Separability of Completely Symmetric States in Multipartite System

Abstract

Symmetry plays an important role in the field of quantum mechanics. In this paper, we consider a subclass of symmetric quantum states in the multipartite system $N^{\otimes d}$, namely, the completely symmetric states, which are invariant under any index permutation. It was conjectured by L. Qian and D. Chu [arXiv:1810.03125 [quant-ph]] that the completely symmetric states are separable if and only if it is a convex combination of symmetric pure product states. In this paper, we proved that this conjecture is true for both bipartite and multipartite cases. And we proved the completely symmetric state $\rho$ is separable if its rank is at most $5$ or $N+1$. For the states of rank $6$ or $N+2$, they are separable if and only if their range contains a product vector. We apply our results to a few widely useful states in quantum information, such as symmetric states, edge states, extreme states, and nonnegative states. We also study the relation of CS states to Hankel and Toeplitz matrices.

arXiv commentThe abstract, introduction and conclusion have been revised. The bibliography was also changed. Some language mistakes have been corrected. Some phrased have been changed to make it more concise and clearer. And the order of the authors' name (First and Second names) have been exchanged. The addresses of the authors has also been verified. arXiv admin note: text overlap with arXiv:1810.03125 by other authors

References in corpus (21)

• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Quantum Correlations in Systems of Indistinguishable Particles
• Separability in 2xN composite quantum systems
• Entanglement and permutational symmetry
• Operational criterion and constructive checks for the separability of low rank density matrices
• Separable balls around the maximally mixed multipartite quantum states
• Four-qubit entangled symmetric states with positive partial transpositions
• Additivity and non-additivity of multipartite entanglement measures
• Separability criteria and entanglement witnesses for symmetric quantum states
• Separability of Bosonic Systems
• Distillability and PPT entanglement of low-rank quantum states
• Quantum states with strong positive partial transpose
• Description of rank four PPT entangled states of two qutrits
• Separability problem for multipartite states of rank at most four
• Dimensions, lengths and separability in finite-dimensional quantum systems
• Properties and construction of extreme bipartite states having positive partial transpose
• Separability and entanglement in 2x3xN composite quantum systems
• Separability of Multipartite Quantum States with Strong Positive Partial Transpose
• Separability of qubit-qudit quantum states with strong positive partial transposes
• Decomposition of completely symmetric states states

Cited by in corpus (7)

• Quantum-Inspired Hierarchy for Rank-Constrained Optimization
• Multipartite entanglement in the diagonal symmetric subspace
• Truncated moment sequences and a solution to the channel separability problem
• Constructing unextendible product bases from multiqubit ones
• The detection power of real entanglement witnesses under local unitary equivalence
• A matrix inequality related to the entanglement distillation problem
• Decomposition of symmetric separable states and ground state energy of bosonic systems