Searching for extremal PPT entangled states

TL;DR

This paper investigates extremal PPT entangled states across various quantum systems, identifying rank conditions for extremality and providing explicit examples, thereby advancing understanding of entanglement structure in quantum states.

Contribution

It introduces a criterion for extremality in PPT states, characterizes rank conditions for extremal PPT entangled states, and explores their configurations in multi-qubit systems.

Findings

Extremal PPT entangled states have ranks (5,5) or (5,6) in qubit--ququart systems.

Explicit examples of extremal states of ranks (5,5) and (5,6) are provided.

Possible rank configurations for extremal states in multi-qubit systems are identified.

Abstract

We study extremality in various sets of states that have positive partial transposes. One of the tools we use for this purpose is the recently formulated criterion allowing to judge if a given state is extremal in the set of PPT states. First we investigate qubit--ququart states and show that the only candidates for extremal PPT entangled states (PPTES) have ranks of the state and its partial transposition (5,5) or (5,6) (equivalently (6,5)). Then, examples of extremal states of (5,5) type and the so--called edge states of type (5,6) are provided. We also make an attempt to explore the set of PPT states with ranks (5,6). Finally, we discuss what are the possible configurations of ranks of density matrices and their respective partial transposition in general three-qubit and four-qubit symmetric states for which there may exist extremal entangled PPT states. For instance in the first case we show that the only possibilities are (4,4,4) and (4,4,5).