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New Journal of Physics

A complete basis of generalized Bell states

Сыч Денис Васильевич, Gerd Leuchs

Published Jan 7, 2009• 90 citations
Publisher

Abstract

A generalization of the Bell states and Pauli matrices to dimensions which are powers of 2 is considered. A basis of maximally entangled multidimensional bipartite states (MEMBS) is chosen very similar to the standard Bell states and constructed of only symmetric and antisymmetric states. This special basis of MEMBS preserves all basic properties of the standard Bell states. We present a recursive and non-recursive method for the construction of MEMBS and discuss their properties. The antisymmetric MEMBS possess the property of rotationally invariant exclusive correlations which is a generalization of the rotational invariance of the antisymmetric singlet Bell state.

DOI
10.1088/1367-2630/11/1/013006

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A generalization of the Bell states and Pauli matrices to dimensions which are powers of 2 is considered. A basis of maximally entangled multidimensional bipartite states (MEMBS) is chosen very similar to the standard Bell states and constructed of only symmetric and antisymmetric states. This special basis of MEMBS preserves all basic properties of the standard Bell states. We present a recursive and non-recursive method for the construction of MEMBS and discuss their properties. The antisymmetric MEMBS possess the property of rotationally invariant exclusive correlations which is a generalization of the rotational invariance of the antisymmetric singlet Bell state.
Entanglement is a powerful resource for processing quantum information. In this context pure, maximally entangled states have received considerable attention. In the case of bipartite qubit-systems the four orthonormal Bell-states are of this type. One of these Bell states, the singlet Bell-state, has the additional property of being antisymmetric with respect to particle exchange. In this contribution we discuss possible generalizations of this antisymmetric Bell-state to cases with more than two particles and with single-particle Hilbert spaces involving more than two dimensions. We review basic properties of these totally antisymmetric states. Among possible applications of this class of states we analyze a new quantum key sharing protocol and methods for comparing quantum states.
Abstract Entanglement is a powerful resource for processing quantum information. In this context pure, maximally entangled states have received considerable attention. In the case of bipartite qubit‐systems the four orthonormal Bell‐states are of this type. One of these Bell states, the singlet Bell‐state, has the additional property of being antisymmetric with respect to particle exchange. In this contribution we discuss possible generalizations of this antisymmetric Bell‐state to cases with more than two particles and with single‐particle Hilbert spaces involving more than two dimensions. We review basic properties of these totally antisymmetric states. Among possible applications of this class of states we analyze a new quantum key sharing protocol and methods for comparing quantum states.
We analyze some special properties of a system of two qubits, and in particular of the so-called Bell basis for this system, and discuss the possibility of extending these properties to higher dimensional systems. We give a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices. However, for none of these bases the special properties of the operation of complex conjugation in Bell basis hold, namely that maximally entangled vectors have up-to-a-phase real coefficients and that factorizable unitaries have real matrix elements.
Abstract. We analyze some special properties of a system of two qubits, and in particular of the so-called Bell basis for this system, which have played an important role in recent papers on entanglement of qubits. In particular, we show which of these properties may be generalized to higher dimension. We give a general construction for bases of maximally entangled vectors in any dimension, but show that none of the properties related to complex conjugation in Bell basis can be realized for higher dimensional analogs. Pacs Nr: 03.67.-a, 03.65.-w, 03.65.Bz 1 I.
We introduce a family of highly symmetric bipartite quantum states in arbitrary dimensions. It consists of all states that are invariant under local phase rotations and local cyclic permutations of the basis. We solve the separability problem for a subspace of these states and show that a sizable part of the family is bound entangled. We also calculate some of the Schmidt numbers for the family in d=3d=3d=3, thereby characterizing the dimensionality of entanglement. Our results allow us to estimate entanglement properties of arbitrary states, as general states can be symmetrized to the considered family by local operations.
Abstract Bell states are of crucial importance for entanglement based methods in quantum information science. Typically, a standard construction of a complete orthonormal Bell-basis by Weyl–Heisenberg operators is considered. We show that the group structure of these operators has strong implication on error correction schemes and on the entanglement structure within Bell-diagonal states. In particular, it implies an equivalence between a Pauli channel and a twirl channel. Interestingly, other complete orthonormal Bell-bases do break the equivalence and lead to a completely different entanglement structure, for instance in the share of positive partial transposition (PPT)-entangled states. In detail, we find that the standard Bell basis has the highest observed share on PPT-states and PPT-entangled states compared to other Bell bases. In summary, our findings show that the standard Bell basis construction exploits a very special structure with strong implications to quantum information theoretic protocols if a deviation is considered.