Error Exponents for Entanglement Transformations From Degenerations

Abstract

This paper explores the trade-off relation between the rate and the strong converse exponent for asymptotic LOCC transformations between pure multipartite states. Any single-copy probabilistic transformation between a pair of states implies that an asymptotic transformation at rate 1 is possible with an exponentially decreasing success probability. However, it is possible that an asymptotic transformation is feasible with nonzero probability, but there is no transformation between any finite number of copies with the same rate, even probabilistically. In such cases it is not known if the optimal success probability decreases exponentially or faster. A fundamental tool for showing the feasibility of an asymptotic transformation is degeneration. Any degeneration gives rise to a sequence of stochastic LOCC transformations from copies of the initial state plus a sublinear number of GHZ states to the same number of copies of the target state. These protocols involve parameters that can be freely chosen, but the choice affects the success probability. In this paper, we characterize an asymptotically optimal choice of the parameters and derive a single-letter expression for the error exponent of the resulting protocol. In particular, this implies an exponential lower bound on the success probability when the stochastic transformation arises from a degeneration.