Resource cost results for one-way entanglement distillation and state merging of compound and arbitrarily varying quantum sources

Abstract

We consider one-way quantum state merging and entanglement distillation under compound and arbitrarily varying source models. Regarding quantum compound sources, where the source is memoryless, but the source state an unknown member of a certain set of density matrices, we continue investigations begun in the work of Bjelaković et. al. [Universal quantum state merging, J. Math. Phys. 54, 032204 (2013)] and determine the classical as well as entanglement cost of state merging. We further investigate quantum state merging and entanglement distillation protocols for arbitrarily varying quantum sources (AVQS). In the AVQS model, the source state is assumed to vary in an arbitrary manner for each source output due to environmental fluctuations or adversarial manipulation. We determine the one-way entanglement distillation capacity for AVQS, where we invoke the famous robustification and elimination techniques introduced by R. Ahlswede. Regarding quantum state merging for AVQS we show by example, that the robustification and elimination based approach generally leads to suboptimal entanglement as well as classical communication rates. * boche@tum.de. † gisbert.janssen@tum.de 2

Distillation of secret-key from a class of compound memoryless quantum sources

“…the state resulting from performing (T p , D) on ρ V according to (7) for each V ∈ V p . Note that the resulting pair (Λ, K) of random variables does not depend on the chosen state ρ V since all state in I p have same sender marginal distribution.…”

“…density matrices from I having sender marginal distribution p. Fix any p ∈ P we suppress the index p for the next lines. Denote by ρ ΛKK ′ E n ,V the state resulting from performing (T p , D) on ρ V according to(7) for each V ∈ V p . Note that the resulting pair (Λ, K) of random variables does not depend on the chosen state ρ V since all state in I p have same sender marginal distribution.…”

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

“…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…”

“…Due to space limitations in this paper, we restrict ourselves to brief proof sketches of the results. The full arguments and further explanations can be gathered in [8] accompanying this work.…”