Symmetry reduction for testing $k$-block-positivity via extendibility

TL;DR

This paper introduces a symmetry-based method to efficiently test $k$-block-positivity by reducing the complexity of semidefinite programs using unitary symmetry, enabling more scalable quantum state analysis.

Contribution

The paper presents a novel symmetry reduction technique for semidefinite programs in testing $k$-block-positivity, significantly decreasing computational complexity.

Findings

SDP size reduced from exponential to polynomial in N

Efficient testing of $k$-block-positivity for larger systems

Demonstrates practical applicability of symmetry exploitation in quantum information

Abstract

We study the problem of testing $k$-block-positivity via symmetric $N$-extendibility by taking the tensor product with a $k$-dimensional maximally entangled state. We exploit the unitary symmetry of the maximally entangled state to reduce the size of the corresponding semidefinite programs (SDP). For example, for $k=2$, the SDP is reduced from one block of size $2^{N+1} d^{N+1}$ to $\lfloor \frac{N+1}{2} \rfloor$ blocks of size $\approx O( (N-1)^{-1} 2^{N+1} d^{N+1} )$.