Partial transpose criteria for symmetric states

TL;DR

This paper reformulates the PPT separability criterion for symmetric multi-qubit states using tensor-based matrix inequalities, providing a clearer experimental interpretation and generalizing known bases for quantum state analysis.

Contribution

It introduces a tensor-based matrix representation for symmetric states that simplifies the PPT criterion and extends the magic basis concept to arbitrary spin-j states.

Findings

Matrix constructed from tensor representation is similar to partial transpose.

Positivity of the matrix is equivalent to positivity of a correlation matrix.

Provides a transparent experimental interpretation of PPT criteria.

Abstract

We express the positive partial transpose (PPT) separability criterion for symmetric states of multi-qubit systems in terms of matrix inequalities based on the recently introduced tensor representation for spin states. We construct a matrix from the tensor representation of the state and show that it is similar to the partial transpose of the density matrix written in the computational basis. Furthermore, the positivity of this matrix is equivalent to the positivity of a correlation matrix constructed from tensor products of Pauli operators. This allows for a more transparent experimental interpretation of the PPT criteria for an arbitrary spin-j state. The unitary matrices connecting our matrix to the partial transpose of the state generalize the so-called magic basis that plays a central role in Wootters' explicit formula for the concurrence of a 2-qubit system and the Bell bases used for the teleportation of a one or two-qubit state.