## SU2 nonstandard bases: case of mutually unbiased bases

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@MISC{Albouy_su2nonstandard,

author = {Olivier Albouy and Maurice R. Kibler},

title = {SU2 nonstandard bases: case of mutually unbiased bases},

year = {}

}

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### Abstract

Abstract. This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU2 corresponding to an irreducible representation of SU2. The representation theory of SU2 is reconsidered via the use of two truncated deformed oscillators. This leads to replacement of the familiar scheme {j 2, jz} by a scheme {j 2, vra}, where the two-parameter operator vra is defined in the universal enveloping algebra of the Lie algebra su2. The eigenvectors of the commuting set of operators {j 2, vra} are adapted to a tower of chains SO3 ⊃ C2j+1 (2j ∈ N ∗), where C2j+1 is the cyclic group of order 2j + 1. In the case where 2j + 1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices. Key words: symmetry adapted bases; truncated deformed oscillators; angular momentum; polar decomposition of su2; finite quantum mechanics; cyclic systems; mutually unbiased bases; Gauss sums 2000 Mathematics Subject Classification: 81R50; 81R05; 81R10; 81R15

### Citations

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Citation Context ...sure for finite quantum systems [24, 25, 26, 27]. To close this section, we note that the (Weyl–Pauli) operators z and vra can be used to generate the (Pauli) group P2j+1 introduced in [29] (see also =-=[30]-=-). The group P2j+1 is a finite subgroup of GL(2j + 1, C) and consists of generalized Pauli matrices. It is spanned by two generators. In fact, the (2j + 1) 3 elements of P2j+1 can be generated by v00 ... |

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Citation Context ...andard Clebsch–Gordan coefficient in the {j2 , jz} scheme and � � 2π qℓ = exp i , ℓ = 1, 2, 3. 2jℓ + 1 The algebra of the new coupling coefficients (3.11) can be developed in a way similar to the one =-=[40]-=- known in the {j 2 , jz} scheme (see [20] for the basic ideas). In particular, following the technique developed in [4, 5], the familiar 6-j and 9-j symbols of Wigner can be expressed in terms of the ... |

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Citation Context ...ormalized bases {|aα〉 : α = 0, 1, . . . , d − 1} and {|bβ〉 : β = 0, 1, . . . , d − 1} of a d-dimensional Hilbert space over C, with an inner product denoted as 〈 | 〉, are said to be mutually unbiased =-=[31, 32, 33, 34, 35, 36, 37, 38, 39]-=- if and only if |〈aα|bβ〉| = δa,bδα,β + (1 − δa,b) 1 √ d . (3.5) The correspondence between equations (3.4) and (3.5) is as follows. In equation (3.4), 2j + 1 corresponds to d while the symbols jra, α,... |

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Citation Context ...Section 4. In a series of appendices, we give some useful relations satisfied by generalized quadratic Gauss sums. 2 A quon realization of the algebra su2 2.1 Two quon algebras Following the works in =-=[12, 13, 14, 15, 16]-=-, we define two quon algebras Ai = {ai−, ai+, Ni} with i = 1 and 2 by ai−ai+ − qai+ai− = 1, [Ni, ai±] = ±ai±, N † i = Ni, (ai±) k = 0, ∀ x1 ∈ A1, ∀ x2 ∈ A2 : [x1, x2] = 0, where � � 2πi q = exp , k ∈ ... |

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Citation Context ...Section 4. In a series of appendices, we give some useful relations satisfied by generalized quadratic Gauss sums. 2 A quon realization of the algebra su2 2.1 Two quon algebras Following the works in =-=[12, 13, 14, 15, 16]-=-, we define two quon algebras Ai = {ai−, ai+, Ni} with i = 1 and 2 by ai−ai+ − qai+ai− = 1, [Ni, ai±] = ±ai±, N † i = Ni, (ai±) k = 0, ∀ x1 ∈ A1, ∀ x2 ∈ A2 : [x1, x2] = 0, where � � 2πi q = exp , k ∈ ... |

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Citation Context ...ormalized bases {|aα〉 : α = 0, 1, . . . , d − 1} and {|bβ〉 : β = 0, 1, . . . , d − 1} of a d-dimensional Hilbert space over C, with an inner product denoted as 〈 | 〉, are said to be mutually unbiased =-=[31, 32, 33, 34, 35, 36, 37, 38, 39]-=- if and only if |〈aα|bβ〉| = δa,bδα,β + (1 − δa,b) 1 √ d . (3.5) The correspondence between equations (3.4) and (3.5) is as follows. In equation (3.4), 2j + 1 corresponds to d while the symbols jra, α,... |

22 |
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Citation Context ...ing algebra of SU2. A possible way to find a realization of vra in terms of the generators j± and jz of SU2 is as follows. The first step is to develop vra on the basis of the Racah unit tensor u (k) =-=[41]-=-. Let us recall that the components u (k) p of u (k) , p = k, k − 1, . . . , −k, are defined by 〈j, m|u (k) p |j, m ′ 〉 = (−1) j−m � j k j −m p m ′ � , where the symbol (· · · ) stands for a 3-jm Wign... |

21 |
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Citation Context ...ormalized bases {|aα〉 : α = 0, 1, . . . , d − 1} and {|bβ〉 : β = 0, 1, . . . , d − 1} of a d-dimensional Hilbert space over C, with an inner product denoted as 〈 | 〉, are said to be mutually unbiased =-=[31, 32, 33, 34, 35, 36, 37, 38, 39]-=- if and only if |〈aα|bβ〉| = δa,bδα,β + (1 − δa,b) 1 √ d . (3.5) The correspondence between equations (3.4) and (3.5) is as follows. In equation (3.4), 2j + 1 corresponds to d while the symbols jra, α,... |

20 |
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Citation Context ...correlation measure for finite quantum systems [24, 25, 26, 27]. To close this section, we note that the (Weyl–Pauli) operators z and vra can be used to generate the (Pauli) group P2j+1 introduced in =-=[29]-=- (see also [30]). The group P2j+1 is a finite subgroup of GL(2j + 1, C) and consists of generalized Pauli matrices. It is spanned by two generators. In fact, the (2j + 1) 3 elements of P2j+1 can be ge... |

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Citation Context ...w that for a d-dimensional Hilbert space, with d prime (d = p) or a power of a prime (d = p e ), with p prime and e positive integer greater than 1, there exists a complete set of d + 1 pairwise MUBs =-=[31, 32, 33, 34, 35, 36, 37, 38, 39, 51, 52, 53, 54, 55, 56, 57, 58, 59]-=-. For d = p = 2j + 1 prime, the bases B0a = {|aα〉 := |jα; 0a〉 : α = 0, 1, . . . , p − 1}, a = 0, 1, . . . , p − 1 satisfy (3.21) and (3.24). Consequently, they constitute an incomplete set of p MUBs. ... |

18 | On SIC-POVMs and MUBs in dimension 6
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Citation Context ...pproaches developed in previous studies through the use of Galois fields and Galois rings, discrete Wigner functions, mutually orthogonal Latin squares, graph theory, and finite geometries (e.g., see =-=[65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]-=- and references cited therein for former works). Our approach to MUBs in the framework of angular momentum should be particularly appropriate for dealing with entanglement of spin states. To close thi... |

16 |
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Citation Context ...The might be useful for a study of the Wigner–Racah agebra of SU2 in a SU2 ⊃ C∗ 2j+1 eigenstates |jα; ra〉 of the set {j2 , vra} can be seen as generalized discrete Fourier transforms (in the sense of =-=[60, 61, 62, 63, 64]-=-) of the eigenstates of the standard set {j2 , jz}. This led us to strongly SABs Bra = {|jα; ra〉 : α = 0, 1, . . . , 2j} which are unbiased with the spherical basis S = {|jm〉 : m = j, j − 1, . . . , −... |

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Citation Context ...pproaches developed in previous studies through the use of Galois fields and Galois rings, discrete Wigner functions, mutually orthogonal Latin squares, graph theory, and finite geometries (e.g., see =-=[65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]-=- and references cited therein for former works). Our approach to MUBs in the framework of angular momentum should be particularly appropriate for dealing with entanglement of spin states. To close thi... |

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Citation Context ... the type |jλΓγ〉 = � |jaΓγ〉 Uaλ, a where the unitary matrix U diagonalizes the matrix v set up on the set {|jaΓγ〉 : a ranging}. Integrity bases to obtain v were given for different subgroups G of SO3 =-=[8, 9, 10]-=-. As a résumé, there are several kinds of physically interesting bases for the irreducible representations of SU2. The standard basis, associated with the commuting set {j 2 , jz}, corresponds to the ... |

13 |
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Citation Context ...alized exclusion principle for particles of fractional spin 1/k (the Pauli exclusion principle corresponds to k = 2). Let us mention that algebras similar to A1 and A2 with N1 = N2 were introduced in =-=[17, 18, 19]-=- to define k-fermions which are, like anyons, objects interpolating between fermions (corresponding to k = 2) and bosons (corresponding to k → ∞). 2.2 Representation of the quon algebras We can find s... |

12 |
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Citation Context ... that equation (3.24) can be proved equally well without using generalized quadratic Gauss sums. The following proof is an adaptation, in the framework of angular momentum, of the method developed in =-=[45]-=- (see also [46, 47, 48, 49]) in order to construct a complete set of MUBs in C d with d prime. Proof. We start from vraz n = vrb, b = a + n, n ∈ Z, which can be derived from equation (2.20). In view o... |

12 |
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Citation Context ...results in Section 3 can be applied to the derivation of MUBs. Proposition 8 provides us with a method for deriving MUBs that differs from the methods developed, used or discussed in [31]–[39] and in =-=[51]-=-–[78]. 4.2.1 d arbitrary Let Va be the matrix of the operator v0a in the computational basis S = {|k〉 : k = d − 1, d − 2, . . . , 0}, (4.2) cf. equation (2.13). This matrix can be expressed in terms o... |

11 |
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Citation Context ...dition, the action of vra on the space Fk is cyclic. More precisely, we can check that (vra) k = e iπ(k−1)(a+r) I, (2.12) where I is the identity operator. From the Schwinger work on angular momentum =-=[22]-=-, we introduce J = 1 2 (n1 + n2) , M = 1 2 (n1 − n2) .sSU2 Nonstandard Bases: Case of Mutually Unbiased Bases 5 Consequently, we can write |n1, n2) = |J + M, J − M). We shall use the notation |J, M〉 :... |

11 | Mutually unbiased bases and finite projective planes
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Citation Context ...w that for a d-dimensional Hilbert space, with d prime (d = p) or a power of a prime (d = p e ), with p prime and e positive integer greater than 1, there exists a complete set of d + 1 pairwise MUBs =-=[31, 32, 33, 34, 35, 36, 37, 38, 39, 51, 52, 53, 54, 55, 56, 57, 58, 59]-=-. For d = p = 2j + 1 prime, the bases B0a = {|aα〉 := |jα; 0a〉 : α = 0, 1, . . . , p − 1}, a = 0, 1, . . . , p − 1 satisfy (3.21) and (3.24). Consequently, they constitute an incomplete set of p MUBs. ... |

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Citation Context ...Section 4. In a series of appendices, we give some useful relations satisfied by generalized quadratic Gauss sums. 2 A quon realization of the algebra su2 2.1 Two quon algebras Following the works in =-=[12, 13, 14, 15, 16]-=-, we define two quon algebras Ai = {ai−, ai+, Ni} with i = 1 and 2 by ai−ai+ − qai+ai− = 1, [Ni, ai±] = ±ai±, N † i = Ni, (ai±) k = 0, ∀ x1 ∈ A1, ∀ x2 ∈ A2 : [x1, x2] = 0, where � � 2πi q = exp , k ∈ ... |

8 |
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Citation Context ...j+1 is the cyclic group of order 2j + 1. In other words, the chain of groups used here depends on the irreducible representation of SU2 to be considered. We shall establish a connection, mentioned in =-=[11]-=-, between the obtained bases and the so-called mutually unbiased bases (MUBs) used in quantum information. The organisation of this paper is as follows. In Section 2, we construct the Lie algebra of S... |

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Citation Context ...(3.24) can be proved equally well without using generalized quadratic Gauss sums. The following proof is an adaptation, in the framework of angular momentum, of the method developed in [45] (see also =-=[46, 47, 48, 49]-=-) in order to construct a complete set of MUBs in C d with d prime. Proof. We start from vraz n = vrb, b = a + n, n ∈ Z, which can be derived from equation (2.20). In view of Proposition 5, the action... |

8 |
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Citation Context ...w that for a d-dimensional Hilbert space, with d prime (d = p) or a power of a prime (d = p e ), with p prime and e positive integer greater than 1, there exists a complete set of d + 1 pairwise MUBs =-=[31, 32, 33, 34, 35, 36, 37, 38, 39, 51, 52, 53, 54, 55, 56, 57, 58, 59]-=-. For d = p = 2j + 1 prime, the bases B0a = {|aα〉 := |jα; 0a〉 : α = 0, 1, . . . , p − 1}, a = 0, 1, . . . , p − 1 satisfy (3.21) and (3.24). Consequently, they constitute an incomplete set of p MUBs. ... |

8 |
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8 |
There is no generalization of known formulas for mutually unbiased bases
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Citation Context ...pproaches developed in previous studies through the use of Galois fields and Galois rings, discrete Wigner functions, mutually orthogonal Latin squares, graph theory, and finite geometries (e.g., see =-=[65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]-=- and references cited therein for former works). Our approach to MUBs in the framework of angular momentum should be particularly appropriate for dealing with entanglement of spin states. To close thi... |

7 |
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7 | Fractional supersymmetric quantum mechanics, Phys
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7 | Planat M. Hjelmslev geometry of mutually unbiased bases
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6 | Fractional supersymmetry and hierarchy of shape invariant potentials
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Citation Context ...alized exclusion principle for particles of fractional spin 1/k (the Pauli exclusion principle corresponds to k = 2). Let us mention that algebras similar to A1 and A2 with N1 = N2 were introduced in =-=[17, 18, 19]-=- to define k-fermions which are, like anyons, objects interpolating between fermions (corresponding to k = 2) and bosons (corresponding to k → ∞). 2.2 Representation of the quon algebras We can find s... |

6 |
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Citation Context ...ts in Section 3 can be applied to the derivation of MUBs. Proposition 8 provides us with a method for deriving MUBs that differs from the methods developed, used or discussed in [31]–[39] and in [51]–=-=[78]-=-. 4.2.1 d arbitrary Let Va be the matrix of the operator v0a in the computational basis S = {|k〉 : k = d − 1, d − 2, . . . , 0}, (4.2) cf. equation (2.13). This matrix can be expressed in terms of the... |

5 | Viewing sets of mutually unbiased bases as arcs in finite projective planes
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5 |
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(Show Context)
Citation Context ...The might be useful for a study of the Wigner–Racah agebra of SU2 in a SU2 ⊃ C∗ 2j+1 eigenstates |jα; ra〉 of the set {j2 , vra} can be seen as generalized discrete Fourier transforms (in the sense of =-=[60, 61, 62, 63, 64]-=-) of the eigenstates of the standard set {j2 , jz}. This led us to strongly SABs Bra = {|jα; ra〉 : α = 0, 1, . . . , 2j} which are unbiased with the spherical basis S = {|jm〉 : m = j, j − 1, . . . , −... |

5 |
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4 |
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Citation Context ...n2 = 0, 1, . . . , k − 1} constitute an orthonormalized basis of Fk. We denote ( | ) the scalar product on Fk so that (n ′ 1, n ′ 2|n1, n2) = δn ′ 1 ,n1 δn ′ 2 ,n2 . 2.3 Two basic operators Following =-=[20, 21]-=-, we define the two linear operators with h = � N1 (N2 + 1), vra = s1s2, (2.6) 1 s1 = q a(N1+N2)/2 a1+ + e iφr/2 [k − 1] q ! (a1−) k−1 , (2.7) 1 s2 = a2−q −a(N1−N2)/2 iφr/2 + e [k − 1] q ! (a2+) k−1 ,... |

4 | M.: Physique quantique - Bellac - 2003 |

3 |
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Citation Context ... j−1 � m=−j q (j−m)a (−1) j−m−1 � j k j −m − 1 1 m � 4j + 1e i2πjr . (3.15) The second step is to express u (k) p in the enveloping algebra of SU2. This can be achieved by using the formulas given in =-=[42, 43]-=-. Indeed, the operator u (k) p acting on ε(j) reads u (k) � p = �1/2 (k − p)! (−1) (k + p)!(2j − k)!(2j + k + 1)! k+p j p � p (2j − p)!(k + p)! + (−1) p!(k − p)! k� � � + (1 − δk,p) (j + jz + p − z + ... |

3 | On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states
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3 |
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3 |
polytopes and finite geometries
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3 |
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3 | in Quantum Theory of Angular Momemtum - Schwinger - 1965 |

2 |
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Citation Context ... quantum chemistry [2, 3]. More generally, state vectors (1.1) with j integer or half of an odd integer are of considerable interest in electronic spectroscopy of paramagnetic ions in finite symmetry =-=[4, 5]-=- and/or in rotational-vibrational spectroscopy of molecules [6, 7]. It is to be noted that in many cases the labels a and γ can be characterized (at least partially) by irreducible representations of ... |

2 |
On bases for irreducible representations of O(3) suitable for systems with an arbitrary finite symmetry group
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(Show Context)
Citation Context ... the type |jλΓγ〉 = � |jaΓγ〉 Uaλ, a where the unitary matrix U diagonalizes the matrix v set up on the set {|jaΓγ〉 : a ranging}. Integrity bases to obtain v were given for different subgroups G of SO3 =-=[8, 9, 10]-=-. As a résumé, there are several kinds of physically interesting bases for the irreducible representations of SU2. The standard basis, associated with the commuting set {j 2 , jz}, corresponds to the ... |

2 |
Invariants polynomiaux des groupes de symétrie moléculaire et cristallographique Group Theoretical Methods
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(Show Context)
Citation Context ... the type |jλΓγ〉 = � |jaΓγ〉 Uaλ, a where the unitary matrix U diagonalizes the matrix v set up on the set {|jaΓγ〉 : a ranging}. Integrity bases to obtain v were given for different subgroups G of SO3 =-=[8, 9, 10]-=-. As a résumé, there are several kinds of physically interesting bases for the irreducible representations of SU2. The standard basis, associated with the commuting set {j 2 , jz}, corresponds to the ... |