Limitation for linear maps in a class for detection and quantification of bipartite nonclassical correlation

TL;DR

This paper proves that linear eigenvalue-preserving maps used for detecting nonclassical correlations are severely limited, with transposition being the only nontrivial map, highlighting the need for nonlinear approaches.

Contribution

It demonstrates that even non-Hermitian-preserving linear EnCE maps cannot extend beyond transposition, emphasizing the limitations of linear maps in quantum correlation detection.

Findings

Only matrix transposition is a nontrivial linear EnCE map.

Linear EnCE maps cannot detect nonclassical correlation beyond transposition.

Nonlinear maps are necessary for broader detection capabilities.

Abstract

Eigenvalue-preserving-but-not-completely-eigenvalue-preserving (EnCE) maps were previously introduced for the purpose of detection and quantification of nonclassical correlation, employing the paradigm where nonvanishing quantum discord implies the existence of nonclassical correlation. It is known that only the matrix transposition is nontrivial among Hermiticity-preserving (HP) linear EnCE maps when we use the changes in the eigenvalues of a density matrix due to a partial map for the purpose. In this paper, we prove that this is true even among not-necessarily HP (nnHP) linear EnCE maps. The proof utilizes a conventional theorem on linear preservers. This result imposes a strong limitation on the linear maps and promotes the importance of nonlinear maps.