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Entropy

On Geometry of p-Adic Coherent States and Mutually Unbiased Bases

Evgenii Igorevich Zelenov

Published Jun 6, 2023• 5 citations
Publisher

Abstract

The paper considers coherent states for the representation of Weyl commutation relations over a field of $p$-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proved that the bases of coherent states corresponding to different lattices are mutually unbiased, and the operators defining the quantization of symplectic dynamics are Hadamard operators.

DOI
10.3390/e25060902

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The paper considers coherent states for the representation of Weyl commutation relations over a field of ppp-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proved that the bases of coherent states corresponding to different lattices are mutually unbiased, and the operators defining the quantization of symplectic dynamics are Hadamard operators.
Coherent States of the ppp-Adic Heisenberg Group and Entropic Uncertainty Relations
Properties of coherent states for the Heisenberg group over a field of ppp -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 SWS_WSW​ is considered and its properties are investigated. In particular, it is proved that SW≥0S_W\geq 0SW​≥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.
Properties of coherent states for the Heisenberg group over a field of ppp-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 SWS_WSW​ is considered and its properties are investigated. In particular, it is proved that SW≥0S_W\geq 0SW​≥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.
The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.
We describe sets of mutually unbiased bases (MUBs) for quantum states defined over the p-adic numbers Q_p, i.e. the states that can be described as elements of the (rigged) Hilbert space L2(Q_p). We find that for every prime p>2 there are at least p+1 MUBs, which is in contrast with the situation for quantum states defined over the real line R for which only 3 MUBs are known. We comment on the possible reason for the difference regarding MUBs between these two infinite dimensional Hilbert spaces.
We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.