Probabilistic aspects of finite frame quantization

arXiv:0803.0077v4 [math-ph] 22 Nov 2009 Finite tight frames and some applications Nicolae Cotfas1 and Jean Pierre Gazeau2 1 Faculty of Physics, University of Bucharest, PO Box 76 - 54, Post Office 76, Bucharest, Romania 2 Laboratoire APC, Université Paris 7-Denis Diderot, 10, rue A. Domon et L. Duquet, 75205 Paris Cedex13, France E-mail: [email protected], [email protected] Abstract. A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications. Finite tight frames and some applications 2 1. Introduction Although, at first glance, a system described by a finite-dimensional Hilbert space looks much simpler than one described by an infinite dimensional space, there is much more knowledge about the latter than the former. The continuous systems of coherent states have many applications [1, 31, 41] but the corresponding discrete version, usually called a frame, seems to be less used in quantum physics. Hilbert space frames, introduced by Duffin and Schaeffer in their work on nonharmonic Fourier series [16], were later rediscovered by Daubechies, Grossmann and Meyer in the fundamental paper [13]. Finite frames [1, 4, 5, 10, 19, 24] are useful in finite-dimensional quantum mechanics [46], particularly in quantum information [33, 34, 3], and play a significant role in signal processing (they give stable signal representations and allow modeling for noisy environments) [14]. Our aim is to present some results concerning the finite frames and their applications in physics, particularly in the context of quantization of finite sets. Particularly also, we try to prove that some mathematical methods used in modeling crystalline or quasicrystalline structures are in fact based on certain finite frames. Each finite frame in a Hilbert space H defines an embedding of H into a higher dimensional Hilbert space (called a superspace), and conversely, each embedding of H into a superspace allows us to define some finite frames. The embedding into a superspace offers the possibility to define some new mathematical objects, useful in certain applications. The construction of coherent states proposed by Perelomov in the case of Lie groups [40] admits a version for finite groups, and leads to some useful finite frames. Certain representations in terms of finite frames can be regarded as Riemann sums corresponding to the integrals occurring in some representations in terms of continuous frames. The description of a physical system in terms of a finite frame allows us to associate a linear operator to a classical observable. The procedure, not necessarily a path to a quantum approach, can be regarded as an extended version of the KlauderBerezin-Toeplitz quantization [6, 29, 30, 32] and represents a change of point of view in considering the physical system [18, 19, 20, 21, 22, 23, 35]. The paper is organized as follows. In section 2 we review some basic elements concerning the notion of tight frame in form suitable for the applications in crystal physics and finite frame quantization we present throughout the paper. We explain how Parseval frames are easily constructed by projection from higher-dimensional spaces, and show how a superspace emerges naturally from the existence of a frame in a given Hilbert space. By following the analogy with the systems of coherent states we introduce the notion of normalized Parseval frame, define its proximity to an orthonormal basis in terms of a natural parameter η and describe some stochastic aspects. A Perelomovlike construction of frames through group representations is described at the end of the section. By taking into consideration the embedding into superspace, we investigate in section 3 the set of the elements which can be represented as a linear combination with integer coefficients of the frame vectors, and present some applications. We show Finite tight frames and some applications 3 in which way some simple crystalline structures in the plane or in space are naturally described with the aid of frames. Section 4 is devoted to what we call frame quantization of discrete variable functions. Frame quantization replaces such functions by matrices, introducing in this way noncommutative algebras of matrices. We present an interesting result issued from the stochastic aspects mentioned in section 2. We also introduce another parameter, ζ, expressing the distance of the “quantum” non-commutative world issued from the frame quantization to the classical commutative one. We then illustrate our results concerning the proximity of the “quantum non-commutativity” to the original “classical” commutativity when the number of elements of a frame is larger by one than the dimension of the vector space. 2. Finite tight frames 2.1. Finite frames Let K be the field R or C, and let H be a N-dimensional Hilbert space over K with M {|ji}N j=1 a fixed orthonormal basis. A system of vectors {|wi i}i=1 is a finite frame for H if there are constants 0 < A ≤ B < ∞ such that 2 A||v|| ≤ The frame operator S|vi = M X i=1 M X i=1 satisfies the relation |hwi|vi|2 ≤ B||v||2 for all |vi ∈ H. |wi ihwi|vi 2 hv|Avi = A||v|| ≤ that is, M X i=1 (1) (2) |hwi|vi|2 = hv|Svi ≤ B||v||2 = hv|Bvi A IH ≤ S ≤ B IH where IH is the identity operator. If A = B, the frame is called an A-tight frame and S = A IH . A frame {|wi i}M i=1 is called an equal norm frame if ||w1 || = ||w2 || = · · · = ||wM ||. A 1-tight frame is usually called a Parseval frame and in this case M X i=1 If {|wii}M i=1 |wiihwi | = IH . is an A-tight frame then { √1A |wii}M i=1 is a Parseval frame. (3) Finite tight frames and some applications 4 2.2. Finite normalized Parseval frames Finite frames play a fundamental role in a wide variety of areas, and generally, each application requires a specific class of frames. In the case of finite frame quantization, we regard a Parseval frame as a finite family of coherent states. In order to improve the correspondence between the two notions we consider Parseval frames which do not contain the null vector and express their vectors in terms of some unit vectors. Let {|wii}M i=1 be a Parseval frame. Denoting 1 κi = hwi |wi i and |uii = √ |wii κi the resolution of identity (3) becomes M X i=1 We have κi |uiihui| = IH . hv|wi = M X i=1 (4) κi hv|uii hui|wi, ||v||2 = M X i=1 κi |hui|vi|2 for any |vi, |wi ∈ H, and the well-known [26, 25, 47] relation N= N X j=1 hj|ji = M N X X j=1 i=1 2 κi |hui|ji| = M X i=1 κi N X j=1 2 |hui|ji| = (5) M X κi . (6) i=1 In this paper, by normalized Parseval frame in H we mean any system of vectors {|uii}M i=1 satisfying the following two conditions: 1) the vectors |uii are unit vectors, that is, hui|ui i = 1, for any i ∈ {1, 2, . . . M} 2) there are {κi }M i=1 positive constants such that M X i=1 κi |uiihui| = IH . (7) √ M M If {|uii}M i=1 is a normalized Parseval frame with the constants {κi }i=1 then { κi |ui i}i=1 1 M is a Parseval frame, and conversely, if {|wii}M i=1 is a Parseval frame then { ||wi || |wi i}i=1 is a normalized Parseval frame with the constants {||wi||2 }M i=1 . In the case κ1 = κ2 = ... = κM , the relations (7) and (5) become [25, 26, 47] respectively M N X |uiihui| = IH M i=1 hv|wi = M N X hv|uii hui|wi, M i=1 (8) ||v||2 = M N X |hui|vi|2 . M i=1 (9) and the frame is called a finite equal norm Parseval frame [7, 8] or a finite normalized tight frame [5]. Finite tight frames and some applications 5 2.3. Normalized Parseval frames versus orthonormal basis and stochastic aspects Let us view the N components of the vector |uii with respect to the orthonormal basis {|ji}N j=1 as the respective conjugates of N functions i 7→ φj (i): |uii = N X φ̄j (i)|ji , (10) j=1 (“bar” means complex conjugate). By using this expansion in the resolution of the unity (7) we find the following orthogonality relations (φj , φk )κ = δjk , (11) with respect to the scalar product defined on the M-dimensional vector space of real or complex valued functions i 7→ φ(i) on the set X = {1, 2, . . . , M} by: ′ def (φ, φ )κ = M X κi φ̄(i) φ′ (i) . (12) i=1 By introducing the N × M matrix L with matrix elements √ √ Lji = κi φ̄j (i) = κi hj|uii , (13) we easily derive from (11) the equation L L† = IH . (14) Let us now express the pair overlaps hui|ui′ i in terms of the functions φj : hui|ui′ i = N X φj (i)φ̄j (i′ ) = K−1/2 L† LK−1/2 j=1 def
ii′ , (15) where K = diag(κ1 , κ2 , . . . , κM ). If M = N, then (14) implies L† = L−1 and so hui|ui′ i = δii′ /κi . The latter orthogonality relations together with (15) implies that κi = 1 for all i since the vectors |ui i’s are all unit. As expected, any family of N vectors satisfying (7) is an orthonormal basis. Let us introduce the real M × M matrix U with matrix elements Uij = |hui|uj i|2 . (16) These elements obey Uii = 1 for 1 ≤ i ≤ M and 0 ≤ Uij = Uji ≤ 1 for any pair (i, j), with i 6= j. Now we suppose that there is no pair of orthogonal elements, i.e. 0 < Uij if i 6= j, and no pair of proportional elements, i.e. Uij < 1 if i 6= j, in the frame. Then from the Perron-Frobenius theorem for (strictly) positive matrices, the rayon spectral r = r(U) is > 0 and is dominant simple eigenvalue of U. There exists a unique vector, vr , kvr k = 1, which is strictly positive (all components are > 0) and Uvr = 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 eigenvalues, one should have r > 1. The value r = 1 represents precisely the limit case Finite tight frames and some applications 6 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 (17) as a kind of “distance” of the frame to the orthonormality. The question is to find the relation between the set {κ1 , κ2 , ..., κM } of weights defining the frame and the distance η. By projecting on each vector |ui i from both sides the frame resolution of the unity (7), we easily obtain the M equations 1 = hui|ui i = def M X j=1 κj |hui|uj i|2 , i.e. Uvκ = vδ , (18) def where t vκ = (κ1 κ2 ... κM ) and t vδ = (1 1 ... 1) is the first diagonal vector in CM . In the “uniform” case for which κi = N/M for all i, i.e. in the case of a finite equal √ norm Parseval frame, which means that vκ = (N/M) vδ , then r = M/N and vr = 1/ M vδ . In this case, the distance to orthonormality is just M −N , (19) η= N a relation which clearly exemplifies what we can expect at the limit N → M. def Another aspect of a frame is the (right) stochastic nature of the matrix P = U K, def evident from (18). The row vector ̟ = t vκ /N = (κ1 /N κ2 /N ... κM /N ) is a stationary probability vector: ̟P = ̟. (20) As is well known, this vector obeys the ergodic property:
κj lim P k ij = ̟j = . k→∞ N (21) 2.4. Parseval frames obtained by projection Let E be a finite-dimensional Hilbert space over K, and let {|ε1 i, |ε2i, ..., |εM i} be an orthonormal basis in E. A large class of tight frames can be obtained by projection [10]. M Theorem 1 If {|φj i}N j=1 is an orthonormal system in E then {|wi i}i=1 , where |wii = N X j=1 |φj ihφj |εi i (22) is a Parseval frame in the subspace H = span {|φ1i, |φ2 i, . . . , |φN i}, that is, ) ( N N X X αj |φj i α1 , α2 , . . . , αN ∈ K . H= K |φj i = j=1 j=1 Proof. We get PM i=1 |wi ihwi | = = PM PN i=1 PN j,k=1 j=1 |φj ihφj |εi i P  P M i=1 hφj |εi ihεi |φk i N k=1 hεi |φk ihφk |   |φj ihφk | = IH .  Finite tight frames and some applications 7 P The operator π = N j=1 |φj ihφj | is the orthogonal projector corresponding to H and |wi i = π|εii. If two orthonormal systems {|φ1 i, |φ2i, ..., |φN i} and {|ψ1 i, |ψ2 i, ..., |ψN i} span the same subspace H then they define the same frame in H. This means that the frame depends on the subspace H we choose, and not on the particular orthonormal system we use. 2.5. Embedding into a superspace defined by a Parseval frame M Let H be a Hilbert space over K, {|ji}N j=1 an orthonormal basis in H, and let {|ei i}i=1 be the canonical basis of KM . The following result, proved independently by Naimark and Han/Larson [10, 27] shows that any finite Parceval frame can be obtained by projection. N Theorem 2 a) If {|wii}M i=1 is a Parseval frame in H then the system {|φj i}j=1 , where |φj i = M X i=1 |ei ihwi|ji = (hw1 |ji, hw2|ji, ..., hwM |ji) (23) is an orthonormal system in KM . b) The Hilbert space H can be identified with the subspace H̃ = span {|φ1 i, |φ2 i, . . . , |φN i} of the superspace KM by using the isometry H −→ H̃ : |vi 7→ |ṽi, where |ṽi = N X j=1 |φj ihj|vi = {|w̃i i}M i=1 M X i=1 |ei ihwi|vi = (hw1|vi, hw2|vi, ..., hwM |vi) (24) c) The frame corresponding to {|wii}M i=1 is the orthogonal projection of the orthonormal basis {|ei i}M i=1 |w̃ii = π|ei i f or any i ∈ {1, 2, ..., M}. P = hj|ki = δjk . Proof. a) From (5) we deduce that hφj |φk i = M i=1 hj|wi ihwi |kiP PN PM PN b) We get |ṽi = j=1 |φj ihj|vi = j=1 i=1 |ei ihwi |jihj|vi = M i=1 |ei ihwi |vi. PN  c) We have π|ei i = j=1 |φj ihj|wii = |w̃i i. (25) The subspace H̃ and the isometry H −→ H̃ have been defined by using an orthonormal basis {|ji}N j=1 but they do not depend on the basis we choose. The representation |ṽi of |vi can be regarded as a discrete counterpart to the usual Fock-Bargmann representation [1]. 2.6. Finite tight frames defined by using groups Some useful frames can be defined in a natural way by using group representations [27]. Let {g : H −→ H | g ∈ G } be an orthogonal (resp. unitary) irreducible representation Finite tight frames and some applications 8 of a finite group G in the real (resp. complex) n-dimensional Hilbert space H, and let |wi ∈ H be a fixed vector. The elements g ∈ G with the property g|wi = α|wi (26) where α is a scalar depending on g, form the stationary group Gw of |wi. Theorem 3 If {gi }M i=1 is a system of representatives of the left cosets of G on Gw then |w1 i = g1 |wi, |w2 i = g2 |wi, ... |wM i = gM |wi form an equal norm tight frame in H, namely M X M ||w||2 IH . |wiihwi | = N i=1 P Proof. The operator Λ : H −→ H, Λ|vi = M i=1 |wi ihwi |vi is self-adjoint ′ hv |(Λ|vi) = M X i=1 (27) (28) hv ′ |wiihwi |vi = (hv ′ |Λ)|vi and therefore, it has a real eigenvalue λ. Since the eigenspace { |vi ; Λ|vi = λ|vi } corresponding to λ is G-invariant M M X X Λ(g|vi) = |wi ihwi|(g|vi) = g|wiihwi|vi = g(Λ|vi) i=1 i=1 and the representation is irreducible we must have Λ|vi = λ|vi for any |vi ∈ H. By using an orthogonal basis {|1i, |2i, ..., |Ni} of H we get Nλ = N X j=1 N M X N X M X X |hj|wii|2 = M ||w||2. hj|wi ihwi|ji = hj|Λ|ji = j=1 i=1 i=1 j=1 One can easily remark that the whole orbit G|wi = { g|wi | g ∈ G } is a tight frame, and more than that, any finite union of orbits is also a tight frame. The relation   2π 2π 2π 2π (29) − α2 sin , α1 sin + α2 cos g(α1, α2 ) = α1 cos n n n n defines a representation of the cyclic group Cn = h g | g n = e i as a group of rotations of the plane, and for example, the orbit   o q  n q   2 2 √1 , √1 √1 , − √1 (30) , − C3 , 0 == , 0 , − 3 3 6 2 6 2 is a Parseval frame in R2 . The relations g(α1, α2 , α3 ) = (−α1 , −α2 , α3 ), h(α1 , α2 , α3 ) = (α2 , α3 , α1 ) (31) define a representation of the tetrahedral group T = h g, h | g 2 = h3 = (gh)3 = e i as a group of rotations of the space, and for example,




(32) T − 12 , 12 , 21 = − 12 , 12 , 21 , 12 , − 21 , 21 , 12 , 21 , − 12 , − 21 , − 12 , − 12 is a Parseval frame in R3 .  Finite tight frames and some applications 9 3. Integer coefficients Let H = RN and let {|wi i}M i=1 , where |w1 i = (w11 , w12 , . . . , w1N ) |w2 i = (w21 , w22 , . . . , w2N ) ..................................... |wM i = (wM 1 , wM 2, . . . , wM N ) be a Parseval frame in RN , that is, M X i=1 |wi ihwi|vi = |vi for any |vi ∈ RN . In view of theorem 2 the vectors |φ1 i = (w11 , w21 , . . . , wM 1 ) |φ2 i = (w12 , w22 , . . . , wM 2 ) ..................................... |φN i = (w1N , w2N , . . . , wM N ) form an orthonormal system in E = RM , and the injective mapping (analysis operator) T : RN −→ RM : |vi 7→ |ṽi = (hw1 |vi, hw2|vi, ..., hwM |vi) which can be written as RN −→ RM , T (α1 , α2 , . . . , αN ) = α1 |φ1 i + α2 |φ2 i + · · · αN |φN i allows us to identify RN with the subspace H̃ = { α1 |φ1 i + α2 |φ2 i + ... + αN |φN i | α1 , α2 , ..., αN ∈ R } of the superspace RM . The one-to-one mapping RN −→ H̃ : |vi 7→ |ṽi is an isometry hṽ|ṽ ′ i = hv|v ′i, ||ṽ|| = ||v|| M and {|w̃ii}M i=1 is a Parseval frame in H̃ corresponding to {|wi i}i=1 M X i=1 |w̃i ihw̃i|ṽi = M X i=1 |w̃i ihwi|vi = M X i=1 T |wiihwi |vi = T |vi = |ṽi. The frame {|w̃i i}M i=1 is the orthogonal projection on H̃ of the canonical basis |e1 i = (1, 0, 0 . . . , 0), |e2 i = (0, 1, 0, . . . , 0), P namely, by denoting π = N j=1 |φj ihφj |, we have |w̃1 i = π|e1 i, |w̃2 i = π|e2 i, ... ... |eM i = (0, 0, . . . , 0, 1) |w̃M i = π|eM i. The matrix of π in terms of the canonical basis {|ei i}M i=1 is   hw1 |w1 i hw1 |w2 i ... hw1 |wM i    hw2 |w1 i hw2 |w2 i ... hw2 |wM i  π=   ... ... ... ...  hwM |w1 i hwM |w2 i ... hwM |wM i (33) Finite tight frames and some applications 10 The linear operator π ⊥ : RM −→ RM , π ⊥ x = x − πx is the orthogonal projector corresponding to the orthogonal complement ) ( M X xi |wi i = 0 . H̃⊥ = x = (x1 , x2 , ..., xM ) (34) i=1 M of H̃ in R , and the vectors |w̃1⊥ i = π ⊥ |e1 i, |w̃2⊥ i = π ⊥ |e2 i, ... ⊥ |w̃M i = π|e⊥ M i. ⊥ form a frame {|w̃i⊥ i}M i=1 in H̃ such that |w̃ii + |w̃i⊥ i = |ei i for any i ∈ {1, 2, ..., M} called the complementary frame [27]. Particularly, one can remark that the complementary frame corresponding to an equal norm frame is an equal norm frame. Each vector |vi ∈ RN can be written as a linear combination of the frame vectors |wi i |vi = N X |wiihwi |vi N X xi |wii i=1 in terms of the frame coefficients hwi |vi. If M > N then the representation of a vector |vi ∈ H as a linear combination of the frame vectors is not unique, and we have |vi = that is, the relation N X i=1 i=1 xi |wi i = N X i=1 |wiihwi |vi which can be written as N X (xi − hwi |vi) |wii = 0 i=1 if and only if (x1 − hw1 |vi, x2 − hw2 |vi, . . . , xM − hwM |vi) ∈ H̃⊥ that is, if and only if (x1 , x2 , . . . , xM ) ∈ (hw1|vi, hw2 |vi, . . . , hwM |vi) + H̃⊥ . From the last relation it follows N X |vi = xi |wii ⇐⇒ π(x1 , x2 , ..., xM ) = (hw1 |vi, hw2 |vi, . . . , hwM |vi) i=1 Finite tight frames and some applications 11 and the inequality obtained by Duffin and Schaeffer [16] M M M X X X 2 |vi = xi |wii =⇒ (xi ) ≥ (hwi |vi)2. i=1 N i=1 (35) i=1 Each vector |vi ∈ R admits a natural representation in terms of frame coefficients hwi |vi, but other representations may offer additional facilities. In certain applications it is advantageous [9] to replace the frame coefficients by quantized coefficients, i.e. by integer multiples of a given δ > 0. In this section we shall present some applications concerning the elements of a Hilbert space which can be written as a linear combination with integer coefficients of the vectors of a fixed frame. 3.1. Orthogonal projection of ZM on a subspace of RM Let E be a vector subspace of RM and let Br (a) = { x ∈ E | kx − ak < r } be the open ball of center a and radius r. A set D ⊂ E is dense in E if the ball Br (a) contains at least a point of D for any a ∈ E and any r ∈ (0, ∞). The set D is relatively dense in E if there is r ∈ (0, ∞) such that the ball Br (a) contains at least a point of D for any a ∈ E. The set D is discrete in E if for each a ∈ D there is r ∈ (0, ∞) such that D ∩ Br (a) = {a}. The set D is uniformly discrete in E if there is r ∈ (0, ∞) such that the ball Br (a) contains at most one point of D for any a ∈ E. The set D is a Delone set in E if it is both relatively dense and uniformly discrete in E. The set D is a lattice in E if it is both an additive subgroup of E and a Delone set in E. In order to describe the orthogonal projection of ZM on E we will use the following result. Theorem 4 [15, 43] Let Φ : RM −→ RL be a surjective linear mapping, where L < M. Then there are subspaces V , V ′ of RL such that a) RL = V ⊕ V ′ b) Φ(ZM ) = Φ(ZM ) ∩ V + Φ(ZM ) ∩ V ′ c) Φ(ZM ) ∩ V ′ is a lattice in V ′ d) Φ(ZM ) ∩ V is a dense subgroup of V . The subspace V in this decomposition is uniquely determined. The theorem 4 allows us to describe subsets ( the M M X X π(ZM ) = Z |w̃i i = ni |w̃i i i=1 of H̃ and ⊥ M π (Z ) = M X i=1 i=1 Z |w̃i⊥ i = ( M X i=1 ni |w̃i⊥ i n1 , n2 , . . . , nM ∈ Z ) n1 , n2 , . . . , nM ∈ Z ) of H̃⊥ . There are subspaces V , V ′ of H̃ and subspaces W , W ′ of H̃⊥ such that H̃ = V ⊕ V ′ H̃⊥ = W ⊕ W ′ π(ZM ) = π(ZM ) ∩ V + π(ZM ) ∩ V ′ π ⊥ (ZM ) = π ⊥ (ZM ) ∩ W +π ⊥ (ZM ) ∩ W ′ (36) Finite tight frames and some applications 12 π(ZM ) ∩ V ′ is a lattice in V ′ , π ⊥ (ZM ) ∩ W ′ is a lattice in W ′ , π(ZM ) ∩ V is a dense subgroup of V and π ⊥ (ZM ) ∩ W is a dense subgroup of W . We say that the starting frame {|wi i}M i=1 is a periodic frame if V = {0}, that is, if ( ) M M X X Z |wi i = ni |wi i n1 , n2 , . . . , nM ∈ Z i=1 i=1 ′ is a lattice in H. The frame {|wi i}M i=1 will be called a quasiperiodic frame if W = {0} and π restricted to ZM is one-to-one. In this case, the collection of spaces and mappings π π⊥ H̃ ←− RM −→ H̃⊥ ∪ ZM (37) is a so-called cut and project scheme [39] and we can define the ∗-mapping π(ZM ) −→ H̃⊥ : x 7→ x∗ = π ⊥ ((π|ZM )−1 x). (38) The projection π restricted to ZM is one-to-one if and only if ZM ∩ H̃⊥ = {0}. The translations of H̃ corresponding to the elements of ZM ∩ H̃ leave the set π ⊥ (ZM ) invariant. If ZM ∩ H̃ contains a basis of H̃ then the starting frame is a periodic frame. 3.2. Honeycomb lattice and diamond structure described in terms of frames The symmetry properties of certain discrete sets can be simpler described by using a frame instead of a basis. Honeycomb lattice (figure 1) is a discrete subset L of the plane such that each point P ∈ L has three nearest neighbours forming an equilateral triangle centered at P . It can be described in a natural way by using the periodic Parseval frame (see (30))     q  2 1 1 1 √1 √ √ √ , |w i = − , , − , 0 , |w i = − |w1 i = 3 2 3 6 2 6 2 as the set [12] L = { n1 |w1 i + n2 |w2 i + n3 |w3 i | (n1 , n2 , n3 ) ∈ L } where the subset L = { n = (n1 , n2 , n3 ) ∈ Z3 | n1 + n2 + n3 ∈ {0, 1} } of Z3 can be regarded as a mathematical model. The nearest neighbours of n ∈ L are n1 = (n1 + ν(n), n2 , n3 ) n2 = (n1 , n2 + ν(n), n3 ) n3 = (n1 , n2 , n3 + ν(n)) where ν(n) = (−1)n1 +n2 +n3 . The six points nij = (ni )j corresponding to i 6= j are the next-to-nearest neighbours, and one can remark that nii = n, nijl = nlji , for any i, j, l ∈ {1, 2, 3}. The mapping d : L × L −→ Z d(n, n′ ) = |n1 − n′1 | + |n2 − n′2 | + |n3 − n′3 | Finite tight frames and some applications 13 is a distance on L, and a point n′ is a neighbour of order l of n if d(n, n′ ) = l. The symmetry group G of the honeycomb lattice is isomorphic with the group of all the isometries of the metric space (L, d), group generated by the transformations L −→ L : (n1 , n2 , n3 ) 7→ (n2 , n3 , n1 ) L −→ L : (n1 , n2 , n3 ) 7→ (n1 , n3 , n2 ) L −→ L : (n1 , n2 , n3 ) 7→ (−n1 +1, −n2 , −n3 ). Honeycomb lattice is a mathematical model for a graphene sheet and the use of the indicated frame leads to a simpler and more symmetric form for the G-invariant mathematical objects occuring in the description of certain physical properties [12]. ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ✡ ❏❏ ✡ ❏❏ ✡✡ ✡ ❏❏ ✡ ❏❏ ✡✡ ✡✡|w2 i ❏❏ ❏ ❏ ❪❏ ❏r ✲✡✡ |w1 i ✡ ✡ ✢|w3 i ❏❏ ❏❏ ✡✡ ✡ ❏❏ ✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ ✡✡ ❏❏ Figure 1. A fragment of the honeycomb lattice Diamond structure can be regarded as the three-dimensional analogue of the honeycomb lattice. Each point P belonging to the diamond structure D has four nearest neighbours forming a regular tetrahedron centered at P . Diamond structure can be described in a natural way by using the periodic Parseval frame (see (32))

|w1 i = − 12 , 21 , 12 , |w2 i = 21 , − 12 , 21 ,

|w4 i = − 12 , − 21 , − 12 |w3 i = 12 , 21 , − 12 , of R3 as the set [11] D = { n1 |w1i + n2 |w2 i + n3 |w3 i + n4 |w4 i | (n1 , n2 , n3 , n4 ) ∈ D } where D = { n = (n1 , n2 , n3 , n4 ) ∈ Z4 | n1 + n2 + n3 + n4 ∈ {0, 1} }. The nearest neighbours of a point n ∈ D are n1 n2 n3 n4 = (n1 + ν(n), n2 , n3 , n4 ) = (n1 , n2 + ν(n), n3 , n4 ) = (n1 , n2 , n3 + ν(n), n4 ) = (n1 , n2 , n3 , n4 + ν(n)) where ν(n) = (−1)n1 +n2 +n3 +n4 . Finite tight frames and some applications 14 The twelve points nij = (ni )j corresponding to i 6= j are the next-to-nearest neighbours, and one can remark that nii = n, nijl = nlji , for any i, j, l ∈ {1, 2, 3, 4}. The mapping d : D×D −→ Z d(n, n′ ) = |n1 −n′1 |+|n2 −n′2 |+|n3 −n′3 |+|n4 −n′4 | is a distance on D, and a point n′ is a neighbour of order l of n if d(n, n′ ) = l. The symmetry group Oh7 of the diamond structure is isomorphic with the group of all the isometries of the metric space (D, d), group generated by the transformations D −→ D : (n1 , n2 , n3 , n4 ) 7→ (n3 , n4 , n2 , n1 ) D −→ D : (n1 , n2 , n3 , n4 ) 7→ (n4 , n2 , n3 , n1 ) D −→ D : (n1 , n2 , n3 , n4 ) 7→ (−n1 +1, −n2 , −n3 , −n4 ). Again the use of a frame leads to a simpler and more symmetric form for the Oh7 -invariant mathematical objects occuring in the description of certain physical properties [11]. 3.3. An application to quasicrystals The group I of all the rotations of R3 leaving a regular icosahedron centered at the origin invariant is called the icosahedral group. The tvelwe points ±(1, τ, 0), ±(−1, τ, 0), ±(−τ, 0, 1), ±(0, −1, τ ), ±(τ, 0, 1), ±(0, 1, τ ) √ where τ = (1 + 5)/2, are the vertices of a regular icosahedron centered at origin. The rotations
α− τ2 β + 21 γ, τ2 α+ 12 β + τ −1 γ, −12 α+ τ −1 β + τ2 γ r(α, β, γ) = τ−1 2 2 2 (39) s(α, β, γ) = (−α, −β, γ). satisfying the relation r 5 = s2 = (rs)3 = IR3 leave this regular icosahedron invariant, and therefore they define a representation of the icosahedral group in R3 . The stationary group Iw of |wi = √ 1 (1, τ, 0) is formed by the rotations g ∈ I 2(τ +2) with g|wi ∈ {|wi, −|wi}, and we can choose the representatives g1 , g2 , ..., g6 of the cosets of I on Iw such that 1 2(τ +2) √ 1 2(τ +2) √ 1 2(τ +2) 1 (−1, τ, 0), 2(τ +2) √ 1 (0, −1, τ ), 2(τ +2) √ 1 (0, 1, τ ). 2(τ +2) |w1 i = g1 |wi = √ (1, τ, 0), |w2 i = g2 |wi = √ |w3 i = g3 |wi = (−τ, 0, 1), |w4 i = g4 |wi = (τ, 0, 1), |w6 i = g6 |wi = |w5 i = g5 |wi = In view of theorem 3 the system {|wi i}6i=1 is a tight frame in R3 . By direct computation one can prove that it is a quasiperiodic Parseval frame 6 X i=1 |wi ihwi| = IR3 . It defines an embedding of H = R3 in the superspace R6 and the set Q = { x ∈ π(Z6 ) | x∗ ∈ π ⊥ ([0, 1]6 ) } defined by using the corresponding ∗-mapping is a quasiperiodic set. (40) Finite tight frames and some applications 15 The diffraction pattern corresponding to Q computed by using the Fourier transform is similar to the experimental diffraction patterns obtained in the case of certain icosahedral quasicrystals [17, 28]. Quasiperiodic sets corresponding to other quasicrystals can be obtained by starting from finite frames, and they help us to better understand the atomic structure of these materials. 3.4. Sequences of finite frames Let (fn )∞ n=0 be the Fibonacci sequence defined by reccurrence as f0 = f1 = 1, fn+1 = fn−1 + fn and let τn = fn+1 /fn . It is well-known that limn→∞ τn = τ . The tetrahedral frame T (1, τ, 0) defined by using the representation (31) coincides with the icosahedral frame I(1, τ, 0) defined by using the representation (39) T (1, τ, 0) = {(1, τ, 0), (−1, τ, 0), (−τ, 0, 1), (0, −1, τ ), (τ, 0, 1), (0, 1, τ ), (−1, −τ, 0), (1, −τ, 0), Therefore (τ, 0, −1), (0, 1, −τ ), (−τ, 0, −1), (0, −1, −τ )} = I(1, τ, 0). limn→∞ T (1, τn , 0) = limn→∞ {(1, τn , 0), (−1, τn , 0), (−τn , 0, 1), (0, −1, τn ), (τn , 0, 1), (0, 1, τn), (−1, −τn , 0), (1, −τn , 0), (τn , 0, −1), (0, 1, −τn), (−τn , 0, −1), (0, −1, −τn )} = I(1, τ, 0) that is, we can approximate the frame I(1, τ, 0) by using the periodic frames T (1, τn , 0). The orbit T ((1 − t)(1, 2, 0) + t(1, τ, 0)) of the tetrahedral group T is a frame in R3 for any t ∈ [0, 1]. It can be regarded as a continuous deformation of the periodic frame T (1, 2, 0) into the icosahedral frame I(1, τ, 0). The relation Rθ (x, y) = (x cos θ − y sin θ, x sin θ + y cos θ) (41) defines an R-irreducible two-dimensional representation of the multiplicative group ( ! ) cos θ − sin θ SO(2) = θ ∈ [0, 2π) sin θ cos θ and the orbit { |θi = (cos θ, sin θ) | θ ∈ [0, 2π) } is a continuous frame Z 1 2π dθ |θihθ| = IR2 . π 0 For any n ∈ N the orbit of Cn corresponding to (1, 0), namely,      2π 2π 2π k = cos k, sin k k ∈ {0, 1, ..., n − 1} Cn (1, 0) = n n n is a finite frame  n−1 2 X 2π 2π k k = I R2 n k=0 n n Finite tight frames and some applications 16 and we have n−1 2 X 2π k n k=0 n   Z n−1 1 2π X 2π 2π 2π n→∞ 1 2π k = k k −→ dθ |θihθ|. n π n k=0 n n π 0 Therefore, we can regard the continuous frame {|θi}θ∈[0,2π] as the limit of the sequence of finite frames ( Cn (1, 0) )∞ n=3. 4. Frame quantization of discrete variable functions 4.1. Finite frame quantization Let X = {a1 , a2 , ..., aM } be a fixed finite set we regard as a set of data concerning a physical system. The space of all the functions ϕ : X −→ K is a Hilbert space with the scalar product M X ϕ(ai ) ψ(ai ) (42) hϕ|ψi = i=1 (evidently, if K = R then ϕ(ai ) = ϕ(ai )) and the isometry l2 (X ) −→ KM : ϕ 7→ (ϕ(a1 ), ϕ(a2 ), ..., ϕ(aM )) (43) allows us to identify the space l2 (X ) with the usual M-dimensional Hilbert space KM . The system of functions {δ1 , δ2 , ..., δM }, where ( 1 if a = ai δi : X −→ K, δi (a) = 0 if a 6= ai is an orthonormal basis in l2 (X ) ϕ= M X i=1 hδi |ϕiδi = M X ϕ(ai )δi . i=1 Let us select among the elements of l2 (X ) an orthonormal set {φ1 , φ2 , ..., φN } such that κi = N X j=1 |φj (ai )|2 6= 0, for all i ∈ {1, 2, ..., M} and let H = span{φ1 , φ2, ..., φN }. In view of theorem 1, the elements N N 1 X 1 X hφj |δi iφj = √ |uii = √ φj (ai ) φj , κi j=1 κi j=1 form a normalized Parseval frame in H, namely, M X κi |uiihui| = IH . i ∈ {1, 2, ..., M} (44) i=1 To each function f : X −→ R which we regard as a classical observable we associate the linear operator M X Af : H −→ H, Af = κi f (ai ) |uiihui|. (45) i=1 Finite tight frames and some applications 17 This can be regarded as a Klauder-Berezin-Toeplitz type quantization [6, 29, 30, 32] of f , the notion of quantization being considered here in a wide sense [18, 19, 20, 21, 22, 23, 35]. 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 (ai )). The function f is called upper (or contravariant) symbol of Af , and the function fˇ : X −→ R, fˇ(ak ) = huk |Af |uk i = is called lower (or covariant) symbol of Af . Since M X k=1 M X i=1 κi f (ai ) |hui|uk i|2 (46) κi |hui|uk i|2 = ||ui||2 = 1 the number fˇ(ak ) is a weighted mean of f (a1 ), f (a2 ), . . . f (aM ), for any k ∈ {1, 2, . . . M}. In terms of the superspace, fˇ(ak ) can be regarded as a scalar product ˇ k ) = h(f (a1 ), . . . , f (aM )), (κ1 |hu1 |uk i|2 , . . . , κM |huM |uk i|2)i. f(a To a certain extent, a quantization scheme consists in adopting a certain point of view in dealing with X . The presented frame quantization f 7→ Af depends on the subspace H ⊂ l2 (X ) we choose. The validity of the frame quantization corresponding to a certain subspace H is asserted by comparing spectral characteristics of Af with data provided by specific protocol in the observation of the considered physical system. An interesting subject of topological study is the triplet ˇ. [M values of f ] ↔ [N ′ eigenvalues of Af , N ′ ≤ N] ↔ [M values of f] 4.2. Probabilistic aspects of finite frame quantization The relations PN PM j=1 |hφj |ui i| 2 = 1 for i ∈ {1, 2, ..., M} (47) 2 i=1 κi |hφj |ui i| = 1 for j ∈ {1, 2, ..., N} show that the considered normalized Parseval frame defines two families of probability distributions. This property can be p interpreted in terms of a Bayesian duality [2]. If ψ ∈ H is such that ||ψ|| = hψ, ψi = 1 then M M X X √ 2 | κi hψ|uii| = κi |hψ|uii|2 = ||ψ||2 = 1 i=1 i=1 √ and hence, adopting the vocabulary of quantum measurement, | κi hψ|uii|2 can be viewed as the probability to find ψ in the state |uii. The trace of the operator Af depends on the lower symbol PN PM P tr Af = N i=1 κi hφk |ui ihui |Af |φk i k=1 k=1 hφk |Af |φk i = = = PM i=1 PM i=1 κi PN k=1 hui |Af |φk ihφk |ui i κi hui|Af |ui i = PM i=1 ˇ i ). κi f(a Finite tight frames and some applications 18 An interesting problem in our finite frame quantization is to compare the starting function f with the lower symbol fˇ. With the stochastic matrix notations of subsection 2.3, the relation ˇ k ) = huk |Af |uk i = f(a is rewritten as M X i=1 κi f (ai ) |hui|uk i|2 f̌ = P f , (48) def def with t f = (f (a1 ) f (a2 ) ... f (aM )) and t f̌ = interesting because it can be iterated: f̌ [k] = P k f , f̌ [k] = P f̌ [k−1] ,
ˇ 2 ) ... f(a ˇ M ) . This formula is fˇ(a1 ) f(a f̌ [1] ≡ f̌ , (49) and so we find from the property (21) of P that the ergodic limit (or “long-term average”) of the iteration stabilizes to the “classical” average of the observable f defined as: f̌ [∞] def = hf icl vδ , hf icl = M X κi i=1 N f (ai ) . (50) 4.3. The classical limit of finite frame quantization We can evaluate the “distance” between the lower symbol and its classical counterpart through the inequality: def ˇ k )| ≤ kI − P k∞ kfk∞ , kf̌ − fk∞ = max |f (ak )− f(a (51) 1≤k≤M where the induced norm [37] on matrix A is kAk∞ = max1≤i≤M present case, because of the stochastic nature of P , we have   kI − P k∞ = 2 1 − min κi . 1≤i≤M PM j=1 |aij |. In the (52) In the uniform case, κi = N/M for all i, we thus have an estimate of how far the two functions f and fˇ are: kf̌ − fk∞ ≤ 2(M − N)/M kfk∞ . In the general case, we can view the parameter def ζ = 1 − min κi 1≤i≤M (53) 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 subsection 2.3. Another way to check that ζ = 1 − N/M → 0 means, in the uniform case κi = N/M for all i, that we go back to the classical spectrum of the observable f results from the following relations. We have N X N N |huj |uk i|2 = ||uk ||2 − |huk |uk i|2 = 1 − M j6=k M M Finite tight frames and some applications 19 and from the relation we get M X X N ˇ k) = N f (aj ) |huj |uk i|2 = f (ak ) + f (aj ) |huj |uk i|2 f(a M j=1 M j6=k     N N N min f (aj ) ≤ fˇ(ak ) − f (ak ) ≤ 1 − max f (aj ). 1− j j M M M Finally, note the estimates: ˇ k )| = |f (ak )− f(a whence M X i=1 κi (f (ak )−f (ai )) |hui|uk i|2 ≤ max |f (ak )−f (ai )| i kf̌ − fk∞ ≤ max |f (ak ) − f (ai )|. i,k (54) 4.4. Frames defined by using eigenvectors of non-commuting operators Let A, B : H −→ H be two operators on a Hilbert space H, which are diagonalizable operators with orthogonal eigenvectors. If AB = BA then there is a basis of H formed by common eigenvectors of A and B, useful in the study of the operators which can be expressed as a function of A and B. Such a basis does not exist if AB 6= BA, but a weaker version of this approach is possible by using a frame. By starting from an orthonormal basis {ϕi }M i=1 formed by eigenvectors of A and an orthonormal basis M {ψj }j=1 formed by eigenvectors of B we can restrict us to a subspace of the form H = span{ ϕ1, ϕ2 , . . . , ϕN } and use the frame {πψj }M j=1 , where π is the orthogonal projector corresponding to H. In order to illustrate this method, let ZM = Z/MZ = {0, 1, . . . , M − 1} and A, B : l2 (ZM ) −→ l2 (ZM ) be the linear operators defined in terms of the canonical basis {δi }M i=1 as Aδj = δj−1 , 2πi Bδj = e M j δj (the elements of ZM are integers considered modulo M, and particularly, −1 = M − 1). The elements of the canonical basis {δi }M i=1 are eigenfunctions of B. The functions φ0 , φ1 , ... φM −1 : ZM −→ C defined as 2πi 1 (55) φj (k) = √ e− M jk M that is, M −1 1 X − 2πi jk e M δk φj = √ M k=0 are eigenfunctions of A M −1 2πi 1 X − 2πi jk √ e M δk−1 = e− M j φj Aφj = M k=0 Finite tight frames and some applications 20 and form an orthonormal basis in l2 (ZM ). Let N ≤ M and H = span{φ0 , φ1 , ... φN −1 }. The elements N −1 1 X 2πi jk j ∈ {0, 1, ..., M − 1} (56) e M φk |uj i = √ N k=0 form a frame in the subspace H such that If j 6= k then M −1 N X |uj ihuj | = IH . M j=0 πi 2πi N −1 (k−j) e M (k−j)(n−1) sin nπ 1 X 2πi (k−j)p 1 1−e M (k−j)N M M huj |uk i = = e . = 2πi π (k−j) N p=0 N 1−e M N sin M (k−j) According to the quantization scheme defined in subsection 4.1, the considered frame allows us to associate to each function f : ZM −→ R the operator Af : H −→ H, M −1 N X f (k) |uk ihuk | Af = M k=0 having the lower symbol fˇ(j) = huj |Af |uj i = N M PM −1 k=0 f (k) |huj |uk i|2 = N f (j) M + N M = N f (j) M + 1 NM P k6=j P f (k) |huj |uk i|2 k6=j f (k) sin2 nπ (k−j) M . π sin2 M (k−j) The entries of the matrix of Af in the orthonormal basis {|φ0 i, |φ1i, ..., |φN −1 i} are and M −1 1 X 2πi k(p−q) eM f (k) hφp |Af |φq i = M k=0 huj |Af |uj i = PN −1 p,q=0 huj |φp i hφp |Af |φq i hφq |uj i = Particularly, we have hφp |Af |φq i = and (57) 1 N PN −1 (58) 2πi (q−p)j M hφp |Af |φq i. p,q=0 e 1 − aM 1 (p−q) M 1 − ae 2πi M M −1 2πi 1  1+e M (p−q) hφp |Af |φq i = M in the case f (k) = ak M −1 in the case f (k) = k ! . It is known that the functions fj : Zm −→ C defined in terms of Hermite polynomials ! r ∞ X π 2π 2 e− M (lM +k) Hj (lM + k) fj (k) = M l=−∞ Finite tight frames and some applications 21 are eigenfunctions of the discrete Fourier transform [36] Therefore M −1 1 X 2πi pk √ e M fj (p) = ij fj (k). M p=0 ij hφp |Afj |φq i = √ fj (p − q). M If the real number x is not a multiple of M then M −1 X 2πi e M kx = k=0 1 − e2πix 2πi 1−eM x . By differentiating this relation we get  
2πi 2πi x 2πix M −1 M 1 − e −Me + e M x 1 − e2πix X 2πi ke M kx =  2 2πi k=0 1−eMx whence hφp |Af |φq i =    M −1 2 if p = q 1 if p 6= q 2πi e M (p−q) −1 in the case f (k) = k. 4.5. Finite quantum systems The study of quantum systems described by finite-dimensional spaces was initiated by Weyl [45] and Schwinger [42] and rely upon the discrete Fourier transform. Let n be a fixed positive integer. The set Zn × Zn × Zn considered together with the multiplication law (θ, α, β)(a′, α′ , β ′) = (θ + θ′ + βα′ , α + α′ , β + β ′ ) where all sums are modulo n, is a group. This group of order n3 is regarded as a discrete version of the Heisenberg group [44]. In any n-dimensional Hilbert space H we can define by choosing an orthonormal basis {|0i, |1i, ..., |n − 1i} the Weyl operators A, B : H −→ H A|ji = |j − 1i, B|ji = e satisfying the relation Aα B β = e 2πi αβ n B β Aα . The mapping (θ, α, β) 7→ e 2πi θ n Aα B β 2πi j n |ji Finite tight frames and some applications 22 defines a unitary irreducible representation of the discrete Heisenberg group in H and Pn−1 νk |ki we have for any vector |vi = k=0 e 2πi θ n Aα B β |vi = e 2πi (θ+αβ) n n−1 X e 2πi βk n k=0 νk+α |ki If we multiply the vectors |u1 i, |u2i, ..., |umi of a frame by arbitrary phase factors we get a new frame eiθ1 |u1 i, eiθ2 |u2 i, ..., eiθm |um i. Pn−1 By choosing a unit vector |ui = k=0 µk |ki with stationary group Gu = Zn × {0} × {0} and neglecting the phase factors we get the frame [46] ) ( n−1 X 2πi (59) |α, βi = e n βk µk+α|ki (α, β) ∈ Zn × Zn k=0 and the resolution of identity n−1 1 X |α, βihα, β| = IH . n α,β=0 In the case n = 3, by choosing |ui = √1 2 (60) |0i + √1 2 |1i we obtain the frame |0, 0i = √12 |0i + √1 2 |1i |0, 1i = √12 |0i + √1 ε |1i 2 |0, 2i = √12 |0i + √1 ε2 |1i 2 |1, 0i = √12 |0i + √1 2 |2i |1, 1i = √12 |0i + √1 ε2 2 |1, 2i = √12 |0i + √1 ε |2i 2 |2, 0i = √12 |1i + √1 2 |2i |2, 1i = √12 ε |1i+ √12 ε2 |2i where ε = e 2πi 3 |2i |2, 2i = √12 ε2 |1i+ √12 ε |2i . The set Zn × Zn can be regarded as a finite version of the phase space, and to each classical observable f : Zn × Zn −→ R we associate the linear operator Af : H −→ H, Af = n−1 1 X f (α, β) |α, βihα, β|. n α,β=0 (61) For example, in the case n = 2 by starting from |ui = 35 |0i + 54 |1i we get the frame |0, 0i = 53 |0i + 45 |1i, |0, 1i = 3 5 |0i − 54 |1i |1, 0i = 54 |0i + 35 |1i, |1, 1i = 4 5 |0i − 53 |1i and to each function f : Z2 × Z2 −→ R we associate the operator P Af = 21 1α,β=0 f (α, β) |α, βihα, β| = 1 50 ( f (0, 0) +f (0, 1) 9 12 12 16 ! + f (1, 0) 9 −12 −12 16 ! 16 12 12 9 + f (1, 1) ! 16 −12 −12 9 !) . Finite tight frames and some applications 23 We have h0, 0|0, 0i = h1, 0|1, 0i = h0, 1|0, 1i = h1, 1|1, 1i = 1 and 7 h0, 0|0, 1i = − 25 h0, 0|1, 0i = h0, 1|1, 0i = 0 24 25 h0, 0|1, 1i = 0 and the lower symbol is h0, 0|Af |0, 0i = 1 2 h0, 1|Af |0, 1i = 1 2 h1, 0|Af |1, 0i = 1 2 h1, 1|Af |1, 1i = 1 2 n n n n h0, 1|1, 1i = 24 25 h1, 0|1, 1i = 7 25 f (0, 0) + f (0, 1) f (0, 0) f (0, 0) f (0, 1)
7 2 25 + f (1, 0)
7 2 25 + f (0, 1) + f (1, 1)
24 2 25 + f (1, 0)
24 2 25 + f (1, 0) + f (1, 1)
7 2 25
24 2 25 o
7 2 25 o
24 2 25 o o + f (1, 1) . One can remark that the lower symbols corresponding to the classical observables we have to analyze depend on the fiducial vector. Therefore, the fiducial vector we use must be a privileged one, for example, a kind of fundamental state. We should also notice the way the values of the observables are “redistributed” along the probability distribution. 4.6. An application of the frame quantization to crystals The set Z × Z can be regarded as a mathematical model for a two-dimensional crystal. By imposing the cyclic boundary condition, the space E = l2 (ZN × ZN ) and the operator H : E −→ E, (Hψ)(n1 , n2 ) = ψ(n1 +1, n2 )+ψ(n1 −1, n2 ) (62) +ψ(n1 , n2 +1)+ψ(n1, n2 −1) allow one to describe the electron evolution inside the crystal in the tight binding approximation [38]. For any k = (k1 , k2 ) ∈ ZN × ZN , the function 2πi ψk : ZN × ZN −→ C, ψk (n1 , n2 ) = e N (k1 n1 +k2 n2 ) (63) is an eigenfunction of H corresponding to the eigenvalue 2πi Ek = e N k1 2πi + e− N k1 2πi +eN k2 2πi + e− N k2 = 2 cos 2π 2π k1 + 2 cos k2 , N N that is, Hψk = Ek ψk . One can remark that Ek = X ψk (n1 , n2 ) (n1 ,n2 )∈C where C is the cluster C = {(1, 0), (−1, 0), (0, 1), (0, −1)} ⊂ ZN × ZN . (64) Finite tight frames and some applications 24 The Hilbert space l2 (C) can be identified with the subspace H = { ϕ : ZN × ZN −→ C | ϕ(n1 , n2 ) = 0 for (n1 , n2 ) 6∈ C }. The N 2 functions { |δ(n1 ,n2 ) i = δ(n1 ,n2 ) : ZN ×ZN −→ C }n1 ,n2 ∈ZN ( 1 if (n′1 , n′2 ) = (n1 , n2 ) δ(n1 ,n2 ) (n′1 , n′2 ) = 0 if (n′1 , n′2 ) 6= (n1 , n2 ) and the N 2 functions { |ψ(k1 ,k2 ) i = ψ(k1 ,k2 ) : ZN ×ZN −→ C }k1 ,k2 ∈ZN 1 2πi (k1 n1 +k2 n2 ) eN (65) N form two orthonormal bases of E related through the discrete Fourier transform. The orthogonal projector corresponding to H is X π= |δ(n1 ,n2 ) ihδ(n1 ,n2 ) | ψ(k1 ,k2 ) (n1 , n2 ) = (n1 ,n2 )∈C and in view of theorem 1, the N 2 functions { |k1 , k2i : ZN ×ZN −→ C }k1 ,k2 ∈ZN 2πi 1 X 1 X |δ(n1 ,n2 ) ihδ(n1 ,n2 ) |ψ(k1 ,k2 ) i = e N (k1 n1 +k2 n2 ) |δ(n1 ,n2 ) i |k1 , k2 i = 2 2 (n1 ,n2 )∈C (n1 ,n2 )∈C form a frame in H N −1 4 X |k1 , k2 ihk1 , k2 | = IH . N 2 k ,k =0 1 2 They satisfy the relation hk1 , k2 |k1′ , k2′ i = 1 4 X 2πi eN [(k1′ −k1 )n1 +(k2′ −k2 )n2 ] (n1 ,n2 )∈C   1 2π ′ 2π ′ cos (k1 − k1 ) + cos (k2 − k2 ) . = 2 N N To a classical observable defined by f : ZN × ZN −→ R we associate the linear operator Af : H −→ H, with the lower symbol Af = N −1 4 X f (k1 , k2 ) |k1, k2 ihk1 , k2 | N 2 k ,k =0 1 (66) 2  2 N −1 1 X 2π ′ 2π ′ ′ ′ hk1 , k2 |Af |k1 , k2 i = 2 f (k1 , k2 ) cos (k1 −k1 )+cos (k2 −k2 ) . N ′ ′ N N k1 ,k2 =0 In the case of the frame quantization we analyze a classical observable by using a suitable smaller dimensional subspace. We can increase the resolution of our analysis by choosing a larger cluster including second order or second and third order neighbours of (0, 0). Finite tight frames and some applications 25 4.7. Quantization with finite tight frames overcomplete by one vector For each positive integer n we consider in the Euclidean space Rn+1 the hyperspace Hn = {x = (x0 , x1 , ..., xn ) | x0 + x1 + · · · + xn = 0 }. The orthogonal projector corresponding to Hn is π : Rn+1 −→ Rn+1 , π(x0 , x1 , . . . , xn ) = nx0 −x1 −···−xn −x0 +nx1 −x2 −···−xn n−1 +nxn , , . . . −x0 −···−x n+1 n+1 n+1 and the orthogonal projections of the vectors of the canonical basis
n 1 1 1 w0 = π(1, 0, 0, . . . , 0) = n+1 , − n+1 , − n+1 , . . . , − n+1
n 1 1 1 , n+1 , − n+1 , . . . , − n+1 w1 = π(0, 1, 0, . . . , 0) = − n+1 ...................................................................................
1 1 1 n wn = π(0, 0, . . . , 0, 1) = − n+1 , − n+1 , . . . , − n+1 , n+1 have the same norm ||w0|| = ||w1|| = · · · = ||wn || = r n . n+1 The corresponding normalized vectors   p n w0 1 1 1 , −√ |u0 i = ||w0|| = , −√ , . . . , −√ n+1 n(n+1) n(n+1) n(n+1)   p 1 1 w1 n 1 |u1 i = ||w1|| = − √ , n+1 , − √ , . . . , −√ n(n+1) n(n+1) n(n+1) ...................................................................................   p n 1 1 1 wn , −√ , . . . , −√ , n+1 |un i = ||wn || = − √ n(n+1) n(n+1) n(n+1) form a normalized tight frame n n X |uk ihuk | = IHn n+1 k=0 such that 1 for k 6= j. n To each function f : {0, 1, . . . , n} −→ R we associate the linear operator n n X Af : Hn −→ Hn , Af = f (k) |uk ihuk |. n+1 huk |uj i = − k=0 The corresponding lower symbol is the function fˇn : {0, 1, . . . , n} −→ R, n X 1 n−1 f (k) f (j) + fˇn (j) = huj |Af |uj i = n n(n + 1) k=0 and if f : {0, 1, 2, . . .} −→ R is a bounded function then we have lim fˇn (j) = f (j), n→∞ for any j ∈ {0, 1, 2, . . .},
Finite tight frames and some applications 26 as expected from the general results given in subsection 4.3. More than that, if f, g : {0, 1, 2, . . .} −→ R are two bounded functions then Pn Pn n2 huj |[Af , Ag ]|uj i = (n+1) 2 l=0 (f (k)g(l) − f (l)g(k)) huj |uk i huk |ul i hul |uj i k=0 P P 1 = − n(n+1) 2 k6=j l6=j (f (k)g(l) − f (l)g(k)) and the lower symbol of the commutator [Af , Ag ] has the property lim huj |[Af , Ag ]|uj i = 0, n→∞ for any j ∈ {0, 1, 2, . . .}. 5. Conclusions In this paper we have presented some elements concerning certain applications of finite frames to crystal/quasicrystal physics and to quantum physics. In order to achieve these two main objectives and inspired by the analogy with standard coherent states, we have introduced the notion of normalized Parseval frame, directly related to the notion of Parseval frame, and analyzed some stochastic aspects. In particular we have defined two types of “distances” , η = r − 1 and ζ = 1 − min1≤i≤M κi , between frames and orthonormal basis in the superspace. For the applications to crystals and quasicrystals, based on the embedding into a superspace defined by a frame, we have analyzed the subset of the elements which can be represented as a linear combination of frame vectors by using only integer coefficients. We have identified in this way two important classes of tight frames, namely the periodic frames and the quasiperiodic frames. We have also presented some convergent sequences of finite frames and an example of continuous deformation of a periodic tetrahedral frame into an icosahedral quasiperiodic frame. Some of these theoretical considerations seem to be new, and might be regarded as a contribution to the finite frame theory. The description of the elements of a vector space based on the use of an overcomplete system is a general method re-discovered several times in different areas of mathematics, science and engineering . For example, in crystallography there exists an alternative description for the hexagonal crystals based on the use of an additional axis. We show that the use of a frame leads to a simpler description of atomic positions in a diamond type crystal. This leads to a simpler description of the symmetry transformations and of the mathematical objects with physical meaning. Some of the most important models used in quasicrystal physics can be generated in a unitary way by using the imbedding into a superpace defined by certain frames. These observations allow a fructuous interchange of ideas and methods between frame theory and quasicrystal physics. Finite frame quantization replaces a real function f defined on a finite set by a self-adjoint operator Af , and the eigenvalues of Af can be regarded as the “quantum spectrum” of f . We compare f with the mean values of Af corresponding to the frame vectors, in the general case and in several particular cases. We have explained the role of the parameter ζ as a kind of distance of the quantum non-commutative world to the classical commutative one. The notion of normalized Parseval frame and Finite tight frames and some applications 27 the corresponding quantization of discrete variable functions is rich of questions which deserve to be thoroughly investigated in the measure that they might shed light on a better understanding of quantum mechanics and quantization. Acknowledgment NC acknowledges the support provided by CNCSIS under the grant IDEI 992 - 31/2007. 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] Badzia̧g P, Bengtsson I, Cabello A, Granström H, and Larsson J 2009 Pentagrams and Paradoxes arXiv:0909.4713v1 [quant-ph] [4] Balazs P 2008 Frames and finite dimensionality: frame transformation, classification and algorithms Applied Mathematical Sciences 2 2131-2144 [5] Benedetto J J and Fickus M 2003 Finite normalized tight frames Advances in Computational Mathematics 18 357-385 [6] Berezin F A 1975 General concept of quantization Commun. Math. Phys. 40 153-74 [7] Casazza P G 2004 Custom building finite frames Contemp. Math. 345 61-86 [8] Casazza P G and Leonhard N 2008 Classes of equal norm Parseval frames Contemp. Math. 451 11-32 [9] Casazza P G, Dilworth S J, Odell E, Schlumprecht T and Zsác A 2008 Coefficient quantization for frames in Banach spaces Journal of Mathematical Analysis and Applications 348 66-86 [10] Christensen O 2003 An introduction to frames and Riesz bases (Boston: Birkhäuser) [11] Cotfas N 1995 A mathematical model for diamond-type crystals J. Phys. A: Math. Gen. 28 1371-9 [12] Cotfas N 2006 On the linear representations of the symmetry groups of single-wall carbon nanotubes J. Phys. A: Math. Gen. 39 9755-65 [13] Daubechies I, Grossmann A and Meyer Y 1986 Painless nonorthogonal expansions J. Math. Phys. 27 1271-1283 [14] Daubechies I 1992 Ten Lectures On Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics) (Philadelphia: SIAM) [15] Descombes D 1986 Eléments de Théorie des Nombres (Paris: PUF) pp. 54-9 [16] Duffin R J and Schaeffer A C 1952 A class of nonharmonic Fourier series Trans. Amer. Math. Soc. 72 341-366 [17] Elser V 1986 The diffraction pattern of projected structures Acta Cryst. A 42 36-43 [18] Garcı́a de Leon P L and Gazeau J P 2007 Coherent state quantization and phase operator Phys. Lett. A 361 301-4 [19] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley-VCH) [20] Gazeau J-P, Garidi T, Huguet E, Lachièze-Rey M and Renauld J 2004 Exemples of Berezin-Toeplitz Quantization: Finite Sets and Unit Interval Proc. CRM and Lecture Notes 34 67-76 [21] Gazeau J-P and Piechocki W 2004 Coherent state quantization of a particle in de Sitter space J. Phys. A: Math. Gen. 37 6977-86 [22] Gazeau J-P, Mourad J and Quéva J 2006 Fuzzy de Sitter space-time via coherent state quantization Proc. 26th ICGTMP (New York) ed J Birman and S Catto (New York: Springer) to appear (Preprint quant-ph/0610222) [23] Gazeau J P, Huguet E, Lachièze-Rey M and Renauld J 2007 Fuzzy sphere from inequivalent coherent states quantizations J. Phys. A: Math. Gen. 40 10225-49 Finite tight frames and some applications 28 [24] Gersten A 1998 Dirac’s representation theory as a framework for signal theory, Ann. Phys. 262 47-72 [25] Goyal V K, Kovačević J and Kelner J A 2001 Quantized frame expansions with erasures Apl. Comput. Harmonic Anal. 10 203-33 [26] Goyal V K, Vetterli and Thao N T 1998 Quantized overcomplete expansions in Rn : Analysis, synthesis, and algorithms IEEE Trans. Inform. Theory 44 16-31 [27] Han D and Larson D R 2000 Frames, bases and group representations, Mem. Amer. Math. Soc. 697 [28] Katz A and Duneau M 1985 Quasiperiodic patterns and icosahedral symmetry J. Physique (France) 47 181-196 [29] Klauder J R 1963 Continuous-representation theory: I. Postulates of continuous-representation theory J. Math. Phys. 4 1055-8 [30] Klauder J R 1963 Continuous-representation theory: II. Generalized relation between quantum and classical dynamics J. Math. Phys. 4 1058-73 [31] Klauder J R and Skagerstam B S 1985 Coherent States - Applications in Physics and Mathematical Physics (Singapore: World Scientific) [32] Klauder J R 1995 Quantization without quantization Ann. Phys. 237 147-60 [33] Klyachko A 2007 Dynamic symmetry approach to entanglement, in J.-P. Gazeau et al. (eds): Proc. NATO Advanced Study Inst. on Physics and Theoretical Computer Science, IOS, Amsterdam [34] Klyachko A A, Can M A, Biniciǒglu S, and Shumovsky A S 2008 A simple test for hidden variables in the spin-1 system Phys. Rev. Lett. 101 020403. [35] Lachièze-Rey M, Gazeau J-P, Huguet E, Renauld J and Garidi T 2003 Quantization of the sphere with coherent states Int. J. Theor. Phys 42 1301-10 [36] Mehta M L 1987 Eigenvalues and eigenvectors of the finite Fourier transform J. Math. Phys. 28 781-5 [37] Meyer C D 2000 Matrix Analysis and Applied Linear Algebra SIAM. [38] Montroll E W 1970 Quantum theory on a network. I. A solvable model whose wavefunctions are elementary functions J. Math. Phys. 2 635-48 [39] Moody R V 2000 Model sets: a survey In: Axel F, Dénoyer F, and Gazeau J-P (editors), From Quasicrystals to More Complex Systems. Editions de Physique & Springer, Berlin, , 145-166 [40] Perelomov A M 1972 Coherent states for arbitrary Lie group Commun. Math. Phys. 26 222-36 [41] Perelomov A M 1986 Generalized Coherent States and their Applications (Berlin: Springer) [42] Schwinger J 1960 Proc. Nat. Acad. Sci. USA 46 570 [43] Senechal M 1995 Quasicrystals and Geometry (Cambridge: Cambridge University Press) pp. 264-6 [44] Štoviček P and Tolar J 1984 Quantum mechanics in discrete space-time Rep. Math. Phys. 20 157-70 [45] Weyl H 1950 Theory of Groups and Quantum Mechanics (New York: Dover Publ.) [46] Zhang S and Vourdas A 2004, Analytic representation of finite quantum systems J. Phys. A: Math. Gen. 37 8349-63 [47] Zimmermann G 2001 Normalized tight frames in finite dimensions Recent Progress in Multivariate Approximation eds. K L W Haußmann and M Reimer (Basel: Birkhäuser) 249-52.