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

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

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Evgeny Zelenov^*

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Submitted:

15 June 2023

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15 June 2023

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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.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction

1.1. Coherent states

Let G be a locally compact group and

U (g), g \in G

be its irreducible unitary representation in a separable complex Hilbert space

H

Fix a unit vector

| ψ 〉 \in H

and consider the subgroup

H \subset G

with the property:

U (h) | ψ 〉 = e^{i ω (h)} | ψ 〉, h \in H .

Let

X = G / H

and

g (x)

be an arbitrary representative from a coset

x \in X

. The following set:

\{| x 〉 = U (g (x)) | ψ 〉, x \in X\}

is the system of (generalized) coherent states [1]. It is easy to see that

| x 〉 〈 x |

does not depend on the choice of a representative

g (x)

from coset x.

If we apply the procedure to the Heisenberg group and vacuum vector as

| ψ 〉

then we obtain the well-known coherent states in quantum theory.

1.2. The Husimi function and the Wehrl entropy

For the system of coherent states constructed above, it is possible to determine the Husimi function and the Wehrl entropy. Namely, let

ρ

be the density operator in the representation space

H

. The Husimi function

Q_{ρ} : X \to C

is defined by the formula:

Q_{ρ} (x) = 〈 x | ρ | x 〉 .

The group G is locally compact, the left-invariant Haar measure on this group is denoted by

μ

. By virtue of invariance, the measure naturally defines a measure on X, which we also denote by

μ

. The Wehrl entropy

S_{W} (ρ)

is given by the formula:

S_{W} (ρ) = - \int_{X} d μ (x) Q_{ρ} (x) log Q_{ρ} (x) .

There is a natural question of the lower bound for the Wehrl entropy:

C = inf_{ρ} S_{W} (ρ) .

The constant

C = C (G)

is a certain characteristic of the group G and its representation U. For example, for the Heisenberg group over a field of real numbers and its standard representation we have

C (H e i s_{R}) = 1

. This conjecture was formulated by Wehrl [2] and subsequently proved by Lieb [3]. Moreover, Lieb proved that the lower bound is achieved on coherent states. Later Carlen [4] proved that the minimum occures only on coherent states.

As an example, we give another interesting result for the group SU(2) and its representation with the highest weight 2J [5]. The following equality is valid:

C (S U (2)) = \frac{2 J}{2 J + 1} .

Thus, the lower bound of the Wehrl entropy does depend in a nontrivial way on the group and its representation.

In this paper we consider the Heisenberg group over the field

Q_{p}

of p-adic numbers and its standard representation. In particular, the following statement is proved:

C (H e i s_{Q_{p}}) = 0,

and the lower bound is achieved on coherent states. The work is motivated by applications in p-adic quantum theory, see [6,7] and references therein.

1.3. Entropic uncertainty relations

Let two orthonormal bases

A = {| a 〉 〈 a |}

and

B = {| b 〉 〈 b |}

be given in the Hilbert space

H

. Each of these bases defines a quantum-classical channel

ρ \to {ρ_{a} = 〈 a | ρ | a 〉}

ρ \to {ρ_{b} = 〈 b | ρ | b 〉}

Denote by

H (ρ_{a, b})

the Shannon entropy for probability distributions

ρ_{a, b}

respectively. Then the following formula is valid:

H (ρ_{a}) + H (ρ_{b}) \geq K,

where the constant K depends only on the bases A and B. The optimal estimate is achieved for a pair of mutually unbiased bases.

Relations of this type are called entropy uncertainty relations. For the first time, the entropy uncertainty relation is obtained by Hirschman [8]. This result was further developed in articles [9,10,11] and many others. For more detailed information, I refer to the review [13].

In this paper, entropy uncertainty relations are considered in the context of p-adic quantum mechanics. In this case, the coherent states form an orthonormal basis. The entropy uncertainty relation for a pair of different bases of coherent states is obtained. Moreover, the obtained estimate is optimal, since the bases of coherent states are mutually unbiased.

1.4. p-adic numbers

We fix a prime number p. Any rational number

x \in Q

is uniquely representable as

x = p^{k} \frac{m}{n}, k, m, n \in Z, p ∤ m, p ∤ n .

Let’s define the norm

{| \cdot |}_{p}

Q

by the formula

{| x |}_{p} = p^{- k}

, Completion of the field of rational numbers with this norm is the field

Q_{p}

of p-adic numbers.The p-adic norm of a rational integer

n \in Z

is always less than or equal to one,

{| n |}_{p} \leq 1

, the completion of rational integers

Z

with the p-adic norm is denoted by

Z_{p}

Z_{p} = x \in Q_{p} : {| x |}_{p} \leq 1

, that is, it is a disk of a unit radius.

For the p-adic norm, the strong triangle inequality holds:

{| x + y |}_{p} \leq {max {| x |}_{p} {, | y |}_{p}} .

The non-Archimedean norm defines totally disconnected topology on

Q_{p}

(the disks are open and closed simultaneously). Two disks either do not intersect, or one lies in the other.

Locally constant functions are continuous, for example:

h_{Z_{p}} (x) = \{\begin{matrix} 1, x \in Z_{p} \\ 0, x \notin Z_{p} \end{matrix}

is a continuous function.

Q_{p}

is Borel isomorphic to the real line

R

. The shift-invariant measure

d x

Q_{p}

is normalized in such a way that

\int_{Z_{p}} d x = 1

For any nonzero p-adic number, the canonical representation holds:

Q_{p} ∋ x = \sum_{k = - n}^{+ \infty} x_{k} p^{k}, n \in Z_{+}, x_{k} \in {0, 1, \dots, p - 1} .

We define integer

{[x]}_{p}

and fractional

{x}_{p}

parts of

x \in Q_{p}^{*}

by the following expressions:

\underset{{x}_{p}}{\underset{︸}{p^{- n} x_{- n} + p^{- n + 1} x_{- n + 1} + \dots + p^{- 1} x_{- 1}}} + \underset{{[x]}_{p}}{\underset{︸}{x_{0} + p x_{1} + \dots + p^{k} x_{k} + \dots}}

The following function, which takes values in a unit circle

T

C

, is the additive character of the field of p-adic numbers.

χ_{p} (x) = exp (2 π i {x}_{p}), χ_{p} (x + y) = χ_{p} (x) χ_{p} (y) .

p-Adic integers

Z_{p}

form a group with respect to addition (a consequence of the non-Archimedean norm) and it is profinite (procyclic) group. This is the inverse limit of finite cyclic groups

Z / p^{n} Z, n \in N

Z / p Z ⟵ \dots ⟵ Z / p^{n} Z ⟵ Z / p^{n + 1} Z ⟵ \dots

Consider the group

{\hat{Z}}_{p}

of characters

Z_{p}

. This group has the form

{\hat{Z}}_{p} = Q_{p} / Z_{p} = Z (p^{\infty}) = \{exp (2 π i m / p^{n}), m, n \in N\} .

This is the Prüfer group. It is a direct limit of finite cyclic groups (i.e. quasicyclic) of order

p^{n}

Z / p Z \to Z / p^{2} Z \to \dots \to Z / p^{n} Z \to \dots .

2. p-Adic coherent states

2.1. Canonical commutation relations

Let

V = Q_{p}^{2}

be a two-dimensional vector space over

Q_{p}

and

Δ

be a non-degenerate symplectic form on this space. On the set

V \times Q_{p}

, we define a group operation according to the rule:

(u, s) (v, t) = (u + v, s + t + Δ (u, v)), u, v \in V, s, t \in Q_{p} .

The set

V \times Q_{p}

equipped with such an operation is the p-adic Heisenberg group

H e i s (Q_{p})

. The set of elements

{(0, s), s \in Q_{p}}

forms a commutative subgroup Z (center) of the Heisenberg group.

Now let’s apply Perelomov’s construction to the groups

H e i s (Q_{p})

and Z as the group G and its subgroup H respectively.

More familiar (completely equivalent) is the language of representations of canonical commutation relations (or Weyl systems).

Let

H

be a separable complex Hilbert space. A map W from V to a set of unitary operators on

H

satisfying the condition

W (u) W (v) = χ_{p} (Δ (u, v)) W (v) W (u), u, v \in V

is called a representation of canonical commutation relations (CCR). We will also require continuity in a strong operator topology and irreducibility. When these conditions are met, such a representation is unique up to unitary equivalence.

Let’s choose an arbitrary unit vector

| ψ 〉 \in H

. The set of vectors in

H

of the form

\{| z 〉 = W (x) | ψ 〉, z \in V\}

is called a system of (generalized) coherent states.

The next element of the construction is the choice of the vector

| ψ 〉

. The Heisenberg group over the field of real numbers is a Lie group. The standard approach is the transition to the corresponding Lie algebra. The vacuum vector is defined as the eigenvector of the annihilation operator. In the p-adic case, there is no structure of a smooth manifold on the Heisenberg group and, accordingly, there is no corresponding Lie algebra. We will use a different approach to the construction of the vacuum vector.

2.2. Vacuum vector

p-Adic integers

Z_{p}

form a ring. Let L be a two-dimensional

Z_{p}

-submodule of the space V. Such submodules will be called lattices.

On the set of lattices, we introduce the operations ∨ and ∧:

L_{1} \lor L_{2} = L_{1} + L_{2} = {z_{1} + z_{2}, z_{1} \in L_{1}, z_{2} \in L_{2}},

L_{1} \land L_{2} = L_{1} \cap L_{2} .

We also define the involution *:

L^{*} = {z \in V : Δ (z, u) \in Z_{p} \forall u \in L} .

It’s easy to see that

{(L_{1} \land L_{2})}^{*} = L_{1} \lor L_{2}

. The lattice L invariant with respect to the involution is called self-dual,

L = L^{*}

We normalize the measure on V in such a way that the volume of a self-dual lattice is equal to one. Symplectic group

S p (V) = S L_{2} (Q_{p})

acts transitively on the set of self-dual lattices.

L

we denote the set of self-dual lattices. On the set

L

, we define metric d by the formula

d (L_{1}, L_{2}) = \frac{1}{2} log # (L_{1} \lor L_{2} / L_{1} \land L_{2})

log everywhere further denotes the logarithm to the base p, # is the number of elements of the set.

Example 1.

Let

{e, f}

be a symplectic basis in

V, Δ (e, f) = 1

.Then the lattices

L_{1} = Z_{p} e \oplus Z_{p} f, L_{2} = p^{n} Z_{p} e \oplus p^{- n} Z_{p} f

are self-dual. If

n \geq 0

, then

\begin{matrix} L_{1} \land L_{2} = p^{n} Z_{p} e \oplus Z_{p} f, L_{1} \lor L_{2} = Z_{p} e \oplus p^{- n} Z_{p} f \\ d (L_{1}, L_{2}) = \frac{1}{2} log # (L_{1} \lor L_{2} / L_{1} \land L_{2}) = \frac{1}{2} log p^{2 n} = n \end{matrix}

Note that for any pair of self-dual lattices, such a basis exists.

The set of self-dual lattices can be represented as a graph. The distance d takes values in the set of non-negative integers. The vertices of the graph are elements of the set

L

, and the edges are pairs of self-dual lattices

{L_{2}, L_{2}} : d (L_{1}, L_{2}) = 1

The graph of self-dual lattices is constructed according to the following rule. Let

K_{p + 1}

denote a complete graph with

p + 1

vertices.The countable family of copies of the graph

K_{p + 1}

is glued together in such a way that each vertex of each graph in this family belongs to exactly

p + 1

graphs

K_{p + 1}

By replacement of each complete graph

K_{p + 1}

by a star graph

S_{p + 1}

we get a Bruhat-Tits tree.

We proceed with the construction of the vacuum vector. Let us choose a self-dual lattice

L \in L

and consider the operator

P_{L} = \int_{L} d z W (z) .

Lemma 1.

The

P_{L}

operator is a one-dimensional projection.

\begin{matrix} P_{L}^{2} = \int_{L} d z W (z) \int_{L} d z^{'} W (z^{'}) = \\ = \int_{l} d z \int_{L} d z^{'} W (z + z^{'}) = \int_{L} d z W (z) = P_{L} . \end{matrix}

The one-dimensionality of the projection

P_{L}

directly follows from the irreducibility of the representation W.

Our desired vacuum state will be this projection. We fix the notation

P_{L} = | 0_{L} 〉 〈 0_{L} |

Definition 1.

The family of vectors

{| z_{L} 〉 = W (z) | 0_{L} 〉, z \in V}

H

is said to be the system of (L-)coherent states.

We denote by

h_{L}

the indicator function of the lattice L,

h_{L} (z) = \{\begin{matrix} 1, z \in L \\ 0, z \notin L \end{matrix}

Theorem 1.

Coherent states satisfy the following relation:

| 〈 z_{L} | z_{L}^{'} 〉 | = h_{L} (z - z^{'}) .

In other words, the coherent states

| z_{L} 〉 〈 z_{L} |

and

| z_{L}^{'} 〉 〈 z_{L}^{'} |

coincide if

z - z^{'} \in L

and are orthogonal otherwise.

Indeed, let

u = z - z^{'}

. Then

| 〈 z_{L} | z_{L}^{'} 〉 | = | χ_{p} (1 / 2 Δ (z, u)) 〈 0_{L} | W (u) 0_{L} 〉 | = | 〈 0_{L} | W (u) 0_{L} 〉 | .

u \in L

the statement of the theorem follows from the definition of a vacuum vector. If

u \notin L

, then by virtue of the self-duality of the lattice L, there exists

v \in L

that

χ_{p} (Δ (u, v)) \neq 1

. We have

\begin{matrix} 〈 0_{L} | W (u) 0_{L} 〉 = 〈 0_{L} | W (- v) W (u) W (v) 0_{L} 〉 = \\ = χ_{p} (Δ (u, v)) 〈 0_{L} | W (u) 0_{L} 〉, \end{matrix}

which is true only if

〈 0_{L} | W (u) 0_{L} 〉 = 0

Therefore, non-matching (and pairwise orthogonal) coherent states are parametrized by elements of the set

V / L = {(Q_{p} / Z_{p})}^{2} ≅ Z (p^{\infty}) \times Z (p^{\infty}) .

This makes the following modification of Definition 1 natural.

Definition 2.

The set

{| α_{L} 〉 = W (α) | 0_{L} 〉, α \in V / L}

is said to be the basis of coherent states for the p-adic Heisenberg group.

3. Entropy

3.1. Lower bound of the Wehrl entropy

Let

ρ

be a density matrix, that is,

ρ

is a positive operator with a unit trace in

H

We will be interested in the following objects:

the Husimi function $Q_{ρ}^{L} (z) = 〈 z | ρ | z 〉$ ,
the Werhl entropy $S_{W}^{L} (ρ) = - \int_{V} Q_{ρ}^{L} (z) log Q_{ρ}^{L} (z) d z$ .

Proposition 1.

The following equality is valid:

Q_{ρ}^{L} (z + z^{'}) = Q_{ρ}^{L} (z), z \in V, z^{'} \in L .

That is the Husimi function is constant on adjacency classes from

V / L

and is thus defined on the set

V / L

Indeed:

\begin{matrix} Q_{ρ}^{L} (z + z^{'}) = 〈 W (z + z^{'}) 0_{L} | ρ | W (z + z^{'}) 0_{L} 〉 = \\ = 〈 χ_{p} (Δ (z, z^{'})) z_{L} | ρ | χ_{p} (Δ (z, z^{'})) z_{L} 〉 = Q_{ρ}^{L} (z) . \end{matrix}

Corollary 1.

For the Wehrl entropy the following equality is valid:

S_{W}^{L} (ρ) = - \sum_{α \in V / L} Q_{ρ}^{L} (α) log Q_{ρ}^{L} (α) .

Proposition 2.

The Husimi function defines the probability distribution on the set

V / L

, that is

Q_{ρ}^{L} (α) \geq 0, \sum_{α \in V / L} Q_{ρ}^{L} (α) = 1 .

Since coherent states form an orthonormal basis, we have

\sum_{α \in V / L} Q_{ρ}^{L} (α) = Tr ρ = 1 .

The following map

ρ \to ρ_{o u t} = Φ_{L} [ρ] = \sum_{α \in V / L} Q_{ρ}^{L} (α) | α_{L} 〉 〈 α_{L} |

defines the quantum-classical channel

Φ_{L}

This is a complete ideal quantum measurement associated with an orthonormal basis of p-adic coherent states.

This measurement is heterodyne, that is, the coordinate and momentum of the particle are measured simultaneously with the maximum accuracy allowed by the uncertainty relation.

Denote by

S (ρ) = - Tr ρ log ρ

the von Neumann entropy. Jensen’s inequality and concavity of

- x log x

gives an inequality

S (ρ) \leq S_{W}^{L} (ρ)

. (as for coherent states for the real (i.e. over the field

R

) Heisenberg group ([2]).

Proposition 3.

The Wehrl entropy has the following property.

S_{W}^{L} (ρ) = S (Φ_{L} [ρ]) .

In other words, the Wehrl entropy of a state ρ is the von Neumann entropy of the measurement result in the basis of coherent states.

This is a consequence of the orthogonality of p-adic coherent states and does not hold in the real case.

In the real case, there is an estimate of

S_{W} (ρ) \leq S (ρ_{o u t})

(Berezin-Lieb inequality [12]).

Theorem 2.

The following lower bound is valid for the Wehrl entropy

0 \leq S_{W}^{L} (ρ),

and

S_{W}^{L} (ρ) = 0

if and only if ρ is a L-coherent state.

In the real case, the inequality

1 \leq S_{W} (ρ)

is valid and equality occurs only on coherent states (the Wehrl-Lieb inequality) [2,3]. This is a non-trivial result.

In the p-adic case, this is a very simple statement, which directly follows from a simple observation that the Husimi function of a coherent state takes only two values – zero and one,

Q_{| β_{L} 〉 〈 β_{L} |}^{L} (α) = δ_{α β} .

3.2. Entropic uncertainty relation

The channel

Φ_{L}

looks very simple in the representation of characteristic functions. The characteristic function

π_{ρ}

of the quantum state

ρ

is given by the relation

π_{ρ} (z) = Tr ρ W (z), z \in V

and defines the state uniquely.

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

Proposition 4.

The channel

Φ_{L}

multiplies the characteristic function of the state by the indicator function of the lattice L:

π_{Φ_{L} [ρ]} (z) = π_{ρ} (z) h_{L} (z), z \in V .

Now let two self-dual lattices

L_{1}

and

L_{2}

be given. These lattices are corresponded to measurements in the bases of coherent states. From the previous sentence, it’s easy to see what sequential measurements look like. Namely:

π_{ρ} \overset{Φ_{L_{1}}}{\to} π_{ρ} h_{L_{1}} \overset{Φ_{L_{2}}}{\to} π_{ρ} h_{L_{1}} h_{L_{2}} = π_{ρ} h_{L_{2}} h_{L_{1}} = π_{ρ} h_{L_{1} \land L_{2}}

The composition of channels is thus naturally denoted by

Φ_{L_{1} \land L_{2}} [ρ]

Proposition 5.

The channel

Φ_{L_{1} \land L_{2}}

is an incomplete ideal measurement. The orthogonal decomposition of the unit corresponding to this measurement is

{P_{a}, a \in V / L_{1} \lor L_{2}}

. All projections

P_{a}

have the same dimension equal to

p^{d (L_{1}, L_{2})}

The entropy of the output state for the channel

Φ_{L_{1} \land L_{2}} [ρ]

is denoted by

S_{W}^{L_{1} \land L_{2}}

. This entropy can no longer be zero, it is bounded from below by the distance between the corresponding lattices.

Corollary 2.

The entropy

S_{W}^{L_{1} \land L_{2}}

satisfies the inequality

d (L_{1}, L_{2}) \leq S_{W}^{L_{1} \land L_{2}} (ρ) .

Proposition 6.

The following inequalities are valid:

S_{W}^{L_{1}} (ρ) \leq S_{W}^{L_{1} \land L_{2}} (ρ) .

In other words, entropy does not decrease with successive measurements:

S (ρ) \leq S (Φ_{L_{1}} [ρ]) \leq S (Φ_{L_{2}} \circ Φ_{L_{1}} [ρ]) .

The following chain of relations is valid:

\begin{matrix} S_{W}^{L_{1} \land L_{2}} (ρ) = S (Φ_{L_{1} \land L_{2}} [ρ]) = S (Φ_{L_{2}} \circ Φ_{L_{1}} [ρ]) = \\ = S_{W}^{L_{2}} (Φ_{L_{1}} [ρ]) \geq S (Φ_{L_{1}} [ρ]) = S_{W}^{L_{1}} (ρ) . \end{matrix}

In the proof, we used Proposition 3 and the inequality between von Neumann entropy and Wehrl entropy. Similarly, we prove that

S_{W}^{L_{1} \land L_{2}} (ρ) \geq S_{W}^{L_{2}} (ρ)

Equality is achieved on coherent states. Let

ρ

be an

L_{2}

-coherent state. Then

ρ = Φ_{L_{2}} [ρ]

. Therefore:

S_{W}^{L_{1}} (ρ) = S_{W}^{L_{1}} (Φ_{L_{2}} [ρ]) = S (Φ_{L_{1}} \circ Φ_{L_{2}} [ρ]) = S_{W}^{L_{1} \land L_{2}} (ρ) .

Theorem 3.

The inequality (entropic uncertainty relation) is valid.

d (L_{1}, L_{2}) \leq S_{W}^{L_{1}} (ρ) + S_{W}^{L_{2}} (ρ) .

Equality occurs for

L_{1}

- or

L_{2}

-coherent states.

Applying the results of Lieb and Frank [14], we can get a stronger estimate:

d (L_{1}, L_{2}) + S (ρ) \leq S_{W}^{L_{1}} (ρ) + S_{W}^{L_{2}} (ρ) .

Equality is also occurs for

L_{1}

- or

L_{2}

-coherent states (these states are pure and the von Neumann entropy is zero on them).

The proof of the above result is based on paper Lieb and Frank [14]. The result of Lieb and Frank is as follows. Let two orthonormal bases be given

{| a_{j} 〉}

and

{| b_{k} 〉}

and

p_{j} = 〈 a_{j} | ρ | a_{j} 〉

q_{k} = 〈 b_{k} | ρ | b_{k} 〉

. Then the estimate is valid

- \sum_{j} p_{j} log p_{j} - \sum_{k} q_{k} log q_{k} \geq S (ρ) - log (sup_{j, k} {| 〈 a_{j} | b_{k} 〉 |}^{2}) .

3.3. Mutually unbiased bases

Entropy uncertainty relations are closely related to mutually unbiased bases [13,16].

Mutually unbiased bases in Hilbert space

C^{D}

are two orthonormal bases

{| e_{1} 〉, \dots, | e_{D} 〉}

and

{| f_{1} 〉, \dots, | f_{D} 〉}

such that the square of the magnitude of the inner product between any basis states

| e_{j} 〉

and

| f_{k} 〉

equals the inverse of the dimension D

| 〈 e_{j} | f_{k} 〉 |^{2} = \frac{1}{D}, \forall j, k \in {1, \dots, D} .

Let

L_{1}

and

L_{2}

be a pair of self-dual lattices,

d (L_{1}, L_{2}) \geq 1

It turns out that the corresponding bases of

L_{1}

- and

L_{2}

-coherent states are mutually unbiased on finite-dimensional subspaces of dimension

p^{d (L_{1}, L_{2})}

Theorem 4.

For bases of

L_{1}

- and

L_{2}

-coherent states

{| α_{L_{1}} 〉, α \in V / L_{1}}

and

{| β_{L_{2}} 〉, β \in V / L_{2}}

the following formula is valid

| 〈 α_{L_{1}} | β_{L_{2}} 〉 |^{2} = p^{- d (L_{1}, L_{2})} h_{L_{1} \lor L_{2}} (α - β) .

The theorem means the following. Our Hilbert space of representation of CCR

H

decomposes into an orthogonal direct sum of finite-dimensional subspaces of the same dimension

p^{d (L_{1}, L_{2})}

H = ⨁_{a \in V / (L_{1} \lor L_{2})} H_{a}, dim H_{a} = p^{d (L_{1}, L_{2})} .

In each of these subspaces, the subbasis of

L_{1}

- and

L_{2}

-coherent states are mutually unbiased.

More information and proof Theorem 4 can be found in [17].

It follows from Theorem 4 that for a pair of bases

L_{1}

- and

L_{2}

-coherent states, the equality is valid

sup_{α \in V / L_{1}, β \in V / L_{2}} {| 〈 α_{L_{1}} | β_{L_{2}} 〉 |}^{2} = p^{- d (L_{1}, L_{2})},

from which the statement of Theorem 3 immediately follows.

References

A. M. Perelomov. Coherent states for arbitrary Lie group. Commun. Math. Phys, 26, 222-236 (1972).
A. Wehrl. On the relation between classical and quantum-mechanical entropy. Reports on Math. physics, 16, 3, 353-358 (1979).
E. H. Lieb. Proof of an entropy conjecture of Wehrl. Commun. Math. Phys. 62, 35-41 (1978).
E, Carlen. Some integral identities and inequalities for entire functions and their application to the coherent state transform. Journal of Functional Analysis 97, 231–249 (1991).
E. H. Lieb, J. P. Solovej. Proof of an entropy conjecture for Bloch coherent spin states and its generalizations. Acta Mathematica. 212 (2), 379–398 (2014).
V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-Adic analysis and mathematical physics, World Scientific, 1994.
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich and E. I. Zelenov, p-Adic mathematical physics: the first 30 years, p-Adic numbers, Ultram. Anal. Appl. 9, 87 (2017).
I. I. Hirschman, Jr. A note on entropy. American Journal of Mathematics 79 (1), 152-156 (1957).
I. Bialynicki-Birula, J. Mycielski. Uncertainty relations for information entropy in wave mechanics. Comm. Math. Phys. 44, 129-132 (1975).
H. Maassen, J. B. M. Uffink. Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988).
K. Kraus. Complementary observables and uncertainty relations. Phys. Rev. D, bf 35, 10, 3070-3075 (1987).
F. A. Berezin. The general concept of quantization. Commun. Math. Phys. 40, 153 (1975).
S. Wehner, A. Winter. Entropic uncertainty relations – a survey. New Journal of Rhysics 12, 025009 (2010).
R. L. Frank, E. H. Lieb. Entropy and uncertainty principle. Ann. Henri Poincare 13, 1711-1717 (2012).
E. I. Zelenov. p-Adic model of quantum mechanics and quantum channels. Proc. Steklov Inst. Math., 285, 132-144 (2014).
M. A. Ballester and S. Wehner. Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases. Phys. Rev. A 75, 022319 (2007).
E. I. Zelenov. On Geometry of p-adic Coherent States and Mutually Unbiased Bases. Preprints.org 2023, 2023051569. [CrossRef]

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

Abstract

1. Introduction

1.1. Coherent states

1.2. The Husimi function and the Wehrl entropy

1.3. Entropic uncertainty relations

1.4. p-adic numbers

2. p-Adic coherent states

2.1. Canonical commutation relations

2.2. Vacuum vector

3. Entropy

3.1. Lower bound of the Wehrl entropy

3.2. Entropic uncertainty relation

3.3. Mutually unbiased bases

References

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