p-Adic model of quantum mechanics and quantum channels

Zelenov, E. I.

doi:10.1134/s0081543814040105

Cited by 9 publications

(4 citation statements)

References 9 publications

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“…The following two statements are given without proof. More detailed information can be found in [15].…”

Section: Entropic Uncertainty Relationmentioning

confidence: 99%

Coherent States of the <em>p</em>-Adic Heisenberg Group and Entropic Uncertainty Relations

Zelenov¹

2023

Preprint

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Properties of coherent states for the Heisenberg group over a field of $p$-adic numbers are investigated. It turns out that coherent states form an orthonormal basis. The family of all such bases is parametrized by a set of self-dual lattices in the phase space. The Wehrl entropy $S_W$ is considered and its properties are investigated. In particular, it is proved that $S_W\geq 0$ and vanishes only on coherent states. For a pair of different bases of coherent states, an entropic uncertainty relation is obtained. It is shown that the lower bound for the sum of the corresponding Wehrl entropies is given by the distance between the lattices. The proof uses the fact that the bases of coherent states corresponding to a pair of lattices are mutually unbiased.

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“…The following two statements are given without proof. More detailed information can be found in [15].…”

Section: Entropic Uncertainty Relationmentioning

confidence: 99%

Coherent States of the <em>p</em>-Adic Heisenberg Group and Entropic Uncertainty Relations

Zelenov¹

2023

Preprint

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“…Example 5.13. In Zelenov's (relatively recent) papers [59,60], Gaussian states on L 2 (Q p ) were defined by χ ρ being the indicator function of a lattice L ⊆ Q p × Q p (multiplied by a suitable character on L). By a lattice, they mean a rank-2 free Z psubmodule of Q p ×Q p , where…”

Section: Systems Admitting Compact Open Subgroupsmentioning

confidence: 99%

“…To overcome this, we advocate the use of Gaussianity in the sense of Bernstein (or B-Gaussianity for short), which is an LCA generalization of Bernstein's classical result: a real probability distribution µ is Gaussian if and only if the sum and difference of two independent µ-distributed random variables are independent [3]. Our notion of B-Gaussian states, valid for any phase space (G, σ), unifies a variety of examples from the literature, including bosonic Gaussian states, discrete Hudson/stabilizer states [20], vacuum states of p-adic oscillator Hamiltonians [52], (classes of) minimal uncertainty states [34], and the (relatively) recently introduced Gaussian states for single mode p-adic systems [59,60].…”

Section: Introductionmentioning

confidence: 99%

Gaussian quantum information over general quantum kinematical systems I: Gaussian states

Bény¹,

Crann²,

Lee³

et al. 2022

Preprint

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We develop a theory of Gaussian states over general quantum kinematical systems with finitely many degrees of freedom. The underlying phase space is described by a locally compact abelian (LCA) group G with a symplectic structure determined by a 2-cocycle on G. We use the concept of Gaussian distributions on LCA groups in the sense of Bernstein to define Gaussian states and completely characterize Gaussian states over 2-regular LCA groups of the form G = F × F endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of n-bosonic modes, n-qudit systems with odd d ≥ 3, and p-adic quantum systems. Our characterization reveals a topological obstruction to Gaussian state entanglement when we decompose the quantum kinematical system into the Euclidean part and the remaining part (whose phase space admits a compact open subgroup). We then generalize the discrete Hudson theorem [20] to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space T n × Z n and fermionic/hard-core bosonic systems with phase space Z 2n 2 (which are not 2-regular), and completely characterize their Gaussian states.

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“…Also, in the last thirty years the p-adic quantum mechanics and the p-adic Schrödinger equations have been studied extensively, see, e.g., [1], [9]- [17], [31], [34]- [36], [46], [49], [54], [57]- [52], [63], among many available references. Here, we present a new perspective: a non-Archimedean quantum mechanics, which includes p-adic quantum mechanics as a particular case, is a q-deformation of the classical quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

Non-Archimedean Quantum Mechanics via Quantum Groups

Zúñiga–Galindo¹

2022

Preprint

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We present a new non-Archimedean realization of the Fock representation of the q-oscillator algebras where the creation and annihilation operators act on complex-valued functions, which are defined on a non-Archimedean local field of arbitrary characteristic, for instance, the field of p-adic numbers. This new realization implies that a large number of quantum models constructed using q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum models. In this framework, we select a q-deformation of the Heisenberg uncertainty relation, and construct the corresponding q-deformed Schrödinger equations. In this way we construct a p-adic quantum mechanics which is a p-deformed quantum mechanics. We also solve the time-independent Schrödinger equations for the free particle, and a particle in a non-Archimedean box. In the last case we show the existence of a discrete sequence of energy levels. We determine the eigenvalues of Schrödinger operator for a general radial potential. By choosing the potential in a suitable form we recover the energy levels of the q-hydrogen atom.

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p-Adic model of quantum mechanics and quantum channels

Cited by 9 publications

References 9 publications

Coherent States of the <em>p</em>-Adic Heisenberg Group and Entropic Uncertainty Relations

Coherent States of the <em>p</em>-Adic Heisenberg Group and Entropic Uncertainty Relations

Gaussian quantum information over general quantum kinematical systems I: Gaussian states

Non-Archimedean Quantum Mechanics via Quantum Groups

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