Universal quantum state merging

Abstract

We consider quantum state merging under uncer tainty of the state held by the merging parties. More precisely we determine the optimal entanglement rate of a merging process when the state is unknown up to membership in a certain set of states. We find that merging is possible at the lowest rate allowed by the individual states.

Resource cost results for entanglement distillation and state merging under source uncertainties

“…It was shown in [7], that the r.h.s. of the equality (2) is independent of the choice of purification.…”

“…The usual notation for entropic quantities extended here to indicate state dependency, we write I(X; Y, ρ) (I c (X Y, ρ), S(X|Y, ρ)) for the quantum mutual information (coherent information, conditional entropy) of a bipartite state ρ shared by parties X and Y . The protocols we consider are build from one-way LOCC channels, which we define concisely in the following (see also the appendix of [7] and references therein). A quantum instrument T on a Hilbert space H is given by a set {T k } K k=1 ⊂ C ↓ (H, K) of trace non-increasing cp maps, such that K k=1 T k is a channel.…”

“…The A → B merging cost for merging of the compound source X is defined analogously and denoted C m,→ (X ) [7], [8].…”

“…Our proof of Theorem 5 has two main ingredients. We use slight generalizations of the results from [7], Theorems 4 and 6 (see [8]) where achievability of the entanglement cost sup ρ∈X S(A|B, ρ) was shown. However, the protocols used there, have classical A → B communication requirements of at least sup ρ∈X S(ρ A ) + sup ρ∈X S(A|B, ρ) which is, in general, strictly greater than the rate given in (3).…”

“…for each i ∈ [N ] and ρ ∈ X i with a positive constant c 2 = c 2 (η). Moreover it is known from [7], Theorem 6 (with some simple modifications, see [8]), that for each i withX i = ∅ and large enough blocklength, there is a (l, k l ,D…”