A characterization of maximally entangled two-qubit states

TL;DR

This paper characterizes maximally entangled two-qubit states by examining the minimal eigenvalues of their partial transposes, establishing a unique property that distinguishes them from other states.

Contribution

It proves that for two-qubit systems, the minimal eigenvalue of the partial transpose being -1/2 precisely characterizes maximally entangled states, a result not extending to higher dimensions.

Findings

Minimal eigenvalue of partial transpose is -1/2 for maximally entangled two-qubit states.

This property does not hold for higher-dimensional bipartite systems.

The result provides a spectral criterion for identifying maximally entangled two-qubit states.

Abstract

As already known by Rana's result \href{https://doi.org/10.1103/PhysRevA.87.054301}{[\pra {\bf87} (2013) 054301]}, all eigenvalues of any partial-transposed bipartite state fall within the closed interval $[-\frac12,1]$. In this note, we study a family of bipartite quantum states whose minimal eigenvalues of partial-transposed states being $-\frac12$. For a two-qubit system, we find that the minimal eigenvalue of its partial-transposed state is $-\frac12$ if and only if such two-qubit state must be maximally entangled. However this result does not hold in general for a two-qudit system when the dimensions of the underlying space are larger than two.