Finite tight frames and some applications

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.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].

Pisot -coherent states quantization of the harmonic oscillator

“…The change of the frame family C produces another quantization, another point of view, possibly equivalent to the previous one, possibly not. The present study lies in the continuity of a series of such explorations, which were already present in the first works by Klauder at the beginning of the sixties of the past century (see for instance [16,33] and references therein), pursued by Berezin [15] in his famous paper of 1975, and more recently extended to various measure sets (see for instance [34][35][36][37][38][39]). …”

An analytic function approach to weak mutually unbiased bases

“…We consider a finite quantum system with variables in Z(d) (the integers modulo d) [2][3][4][5][6][7][8]. Let |X; n the basis of position states in the d-dimensional Hilbert space H(d), and |P ; n the basis of momentum states:…”

“…We give an example of two lines through the origin in G (21), which have three points in common. The lines L(1, 8) = L(2, 3) and L(1, 11) = L(2, 5) have in common the three points (0, 0), (7,14), (14,7), and they are shown in Fig.1. This example shows that our geometry is a non-near-linear geometry.…”

“…After the pioneering work by Schwinger [1], there has been a lot of work on various aspects of a quantum system Σ(d) with variables in Z(d) (the ring of integers modulo d), described with a d-dimensional Hilbert space H(d). The work combines Quantum Physics with Discrete Mathematics and has applications to areas like quantum information, quantum cryptography, quantum coding, etc (for reviews see [2][3][4][5][6][7][8]).…”

Positive-Operator Valued Measure (POVM) Quantization

“…which means that the set { √ ν i |x i } is a Parseval frame [13][14][15][16]. Such an identity is possible if N ≥ n;…”