Miscellaneous Applications of Quons

This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We show the interest of such algebras for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU_2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).


Introduction
Deformed oscillator algebras are spanned by generalized bosons or fermions which are often referred to as quons, q-bosons, parafermions (parabosons) or k-fermions [1,2,3,4,5,6,7,8,9,10]. Such objects are of paramount importance for generating q-deformed Lie algebras. In this respect, quons can be used to derive q-deformed Lie (super)algebras associated with q-deformed models in atomic, molecular and condensed matter physics as well as in nuclear and particle physics.
It is the aim of the present work to show the interest of quons, when q is a kth primitive root of unity, for fractional supersymmetric quantum mechanics of order k and for the determination of mutually unbiased bases used in quantum information. This paper takes it origin in an invited talk at the 3 rd International Microconference "Analytic and Algebraic Methods in Physics". It presents a review character although it exhibits some original aspects partly presented in schools and conferences.
The organisation of the paper is as follows. Section 2 is devoted to some generalities on quon algebras and k-fermions which are objects interpolating between bosons and fermions. Section 3 deals with fractional supersymmetric quantum mechanics of order k. In Section 4 we show how to construct the Lie algebra of SU 2 from two quon algebras. This leads to a polar decomposition which is used in Section 5 for generating mutually unbiased bases (MUBs) in d-dimensional Hilbert spaces with d prime. The approach followed for MUBs is also suggested for deriving certain positive operator valued measures (POVMs) in finite-dimensional Hilbert spaces.
Regarding the notations, let us mention that δ a,b stands for the Kronecker symbol for a and b, the bar indicates complex conjugation, A † denotes the adjoint of the operator A, I is the identity operator, and [A, B] q := AB − qAB so that [A, B] 1 (respectively [A, B] −1 ) is the commutator [A, B] (respectively anticommutator {A, B}) of the operators A and B.

Quons and k-fermions
We first define a quon algebra or q-deformed oscillator algebra for q a root of unity. Definition 1. The three operators a − , a + and N a such that where define a quon algebra or q-deformed oscillator algebra denoted as A q (a − , a + , N a ). The operators a − and a + are referred to as quon operators. The operators a − , a + and N a are called annihilation, creation and number operators, respectively.
Definition 1 differs from the one by Arik and Coon [1] in the sense that we take q ∈ S 1 instead of 0 < q < 1. For k = 2 (respectively k → ∞), the quon operators coincide with the ordinary fermion (respectively boson) operators.
For arbitrary k, the quon operators a − and a + are not connected via Hermitian conjugation. It is only for k = 2 or k → ∞ that we may take a + = a † − . In general (i.e., for k = 2 or k → ∞), we have a † ± = a ∓ . Therefore, it is natural to consider the so-called k-fermionic algebra Σ q with the generators a − , a + , a + + := a † + , a + − := a † − , and N a [9,10]. The defining relations for Σ q correspond to the ones of A q (a − , a + , N a ) and Aq(a + + , a + − , N a ) complemented by the relation Observe that for k = 2 or k → ∞, the latter relation corresponds to an identity. The operators a − , a + , a + + and a + − are called k-fermion operators and we also use the terminology k-fermions in analogy with fermions and bosons [9]. They clearly interpolate between fermions and bosons.
In a way similar to the one used for ordinary fermions and ordinary bosons, we can define the coherence factor g (m) for an assembly of m k-fermions. Such a definition reads stands for the average value of the operator X on the k-fermionic coherent state |z) defined in [10]. A simple calculation in Σ q shows that From equation (3), we see once again that k = 2 corresponds to ordinary fermions and k → ∞ to ordinary bosons. Equation (3) is in agreement with a generalized Pauli exclusion principle according to which, for a many-particle system, a k-fermionic state (corresponding to a spin 1/k) cannot be occupied by more than k − 1 identical k-fermions [9].
To close this section, let us mention that the k-fermions introduced in [9,10] share some common properties with the parafermions of order k − 1 discussed in [35,36,37,38,39,40,41]. Indeed, a parafermionic algebra of order k−1 corresponds to a fractional supersymmetric algebra of order k.
3 Quons and fractional supersymmetry 3.1 Fractional supersymmetric system Following Rubakov and Spiridonov [35], who initially considered the case k = 3, we start with the definition of a fractional supersymmetric system of order k [42].
Definition 2. For fixed k in N \ {0, 1}, a fractional supersymmetric quantum system of order k, or k-SUSY system in short, is a doublet (H, Q) k of linear operators H and Q, acting on a separable Hilbert space H, such that H is self-adjoint and The operators H and Q ± are called the Hamiltonian and the supercharges of the k-SUSY system.
By way of illustration, let us show that the case k = 2 corresponds to Witten's approach of ordinary supersymmetric quantum mechanics. According to Witten [43], a triplet of linear operators (H, P, S) of linear operators H, P and S, with P bounded, defines a supersymmetric quantum system if By putting we get the relations which correspond to a (H, Q) 2 system or ordinary supersymmetric system with a Z 2 -grading involving fermionic and bosonic states. Going back to the general case, the doublet (H, Q) k , with arbitrary k, defines a k-SUSY system, with a Z k -grading, for which the Hamiltonian H and the two (nonindependent) supercharges Q ± are up to now formal operators. A natural question arises: How to find realizations of H and Q ± ? In this respect, the definition of a generalized Weyl-Heisenberg algebra is essential [42].

Generalized Weyl-Heisenberg algebra
1} is a set of k functions, be the algebra spanned by the four linear operators X + , X − , N and K acting on the space H and satisfying and [N, where q := exp(2πi/k) and k ∈ N \ {0, 1}.
It should be realized that, for fixed k, equations (4)-(6) define indeed a family of generalized Weyl-Heisenberg algebras W k (f ). The various members of the family are distinguished by the various sets f .
It is possible to find a realization of such that any k-fermion operator commutes with any generalized boson operator. Indeed, by introducing

Realizations of k-SUSY systems
As an important result, the following proposition shows that, for fixed k and given f , it is possible to associate a k-SUSY system, characterized by a specific doublet (H, Q) k , with the algebra W k (f ).

Proposition 1. For a fixed value of k and a given set f , the relations
In the light of Proposition 1, we foresee that a k-SUSY system with a Z k -grading can be considered as a superposition of k − 1 ordinary supersymmetric subsystems (corresponding to k = 2) with a Z 2 -grading [42].
As an example, we consider the case of the Z 3 -graded supersymmetric oscillator corresponding to The corresponding algebra W 3 (f ) can be represented by in terms of 3-fermions (f − , f + ) and ordinary bosons (b − , b + ). The system (H, Q) 3 associated with W 3 (f ) is defined by and with Π 3 ≡ Π 0 . In terms of 3-fermions and ordinary bosons, we have Finally, the energy spectrum of H reads a symbolic writing to mean that it contains equally spaced levels with a nondegenerate ground state (denoted as 1), a doubly degenerate first excited state (denoted as 2) and an infinite sequence of triply degenerate excited states (denoted as 3).
Other sets f lead to other fractional supersymmetric dynamical systems [44]. For instance, the case corresponds to translational shape-invariant systems as for example the harmonic oscillator system (for a = 0 and b > 0), the Morse system (for a < 0 and b > 0) and the Pöschl-Teller system (for a > 0 and b > 0). Furthermore, the case

Quons and polar decomposition
The approach presented in this section is a review based on the original developments given in [45,46,47,48,49]. Here, we shall limit ourselves to those aspects which are relevant for Section 5.
We start with two commuting quon algebras A q (a − , a + , N a ) ≡ A q (a) with a = x, y corresponding to the same value of the deformation parameter q. Their generators satisfy equations (1) and (2) with a = x, y and [X, Y ] = 0 for any X in A q (x) and any Y in A q (y). Then, let us look for Hilbertian representations of A q (x) and A q (y) on k-dimensional Hilbert spaces F x and F y spanned by the orthonormal bases {|n 1 ) : n 1 = 0, 1, . . . , k − 1} and {|n 2 ) : n 2 = 0, 1, . . . , k − 1}, respectively. We easily obtain the representations defined by x + |n 1 ) = |n 1 + 1), x + |k − 1) = 0, x − |n 1 ) = [n 1 ] q |n 1 − 1), x − |0) = 0, N x |n 1 ) = n 1 |n 1 ) and y + |n 2 ) = [n 2 + 1] q |n 2 + 1), y + |k − 1) = 0, y − |n 2 ) = |n 2 − 1), for A q (x) and A q (y), respectively. The cornerstone of this approach is to define the two linear operators The operators h and v ra act on the states We now adapt the trick used by Schwinger in his approach to angular momentum via a coupled pair of harmonic oscillators. This can be done by introducing two new quantum numbers J and M defined by We guess that ǫ(j) is a space of constant angular momentum j. As a matter of fact, we can check that ǫ(j) is stable under h and v ra . Moreover, by defining the operators j ± and j z through for any r in R and any a in R. The latter commutation relations correspond to the Lie algebra of SU 2 . We have here a polar decomposition of j ± . Thus, from two q-deformed oscillator algebras we obtained a polar decomposition of the nondeformed Lie algebra of SU 2 . In addition, the complete set of commuting operators {j 2 , j z }, where j 2 is the Casimir of SU 2 can be replaced by another complete set of commuting operators, namely {j 2 , v ra }, for which we have the following result. It is to be noted that the operators z, defined through z|j, m = q j−m |j, m , and v ra can be used to generate the Pauli group P 2j+1 , a subgroup of order (2j + 1) 3 of the linear group GL(2j + 1, C) [49].
For fixed a, r and j, the common orthonormalized eigenvectors of the commuting operators j 2 and v ra can be taken in the form or alternatively In other words, for fixed a, r and j, the space ǫ(j) can be spanned by the basis B ra = {|jα; ra : α = 0, 1, . . . , 2j}.
The replacement of the spherical basis S, adapted to the chain SO 3 ⊃ SO 2 and to the set {j 2 , j z }, by the basis B ra , adapted to the chain SO 3 ⊃ Z 2j+1 and to the set {j 2 , v ra }, leads to a new form of the Wigner-Racah algebra of SU 2 [46]. Note that the notation in (8) is particularly appropriate for quantum information theory (the qudits |0 , |1 , . . . , |d − 1 in the expansion (8) constitute the computational or canonical basis of a d-dimensional Hilbert space).
We shall see in Section 5 how the parameter a can be chosen in such a way to generate MUBs.
Each of the operators Π aα acting on C d can be considered as a vector in C d 2 endowed with the Hilbert-Schmidt inner product. Thus, Π aα can be developed as where The calculation of Tr Π † aα Π bβ leads to By defining the vectors s(aα) := (s 1 (aα), s 2 (aα), . . . , s d 2 (aα)) with where s(aα) · s(bβ) stands for the usual inner product d 2 i=1 s i (aα)s i (bβ) in C d 2 . Note that the relation (12) is independent of the basis chosen for developing the operators Π aα .
As a result, the determination of the d(d + 1) vectors |aα in C d , for d a power of a prime, amounts to find d(d + 1) operators Π aα acting on C d or to find d(d + 1) vectors s(aα) in C d 2 satisfying (12).

POVMs
An approach similar to the one developed for MUBs can be set up for symmetric informationally complete (SIC) POVMs [61,62,63,64,65,66,67,68]. We shall briefly discuss the starting point for a study of SIC-POVMs along the lines of Section 5.1 (see [69] for more details). A SIC-POVM in dimension d can be defined as a set of d 2 nonnegative operators P x acting on C d and satisfying Let be the development of P x in terms of the operators E mm ′ and let r(x) := (r 1 (x), r 2 (x), . . . , r d 2 (x)) be the vector in C d 2 of components It is immediate to show that r(x) · r(y) = 1 d + 1 (dδ x,y + 1), a relation (independent of the basis chosen for developing P x ) to be compared with (12). The determination of the d 2 operators P x satisfying (15) amounts to find d 2 vectors |Φ x in C d or d 2 vectors r(x) in C d 2 satisfying (16). The search for solutions of (16) is presently under progress.

Open questions
To close this paper, we would like to address a few questions which arose during the conference. Possible future developments concern (i) a field theory approach to k-fermions and their relation to anyons, (ii) the classification, in terms of the sets f , of the Weyl-Heisenberg algebras W k (f ) that lead to integrable systems, (iii) the passage from fractional supersymmetric quantum mechanics to fractional supersymmetric nonHermitian quantum mechanics, as for example along the lines of PT-symmetric regularizations [70], (iv) the Wigner-Racah algebra of the group SU 2 in the {j 2 , v ra } scheme, and (v) the construction of MUBs and SIC-POVMs in an unified way. Some comments regarding points (iv) and (v) are in order. The {j 2 , v ra } scheme described in Section 4 constitutes an alternative to the familiar {j 2 , j z } scheme of angular momentum theory. As a further step, it would be interesting to find differential realizations of the operator v ra as well as realizations of the bases B ra on the sphere S 2 for j integer and on Fock-Bargmann spaces (in 1 and 2 dimensions) for j integer or half of an odd integer.
The preliminary study reported in Section 5 for MUBs and POVMs is based on the replacement of | Φ x |Φ y | = 1 √ d + 1 dδ x,y + 1, corresponding to inner products in C d , by r(x) · r(y) = 1 d + 1 (dδ x,y + 1) , corresponding to inner products in C d 2 , respectively. We may ask the question: Is it easier to find solutions of (17), (18) than to find solutions of (19), (20)? We do not have any answer. In the case of MUBs, we have a general solution (see Proposition 3) of (19) and thus (17) for d prime in the framework of angular momentum theory. It would be interesting to extend this result for d a power of a prime. In the case of SIC-POVMs, to prove or disprove the conjecture according to which SIC-POMVs exist in any dimension amounts to prove or disprove that (20) has a solution in any dimension.
Note. After the submission (July 23, 2007) of the present paper for publication in SIGMA, the author became aware of a preprint dealing with the existence of SIC-POVMs posted (July 20, 2007) on arXiv [71]. The main result in [71] is that SIC-POVMs exist in all dimensions. As a corollary of this result, equation (20) admits solutions in any dimension.