Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0510200 · Physical Review A

Equientangled bases in arbitrary dimensions

Vahid Karimipour, Laleh Memarzadeh

Published Jan 20, 2006• 12 citations
arXivHTMLPDF

Abstract

For the space of two identical systems of arbitrary dimensions, we introduce a continuous family of bases with the following properties: (i) the bases are orthonormal, (ii) in each basis, all the states have the same values of entanglement, and (iii) they continuously interpolate between the product basis and the maximally entangled basis. The states thus constructed may find applications in many areas related to the quantum information science including quantum cryptography, optimal Bell tests, and the investigation of the enhancement of channel capacity due to entanglement.

DOI
10.1103/physreva.73.012329

Related papers

Equientangled Bases in Arbitrary Dimensions and Quadratic Gauss Sums

Vlad Gheorghiu

Recently [Karimipour and Memarzadeh, PhysRevA 73, 012329 (2006)] posed the problem of finding a continuous family of orthonormal bases in a bipartite space of two identical systems with the following properties: i) in each basis, all states have to be equally entangled, and ii) the family continuously interpolate between the product basis and the maximally entangled basis. The authors provided a necessary condition and simple examples for relatively small dimensions, but questioned the existence of a general solution for arbitrary dimensions. Employing the properties of quadratic Gauss sums, we prove that such a family of bases exists for all dimensions and provide an explicit simple parametrization. We illustrate the behaviour of our solution with particular examples.

arXiv:1003.4424•Mar 23, 2010•0 citations

Construction of equally entangled bases in arbitrary dimensions via quadratic Gauss sums and graph states

Vlad Gheorghiu, Shiang Yong Looi

Recently, Karimipour and Memarzadeh [Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space, each of dimension $D$, with the following properties: (i) The family continuously interpolates between a product basis and a maximally entangled basis as some parameter $t$ is varied, and (ii) for a fixed $t$, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for $D\ensuremath{\leqslant}5$, but the existence of a solution for arbitrary dimensions remained an open problem. We prove that such families exist in arbitrary dimensions by providing two simple solutions, one that employs the properties of quadratic Gauss sums and the other that uses graph states. The latter can be generalized to more than two parties---we can construct equally entangled families that vary continuously between a product basis and a graph-state basis.

arXiv:1004.1633•Physical Review A•Jun 25, 2010•6 citations

Unextendible maximally entangled bases and mutually unbiased bases

Bin Chen, Shao-Ming Fei

We study an unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of ${d}^{2}$ orthonormal maximally entangled states in ${\mathbb{C}}^{d}⨂{\mathbb{C}}^{{d}^{\ensuremath{'}}}(\frac{{d}^{\ensuremath{'}}}{2}<d<{d}^{\ensuremath{'}})$ is provided. The complementary space of the set of these ${d}^{2}$ orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in ${\mathbb{C}}^{2}⨂{\mathbb{C}}^{3}$ which are mutually unbiased.

arXiv:1309.3356•Physical Review A•Sep 4, 2013•45 citations

Stabilizer codes and equientangled bases from phase states

Mostafa Mansour, Mohammed Daoud A

We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.

International Journal of Modern Physics B•Mar 29, 2017•7 citations

Why Two Qubits Are Special

K. G. H. Vollbrecht, R. F. Werner

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.

•Oct 15, 1999•43 citations

Why two qubits are special

K. G. H. Vollbrecht, R. F. Werner

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.

arXiv:quant-ph/9910064•Journal of Mathematical Physics•Oct 1, 2000•45 citations
For the space of two identical systems of arbitrary dimensions, we introduce a continuous family of bases with the following properties: (i) the bases are orthonormal, (ii) in each basis, all the states have the same values of entanglement, and (iii) they continuously interpolate between the product basis and the maximally entangled basis. The states thus constructed may find applications in many areas related to the quantum information science including quantum cryptography, optimal Bell tests, and the investigation of the enhancement of channel capacity due to entanglement.
Recently [Karimipour and Memarzadeh, PhysRevA 73, 012329 (2006)] posed the problem of finding a continuous family of orthonormal bases in a bipartite space of two identical systems with the following properties: i) in each basis, all states have to be equally entangled, and ii) the family continuously interpolate between the product basis and the maximally entangled basis. The authors provided a necessary condition and simple examples for relatively small dimensions, but questioned the existence of a general solution for arbitrary dimensions. Employing the properties of quadratic Gauss sums, we prove that such a family of bases exists for all dimensions and provide an explicit simple parametrization. We illustrate the behaviour of our solution with particular examples.
Recently, Karimipour and Memarzadeh [Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space, each of dimension DDD, with the following properties: (i) The family continuously interpolates between a product basis and a maximally entangled basis as some parameter ttt is varied, and (ii) for a fixed ttt, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for $D\ensuremath{\leqslant}5$, but the existence of a solution for arbitrary dimensions remained an open problem. We prove that such families exist in arbitrary dimensions by providing two simple solutions, one that employs the properties of quadratic Gauss sums and the other that uses graph states. The latter can be generalized to more than two parties---we can construct equally entangled families that vary continuously between a product basis and a graph-state basis.
We study an unextendible maximally entangled basis in arbitrary bipartite spaces. A systematic way of constructing a set of d2{d}^{2}d2 orthonormal maximally entangled states in ${\mathbb{C}}^{d}⨂{\mathbb{C}}^{{d}^{\ensuremath{'}}}(\frac{{d}^{\ensuremath{'}}}{2}<d<{d}^{\ensuremath{'}})$ is provided. The complementary space of the set of these d2{d}^{2}d2 orthonormal maximally entangled states contains no maximally entangled states that are orthogonal to all of them. Furthermore, we investigate mutually unbiased bases in which all the bases are unextendible maximally entangled ones. We present two unextendible maximally entangled bases in C2⨂C3{\mathbb{C}}^{2}⨂{\mathbb{C}}^{3}C2⨂C3 which are mutually unbiased.
We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.
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 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.