Finite quantum systems and frame quantization

Finite quantum systems and frame quantization Nicolae Cotfas1 , Jean Pierre Gazeau2 and Apostol Vourdas3 1

Faculty of Physics, University of Bucharest, PO Box 76 - 54, Post Office 76, Bucharest, Romania 2 Laboratoire APC, Universit´e Paris Diderot, 10, rue A. Domon et L. Duquet, 75205 Paris Cedex13, France 3 Department of Computing, University of Bradford, Bradford BD7 1DP, United Kingdom E-mail: [email protected], [email protected], [email protected] Abstract. The quantum observables used in the case of quantum systems with finitedimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Some finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder-Berezin-Toeplitz type quantization.

Finite quantum systems

2

1. Introduction Weyl’s formulation of quantum mechanics [41] opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl’s approach, generalized by Schwinger [33], several authors (e.g., [30, 31, 34, 35, 10, 19, 42, 24, 23, 36, 37, 38, 39, 32]) investigated the analogue of the harmonic oscillator in finite-dimensional Hilbert spaces. This work combines concepts from quantum mechanics with discrete mathematics (number theory, finite sums [8], combinatorics, etc). The general formalism of coherent states [28, 29, 17, 43] can also be used in the context of finite-dimensional quantum systems. This powerful method defines continuous systems of coherent states as well as finite systems of coherent states, usually called finite tight frames [3, 11, 12, 18]. In finite quantum systems mainly continuous systems of coherent states have been studied. Among recent explorations of finite systems of coherent states, see for instance [13, 15]. In this paper we study finite frames in the context of finite quantum systems. This is a finite frame quantization, and it is a finite version of a conventional Klauder-Berezin-Toeplitz type coherent state quantization. In section 2 we briefly present topics from the theory of finite quantum systems, which are needed later. In section 3 we present the general formalism for finitedimensional tight frames and the related quantization. We define covariant and contravariant symbols and we develop some probabilistic aspects of the formalism. We also define Wigner functions and Weyl functions in this context. In section 4 we present an example of tight frames. They depend on a ‘fiducial’ vector, and we use an eigenvector of the discrete Fourier transformation which is related to the harmonic oscillator vacuum through a Zak [45] or Weil [40] transform. The example has been previously defined in [44] in terms of the Perelomov method by starting from a discrete version of the Heisenberg group. In section 5 we show how the formalism can be used for practical calculations (matrix elements, spectra, etc) in the quantization context. We conclude in section 6 with a discussion of our results. 2. Finite quantum systems We consider a quantum system where position and momentum take values in the ring Zd = Z/dZ = {0, 1, ..., d−1} of the integers modulo d, where d is a fixed positive integer. The Hilbert space H of this system is d-dimensional, and we describe it by using the orthonormal basis of ‘position states’ {|e0 i, |e1 i, ..., |ed−1 i}. The finite Fourier transform F : H −→ H,

d−1 1 X 2πi kn e d |ek ihen | F =√ d n,k=0

(1)

Finite quantum systems

3

allows us to consider the directly related orthonormal basis of ‘momentum states’ {|f0 i, |f1 i, ..., |fd−1 i} satisfying the relations d−1

1 X 2πi kn e d |en i , |fk i = F |ek i = √ d n=0

d−1

1 X − 2πi nk |en i = F + |fn i = √ e d |fk i . d k=0

Each state |ψi ∈ H can be expanded as |ψi =

d−1 X

ψn |en i =

n=0

d−1 X

ψ˜k |fk i

k=0

where the functions ψ : Zd −→ C : n 7→ ψn and ψ˜ : Zd −→ C : k 7→ ψ˜k satisfying d−1

d−1

1 X − 2πi kn ψ˜k = √ e d ψn . d n=0

1 X 2πi nk ˜ ψn = √ e d ψk , d k=0

(2)

are the corresponding ‘wavefunctions’ in the position and momentum representations. The parity operator with respect to the origin is defined as P (0, 0) = F 2 and P (0, 0)2 = F 4 = IH . The self-adjoint ‘position’ and ‘momentum’ operators x, p : H −→ H, x=

d−1 X

n |en ihen | ,

p=

n=0

d−1 X

k |fk ihfk | ,

(3)

k=0

are defined modulo d and satisfy the relations 2πi n(k−j) d

F xF + = p ,

x|en i = n|en i ,

x|fk i =

1 d

Pd−1

ne

F pF + = −x ,

p|fk i = k|fk i ,

p|en i =

1 d

Pd−1

ke

n,j=0

k,m=0

|fj i ,

2πi k(m−n) d

|em i .

The function [37] d−1

1 X 2πi nx ed , ∆0 (x) = d n=0

∆0 : R −→ C,

(4)

and its derivatives d−1

ds 1 X 2πi s 2πi nx ∆s (x) = s ∆0 (x) = ( n) e d , dx d n=0 d

∆s : R −→ C ,

are useful in the calculation of matrix elements  s d−1 1 X s 2πi l(n−m) d hen |x |em i = n δnm , hen |p |em i = led = ∆s (n−m) , d l=0 2πi s

s

s

 s d−1 d 1 X s − 2πi l(k−j) le d = ∆s (k−j) . hfj |p |fk i = k δjk , hfj |x |fk i = d l=0 2πi s

s

s

It is easily seen that ∆0 (x) =

1 d

d−1 X n=0

e

2πi nx d

  if x ∈ dZ ,  1 = 1 e2πix − 1  if x 6∈ dZ .  x d e 2πi d −1

(5)

Finite quantum systems

4

When x is an integer which      d−1  X 1 2πi nx 2πi d ∆1 (x) = n e =  d n=0 d    

is not a multiple of d then ∆0 (x) = 0. Also πi (d − 1) if x ∈ dZ , d  2πi 
2πi 2πie2πix e d x − 1 − 2πi e d x e2πix − 1 d if x ∈ 6 dZ . 2  2πi x d −1 d e

Therefore  d−1   if n = m , d 2 hen |p|em i = ∆1 (n−m) = 1  2πi  2πi if n 6= m , (n−m) ed −1

(6)

and hen |[x, p]|em i = (n−m) hen |p|em i =

   0  

e

if n = m , n−m 2πi (n−m) d

−1

(7)

if n 6= m .

The displacement operators [33, 34, 44] A , B : H −→ H ,

A=e

2πi p d

,

B =e

2πi x d

,

(8)

are single-valued and A|en i = |en−1 i , B|en i = e

2πi n d

A|fk i = e

|en i ,

2πi k d

|fk i ,

B|fk i = |fk+1 i ,

Ad = B d = IH , Aα B β = e

2πi αβ d

B β Aα .

General displacements operators with respect to (α, β) ∈ Zd × Zd are given by πi

D(α, β) = Aα B β e− d αβ ;

[D(α, β)]† = D(−α, −β) .

(9)

For an arbitrary operator Θ, it has been shown [39] that d−1 1 X D(α, β)Θ[D(α, β)]† = IH TrΘ . d α,β=0

(10)

The parity operator with respect to (α, β) ∈ Zd × Zd is given by P (α, β) = D(α, β)P (0, 0)[D(α, β)]† ,

[P (α, β)]2 = IH

(11)

and it is related to the displacement operators through the Fourier transform d−1 1 X 2πi (αδ−βγ) D(α, β) = ed P (γ, δ) . d γ,δ=0

(12)

Finite quantum systems

5

3. Tight finite frames and finite frame quantization Let H be a d-dimensional Hilbert space and {|e0 i, |e1 i, ..., |ed−1 i} an orthonormal basis of H. Let X = {a0 , a1 , ..., aM −1 } be a set of ‘parameters’ or ‘indices’, with M ≥ d. An example will be given later with X = Zd × Zd for which M = d2 . We consider an orthonormal system of d functions {φk : X −→ C}d−1 k=0 : M −1 X

φj (an ) φk (an ) = δjk

(13)

n=0

such that κn =

d−1 X

|φk (an )|2 6= 0

for any n ∈ {0, 1, ..., M − 1} .

(14)

k=0

We immediately note that M −1 X

κn = d and κn ≤ 1 .

n=0

The latter inequality is strict if d < M . We then consider the map X 3 an 7→ |an i ∈ H defined by d−1 1 X |an i = √ φk (an ) |ek i , κn k=0

n = 0, ..., M − 1 .

(15)

The M vectors |an i overlap as: d−1 X 1 φk (an ) φk (am ) . han |am i = √ κn κm k=0

(16)

They solve the unity in H in the following way: M −1 X

κn |an ihan | = IH .

(17)

n=0

Therefore the vectors |an i form a tight finite frame in H [11, 12]. It follows that any state |χi ∈ H can be expanded as M −1 X

κn |an ihan |χi = |χi ,

(18)

n=0

and the scalar product of two states in H can be written as hχ|ψi =

M −1 X

κn hχ|an ihan |ψi .

(19)

n=0

For later use we give the relation d−1

X 2πi πi 1 D(α, β)|an i = √ e− d αβ φk (an ) e d βk |ek−α i. κn k=0

(20)

−1 The considered tight frame {|an i}M n=0 defines two probability distributions which can be interpreted in terms of a Bayesian duality [2]:

Finite quantum systems

6

(i) A prior distribution on the set of indices k ∈ {0, ..., d − 1}, with parameter an ∈ X , k 7→

|φk (an )|2 = |hek |an i|2 , κn

(21)

with d−1 X

|hek |an i|2 = 1 for n ∈ {0, 1, ..., M −1} .

k=0

Adopting a quantum mechanical language, this probability could be considered as concerning experiments performed on the system within some experimental protocol in order to measure the spectral values of a certain self-adjoint operator (a “quantum observable”) A acting in H (e.g. the position x in (3) and having P the spectral resolution A = k λk |ek ihek | (e.g. the measured positions “k” in (3). Precisely, |hek |an i|2 is the probability to get the value λk from a measurement of A which is performed on the system when the latter is prepared in the “state” |an i. (ii) A posterior distribution on the original set of parameters an ∈ X , equipped with uniform (discrete) measure, and with parameter k ∈ {0, ..., d − 1}, an 7→ |φk (an )|2 ,

(22)

with M −1 X

√ | κn |φk (an )|2 = 1 .

n=0

The Bayesian duality stems in the two interpretations: the resolution of the unity verified by the states |an i, introduces a preferred prior measure on X , which is the |φk (an )|2 , with this distribution itself playing set of parameters of the distribution k 7→ κn the role of the likelihood function. The associated distributions an 7→ |φk (an )|2 on the original set X , indexed by k, become the related conditional posterior distributions. 3.1. Covariant and contravariant symbols To each function f : X −→ C we associate the operator Af : H −→ H,

Af =

M −1 X

κn f (an ) |an ihan | .

(23)

n=0

The operator Af corresponding to a real function f is self-adjoint, and this can be regarded as a Klauder-Berezin-Toeplitz type quantization [4, 5, 6, 7, 20, 21, 22] of f , the notion of quantization being considered here in a wide sense [15]. The eigenvalues of the matrix Af form the “quantum spectrum” of f (by opposition to its “classical spectrum” that is the set of its values f (an )). Given an operator A = Af , the function f , possibly not unique, is called upper [25] or contravariant symbol [6] (or P -function) of A. We also introduce the function

Finite quantum systems

7

fˇ= Aˇf : X −→ R, such that fˇ(an ) = han |Af |an i =

M −1 X

κm f (am ) |han |am i|2

m=0

1 = κn

2 d−1 X f (am ) φk (an ) φk (am ) , m=0

M −1 X

(24)

k=0

which is called lower [25] or covariant symbol [6] (or Q-function) of Af . It can be viewed as finite version of the so-called Berezin transform of the original function f (see for instance [14]). The matrix elements of Af with respect to the orthonormal basis {|ek i} are hek |Af |ej i =

M −1 X

κn f (an ) hek |an ihan |ej i =

n=0

M −1 X

f (an ) φk (an ) φj (an ) . (25)

n=0

Using them in conjunction with fˇ(an ) =

d−1 X

han |ek ihek |Af |ej ihej |an i ,

k,j=0

we get again Eq.(24). We note that tr Af =

M −1 X

κn fˇ(an ) =

n=0

M −1 X

κn f (an ) .

(26)

n=0

An interesting problem in our finite frame quantization is to compare the starting function f with the lower symbol fˇ. Before proceeding with this study, more informations on frames are needed. 3.2. Stochastic side of a finite tight frame Let us introduce the real M × M matrix U with matrix elements Umn = |ham |an i|2 .

(27)

These elements obey Unn = 1 for 0 ≤ n ≤ M − 1 and 0 ≤ Umn = Unm ≤ 1 for any pair (m, n), with m 6= n. Now we suppose that there is no pair of orthogonal elements, i.e. 0 < Umn if m 6= n, and no pair of proportional elements, i.e. Umn < 1 if m 6= n, in the frame. Then from the Perron-Frobenius theorem for (strictly) positive matrices, the spectral radius r = r(U ) is > 0 and is dominant simple eigenvalue of U . It is proved (Collatz-Wielandt formula) that (U v)n def where N = {v | v ≥ 0 v 6= 0} . v∈N 0≤n≤M −1 ,vn 6=0 vn There exists a unique vector, vr , kvr k = 1, which is strictly positive (all components are > 0 and can be interpreted as probabilities) and U vr = rvr . All other eigenvalues α of U lie within the open disk of radius r : |α| < r. Since tr U = M , and that U has M r = max

min

Finite quantum systems

8

eigenvalues, one should have r > 1. The value r = 1 represents precisely the limit case in which all eigenvalues are 1, i.e. U = I and the frame is just an orthonormal basis of CM . It is then natural to view the number def

η = r−1

(28)

as a kind of “distance” of the frame to the orthonormality. The question is to find the relation between the set {κ0 , κ1 , ..., κM −1 } of weights defining the frame and the distance η. By projecting on each vector |an i from both sides the frame resolution of the unity (17), we easily obtain the M equations 1 = ham |am i =

M −1 X

κn |ham |an i|2 ,

i.e. U vκ = vδ ,

(29)

n=0 def

def

where t vκ = (κ0 κ1 ... κM −1 ) and t vδ = (1 1 ... 1) is the first diagonal vector in CM . Note that if U is not singular we have vκ = U −1 vδ . In the “uniform” case for which κn = d/M for all n, i.e. in the case of a finite equal √ norm Parseval frame, which means that vκ = (d/M ) vδ , then r = M/d and vr = 1/ M vδ . In this case, the distance to orthonormality is just M −d η= , (30) d a relation which clearly exemplifies what we can expect at the limit d → M . def def Let us now examine the matrix P = U K, where K = diag(κ0 , κ1 , . . . , κM −1 ). def t Its  (right) stochastic  nature is evident from (29). The row vector $ = vκ /d = κM − 1 κ0 κ1 ··· is a stationary probability vector: d d d $P =$.

(31)

As is well known, this vector obeys the ergodic property:
κn . lim P k mn = $n = k→∞ d

(32)

3.3. Semi-classical aspects of finite frame quantization through lower symbols With the stochastic matrix approach developed above, the relation between upper and lower symbols, fˇ(an ) = han |Af |an i =

M −1 X

κm f (am ) |han |am i|2 ,

m=0

is rewritten as ˇ f = Pf ,

(33)
def def with t f = (f (a0 ) f (a1 ) ... f (aM −1 )) and tˇ f = fˇ(a0 ) fˇ(a1 ) ... fˇ(aM −1 ) . This formula is interesting because it can be iterated: ˇ f [k] = P k f ,

ˇ f [k] = P ˇ f [k−1] ,

ˇ f [1] ≡ ˇ f,

(34)

Finite quantum systems

9

and so we find from the property (32) of P that the ergodic limit (or “long-term average”) of the iteration stabilizes to the “classical” average of the observable f defined as: def

ˇ f [∞] = hf icl vδ ,

where hf icl =

M −1 X n=0

κn f (an ) . d

(35)

We can conclude from this fact that the chain of maps P : fˇ[k−1] 7→ fˇ[k] corresponds to a sort of increasing regularization of the original function f . We can evaluate the “distance” between the lower symbol and its classical counterpart through the inequality: def kˇ f − f k∞ =

max

0≤n≤M −1

|f (an )− fˇ(an )| ≤ kI − P k∞ kf k∞ ,

where the induced norm [26] on matrix A is kAk∞ = max0≤m≤M −1 present case, because of the stochastic nature of P , we have   kI − P k∞ = 2 1 − min κn .

(36)

PM −1 n=0

0≤n≤M −1

|amn |. In the

(37)

In the uniform case, κn = d/M for all n, we thus have an estimate of how far the two functions f and fˇ are: kˇ f − f k∞ ≤ 2(M − d)/M kf k∞ . In the general case, we can view the parameter def

ζ = 1−

min

0≤n≤M −1

κn

(38)

as a “distance” of the “quantum world” to the classical one, of non-commutativity to commutativity, or again of the frame to orthonormal basis, like the distance η = r − 1 introduced in (28). Another way to check that ζ = 1 − d/M → 0 means, in the uniform case κn = d/M for all n, that we go back to the classical spectrum of the observable f results from the following relations. We have d d d X |ham |an i|2 = ||an ||2 − |han |an i|2 = 1 − , M m6=n M M and, from the relation M −1 X d X d ˇ f (an ) = f (am ) |ham |an i|2 = f (an ) + f (am ) |ham |an i|2 , M m=0 M m6=n

we get     d d d 1− min f (an ) ≤ fˇ(am ) − f (am ) ≤ 1 − max f (an ) . (39) n n M M M From the relation −1 M X |f (an )− fˇ(an )| = κm (f (an )−f (am )) |han |am i|2 ≤ max |f (an )−f (am )| m m=0

we also get the alternative estimate on the proximity between lower and upper symbols: max |f (an )− fˇ(an )| ≤ max |f (an )−f (am )| . n

n,m

(40)

Finite quantum systems

10

3.4. Wigner and Weyl functions in the context of frame quantization ff (α, β) of Af is The Weyl (or ambiguity) function W ff (α, β) = Tr[D(α, β)Af ] = W αβ − πi d

=e

d−1 X

e

2πi rβ d

M −1 X

κn f (an )han |D(α, β)|an i

n=0 M −1 X

r=0

f (an )φr−α (an )φr (an ) .

(41)

n=0

The Wigner function Wf (α, β) of Af is Wf (α, β) = Tr[P (α, β)Af ] = = e−

4πi αβ d

d−1 X

e

M −1 X

κn f (an )han |P (α, β)|an i

n=0 M −1 X − 4πi rβ d

r=0

f (an )φ−r−2α (an )φr (an ) .

(42)

n=0

If Af is a Hermitian operator, then the Wigner function is real. Using Eq.(12) we prove that the Weyl and Wigner functions are related through the Fourier transform X 2πi ff (α, β) = 1 (43) e d (αδ−βγ) Wf (γ, δ). W d γ,δ The operator Af can be written as X X ff (−α, −β)D(α, β) Af = Wf (α, β)P (α, β) = W α,β

(44)

α,β

The proof is based on the expression (25) for the matrix elements of Af with respect to the orthonormal basis |ej i. 4. An example of tight frame We consider an example of the general tight frames discussed earlier, where X = Zd ×Zd and πi 2πi 1 φn : Zd × Zd −→ C, φn (α, β) = √ e− d αβ e− d βn w¯α+n . (45) d P 2 Here the wk ’s where k ∈ Zd are d complex numbers normalized so that d−1 k=0 |wk | = 1. The spectral radius of the matrix U is r = d and “distances” of such a frame to orthonormality are respectively η = d − 1 and ζ = 1 − 1/d. It is easily proved that the functions φn form an orthonormal system for the inner product (13). Hence our general formalism leads to the d2 unit vectors |α, βi = e

πi αβ d

d−1 X

e

2πi βn d

wα+n |en i = D(α, β)|si,

(α, β) ∈ Zd × Zd (46)

n=0

P where |si = wn |en i is a normalized ‘fiducial vector’. By construction, they form a finite tight frame in the Hilbert space H with uniform distribution καβ = d/d2 = 1/d: d−1 1 X |α, βihα, β| = IH . d α,β=0

(47)

Finite quantum systems

11

Also we have the overlap between two elements of the frame: hα, β|γ, δi = e

iπ (γδ−αβ) d

d−1 X

e

2πi (δ−β)n d

wα+n wγ+n .

(48)

n=0

Note that the Fourier transform of wα+n wγ+n appears here. We can now prove the following relations F |α, βi = |β, −αi , iπ

D(γ, δ)|α, βi = e d (βγ−αδ) |α + γ, β + δi .

(49)

In Ref.[44] is obtained a similar frame by starting from a discrete version of the Heisenberg group [34, 36, 38] defined in terms of the displacement operators. Then the resolution of the identity of Eq.(47) follows from the general Eq.(10). In this section we derive some of their properties through our approach, rather than through the properties of the displacement operators. wk 6 q 0.5 q

wk 6

q q

0.25

q q

1

0.5

q q q q q q q q q q q q q

qq

q

0.25

q

10

19

k -

2

q

q

qq

qqqqqqqqqqqqqqqqqqqqqqqqqqqq 20

q

q

q

q

q

39

k -

Figure 1. The wk of Eq.(50) in the cases d = 20 (left) and d = 40 (right).

We choose as wk the following numbers: ∞ ∞ X 2π 2 1 X − π (αd+k)2 d wk = √ e ; N0 = e− d γ . N0 α=−∞ γ=−∞

(50)

This is similar to a Zak [45] or Weil [40] transform of the Gaussian function. Using the Jacobi theta function P iπτ n2 +2inz (51) θ3 (z; τ ) = ∞ n=−∞ e we rewrite wk as
i ; 1 θ3 πk wk = √ q d d
. d θ3 0; 2i

(52)

d

The components wk have significant values only near 0 (modulo d). This is expected 2 because wk is related to the Gaussian function e−x /2 (which is the ground state wave function in the harmonic oscillator formalism).

Finite quantum systems

12

Table 1. Eigenvalues λn of Aq and En of A q2 +p2 in the case d = 20. 2

n

λn

En

n

λn

En

0 1 2 3 4 5 6 7 8 9

2.41822 3.04403 3.21825 4.00253 5.00008 6. 6.83772 7. 8. 9.

18.9479 31.7393 43.5883 54.9417 65.9467 76.6685 87.1869 97.6208 108.025 118.172

10 11 12 13 14 15 16 17 18 19

10. 11. 12. 12.1623 13. 13.9999 14.9975 15.7817 15.9560 16.5818

126.949 133.955 143.126 153.735 162.329 171.491 185.562 203.131 225.959 260.924

5. Quantum mechanics for finite systems based on finite frame quantization The set Zd ×Zd is now considered as a finite version of a phase space, i.e. the mechanical phase space for the motion of a particle on the periodic set Zd . In accordance with the content of Section 3 we associate to each classical observable f : Zd × Zd −→ R the linear operator Af : H −→ H ,

d−1 1 X Af = f (α, β) |α, βihα, β| , d α,β=0

and the lower symbol fˇ= Aˇf : Zd × Zd −→ R ,

fˇ(a, b) = ha, b|Af |a, bi ,

(53)

(54)

satisfying the relations fˇ(a, b) =

1 d

P 2 d−1 2πi (β−b)n d e f (α, β) w w , α+n a+n n=0 α,β=0

Pd−1

(55)

and P 2πi b(m−n) d fˇ(a, b) = d−1 wa+n wa+m hen |Af |em i . n,m=0 e

(56)

We know from (36) that the deviation in the sense of the norm k · k∞ is bounded by 1 (57) kf − ˇf k∞ ≤ 2ζ kf k∞ , with ζ = 1 − . d 5.1. Example The linear operators corresponding to coordinate and momentum functions q : Zd × Zd −→ {0, 1, . . . d−1} , q(α, β) = α (mod d) , p : Zd × Zd −→ {0, 1, . . . d−1} ,

(58)

p(α, β) = β (mod d) ,

are Aq , Ap : H −→ H, d−1 1 X α |α, βihα, β| , Aq = d α,β=0

d−1 1 X Ap = β |α, βihα, β| , d α,β=0

(59)

Finite quantum systems

13

λn 6

En 6

q

250

200

15

10

5

q q q q

q

q q q

1

q

q

q

q

q q q q

q

q q q

100

-n

10

q

q

q

q

q

q

q

q

q

19

q q

q

q

q q

q

q

q

q

-n

10

19

Figure 2. Eigenvalues λn of Aq and En of A q2 +p2 in the case d = 20. 2

and ( P d−1 hen |Aq |em i =

α=0

0

2 if n = m , α wα+n if n = 6 m,

 d−1   if n = m ,  2 d−1 hen |Ap |em i = X 1   wα+n wα+m if n 6= m .  2πi (n−m) − 1 α=0 ed

(60)

So Aq is diagonal in the orthonormal basis of position states. More generally, for each s ∈ {0, 1, 2, ...}, we have Pd−1 s 2 hen |Aqs |em i = δnm α=0 α wα+n , (61)
Pd−1 s d hen |Aps |em i = 2πi wα+n wα+m , ∆s (n−m) α=0 and one can compare these matrix elements with s

s

hen |x |em i = n δnm

and



d hen |p |em i = 2πi s

s ∆s (n−m) .

As pointed out, the ‘position states’ |e0 i, |e1 i, ..., |ed−1 i are eigenvectors of Aq , namely Aq |en i = λn |en i

where λn =

d−1 X

2 , α wα+n

(62)

α=0

and hen |[Aq , Ap ]|em i =

   0  

e

if n = m , λn − λm 2πi (n−m) d

−1

d−1 X

wα+n wα+m if n 6= m .

(63)

α=0

Also Aqs = F Aps F +

(64)

Finite quantum systems

14

These relations show that Aqs and Aps have the same spectrum. Mutatis mutandis, the ‘momentum states’ |f0 i, |f1 i, ..., |fd−1 i are eigenvectors of Ap corresponding to the eigenvalues λ0 , λd−1 , λd−2 , ..., λ1 , namely, we have Ap |fk i = λd−k |fk i.

(65)

The distribution of the eigenvalues λn and En of the operators Aq and A q2 +p2 is presented 2

in table 1 and figure 2. One can remark a tendency, as M = d2 becomes larger and larger w.r.t. d, to have λn = n, and a tendency of the levels En to become equidistant levels.

ˇq (a,b) 6 A

ˇ 2 (a,b) 6 A q 700

500

20 16 12 8

q q

q

4

qqqqqqq

qq

qq qqq

qqq

q

qq qqq

q

qq qqq

qqqqq

q

q

q

q

q q q

100

2

20

39

a -

qq qqqq qq

qq

q

q qq

2

qq

qq

q

q

q

q

q

q

q

q

20

q

q

q

qq q q

q

q q

39

a -

Figure 3. The a-dependence of Aˇq (a, b) and Aˇq2 (a, b) in the case d = 40, i.e. M = 1600.

As expected the lower symbol Aˇqs (a, b) does not depend on b Aˇqs (a, b) =

d−1 X

2 wa+n

n=0

d−1 X

2 αs wα+n ,

(66)

α=0

and the lower symbol Aˇps (a, b) does not depend on a Aˇps (a, b) =

d−1 X

2 wb+n

n=0

d−1 X

2 αs wα+n .

(67)

α=0

In particular, we have qˇ(a, b) =

d−1 X

2 λn wa+n

,

pˇ(a, b) =

n=0

d−1 X

2 λn wb+n ,

(68)

n=0

and 2

ha, b|(Aq ) |a, bi =

d−1 X n=0

2 λ2n wa+n

,

2

ha, b|(Ap ) |a, bi =

d−1 X n=0

2 λ2n wb+n .

(69)

Finite quantum systems

15

The a-dependence of qˇ(a, b) and Aˇq2 (a, b) in the case d = 40, i.e. M = 1600, is presented in figure 3. Since kqk∞ = d − 1 = kpk∞ , the bound to deviation (57) gives for the position and momentum: (d − 1)2 (d − 1)2 ˇ k∞ ≤ 2 , kp − p , (70) d d which is clearly not an optimal bound in view of the figure 3 (actually we should think in terms of relative deviation). We can also show that ! d−1 d−1 X X 2 2 (∆Aq )2 = ha, b|(Aq )2 |a, bi − (ha, b|Aq |a, bi)2 = λn − λm wa+m wa+n . (71) ˇ k∞ ≤ 2 kq − q

n=0

m=0

The lower symbols of x and p can be expressed in terms of the eigenvalues of Aq and Ap P 2 xˇ(α, β) = hα, β|x|α, βi = d−1 n=0 nwα+n = λα , (72) pˇ(α, β) = hα, β|p|α, βi = λ−β = λd−β . 6. About the non-uniqueness of the upper symbol The function f : Zd × Zd −→ R is an upper symbol of the position operator x, d−1 d−1 X 1 X n|en ihen | f (α, β) |α, βihα, β| = d α,β=0 n=0

if and only if the values f (α, β) satisfy the system of d2 equations d−1 2πi 1 X f (α, β) e d β(m−n) wα+n wα+m = n δnm . d α,β=0

Generally the upper symbol corresponding to a linear operator is not unique. Looking for a function f depending only on α, that is, a function of the form f (α, β) = ϕ(α) the previous system of equations becomes d−1 2πi 1 X ϕ(α) e d β(m−n) wα+n wα+m = n δnm d α,β=0

and is equivalent to d−1 X

2 ϕ(α) wα+n =n

(73)

α=0

that is,     

w02 w12 ... 2 wd−1

w12 w22 ... w02

w22 w32 ... w12

... ... ... ...

2 wd−2 2 wd−1 ... 2 wd−3

2 wd−1 w02 ... 2 wd−2

    

ϕ(0) ϕ(1) .. . ϕ(d−1)





    =  

0 1 .. . d−1

   . 

Finite quantum systems

16

6 ϕ(α) q

1500

q

q q 1

q −1000

q

q

q q

q q

q

q q

10

q

q

q q

19

α -

q q

Figure 4. The upper symbol ϕ(α) of x obtained by solving (73) in the case d = 20.

The matrix of the system is (up to a permutation of rows) a circulant matrix with the absolute value of the derminant d−1   Y 2πi 2πi 2πi 2 . (74) w02 + e d n w12 + e d 2n w22 + . . . + e d (d−1)n wd−1 n=0

A discussion of necessary and sufficient conditions for circulant matrices to be nonsingular is presented in [16]. The upper symbol f (α, β) = ϕ(α) of the position operator x, obtained by solving (73) in the case d = 20, is presented in figure 4. 7. Concluding remarks The finite-dimensional quantum systems represent an essential ingredient in the development of several fields including spin systems, ensembles of two-level atoms, quantum dots, quantum optics and quantum information [1, 9, 27, 38]. In the present paper we have studied tight frames in this context and related phase space quantities (covariant and contravariant symbols and Wigner and Weyl functions). The general theory has been exemplified with a particular tight frame (in section 4) which has been used for calculations of matrix elements, spectra, etc. If we use a different fiducial vector, or if we perform unitary transformations on |α, βi, we get other tight frames. But it is an interesting problem for further work to give examples of tight frame, that they be uniform or not. There are several other problems related to finite quantum systems: symplectic transformations and tomography, mutually unbiased bases, multipartite systems comprised of many finite quantum systems and their entanglement, etc. In future works we also plan to approach these questions from a finite tight frame point of view. Acknowledgment NC acknowledges the support provided by CNCSIS under the grant IDEI 992 - 31/2007.

Finite quantum systems

17

References [1] Ali S T, Antoine J-P and Gazeau J-P 2000 Coherent States, Wavelets and their Generalizations (Graduate Texts in Contemporary Physics)(New York: Springer) [2] Ali S T, Gazeau J-P and Heller B 2008 Coherent states and Bayesian duality J. Phys. A: Math. Theor. 41 365302 [3] Benedetto J J and Fickus M 2003 Finite normalized tight frames Advances in Computational Mathematics 18 357-385 [4] Berezin F A 1974 Quantization Math. USSR Izv. 8 1109 [5] Berezin F A 1975 Quantization of complex symmetric spaces Math. USSR Izv. 9 341 [6] Berezin F A 1975 General concept of quantization Commun. Math. Phys. 40 153-74 [7] Berezin F A 1978 A connection between co- and contravariant operator symbols on classical complex symmetric spaces Sov. Math. Dokl. 19 786 [8] Berndt B.C., Evans R.J., Williams K.S. 1998 Gauss and Jacobi sums (Wiley, NY) [9] Bouwmeester D, Ekert A K and Zeilinger A (ed) 2000 The Physics of Quantum Information (Berlin: Springer) [10] Chaturvedi S 2002 Aspects of mutually unbiased bases in odd-prime-power dimensions Phys. Rev. A 65 044301 [11] Christensen O 2003 An introduction to frames and Riesz bases (Boston: Birkh¨auser) [12] Cotfas N and Gazeau J P 2010 Finite tight frames and some applications J. Phys. A: Math. Theor. 43 193001 [13] Garc´ıa de L´eon P L and Gazeau J P 2007 Coherent state quantization and phase operator Phys. Lett. A361 301 [14] Engliˇs M 1999 Compact Toeplitz operators via the Berezin transform on bounded symmetric domains Integral Equations Operator Theory 33 426 [15] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley-VCH) [16] Geller D, Kra I, Popescu S, and Simanca S On circulant matrices preprint [ http://www.math.sunysb.edu/ sorin/eprints/circulant.pdf ]. [17] Gilmore R 1972 Geometry of symmetrized states Ann. Phys. (NY) 74 391 [18] Han D and Larson D R 2000 Frames, bases and group representations Mem. Amer. Math. Soc. 697 [19] Kibler M 2009 An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group, J. Phys. A: Math. Gen. 42 353001 [20] Klauder J R 1963 Continuous-representation theory: I. Postulates of continuous-representation theory J. Math. Phys. 4 1055-8 [21] Klauder J R 1963 Continuous-representation theory: II. Generalized relation between quantum and classical dynamics J. Math. Phys. 4 1058-73 [22] Klauder J R 1995 Quantization without quantization Ann. Phys. 237 147-60 [23] Klimov A.B., Sanchez-Soto L.L., de Guise H. 2005 Multicomplementary operators via finite Fourier transform J. Phys. A: Math. Gen. 38 2747 [24] Leonhardt U. 1996, Discrete Wigner function and quantum state tomography Phys. Rev. A 53 2998 [25] Lieb E H 1973 Commun. Math. Phys. 31 327 [26] Meyer C D 2000 Matrix Analysis and Applied Linear Algebra SIAM. [27] Nielsen M A and Chuang I L 2001 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) [28] Perelomov A M 1972 Coherent states for arbitrary Lie group Comm. Math. Phys. 26 222-236 [29] Perelomov A M 1986 Generalized Coherent States and their Applications (Berlin: Springer) [30] Santhanam T S and Tekumalla A R 1976 Quantum mechanics in finite dimensions Found. Phys. 6 583-589

Finite quantum systems

18

[31] Santhanam T S 1976 Canonical commutation relation for operators with bounded spectrum Phys. Lett. A 56 345. [32] Saniga M and Planat M 2006 Hjelmslev geometry for mutually unbiased bases J. Phys. A: Math. Gen. 39 435 [33] Schwinger J 1960 Unitary operator bases Proc. Natl. Acad. Sci. (USA) 46 570 ˇ [34] Stoviˇ cek P and Tolar J 1984 Quantum mechanics in discrete space-time Rep. Math. Phys. 20 157-70 [35] Tolar J, Hadzitaskos G. 2009 Feynman’s path integral and mutually unbiased bases J. Phys. A: Math. Gen. 42 245306 [36] Vourdas A and Bendjaballah C 1993 Duality measurements and factorization in finite quantum systems Phys. Rev. A 47 3523 [37] Vourdas A. 1996 The angle-angular momentum quantum phase space J. Phys. A: Math. Gen. 29 42754288 [38] Vourdas A 2004 Quantum systems with finite Hilbert space Rep. Prog. Phys. 67 267-320 [39] Vourdas A 2007 Quantum systems with finite Hilbert space: Galois fields in quantum mechanics J. Phys. A: Math. Gen. 40 R285 [40] Weil A 1954 Sur certains groupes d’op´erateurs unitaires Acta 111 143-211 [41] Weyl H 1931 Theory of Groups and Quantum Mechanics (New Tork: Dover) [42] Wootters W 1987 A Wigner function formulation of finite state quantum mechanics Ann. Phys. (NY) 176 1 [43] Zhang W M, Feng D H and Gilmore R 1990 Coherent states: Theory and some applications Rev. Mod Phys. 62 867 [44] Zhang S and Vourdas A 2004 Analytic representation of finite quantum systems J. Phys. A: Math. Gen. 37 8349-63 [45] Zak J 1967 Finite translations in solid state physics Phys. Rev. Lett. 19 1385