Entropy Gain in $$p$$ -Adic Quantum Channels
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tropy Gain in p -Adic Quantum Channels E. I. Zelenov* Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991 Russia *e-mail: [email protected] Received December 20, 2019; revised January 16, 2020; accepted January 29, 2020
Abstract—A configuration of a p -adic quantum linear bosonic Gaussian channel is proposed. The entropy gain of such a channel is calculated. An adelic formula for the entropy gain is derived. DOI: 10.1134/S1063779620040814
1. INTRODUCTION Starting from [1, 2], non-Archimedean analysis has been actively used to build physical models. Thereby, the branch in mathematical physics ( p -adic mathematical physics) has arisen. One can find a bibliography on this subject in the book [3] and reviews [4, 5]. The article is organized as follows. Firstly, we give the necessary facts of the p -adic analysis. Secondly, we give a definition of the p -adic Gaussian state and the p -adic Gaussian linear bosonic channel and also give their properties. Basically, this information follows [6]; the results are presented without proof. New results are presented below. Thirdly, we prove a formula for the magnitude of the entropy gain in the p -adic Gaussian channel. Fourthly, we give the adelic formula for the entropy gain and its possible applications.
tice L* means a subset of the space F in the form u ∈ L* if and only if condition Δ(u, v ) ∈ Z p is satisfied for all v ∈ L .
2. p -ADIC NUMBERS AND SYMPLECTIC GEOMETRY In this section, a number of known facts concerning the geometry of lattices in a two-dimensional symplectic space over the field Q p of p -adic numbers are presented without proof. The necessary information from the p -adic analysis is contained, for example, in [7]. Most statements regarding the geometry of lattices can be found in [8]. Let F be a two-dimensional vector space over the field Q p . A nondegenerate symplectic form Δ : F × F → Q p is given in the space F . A free module of rank two over the ring Z p of integer p -adic numbers considered as a subset of F will be called a lattice in F . A lattice is a compact set in the natural topology in the space F . We introduce the duality relation on the set of lattices. Let L be a lattice; then the notion of the dual lat-
In the space F , there exists a unique translationinvariant measure (Haar measure) up to normalization. We will normalize the measure in such a way that the measure of the unit ball is equal to unity. The action of the symplectic group preserves the measure; therefore, the measure of any self-dual lattice is equal to unity. The measure of the lattice L will be denoted by L . If L is a self-dual lattice, then, as noted above, L = 1; the converse is also true, if L = 1, then the lattice L is self-dual. It is easy to verify the validity of the relation L L* = 1.
If L coincides with L*, then the lattice L will be called a self-dual lattice. For any self-dual lattice L , there exists a symplectic basis ( e1, e2 ) in the space F such that the lattice L has the form
L = Z pe1 ⊕ Z pe2, that is, L is the unit b
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