When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations. These Bell states are in one to one correspondence with the N2elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function. Thanks to the properties of generalised Bell states and of quadratic extensions of finite fields, we derive a particular solution for the Mean King's problem. This solution is in turn shown to be in one to one correspondence with a set of N2self-adjoint operators that provides a discrete realisation of the Wigner quasi-distribution.When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations. These Bell states are in one to one correspondence with the N2elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function. Thanks to the properties of generalised Bell states and of quadratic extensions of finite fields, we derive a particular solution for the Mean King's problem. This solution is in turn shown to be in one to one correspondence with a set of N2self-adjoint operators that provides a discrete realisation of the Wigner quasi-distribution.

International Journal of Modern Physics BMay 20, 200615 citations

A new expression for mutually unbiased bases in prime power dimensions

Thomas Durt

Mutually unbiased bases generalize the X, Y and Z qubit bases. They possess numerous applications in quantum information science. It is well-known that in prime power dimensions N=p^m (with p prime and m a positive integer) there exists a maximal set of N+1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic in odd prime dimensions (J. Phys. A, 14, 3241 (1981)), and Wootters and Fields (Ann. Phys. 191, 363 (1989)) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalised Pauli group in terms of the additive characters of the field and we derive an exact groupal composition law inside the elements of the commuting subsets of the generalised Pauli group, renormalised by a well-chosen phase-factor.Mutually unbiased bases generalize the X, Y and Z qubit bases. They possess numerous applications in quantum information science. It is well-known that in prime power dimensions N=p^m (with p prime and m a positive integer) there exists a maximal set of N+1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic in odd prime dimensions (J. Phys. A, 14, 3241 (1981)), and Wootters and Fields (Ann. Phys. 191, 363 (1989)) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalised Pauli group in terms of the additive characters of the field and we derive an exact groupal composition law inside the elements of the commuting subsets of the generalised Pauli group, renormalised by a well-chosen phase-factor.

arXiv:quant-ph/0409090Sep 15, 20049 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

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

About mutually unbiased bases in even and odd prime power dimensions

Thomas Durt

Mutually unbiased bases generalize the X , Y and Z qubit bases. They possess numerous applications in quantum information science. It is well known that in prime power dimensions N = p m (with p prime and m a positive integer), there exists a maximal set of N + 1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic ( 1981 J. Phys. A: Math. Gen. 14 3241 ) in odd prime dimensions, and Wootters and Fields (1989 Ann. Phys. 191 363) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalized Pauli group in terms of the additive characters of the field, and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized Pauli group, renormalized by a well-chosen phase-factor.Mutually unbiased bases generalize the X , Y and Z qubit bases. They possess numerous applications in quantum information science. It is well known that in prime power dimensions N = p m (with p prime and m a positive integer), there exists a maximal set of N + 1 mutually unbiased bases. In the present paper, we derive an explicit expression for those bases, in terms of the (operations of the) associated finite field (Galois division ring) of N elements. This expression is shown to be equivalent to the expressions previously obtained by Ivanovic ( 1981 J. Phys. A: Math. Gen. 14 3241 ) in odd prime dimensions, and Wootters and Fields (1989 Ann. Phys. 191 363) in odd prime power dimensions. In even prime power dimensions, we derive a new explicit expression for the mutually unbiased bases. The new ingredients of our approach are, basically, the following: we provide a simple expression of the generalized Pauli group in terms of the additive characters of the field, and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized Pauli group, renormalized by a well-chosen phase-factor.

Journal of Physics A Mathematical and GeneralMay 25, 200584 citations

Mutually unbiased bases and discrete Wigner functions

Gunnar G. Bjork, José Luis Romero, A. B. Klimov +1 more

Mutually unbiased bases and discrete Wigner functions are closely but not uniquely related. Such a connection becomes more interesting when the Hilbert space has a dimension that is a power of a prime N=dn, which describes a composite system of n qudits. Hence, entanglement naturally enters the picture. Although our results are general, we concentrate on the simplest nontrivial example of dimension N=8=23. It is shown that the number of fundamentally different Wigner functions is severely limited if one simultaneously imposes translational covariance and that the generating operators consist of rotations around two orthogonal axes, acting on the individual qubits only.Mutually unbiased bases and discrete Wigner functions are closely but not uniquely related. Such a connection becomes more interesting when the Hilbert space has a dimension that is a power of a prime N=dn, which describes a composite system of n qudits. Hence, entanglement naturally enters the picture. Although our results are general, we concentrate on the simplest nontrivial example of dimension N=8=23. It is shown that the number of fundamentally different Wigner functions is severely limited if one simultaneously imposes translational covariance and that the generating operators consist of rotations around two orthogonal axes, acting on the individual qubits only.

arXiv:quant-ph/0608173Journal of the Optical Society of America BJan 26, 200735 citations

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.