arXiv:0910.3035 · Physical Review A

Maximally entangled states via mutual unbiased collective bases

M. Revzen

Published Jan 21, 2010• 25 citations

Abstract

Relative and center-of-mass coordinates are used to generalize mutually unbiased bases (MUB) and define mutually unbiased collective bases (MUCB). Maximally entangled states are given as product states in the collective variables. These states are analyzed in terms of the Wigner representative function of the states and shown to display a discontinuous attribute of the entanglement. Finite Hilbert space dimensionality collective coordinates are introduced and provide a framework for the analysis.

DOI: 10.1103/physreva.81.012113

Related papers

Maximal Entanglement via Collective Coordinates

M. Revzen

arXiv:1206.3884Jun 18, 20120 citations

Maximally Entangled States

M. Revzen

arXiv:1404.6970Apr 28, 20141 citation

Maximal entanglement, collective coordinates and tracking the King

M Revzen

arXiv:1209.1704Journal of Physics A Mathematical and TheoreticalJan 29, 201314 citations

Mutually unbiased phase states, phase uncertainties, and Gauss sums

Michel R. P. Planat, Haret Codratian Rosu

Jun 16, 200533 citations

Partially unbiased entangled bases

Amir Kalev, F. C. Khanna, M. Revzen

arXiv:0812.0636Physical Review AAug 17, 20098 citations

Quantum phase uncertainty in mutually unbiased measurements and Gauss sums

Michel R. P. Planat, Haret Codratian Rosu

arXiv:quant-ph/0502167Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEMay 23, 20052 citations

arXiv:0910.3035 · Physical Review A

Maximally entangled states via mutual unbiased collective bases

M. Revzen

Published Jan 21, 2010• 25 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.81.012113

Related papers

Maximal Entanglement via Collective Coordinates

M. Revzen

Maximal entangled states (MES) provide a basis to 2d-dimensional particles Hilbert space, d=prime $\ne2$ . These states allow generalization of the Mean King Problem. The states may be viewed as build of points each underpins a product state carrying a mutual unbiased bases (MUB) label or, alternatively, as product states labeled with center of mass and relative coordinates. The coordinate-like label of the center of mass and the momentum-like of the relative coordinates provides a MES account of the Hilbert space in close analogy with the single particle phase space coordinates.Maximal entangled states (MES) provide a basis to 2d-dimensional particles Hilbert space, d=prime $\ne2$. These states allow generalization of the Mean King Problem. The states may be viewed as build of points each underpins a product state carrying a mutual unbiased bases (MUB) label or, alternatively, as product states labeled with center of mass and relative coordinates. The coordinate-like label of the center of mass and the momentum-like of the relative coordinates provides a MES account of the Hilbert space in close analogy with the single particle phase space coordinates.

arXiv:1206.3884Jun 18, 20120 citations

Maximally Entangled States

M. Revzen

Every Maximally Entangled State (MES) of two d-dimensional particles is shown to be a product state of suitably chosen collective coordinates. The state may be viewed as defining a "point" in a "phase space" like d^2 array representing d^2 orthonormal Maximally Entangled States basis for the Hilbert space. A finite geometry view of MES is presented and its relation with the afore mentioned "phase space" is outlined: "straight lines" in the space depict product of single particle mutually unbiased basis (MUB) states, inverting thereby Schmidt's diagonalization scheme in giving a product single particle states as a d-terms sum of maximally entangled states. To assure self sufficiency the essential mathematical results are summarized in the appendices.Every Maximally Entangled State (MES) of two d-dimensional particles is shown to be a product state of suitably chosen collective coordinates. The state may be viewed as defining a "point" in a "phase space" like d^2 array representing d^2 orthonormal Maximally Entangled States basis for the Hilbert space. A finite geometry view of MES is presented and its relation with the afore mentioned "phase space" is outlined: "straight lines" in the space depict product of single particle mutually unbiased basis (MUB) states, inverting thereby Schmidt's diagonalization scheme in giving a product single particle states as a d-terms sum of maximally entangled states. To assure self sufficiency the essential mathematical results are summarized in the appendices.

arXiv:1404.6970Apr 28, 20141 citation

Maximal entanglement, collective coordinates and tracking the King

M Revzen

Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime $\ne 2$ . The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics, notions used are reviewed in an attempt to have the paper self contained.Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime $\ne 2$. The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics, notions used are reviewed in an attempt to have the paper self contained.

arXiv:1209.1704Journal of Physics A Mathematical and TheoreticalJan 29, 201314 citations

Mutually unbiased phase states, phase uncertainties, and Gauss sums

Michel R. P. Planat, Haret Codratian Rosu

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/sqrt{d), with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d+1 have been derived for prime power dimensions d=p^m using the tools of abstract algebra. Presumably, for non prime dimensions the cardinality is much less. Here we reinterpret MUBs as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states to additive characters of Galois fields (in odd characteristic p) and to Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspectsMutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/sqrt{d), with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d+1 have been derived for prime power dimensions d=p^m using the tools of abstract algebra. Presumably, for non prime dimensions the cardinality is much less. Here we reinterpret MUBs as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states to additive characters of Galois fields (in odd characteristic p) and to Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspects

Jun 16, 200533 citations

Partially unbiased entangled bases

Amir Kalev, F. C. Khanna, M. Revzen

In this contribution we group the operator basis for ${d}^{2}$ -dimensional Hilbert space in a way that enables us to relate bases of entangled states with single-particle mutually unbiased state bases (MUB), each in dimensionality $d$ . We utilize these sets of operators to show that an arbitrary density matrix for this ${d}^{2}$ -dimensional Hilbert-space system is analyzed via ${d}^{2}+d+1$ measurements, ${d}^{2}\ensuremath{-}d$ of which involve those entangled states that we associate with MUB of the $d$ -dimensional single-particle constituents. The number ${d}^{2}+d+1$ lies in the middle of the number of measurements needed for bipartite state reconstruction with two-particle MUB $({d}^{2}+1)$ and those needed by single-particle MUB $[{({d}^{2}+1)}^{2}]$ .In this contribution we group the operator basis for ${d}^{2}$-dimensional Hilbert space in a way that enables us to relate bases of entangled states with single-particle mutually unbiased state bases (MUB), each in dimensionality $d$. We utilize these sets of operators to show that an arbitrary density matrix for this ${d}^{2}$-dimensional Hilbert-space system is analyzed via ${d}^{2}+d+1$ measurements, ${d}^{2}\ensuremath{-}d$ of which involve those entangled states that we associate with MUB of the $d$-dimensional single-particle constituents. The number ${d}^{2}+d+1$ lies in the middle of the number of measurements needed for bipartite state reconstruction with two-particle MUB $({d}^{2}+1)$ and those needed by single-particle MUB $[{({d}^{2}+1)}^{2}]$.

arXiv:0812.0636Physical Review AAug 17, 20098 citations

Quantum phase uncertainty in mutually unbiased measurements and Gauss sums

Michel R. P. Planat, Haret Codratian Rosu

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse $1/\sqrt{d}$ , with $d$ the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality $d+1$ have been derived for prime power dimensions $d=p^m$ using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse $1/\sqrt{d}$, with $d$ the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality $d+1$ have been derived for prime power dimensions $d=p^m$ using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.

arXiv:quant-ph/0502167Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEMay 23, 20052 citations

Maximal entangled states (MES) provide a basis to 2d-dimensional particles Hilbert space, d=prime

\ne2

. These states allow generalization of the Mean King Problem. The states may be viewed as build of points each underpins a product state carrying a mutual unbiased bases (MUB) label or, alternatively, as product states labeled with center of mass and relative coordinates. The coordinate-like label of the center of mass and the momentum-like of the relative coordinates provides a MES account of the Hilbert space in close analogy with the single particle phase space coordinates.

Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime

\ne 2

. The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics, notions used are reviewed in an attempt to have the paper self contained.

In this contribution we group the operator basis for

{d}^{2}

-dimensional Hilbert space in a way that enables us to relate bases of entangled states with single-particle mutually unbiased state bases (MUB), each in dimensionality

d

. We utilize these sets of operators to show that an arbitrary density matrix for this

{d}^{2}

-dimensional Hilbert-space system is analyzed via

{d}^{2}+d+1

measurements, ${d}^{2}\ensuremath{-}d$ of which involve those entangled states that we associate with MUB of the

d

-dimensional single-particle constituents. The number

{d}^{2}+d+1

lies in the middle of the number of measurements needed for bipartite state reconstruction with two-particle MUB

({d}^{2}+1)

and those needed by single-particle MUB

[{({d}^{2}+1)}^{2}]

Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse

1/\sqrt{d}

, with

d

the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality

d+1

have been derived for prime power dimensions

d=p^m

using the tools of abstract algebra (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for physical states and find them related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in our quest of minimal uncertainty in quantum information primitives.