined on D (not necessary bounded) and ψ x is a unit vector in H (not depending on ϕ). This means that for any ϕ ∈ D, A ∈ B(H) and F ∈ F such that F ⊆ F s . It follows that for an arbitrary a priori state ρ the posteriori state is ρ x = |ψ x ψ x | for µ ρ -almost all x ∈ F s (that is, ρ x does not depend on ρ).
This assertion can be obtained by using the arguments from the proof of the implication (ii) ⇒ (iv) in Proposition 2 with dim K = 1. The only difference appears at the point where Lemma 5 is used, since in this case we can not exclude the possibility of rankV k (x) = 1. It is this possibility that prevents to prove that any irreducible instrument is efficient. Proof. Let M be a instrument in a Hilbert space H with the outcome set {X, F }, which is irreducible but not efficient. By Proposition 5 there exists a positive σ-finite measure µ on {X, F }, a subset F of X such that M(F ) = 0, a dense domain D ⊆ H, a family {x → ω k (x)} k of functions on F , such that ω k (x) is a linear functional defined on D for each x ∈ F , and a function x → ψ x on F such that ψ x is a unit vector in H for each x ∈ F , for which relation (28) holds. By using the polarization identity one can show that where denotes the Bochner integral.
It follows that the operator M(F ) ⊗ Id K (ω) is separable. Since an arbitrary pure state in S(H⊗K) can be approximated by a sequence of above-considered states, the operator M(F )⊗Id K (ω) is separable for any state ω ∈ S(H⊗K). Thus the operation M(F ) is entanglement-breaking. Example 1. Let P be a spectral projector valued measure on a measurable space {X, F } and |ψ 0 be a fixed unit vector in H. Then the instrument M(F )[ρ] = [TrP (F )ρ]|ψ 0 ψ 0 | is obviously irreducible, but it is efficient if and only if the spectral measure P has (uniform) multiplicity one, since it is easy to see that the spectral measure from representation (13) coincides with P . Note that the channel M(X) : ρ → |ψ 0 ψ 0 | is entanglement-breaking.

Two auxiliary lemmas
Lemma 5. Let L 1 = lin({ϕ i } i∈N ) be a linear space and L 0 1 be a countable subset of L 1 consisting of finite linear combinations of the vectors ϕ 1 , ϕ 2 , ... with rational coefficients. Let {A k } be a finite or countable family of nonzero linear operators from L 1 to a linear space L 2 such that the set {A k (ϕ)} ⊂ L 2 consists of collinear vectors for any ϕ ∈ L 0 1 . If at least one operator in the family {A k } has rank > 1 then A k = λ k A 1 for all k, where {λ k } is a set of nonzero scalars.
Thus we have Hence the vectors A k (ϕ 1 ) and A k (ϕ 2 ) are not collinear. By repeating the above arguments with A k instead of A 1 and A 1 instead of A k we obtain It follows from (30) and (31) that A k (ϕ i ) = λ k A 1 (ϕ i ) for all i ∈ N and hence A k = λ k A 1 .
Lemma 6. Let {π α , ρ α } be a countable or continuous ensemble of states in S(H) such that ρ α ∼ =ρ for all α, whereρ is the average state of this ensemble. Then ρ α =ρ for all α.
The assertion of this lemma follows from existence of a finite strictly convex function on the set S(H) depending only on the spectrum of a state. As the simplest example one can consider the function f (ρ) = Trρ 2 .
I am grateful to A.S.Holevo and the participants of his seminar "Quantum probability, statistic, information" for useful discussion. I am also grateful to the organizers of the program "Quantum information theory" in the Institut Mittag-Leffler (Djursholm, Sweden), where the work was completed, and to the participants of this program, especially, to E.Effros, M.Wolf and C.Palazuelos for the help in solving the particular problems.
This work is partially supported by the program "Mathematical control theory" of Russian Academy of Sciences, by the federal target program "Scientific and pedagogical staff of innovative Russia" (program 1.2.1, contract P 938) and by RFBR grant 09-01-00424-a.