Positive reduction from spectra

TL;DR

This paper investigates conditions under which bipartite quantum states with a given spectrum remain positive after reduction, providing explicit criteria and new entanglement tests related to spectra and separability.

Contribution

It introduces necessary and sufficient linear inequalities for spectra to ensure positivity under reduction, especially for qubit subsystems, and proposes new entanglement criteria based on spectra.

Findings

Derived explicit linear inequalities for spectra under reduction.

Established conditions for qubit and specific subsystem cases.

Proposed new entanglement criteria related to spectra and PPT.

Abstract

We study the problem of whether all bipartite quantum states having a prescribed spectrum remain positive under the reduction map applied to one subsystem. We provide necessary and sufficient conditions, in the form of a family of linear inequalities, which the spectrum has to verify. Our conditions become explicit when one of the two subsystems is a qubit, as well as for further sets of states. Finally, we introduce a family of simple entanglement criteria for spectra, closely related to the reduction and positive partial transpose criteria, which also provide new insight into the set of spectra that guarantee separability or positivity of the partial transpose.