Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace

TL;DR

This paper investigates the relationship between absolute separability and positive partial transposition in symmetric multiqubit states, revealing they are not equivalent through explicit counterexamples.

Contribution

It demonstrates that symmetric absolutely PPT states are not necessarily symmetric absolutely separable, providing explicit counterexamples for five and more qubits.

Findings

Symmetric absolutely PPT states can be entangled.

Counterexamples exist for five and more qubits.

Symmetry-preserving unitaries do not guarantee absolute separability.

Abstract

The equivalence between absolutely separable states and absolutely positive partial transposed (PPT) states in general remains an open problem in quantum entanglement theory. In this work, we study an analogous question for symmetric multiqubit states. We show that symmetric absolutely PPT (SAPPT) states (symmetric states that remain PPT after any symmetry-preserving unitary evolution) are not always symmetric absolutely separable by providing explicit counterexamples. More precisely, we construct a family of entangled five-qubit SAPPT states. Similar counterexamples for larger odd numbers of qubits are identified.