Rank Reduction for the Local Consistency Problem

TL;DR

This paper demonstrates that for compatible local quantum states, there exists a low-rank global state satisfying the local constraints, with applications to fermionic and bosonic N-representability problems and local channels.

Contribution

It proves the existence of low-rank global density operators for compatible local states and extends this to local channels via channel-state duality.

Findings

Compatible local density operators can be satisfied by low-rank global density operators.

Low Kraus rank global channels can generate compatible local channels.

Results apply to fermionic and bosonic N-representability problems.

Abstract

We address the problem of how simple a solution can be for a given quantum local consistency instance. More specifically, we investigate how small the rank of the global density operator can be if the local constraints are known to be compatible. We prove that any compatible local density operators can be satisfied by a low rank global density operator. Then we study both fermionic and bosonic versions of the N-representability problem as applications. After applying the channel-state duality, we prove that any compatible local channels can be obtained through a global quantum channel with small Kraus rank.