Properties of Local Quantum Operations with Shared Entanglement

TL;DR

This paper explores the structure and complexity of multi-party local quantum operations with shared entanglement or randomness, establishing geometric properties, complexity results, and characterizations in terms of positive linear functionals.

Contribution

It introduces a geometric ball of local operations with shared randomness, proves NP-hardness of the weak membership problem for entangled operations, and characterizes these operations via positive linear functionals on a specific cone.

Findings

Existence of a ball of local operations with shared randomness within no-signaling operations

NP-hardness of the weak membership problem for local operations with shared entanglement

Characterization of local operations with shared entanglement and randomness via positive linear functionals

Abstract

Multi-party local quantum operations with shared quantum entanglement or shared classical randomness are studied. The following facts are established: (i) There is a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. (ii) The existence of the ball of local operations with shared randomness is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. (iii) Local operations with shared entanglement are characterized in terms of linear functionals that are ``completely'' positive on a certain cone K of separable Hermitian operators, under a natural notion of complete positivity appropriate to that cone. Local operations with shared randomness (but not entanglement) are also characterized in terms of linear functionals that are merely positive on that same cone K. (iv) Existing characterizations of no-signaling operations are generalized to the multi-party setting and recast in terms of the Choi-Jamiolkowski representation for quantum super-operators. It is noted that the standard nonlocal box is an example of a no-signaling operation that is separable, yet cannot be implemented by local operations with shared entanglement.