An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group

Kibler, M.

doi:10.1088/1751-8113/42/35/353001

Cited by 57 publications

(82 citation statements)

References 145 publications

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“…The literature contains, beside the aforementioned constructions that use orthogonal unitary matrices 35 112, 182 Nearly all of these constructions rely on properties of primes and prime powers and on an underlying finite field. 85,181,[183][184][185][186][187] After some translation the problem is equivalent to that of finding mutually orthogonal Cartan subalgebras in the Lie algebra of SL(N ). 76,188 The problem also occurs in radar science,…”

Section: Brief Summary and Concluding Remarksmentioning

confidence: 99%

On Mutually Unbiased Bases

Durt

Englert

Bengtsson

et al. 2010

Int. J. Quantum Inform.

668

796

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Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p m , with p prime and m a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12, . . . ).In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, . . . , N − 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two-q-bit Bell states) that are related to the mutually unbiased bases.There is a link to the mathematics of finite affine planes. We also exploit the one-toone correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist -one can find no more than three pairwise unbiased bases -although there is currently no clear-cut demonstration of the case.

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Section: Brief Summary and Concluding Remarksmentioning

confidence: 99%

On Mutually Unbiased Bases

Durt

Englert

Bengtsson

et al. 2010

Int. J. Quantum Inform.

668

796

View full text Add to dashboard Cite

show abstract

“…For instance, we obtain where I d is the unit matrix of dimension d × d with d given by (14). Similar vector phase states can be obtained for E 2d by permuting the n and l quantum numbers occurring in the derivation of the vector phase states for E 1d .…”

Section: Vector Phase States For E 1d and E 2dmentioning

confidence: 77%

“…[12][13][14] In this regard, note that the |aα state in (59) can be identified with the |aα; r state with r = 0 discussed recently in the framework of the quadratic discrete Fourier transform. 15 In fact, we obtain…”

Section: Application To Mutually Unbiased Basesmentioning

confidence: 93%

Phase operators, phase states and vector phase states for SU3 and SU2, 1

Daoud

Kibler

2011

Journal of Mathematical Physics

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About a new family of coherent states for some SU(1,1) central field potentials J. Math. Phys. 54, 032103 (2013); 10.1063/1.4795137 Q-Deformed SU(1,1) and SU(2) squeezed and intelligent states and quantum entanglement AIP Conf.This paper focuses on phase operators, phase states, and vector phase states for the sl 3 Lie algebra. We introduce a one-parameter generalized oscillator algebra A κ (2) which provides a unified scheme for dealing with su 3 (for κ < 0), su 2,1 (for κ > 0), and h 4 ⊗ h 4 (for κ = 0) symmetries. Finite-and infinite-dimensional representations of A κ (2) are constructed for κ < 0 and κ ≥ 0, respectively. Phase operators associated with A κ (2) are defined and temporally stable phase states (as well as vector phase states) are constructed as eigenstates of these operators. Finally, we discuss a relation between quantized phase states and a quadratic discrete Fourier transform and show how to use these states for constructing mutually unbiased bases. C

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“…Construction of MUBs associated with a finite-dimensional Hilbert space was considered, using different methods such as, for instance, ones based on discrete Fourier analysis over Galois fields and Galois rings, generalized Pauli matrices, angular momentum theory, etc. (for a recent review and other methods of MUBs construction, see, for instance, [26,27]). The states (15) are unbiased to the computational basis…”

Section: Generation Of Mutually Unbiased Basesmentioning

confidence: 99%

Quantum encoding based on optical phase states and mutually unbiased bases in the presence of several eavesdroppers

Daoud

Ez‐Zahraouy

2012

J Russ Laser Res

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We investigate quantum key distribution where information is carried by a d-level optical quantum system. The sender uses the mutually unbiased bases generated from the phase states of the quantum system. We consider the effects of several intercept and resend attacks. We derive explicitly the mutual information between the legitimate parties as well as the mutual information between the sender and the different eavesdroppers in the communication channel. We discuss the security of the cryptographic scheme and present some numerical analysis.

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An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group

Cited by 57 publications

References 145 publications

On Mutually Unbiased Bases

On Mutually Unbiased Bases

Phase operators, phase states and vector phase states for SU3 and SU2, 1

Quantum encoding based on optical phase states and mutually unbiased bases in the presence of several eavesdroppers

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