Quantum mechanics in a discrete space-time

Šťovı́ček, P.; Tolar, J.

doi:10.1016/0034-4877(84)90030-2

Cited by 76 publications

(40 citation statements)

References 13 publications

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“…We know that the number of MUBs in C d is lesser or equal to d + 1 and that the limit d + 1 is reached when d is a power of a prime [50,51,52,53,54,55,56,57,58,59]. Let us suppose that d is the power of a prime.…”

Section: Mubsmentioning

confidence: 99%

Miscellaneous Applications of Quons

Kibler

2007

SIGMA

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Abstract. This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We motivate why such algebras are interesting for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU 2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).

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Section: Mubsmentioning

confidence: 99%

Miscellaneous Applications of Quons

Kibler

2007

SIGMA

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“…The Wigner function W |Ψ (x, p) is a quasi-probability distribution in phase space and is normalized so that 1 2π 27) where dx dp/2π is the invariant measure in phase space (here we have takenh = 1). With these definitions we can represent a regularized version of an eigenstate of the position operatorx with mean value equal to zero, |x 0 , as a state described by the Gaussian Wigner function: 28) for which the variances of the position and momentum operators are (∆x) 2 = 1 2 e −2ξ and (∆p) 2 = 1 2 e 2ξ , respectively. The state (2.28) is a minimum uncertainty state, i.e., (∆x)(∆p) = 1 2 irrespective of the value of the squeezing parameter ξ.…”

Section: A Continuous Limitmentioning

confidence: 99%

Quantum-information distributors: Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit

2001

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We show that for any Hilbert-space dimension, the optimal universal quantum cloner can be constructed from essentially the same quantum circuit, i.e., we find a universal design for universal cloners. In the case of infinite dimensions (which includes continuous variable quantum systems) the universal cloner reduces to an essentially classical device. More generally, we construct a universal quantum circuit for distributing qudits in any dimension which acts covariantly under generalized displacements and momentum kicks. The behavior of this covariant distributor is controlled by its initial state. We show that suitable choices for this initial state yield both universal cloners and optimized cloners for limited alphabets of states whose states are related by generalized phase-space displacements.03.67.-a, 03.65.Bz

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“…Its geometric interpretation as the simplest quantum kinematic on a finite discrete configuration space formed by a periodic chain of N points was elaborated by J. Schwinger [9]. In [10] we proposed a group theoretical formulation of this quantum model as well as a finite-dimensional analogue of quantum evolution operator for a free particle.…”

Section: Quantum Mechanics In Finite-dimensional Hilbert Spacesmentioning

confidence: 99%

Feynman's path integral and mutually unbiased bases

Tolar

Chadzitaskos

2009

J. Phys. A: Math. Theor.

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Abstract. Our previous work on quantum mechanics in Hilbert spaces of finite dimensions N is applied to elucidate the deep meaning of Feynman's path integral pointed out by G. Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N × N -approximation of quantum mechanics in L 2 (R) applied to our finite-dimensional analogue of a free quantum particle.

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Quantum mechanics in a discrete space-time

Cited by 76 publications

References 13 publications

Miscellaneous Applications of Quons

Miscellaneous Applications of Quons

Quantum-information distributors: Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit

Feynman's path integral and mutually unbiased bases

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