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Coherent States of the p-Adic Heisenberg Group and Entropic Uncertainty Relations

Evgenii Igorevich Zelenov

Published Jun 15, 2023• 0 citations
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Abstract

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.

DOI
10.20944/preprints202306.1116.v1

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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.
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.
It was conjectured by Wehrl and proven by Lieb that the Shannon differential entropy of the Husimi function of a continuous-variable quantum state is lower bounded by the value it takes on coherent states. This corroborates the intuition, familiar from Heisenberg's uncertainty relations, that coherent states are minimal uncertainty states. The analogous statement for the Wigner function would be that the Wigner differential entropy of states with a positive Wigner function is bounded below by the entropy of pure Gaussian states. This was conjectured a decade ago, but recently disproven. Here, we prove the analogous statement for Kirkwood-Dirac distributions with mutually unbiased bases and for the Gross-Wigner distributions. For these discrete systems, the entropy-minimizing role is played by the pure states with positive quasiprobability distributions. Further, we prove that Kirkwood-Dirac distributions with large nonpositivity have large uncertainty.
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.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
Entropic uncertainty relations, based on sums of entropies of probability distributions arising from different measurements on a given pure state, can be seen as a generalization of the Heisenberg uncertainty relation that is in many cases a more useful way to quantify incompatibility between observables. Of particular interest are relationships between `mutually unbiased' observables, which are maximally incompatible. Lower bounds on the sum of entropies for sets of two such observables, and for complete sets of observables within a space of given dimension, have been found. This paper explores relations in the intermediate regime of large, but far from complete, sets of unbiased observables.
Wehrl entropy is an entropy associated with the Husimi quasiprobability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Bia\l{}ynicki-Birula--Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.