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arXiv:quant-ph/0209062 · Fortschritte der Physik

Antisymmetric multi‐partite quantum states and their applications

I. Jex, G. Alber, S.M. Barnett, A. Delgado

Published Feb 26, 2003• 20 citations
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Abstract

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.

DOI
10.1002/prop.200310021

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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.
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.
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.
Abstract This review is about single‐particle entanglement. Entanglement occurs when the state of a quantum system with at least two degrees of freedom has a particular non‐separable form. In the case of single‐particle entanglement, this quantum correlation is shared by the same particle being it a photon, a neutron, an ion, or an atom. Here, the basics of quantum entanglement are discussed focusing on the case it is related to the degrees of freedom of a single particle. It is discussed how the violation of peculiar inequalities in this context rules out any realistic non‐contextual hidden variable theory alternative to quantum mechanics. Moreover, experiments that demonstrate single‐particle entanglement for photons, neutrons, and atoms are discussed. Finally, the applications of single‐particle entanglement as a resource for quantum information are discussed and specifically quantum key distribution is detailed, where the use of single‐particle entangled photons allows to improve the security of the BB84 protocol.
We investigate absolutely maximally entangled (AME) states, which are multipartite quantum states that are maximally entangled with respect to any possible bipartition. These strong entanglement properties make them a powerful resource for a variety of quantum information protocols. In this paper, we show the existence of AME states for any number of parties, given that the dimension of the involved systems is chosen appropriately. We prove the equivalence of AME states shared between an even number of parties and pure state threshold quantum secret sharing (QSS) schemes, and prove necessary and sufficient entanglement properties for a wider class of ramp QSS schemes. We further show how AME states can be used as a valuable resource for open-destination teleportation protocols and to what extend entanglement swapping generalizes to AME states.
One way to explore multiparticle entanglement is to ask for maximal entanglement with respect to different bipartitions, leading to the notion of absolutely maximally entangled states or perfect tensors. A different path uses unitary invariance and symmetries, resulting in the concept of multiparticle singlet states. We show that these two concepts are incompatible in the sense that the space of pure multiparticle singlet states does not contain any state for which all partitions of two particles vs the rest are maximally entangled. This puts restrictions on the construction of quantum codes and contributes to discussions in the context of the anti-de Sitter/conformal field theory correspondence and quantum gravity.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud; 03.65.Bz Quantum entanglement is an important resource to be implemented in quantum information processing and quantum communication protocols. [1] The interests in studying multi-dimensional entangled states come from both the foundations of quantum mechanics and the development of new protocols in quantum communication. For example, it was shown that maximally entangled states of two quantum systems in a high-dimensional Hilbert space, qudits, violate local realism stronger than qubits, and entangled qudits are less affected by noise than entangled qubits. [2,3] In quantum cryptography [4], the use of entangled qutrits [4,5] or qudits [6,7] instead of qubits is more secure against eavesdropping attacks. Furthermore, the protocols for quantum teleportation or for quantum cryptography, work best with maximally entangled states. All these facts motivate the development of techniques to generate entangled states among quantum systems in higher dimensional Hilbert space with a good quality of entanglement.