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arXiv:0812.4453 · Physical Review Letters

Entanglement and Permutational Symmetry

Géza Tóth, Otfried Gühne

Published May 1, 2009• 158 citations
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Abstract

We study the separability of permutationally symmetric quantum states. We show that for bipartite symmetric systems most of the relevant entanglement criteria coincide. However, we provide a method to generate examples of bound entangled states in symmetric systems, for the bipartite and the multipartite case. These states shed some new light on the nature of bound entanglement.

DOI
10.1103/physrevlett.102.170503

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We study the separability of permutationally symmetric quantum states. We show that for bipartite symmetric systems most of the relevant entanglement criteria coincide. However, we provide a method to generate examples of bound entangled states in symmetric systems, for the bipartite and the multipartite case. These states shed some new light on the nature of bound entanglement.
One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.
Entanglement of multipartite systems is studied based on exchange symmetry under the permutation group SN{S}_{N}SN​. With the observation that symmetric property under the exchange of two constituent states and their separability are intimately linked, we show that antisymmetric (fermionic) states are necessarily globally entangled, while symmetric (bosonic) states are either globally entangled or fully separable and possess essentially identical states in all the constituent systems. It is also shown that there cannot exist a fully separable state which is orthogonal to all symmetric states, and that full separability of states does not survive under total symmetrization unless the states are originally symmetric. Besides, anyonic states permitted under the braid group BN{B}_{N}BN​ must also be globally entangled. Our results reveal that exchange symmetry is actually sufficient for pure states to become globally entangled or fully separable.
We propose a technique to investigate multipartite entanglement in the symmetric subspace. Our approach is to map an N -qubit symmetric state onto a bipartite symmetric state of higher local dimension. We show that this mapping preserves separability and allows to characterize the entanglement of the original multipartite state. In particular, we establish a connection between the border rank and the Schmidt rank, and derive lower bounds on entanglement measures. Finally, we reveal the existence of entangled symmetric subspaces, where all bipartite states are entangled.
Symmetry plays an important role in the field of quantum mechanics. We consider a subclass of symmetric quantum states in the multipartite system ${N}^{\ensuremath{\bigotimes}d}$, namely, the completely symmetric states, which are invariant under any index permutation. It was hypothesized 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. We prove that this conjecture is true for the both bipartite and multipartite cases. Further, we prove that the completely symmetric state $\ensuremath{\rho}$ is separable if its rank is at most 5 or N+1N+1N+1. For the states of rank 6 or N+2N+2N+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 non-negative states. We also study the relation of completely symmetric states to Hankel and Toeplitz matrices.