Conditions for uniqueness of product representations for separable quantum channels and separable quantum states

Abstract

We give a sufficient condition that an operator sum representation of a separable quantum channel in terms of product operators is the unique product representation for that channel, and then provide examples of such channels for any number of parties. This result has implications for efforts to determine whether or not a given separable channel can be exactly implemented by local operations and classical communication. By the Choi-Jamiolkowski isomorphism, it also translates to a condition for the uniqueness of product state ensembles representing a given quantum state. These ideas follow from considerations concerning whether or not a subspace spanned by a given set of product operators contains at least one additional product operator.

Latest version has a new title, and is completely reorganized to emphasize the quantum information applications as opposed to the mathematical results. Version 3 has revised results for more than two parties, having discovered an error in earlier versions (see Conclusions for an explanation). Bipartite results are unchanged

References in corpus (3)

• Separable states with unique decompositions
• Local quantum protocols for separable measurements with many parties
• Necessary condition for local quantum operations and classical communication with extensive violation by separable operations

Cited by in corpus (4)

• On the Schmidt-rank-three bipartite and multipartite unitary operator
• Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose
• Necessary condition for local quantum operations and classical communication with extensive violation by separable operations
• Truncated moment sequences and a solution to the channel separability problem