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

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

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Evgeny I. Zelenov^*

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22 May 2023

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23 May 2023

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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.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction. MUBs.

Mutually unbiased bases (MUBs) [1] 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} .

Such bases have numerous applications in quantum information theory (quantum key distribution [2,3,4], quantum state tomography [5], detection of quantum entanglement [6], etc).

The problem is to describe the MUBs set for an arbitrary D.

Within this general statement of the problem, there is a range of subtasks.

Denote by

M (D)

the maximum number of MUBs in

C^{D}

The first problem is what is

M (D)

equal to. In general, to find

M (D)

is a very difficult task, for example,

M (6)

has not been found to date, despite considerable efforts [7]. The answer is known when the dimension D is the power of a prime number, namely

M (p^{n}) = p^{n} + 1

[8].

It is possible to get the following estimation [8]:

p_{1}^{n_{1}} + 1 \leq M (D) \leq D + 1,

where

D = p_{1}^{n_{1}} p_{2}^{n_{2}} \dots p_{k}^{n_{k}}, p_{1}^{n_{1}} < p_{2}^{n_{2}} < \dots < p_{k}^{n_{k}}

is prime numbers decomposition of D.

The amazing thing is that this is almost all that is known by now.

The problem of finding

M (D)

is closely related to the well-known Winnie-the-Pooh conjecture [9]. Let us consider the Lie algebra

{sl}_{D} (C)

D \times D

matrices with zero trace. The problem of decomposition of this algebra into a direct sum of Cartan subalgebras pairwise orthogonal with respect to the Killing form is posed.

The conjecture is as follows:

{sl}_{D} (C)

is orthogonally decomposable if and only if

D = p^{n}

for some prime p.

The corresponding conjecture for MUB looks like this: a complete collection of MUBs exists only in prime power dimension D [10].

Let

B

be an orthonormal basis in

C^{D}

. Let’s call a matrix A complex Hadamard if

B

and

A (B)

are mutually unbiased bases.

Two Hadamard matrices A and C are equivalent if there exist monomial matrices

M_{1}

and

M_{2}

such that the condition is satisfied:

A = M_{1} C M_{2} .

The problem is to describe the sets of equivalence classes of Hadamard matrices.

There is a complete description only for the case

D \leq 5

, and for

D = 2, 3, 5

the number of Hadamard matrices is finite, for

D = 4

there exists a one-dinensional family. For the case

D = 6

, the existence of a complex 4-dimensional family of Hadamard matrices is proved [11], for

D = 7

, the existence of a one-dimensioin family is proved [12].

There are difficulties with definition of mutually unbiased bases in the case of an infinite-dimensional Hilbert space [13]. In this paper we will give such a definition. Despite its seeming naivety, the definition mentioned above naturally arises in the context of p-adic quantum mechanics.

I also want to note that the above brief overview and bibliographic references do not pretend to be complete, many important and interesting articles are not mentioned.

2. p-Adics numbers.

Just a few words about p-adic numbers, in order to introduce the necessary notation. For more information about p-adic numbers, see, for example, [14].

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, n > 0, 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}

(the Haar measure) 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} .

Using the canonical representation, we define the integer

{[x]}_{p}

and fractional

{x}_{p}

parts of the number

x \in Q_{p}

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

3. Representations of CCR. Coherent states.

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.

Let

H

be a separable complex Hilbert space. A map 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.

p-Adic integers

Z_{p}

form a ring. Let L be a two-dimensional (compact)

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.

Indeed, we have:

\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 states

{| 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 p-adic (L-)coherent states.

Remark 1.

The CCR representations are closely related to the representations of the Heisenberg group. In the language of representation theory, p-adic coherent states are nothing but coherent states for the p-adic Heisenberg group.

4. Main result.

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 2.

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 subbases of

L_{1}

- and

L_{2}

-coherent states are mutually unbiased.

Let’s prove Theorem 2.

The following formula is valid.

| 〈 0_{L_{1}} | W (β) 0_{L_{2}} 〉 | = \{\begin{matrix} | 〈 0_{L_{1}} | 0_{L_{2}} 〉 |, β \in L_{1} \lor L_{2}, \\ 0, β \notin L_{1} \lor L_{2} . \end{matrix}

(1)

Let

β \in L_{1} \lor L_{2}

, then

β = β_{1} + β_{2}, β_{1} \in L_{1}, β_{2} \in L_{2}

and

| 〈 0_{L_{1}} | W (β) 0_{L_{2}} 〉 | = | 〈 W (- β_{1}) 0_{L_{1}} | W (β_{2}) 0_{L_{2}} 〉 | = | 〈 0_{L_{1}} | 0_{L_{2}} 〉 | .

β \notin L_{1} \lor L_{2}

, then there exists

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

such that

χ_{p} (Δ (γ, β) \neq 1

and

〈 0_{L_{1}} | W (β) 0_{L_{2}} 〉 = 〈 W (γ) 0_{L_{1}} | W (β) W (γ) 0_{L_{2}} 〉 = χ_{p} (Δ (γ, β) 〈 0_{L_{1}} | W (β) 0_{L_{2}} 〉 .

From the latter equality, it obviously follows that

〈 0_{L_{1}} | W (β) 0_{L_{2}} 〉 = 0, β \notin L_{1} \lor L_{2}

Now let’s use formula (1) and the Parseval-Steklov identity:

\begin{matrix} 1 = \sum_{β \in V / L_{2}} {|〈 0_{L_{1}} | W (β) 0_{L_{2}} 〉|}^{2} = \\ = {|〈 0_{L_{1}} | 0_{L_{2}} 〉|}^{2} \sum_{β \in L_{1} \lor L_{2} / L_{2}} 1 = {|〈 0_{L_{1}} | 0_{L_{2}} 〉|}^{2} p^{d (L_{1}, L_{2})} . \end{matrix}

(2)

The following equation follows from formula (2):

{|〈 0_{L_{1}} | 0_{L_{2}} 〉|}^{2} = p^{- d (L_{1}, L_{2})} .

(3)

Taking into account formula (1) and the equality (3), we obtain a proof of Theorem 2.

In the case of

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

the subspaces

H_{a}, a \in V / (L_{1} \lor L_{2})

have dimension p. As it can be seen from the construction of the graph of lattices, there are exactly

p + 1

pieces of self-dual lattices with unit pairwise distances (the complete graph

K_{p + 1}

). These lattices define a complete set of MUB in each subspace

H_{a}

In the case of

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

the subspaces

H_{a}, a \in V / (L_{1} \lor L_{2})

have dimension

p^{2}

. As it can be seen from the construction of the graph of lattices, there are exactly

p (p + 1)

pieces of self-dual lattices lying at a distance of 2 from the lattice

L_{1}

. The bases corresponding to these lattices are not mutually unbiased. However, among this set there are families consisting of

p + 1

pieces of mutually unbiased bases. These bases saturate the entropic uncertainty relations [15,16].

Remark 2.

Instead of the field

Q_{p}

, we can consider its algebraic extension of degree n. Such extensions exist for any n. In this case, the elementary building block of the lattice graph will be the complete graph

K_{p^{n} + 1}

, which has

p^{n} + 1

vertex. The coherent state bases corresponding to the vertices of this graph form a complete set of mutually unbiased bases in

p^{n}

-dimensional space.

The theorem makes the following definitions natural.

Definition 3.

Let

H

be an infinite-dimensional Hilbert space. Orthonormal bases

{| e_{i} 〉}

and

{| f_{j} 〉}

are mutually unbiased if there exists a decomposition

H = \oplus H_{k}, dim H_{k} = n_{k} < \infty,

such that the subbases

{| e_{i} 〉 {} |}_{H_{k}}

and

{| f_{j} 〉 {} |}_{H_{k}}

are mutually unbiased for all k.

Let’s make two remarks. First, the definition assumes that the bases are divided into finite blocks of size

n_{k}

, each of which forms a subbases in the corresponding subspace

H_{k}

. Secondly, in the case under consideration of the representation of CCR over a field of p-adic numbers of the dimension of subspaces

H_{k}

there are powers of p. The construction can be extended to the case of CCR over Vilenkin groups, in which case the above dimensions can be arbitrary natural numbers.

5. p-Adic dynamics. Hadamard operators.

The proposed definition of mutually unbiased bases for the case of an infinite-dimensional Hilbert space makes it possible to introduce the concept of the Hadamard operator for such spaces in a similar way.

Definition 4.

The operator A in the Hilbert space

H

is called the Hadamard operator if for some orthonormal basis

{| e_{i} 〉}

H

the bases

{| e_{i} 〉}

and

A ({| e_{i} 〉})

are mutually unbiased.

In other words, the Hadamard operator is given by an infinite block-diagonal matrix whose diagonal blocks are ordinary finite Hadamard matrices.

It turns out that the dynamics of a p-adic quantum system is determined by Hadamard operators. More detailed information about p-adic quantum theory can be found, for example, in [17,18]. The dynamics of a classical system is given by a linear symplectic transformation

g \in S p (V)

of the phase space V. A one-parameter family

g_{t}

of such transformations can be specified, in which case the parameter

t \in Q_{p}

is interpreted as time. For example, the dynamics of a free particle of unit mass is given by the family

g_{t}, t \in Q_{p}

, which in some fixed basis of space V has the form:

g_{t} = (\begin{matrix} 1 & t \\ 0 & 1 \end{matrix}), t \in Q_{p} .

If the dynamics of a classical system is determined by the action of a symplectic group on the phase space, then the dynamics of the corresponding quantum system is given by the so-called metaplectic representation of the symplectic group in the Hilbert space of the representation of CCR. The existence of such a representation follows from the uniqueness of irreducible representations of CCR.

Let

(W, H)

be a representation (irreducible) of CCR and

g \in S p (V)

. Then, by virtue of the uniqueness of the representation, the representations

(W, H)

and

(W_{g}, H), W_{g} (z) = W (g z), z \in V

are unitarily equivalent, that is, there is a unitary operator satisfying the condition

U (g) W (z) = W_{g} (z) U (g), z \in V .

The operators

U (g), g \in S p (V)

define a metaplectic representation of

S p (V)

Theorem 3.

Let L be a lattice in V such that

d (L, g L) \geq 1

. Then

U (g)

is the Hadamard operator for bases

{{| α 〉}_{L}}

and

{{| β 〉}_{g L}}

As mentioned above, the symplectic group acts transitively on the set of self-dual lattices. Thus, if a self-dual lattice L is given, then its image gL under the action of the symplectic transformation g is also a self-dual lattice. The validity of Theorem 3 follows from Theorem 2 for a pair of lattices L and gL.

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

Abstract

1. Introduction. MUBs.

2. p-Adics numbers.

3. Representations of CCR. Coherent states.

4. Main result.

5. p-Adic dynamics. Hadamard operators.

References

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