Background

Recently, Matsumoto, Shimono, and Winter (MSW) [1] pointed out an important connection between the channel capacity and entanglement of formation that allows one to draw some conclusion about the additivity of the latter from that of the former. There has also been speculation that the additivities of channel capacity and of entanglement of formation are equivalent. A connection between the additivity of entanglement of formation and the multiplicativity of the p-norm measure of purity has also been given by Audenaert and Braunstein [2].

In view of this it is natural to ask whether every bipartite state _AB can be associated with a channel in the sense of MSW. In particular, is its reduced density matrix, _A, the optimal average output of a channel whose capacity is related to the entanglement of formation of _AB, as described in [1]? We give a more precise formulation of this statement and show that the answer is negative.

Recall that a state is represented by a density matrix , i.e., a positive semidefinite operator with Tr = 1. The Von Neumann entropy of is S() = -Tr log . By an ensemble = {_i, _i }, we mean a set of density matrices _i and associated probabilities _i. By a channel , we mean a completely positive, trace-preserving (CPT) map. The Holevo capacity of the map is

(1)

where the supremum is taken over all ensembles = {_i, _i } and = ∑_{i ii}. The optimal average input is the state _opt = ∑_{i ii} associated with the ensemble which attains the supremum in (1). The optimal average output is then (_opt).

The key to the MSW construction is the following result of Stinespring [3], which was subsequently used by Lindblad [4] in his work on relative entropy.

Given a CPT map on (), one can find an auxiliary space _B, a density matrix _B, and a unitary map U_AB on _{A B} such that

() = TrB[U†AB BUAB],

(2)

where Tr_B denotes the partial trace and we have identified with _A.

Although we are interested in the case in which is a density matrix, representation (2) is valid for all operators in (). When () = ∑_k A_k^†A_k, it is easy to construct a representation of the form (2), as discussed in Section III.D of [5]. In fact, U^†_{AB B}U_AB is the block matrix that has the form ∑_jk A_j^†A_k |j><k|.

Given a bipartite state _AB on _{A B}, its entanglement of formation (EoF) satisfies

(3)

Although it is customary to define the EoF using ensembles for which all _k are pure states, there is no loss of generality in allowing the presence of arbitrary states.

Let be a CPT map with a representation _B, _B, U_AB as in Theorem 1. The Holevo capacity of satisfies

(4)

where is a density matrix on = _A. Moreover, the state _AB which attains this supremum satisfies Tr_{B AB} = (_opt), where _opt is the optimal input of .

It follows immediately that for the state _AB,

EoF(AB) = S[(opt)] – CHolv().

(5)

It is thus natural to ask the following.

Given a bipartite state _AB on _{A B}, is there a CPT map on (_A) such that Tr_BAB is the optimal average output state of and

EoF(AB) = S(TrBAB) – CHolv()?

(6)

We show that the answer to this question is negative.

Our counterexample is based on a result in [6] and [7] which comes from the fact that when
= ∑_{k kk},

(7)

where the relative entropy is defined as

H(, ) = Tr[log – log ].

(8)

Let be a CPT map with optimal average input _opt. Then,

(9)

It follows as an immediate corollary that if = {_k, _k } is any optimal ensemble for the channel , then H[(_k), (_opt)] = C_Holv() is independent of k; i.e., the optimal average output is “equidistant” in the sense of relative entropy from all outputs (_k) in the ensemble.

Counterexample

An affirmative answer to Question 3 above would imply that if the ensemble {_k, |_k><_k|} is optimal for EoF(_AB), then {_i,Tr_B|_k><_k|} would be an optimal output ensemble for the corresponding map . It would then follow from the equidistance corollary to Theorem 4 that

H(k, A) = –S(k) – TrAk log A = C,

(10)

where _k =Tr_B|_k><_k| and C is a constant that is independent of k. (In fact, C is the Holevo capacity of if such a channel exists.) We next present an example of a bipartite qubit state which does not satisfy (10).

It may be that counterexamples to (10) can already be found in the literature on entanglement. However, we take advantage of a result of Wootters [8] to construct a rather simple qubit counterexample that does not require extensive numerical computation. In [8], Wootters obtained an explicit formula for the EoF of a bipartite qubit state using a quantity designated as the concurrence. Moreover, he showed that the EoF could be achieved using an ensemble of at most four pure states |_k><_k|, for which S(Tr_B|_k><_k|) =EoF(_AB). Thus, for such an ensemble, (10) holds if and only if

TrA[(TrB|k><k|) log A]

(11)

is independent of k. We can assume without loss of generality that _A is diagonal, with eigenvalues ½(1 ± x). Then Tr log _A depends only on the diagonal elements of which can be written as ½(1 ± d). In fact,

(12)

which depends linearly on d for a fixed value of x not equal to 0. Thus, when _A ≠ ½I, (11) is independent of k if and only if all _k = Tr_B|_k><_k| have the same diagonal elements. However, we also know that S(_k) = EoF(_AB) is independent of k, which implies that all _k have the same eigenvalues. Thus, all of the reduced density matrices Tr_B|_k><_k| must have the form

for some fixed value of t. It is not hard to find an example for which this does not hold.

Let |β₀> = (1/2)(|00> + |11>) and |β₃> = (_z I)|β₀> = (1/2)(|00> – |11>) be the indicated maximally entangled Bell states, and

(13)

The reduced density matrix is _A = Tr_B = ½(I + (3/16)_z) ≠ ½I. One can show that the optimal EoF decomposition of (13) has the form _AB = ∑⁴_{k=1 k}|_k><_k|, with |₁> = a₀|β₀> + a₃|β₃> so that ₁ = Tr_B(|₁><₁|) is diagonal in the basis |0>, |1> (i.e., t = 0 in the matrix above). However, the remaining _k are superpositions which contain Bell states |β_k> and the product states |01>, |10> in a form which necessarily yields a reduced density matrix _k that is not diagonal. By the discussion above, this implies a negative answer to Question 3.

To actually find the entanglement of formation and optimal decomposition of _AB, let

(14)

be the density matrix with all spins flipped. The concurrence µ can be expressed in terms of the eigenvalues of ( )^1/2. Following Wootters [8], one finds

Let

(15)

Then, proceeding as described in [8], one finds

The optimal ensemble has weights

₁ = 0.1527, ₂ = ₃ = ₄ = 0.2824

associated with the projections for the following pure states:

One can easily verify that the diagonal elements of ₁ = Tr_B|₁><₁| are not equal to those of
_k =Tr_B|_k><_k| for k = 2, 3, 4. One can also compute the reduced density matrices _k and infer that ₁ is diagonal in the basis |0>, |1>, but the others are not. Alternatively, one can observe that if a reduced density matrix is diagonal in the basis |0>, |1>, then any purification must have the form

(16)

with |₁>, |₂> orthogonal. By rewriting

one sees that ₂ does not have the form (16) with orthogonal |_k>. The actual reduced density matrices have the form

Remarks on representations

In the canonical method of constructing a representation of the form (2), the reference state _B is pure, and d_B, the dimension of _B, is equal to the number of Kraus operators. Then the lifted state _AB = U_{AB B}U^†_AB, for which () = Tr_BAB has rank at most d = d_A, the dimension of the original Hilbert space. This is clearly a very restricted class of bipartite states. Moreover, generically, d_B > d, since many maps require the maximum number d² of Kraus operators. Thus, the canonical representation yields only bipartite states that are far more singular than typical states.

Some maps () can be represented in the form (2) using a mixed reference state _B. Indeed, given a mixed bipartite state _B and unitary operator U_AB, (2) can be used to define a channel . Unfortunately, relatively little is known about such mixed-state representations. It has been suggested that one might be able to represent a channel in the form (2) using a space of dimension d_B = d if mixed states are used. However, this is known not to be true in general [9, 10].

Now suppose that a qubit channel can be represented (possibly using a mixed reference state _B) in the form (2) using an auxiliary qubit space _B so that d_B = 2. Then the argument given before (11) can be used to show that for the optimal input distribution {_kk},

Tr(k) log (opt),

(17)

must be independent of k. One can easily check that the three-state examples given in [11] do not have this property. Most non-unital qubit maps for which the translation of the image of the Bloch sphere lacks symmetry also violate (17). This implies that such channels require an auxiliary space with d_B > 2. One expects d_B = 4, consistent with the fact that such maps also require four Kraus operators. This gives another, somewhat indirect, proof that some CPT maps require an auxiliary space with d_B > d.

It would be of some interest to characterize the maps which admit mixed-state representations with d_B = d, as well as the bipartite states corresponding to the optimal average output.

Acknowledgments

It is a pleasure to thank Professor Christopher King for stimulating and helpful discussions. This work was partially supported by the U.S. National Science Foundation under Grant No. DMS-03-14228.

References