A problem of existence of bound entangled states with non-positive partial transpose and the Hilbert's 17th problem

TL;DR

This paper explores the existence of bound entangled states with non-positive partial transpose, linking it to Hilbert's 17th problem, and investigates polynomial positivity and sum-of-squares representations in quantum states.

Contribution

It establishes a novel connection between bound entangled states with NPPT and Hilbert's 17th problem, analyzing polynomial positivity in quantum state expectation values.

Findings

Expectation value of Werner states computed for NPPT and undistillable regions

Identified positive polynomials not expressible as sum of squares

Proposed a remedy for the pathological behavior in polynomial representation

Abstract

It is found that the problem of existence of bound entangled states with non-positive partial transpose (NPPT) has the intriguing relation to the Hilbert's 17th problem. More precisely, we compute the expectation value of the partially transposed Werner states by Schmidt rank-2 vectors for NPPT and 1-copy undistillable region. It is the positive polynomial but shown not to be expressed as the sum of squares of polynomials. A remedy for such pathological behavior as well as a similar but different approach to the problem is also mentioned.