Partial transpose criteria for symmetric states

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.

arXiv comment8 pages

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• Absolute separability witnesses for symmetric multiqubit states
• Genuinely entangled symmetric states with no $N$-partite correlations
• Maximum entanglement of mixed symmetric states under unitary transformations
• Optimal measurement strategies for fast entanglement detection
• Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace
• Polytopes of Absolutely Wigner Bounded Spin States
• Geometric Multiaxial Representation of N-qubit Mixed Symmetric Separable States
• Decomposition of symmetric separable states and ground state energy of bosonic systems
• Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement