Optimal reference states for maximum accessible entanglement under the local-particle-number superselection rule

Abstract

Global conservation laws imply superselection rules (SSRs) which restrict the operations that are possible on any given state. Imposing the additional constraint of local operations and classical communication forbids the transfer of quantum systems between spatially separated sites. In the case of particle conservation this imposes a SSR for local particle number. That is, the coherences between subspaces of fixed particle number at each site are not accessible and any state is therefore equivalent to its projection onto these subspaces. The accessible entanglement under the SSR is less than (or equal to) that available in the absence of the SSR. An ancilla can be used as a reference system to increase the amount of accessible entanglement. We examine the relationship between local-particle-number uncertainty and the accessible entanglement and consider the optimal reference states for recovering entanglement from certain systems. In particular we derive the optimal ancilla state for extracting entanglement for a single-shared particle and make steps toward the optimum for general systems. We also show that a reference for phase angle is fundamentally different to a reference for the SSR associated with particle conservation.