Separable Operations on Pure States

We show that the possible ensembles produced when a separable operation acts on a single pure bipartite entangled state are completely characterized by a majorization condition, a collection of inequalities for Schmidt coefficients, which is identical to that already known for the particular case of local operations and classical communication (LOCC). As a consequence, various known results for LOCC, including some involving monotonicity of entanglement, can be extended to the class of all separable operations.

A separable operation Λ on a bipartite quantum system is a transformation of the form where ρ is an initial density operator on the Hilbert space H A ⊗ H B . The Kraus operators A k ⊗ B k are arbitrary product operators satisfying the closure condition with I A and I B the identity operators. The extension to multipartite systems is obvious, but here we will only consider the bipartite case. To avoid technical issues the sums in (1) and (2) as well as the dimensions D A and D B of H A and H B are assumed to be finite. Local operations with classical communication (LOCC) form a subset of separable operations in which the Kraus operators A k ⊗ B k are restricted by the requirement that they be generated in the following fashion. Alice carries out an operation {A (1) in the usual way with the help of an ancilla, the measurement of which yields the value of i, which is then transmitted to Bob. He uses i to choose an operation {B (2,i) j }, the result j of which is transmitted back to Alice, whose next operation can depend on j as well as i, and so forth. While it is (fairly) easy to see that the end result after an arbitrary number of rounds is of the form (1), it is difficult to characterize in simple mathematical or physical terms precisely what it is that distinguishes LOCC from more general separable operations. Examples show that separable operations can be more effective than LOCC in distinguishing certain sets of orthogonal states [1], even in a system * Electronic address: vgheorgh@andrew.cmu.edu as simple as two qubits [2], but apart from this little is known about the difference.
What we demonstrate in Sec. II of this paper is that the ensemble {p k , |φ k } produced by a separable operation acting on a pure state |ψ , see (5), satisfies a majorization condition (7), which is already known to be a necessary and sufficient condition for producing the same ensemble from the same |ψ by LOCC. Among the consequences discussed in Sec. III are: a separable operation acting on a pure state can be "simulated" by LOCC; a necessary condition for a deterministic transformation |ψ → |φ given in [3] where D = D B , and we assume without loss of generality that D A ≥ D B . Here {|a j } and {|b j } are orthonormal bases chosen so that the Schmidt weights (coefficients) λ j are in increasing order, i.e.
The separable operation acting on |ψ will produce an ensemble and In [4] it was shown that such an ensemble {p k , |φ k } N k=1 can be produced from |ψ by a suitable LOCC if and only if the majorization inequalities and similarly for the |φ k . Here Tr A (|ψ ψ|) is the reduced density operator of |ψ ψ| on Bob's side, and χ n (·) is defined to be the sum of the first n smallest eigenvalues of its argument. Note that we are assuming that were greater than D A the extra zero eigenvalues in Tr A |ψ ψ| would cause confusion when using χ n . Our main result is the following.
can be produced by a bipartite separable operation acting on the normalized state |ψ if and only if the majorization condition defined by the collection of inequalities in (7) is satisfied.
Proof. To simplify the proof we assume that D A = D B = D. If D A is larger, one always modify each A k by following it with a suitable local unitary which has the result that as long as the Kraus operators are acting on a fixed |ψ the action on the A side takes place in a subspace of H A of dimension D. These local unitaries do not change the Schmidt weights of the |φ k or alter the closure condition (2). For more details about this "decoupling" see [3].
When the majorization condition (7) holds the result in [4] guarantees the existence of an LOCC (hence separable operation) which will produce the ensemble out of |ψ . The reverse inference, that the ensemble {p k , |φ k } N k=1 defined in (5) and (6) satisfies (7), follows from noting that and applying Theorem 2 below with R = I A ⊗ I B , corresponding to (2), so R = 1.

B. A majorization theorem
where R = sup ω =1 R|ω is the largest eigenvalue of the positive operator Proof. By map-state duality [3,5,6], using the Schmidt bases of |ψ , we transform the state A k ⊗ B k |ψ to a map denotes an operator mapping H B to H A , andB k = B T k is the transpose of B k . The matrix of ψ using the Schmidt bases of |ψ is diagonal, with the entries on the diagonal in increasing order. (See Sec. II of [3] for more details on map-state duality.) Upon writing the partial traces as For some n between 1 and D write the diagonal matrix ψ as where ψ n is the same matrix but with λ n+1 , λ n+2 , . . . set equal to zero, whileψ n is obtained by setting λ 1 , λ 2 , . . . λ n equal to zero. Lemma 1, below, tells us that for each k, By map-state duality, where |ψ n , the counterpart of ψ n , is given by (3) with D replaced by n. Inserting (17) in (16) and summing over k, see (11), yields This establishes (14), which is equivalent to (10).
Lemma 1. Let A, B, and ψ be D × D matrices, where ψ is diagonal with nonnegative diagonal elements in increasing order, and for some 1 n D let ψ n be obtained from ψ by setting all but the n smallest diagonal elements equal to 0, as in (15). Then Proof. The inequality where P n is a projector (orthogonal projection operator) of rank at least n, follows from the fact that for any Hermitian operator T the sum of its n smallest eigenvalues is the minimum of Tr(P n T P n ) over such P n , see page 24 of [7]. Choose P n to be the projector onto the orthogonal complement of the range of Aψ n , whereψ n = ψ − ψ n , as in (15). The rank of Aψ n is no larger than the rank of ψ n , which is smaller than or equal to D − n. Thus the dimension of the range of Aψ n cannot exceed D − n, so the rank of P n is at least n. By construction, P n Aψ n = 0, so P n Aψ = P n A(ψ n +ψ n ) = P n Aψ n .
Thus with this choice of P n the right side of (20) is and this implies (19), since P n I and Aψ n BB † ψ † n A † is positive.

III. CONSEQUENCES
The following are some consequences of Theorem 1. For more details about the relation between the {A k , B k } N k=1 set and the {Â l ⊗ U l } M k=1 set see [8].
iii) A deterministic transformation |ψ → |φ by a separable operation is possible if and only if E n (|φ ) E n (|ψ ) for every n between 1 and D, with E n (.) defined in (8) This is often written as λ ψ ≺ λ φ , where λ ψ and λ φ are vectors of the corresponding Schmidt weights. (This extends Theorem 1 in [3].) iv) The maximum probability of success for the transformation |ψ → |φ by a separable operation is given by where E n (·) was defined in (8).
v) An entanglement measures E defined on pure bipartite states is nonincreasing on average under separable operations, which is to say if and only if it is similarly nonincreasing under LOCC.
vi) LetÊ with the infimum over all ensembles {p i , |ψ i } yielding the density operator ρ = i p i |ψ i ψ i |, be the convex roof extension of a pure state entanglement measure E that is monotone on pure states in the sense of (24). ThenÊ is monotone on mixed states in the sense that for any ensemble {p k , σ k } produced from ρ by separable operations.
The result (i) is an immediate consequence of Theorem 1, as the same majorization condition applies for both separable and LOCC. Then (ii), (iii), and (iv) are immediate consequences of known results, in [8], [9], and [10], respectively, for LOCC. The result (v) is an obvious consequence of (i), whereas (vi) follows from general arguments about convex roof extensions; see Sec. XV.C.2 of [11].

IV. CONCLUSION
We have shown that possible ensembles of states produced by applying a separable operation to a bipartite entangled pure state can be exactly characterized through a majorization condition, the collection of inequalities (7) for different n. These have long been known to be necessary and sufficient conditions for producing such an ensemble using LOCC, so their extension to the full class of separable operations is not altogether surprising, even if our proof is not altogether straightforward.
Connecting the full set of separable operations with the more specialized LOCC class immediately yields several significant consequences for the former, as indicated in the list in Sec. III, because much is already known about the latter. Of particular significance is that various entanglement measures are monotone, meaning they cannot increase, under separable operations-something expected on intuitive grounds, but now rigorously proved. Since such monotonicity under LOCC has long been considered a necessary, or at least a very desirable condition for any "reasonable" entanglement measure on mixed states (see Sec. XV.B of [11]), one wonders whether monotonicity under separable operations, in principle a stronger condition, might be an equally good or even superior desideratum.
Our results apply only to bipartite states, but separable operations and the LOCC subclass can both be defined for multipartite systems. Might it be that in the multipartite case the ensemble produced by applying a separable operation to a pure entangled state could also be produced by some LOCC applied to the same state? It might be, but proving it would require very different methods than used here. There are no simple multipartite analogs of the Schmidt representation (3), the majorization condition (7), or map-state duality.
Even in the bipartite case we still know very little about separable operations which are not LOCC, aside from the fact that they exist and can be used to distinguish certain collections of orthogonal states more effectively than LOCC. The results in this paper contribute only indirectly to a better understanding of this matter: looking at what a separable operation does when applied to a single entangled state will not help; one must ask what it does to several different states.