Negative eigenvalues of partial transposition of arbitrary bipartite states

TL;DR

This paper generalizes the bounds on negative eigenvalues of the partial transposition of bipartite quantum states, showing that the number of negative eigenvalues is limited by the dimensions of the subsystems and all eigenvalues lie within [-1/2,1].

Contribution

It extends previous results from two-qubit states to arbitrary bipartite states, establishing a maximum number of negative eigenvalues based on subsystem dimensions.

Findings

Maximum negative eigenvalues are limited to (m-1)(n-1) for m×n states.

All eigenvalues of partial transposition are within [-1/2,1].

Explicit examples demonstrate the tightness of the bounds.

Abstract

The partial transposition of a two-qubit state has at most one negative eigenvalue and all the eigenvalues lie in [-1/2,1]. In this Brief Report, we extend this result by Sanpera et al. [A. Sanpera, R. Tarrach and G. Vidal, Phys. Rev. A 58, 826 (1998)] to arbitrary bipartite states. We show that partial transposition of an $m\otimes n$ state can not have more than (m-1)(n-1) number of negative eigenvalues. Low-dimensional states have been studied to show the tightness of this result and explicit examples have been provided for $mn\le 9$. It is also shown that all the eigenvalues of partial transposition lie within [-1/2,1]. Some possible applications are also discussed.