Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States.

A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl-Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of N = d - 1 symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl-Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a d-dimensional space) is describable by a N-qubit vector (in a N-dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the N qubits makes it possible to measure the entanglement between the N qubits forming the qudit. This is confirmed by a Fubini-Study metric analysis. A new parameter, proportional to the permanent and called perma-concurrence, is introduced for characterizing the entanglement of a symmetric qudit arising from N qubits. For d = 3 ( ⇔ N = 2 ), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for d = 4 and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article.

1. Introduction

Geometrical representations are of particular interest in various problems of quantum mechanics. For instance, the Bloch representation is widely used in the context of characterizing quantum correlations in multiqubit systems [1,2,3]. This representation is based on the idea of Majorana to visualize a j-spin as a set of points in a sphere [4]. The Bloch sphere was used in the study of entanglement quantification and classification in multiqubit systems [5,6,7]. The investigation and the understanding of quantum correlations in multipartite quantum systems are essential in several branches of quantum information such as quantum cryptography [8], quantum teleportation [9] and quantum communication [10,11].

The separability for two-qubit states can be addressed with the concept of the Wootters concurrence [12,13]. However, for multiqubit quantum systems, the measure of quantum correlations is very challenging. Several ways to understand the main features of entangled multiqubit states were employed in the literature [14,15].

Algebraic and geometrical methods were intensively used in quantum mechanics [16,17,18,19] and continue nowadays to contribute to our understanding of entanglement properties in multipartite quantum systems (for instance, see [20,21,22,23]). In this spirit, several works were devoted to geometrical analysis of entangled multipartite states to find the best measure to quantify the amount of entanglement in a multiqubit system [5,20,24,25,26,27,28,29,30]. The classification of multipartite entangled states was investigated from several perspectives using different geometrical tools [31,32,33] to provide the appropriate way to approach the quantum correlations in multiqubit states. Among these quantum states, j-spin coherent states are of special interest [34]. Indeed, they are the most classical (in contrast to quantum) states and can be viewed as -qubit states which are completely separable. In this sense, spin coherent states can be used to characterize the entanglement in totally symmetric multiqubit systems [35].

The multipartite quantum states, invariant under permutation symmetry, have attracted a considerable attention during the last decade. This is essentially motivated by their occurrence in the context of multipartite entanglement [7,27,36,37,38,39,40,41] and quantum tomography [42,43,44]. In fact, the dimension of the Hilbert space for an ensemble of N qubits system reduces to when the whole system possesses the exchange symmetry. The appropriate representations to deal with the totally symmetric states are the Dicke basis [45,46,47] and Majorana representation [4].

In the present paper, we consider a realization of the generalized Weyl–Heisenberg algebra, introduced in [48,49,50], by means of an ensemble of two-qubit operators. We investigate the correspondence between the vectors of the representation space of the Weyl–Heisenberg algebra and the Dicke states. Using the decomposition properties of Dicke states, we show that the separable states are necessarily the Perelomov coherent states associated with the generalized Weyl–Heisenberg algebra. The coherent states are written as tensor products of single qubit coherent states. We also discuss the separability in terms of the permanent of the matrix of the overlap between spin coherent states.

2. Qubits and Generalized Weyl–Heisenberg Algebra

2.1. Bosonic and Fermionic Algebras

The study of bosonic and fermionic many particle states is simplified by considering the algebraic structures of the corresponding raising and lowering operators. On the one hand, for bosons, the creation operators and the annihilation operators satisfy the commutation relations where stands for the identity operator. On the other hand, fermions are specified by the following anti-commutation relations of the creation operators and the annihilation operators . The properties of Fock states follow from the commutation and anti-commutation relations which impose only one particle in each state for fermions (in a two-dimensional space) and an arbitrary number of particles for bosons (in an infinite-dimensional space). Following Wu and Lidar [51], there is a crucial difference between fermions and qubits (two level systems). In fact, a qubit is a vector in a two-dimensional Hilbert space as for fermions and the Hilbert space of a multiqubit system has a tensor product structure similar to for bosons. In this respect, the raising and lowering operators commutation rules for qubits are neither specified by relations of bosonic type (1) nor of fermionic type (2).

2.2. Qubit Algebra

The algebraic structure relations for qubits are different from those defining Fermi and Bose operators. Indeed, denoting by and the states of a two-level system (qubit), the lowering (), raising (), and number (K) operators defined by satisfy the relations (we use to denote the adjoint of A). Furthermore, the creation and the annihilation operators satisfy the nilpotency conditions as in the case of fermions.

We note that the commutation relations in Equation (6) coincide with those defining the algebra introduced in [52] to provide an alternative algebraic description of qubits instead of the parafermionic formulation considered in [51]. In addition, the generalized oscillator algebra introduced in [49] as a particular case of the generalized Weyl–Heisenberg algebra of bosonic type [48] provides an alternative description of qubits (in [50], the algebra is also denoted as in view of its extension to ). In fact, Equation (6) corresponds to .

2.3. Qudit Algebra

To give an algebraic description of d-dimensional quantum systems , we consider a set of qubits. We denote as , , and the raising, lowering, and number operators associated with the i-th qubit. They satisfy relations similar to Equation (6), namely, and for .

Let us denote as the two-dimensional Hilbert space for a single qubit. An orthonormal basis of is given by the set

The multiqubit -dimensional Hilbert space for the N qubits has the following tensor product structure (with factors), similar to for bosons. In other words, the set where constitutes an orthonormal basis of . The Dicke states shall be defined in Section 3 as linear combinations of the states .

We define the collective lowering, raising and number operators in the Hilbert space as follows in terms of the annihilation, creation, and number operators , , and , respectively. In Equation (10), should be understood as the operator , where stands, among the N operators, at the i-th position from the left. It is trivial to check that

The action of and on vectors involving qubits and , as for Dicke states, shall be considered in Section 3.

By using Equations (7), (9), and (10), we obtain for . In particular, for , the relations (11) give which lead to the nilpotency relations

Equation (12) for () gives back Equation (7) which is reminiscent of the Pauli exclusion principle for fermions.

In view of Equations (8) and (10), the qudit operators , , and K satisfy the commutation rules which are similar to the relations defining the generalized Weyl–Heisenberg algebra introduced in [49]. More precisely, let us put

Then, we have the relations where the parameter is

Therefore, the operators , , and K generate the algebra with . This shows that the algebra can be described by a set of N qubits. According to the analysis in [49], since , the algebra admits finite-dimensional representations. Indeed, we shall show that the representation constructed on the basis of the Dicke states (see Section 3) is of dimension .

Note that the lowering and raising operators and close the following trilinear commutation relations similar to in a para-fermionic algebra [53]. Note that the definition in Equation (10) is also identical to the decomposition used by Green for defining para-fermions from ordinary fermions [54].

3. Dicke States

3.1. Definitions

The Hilbert space can be partitioned as where the sub-space is spanned by the orthonormal set

Each vector of contains qubits and k qubits . The dimension of the space is given by in terms of the binomial coefficient , and satisfies

Clearly, is invariant under any of the permutations of the N qubits. The orthogonal decomposition in Equation (17) of turns out to be useful in the definition of Dicke states.

To each , it is possible to associate a Dicke state which is the sum (up to a normalization factor) of the various states of . To be more precise, let us define the Dicke state as follows [46,47] where the numbers 0 and 1 in the vector are and k, respectively. Furthermore, the summation over runs on the permutations of the symmetric group restricted to the identity permutation and the permutations between the 0s and 1s (the permutations between the various 0s as well as those between the various 1s are excluded, only the permutations between the 0s and 1s leading to distinct vectors are permitted). Each vector in Equation (18) involves qubits. A Dicke state is thus a normalized symmetrical superposition of the states of . More precisely, Equation (18) means

Indeed, each Dicke state and, more generally, any linear combination of the Dicke states (with ) transform as the totally symmetric irreducible representation of the group of the permutations of the qubits.

As a trivial example, for the Dicke states and are nothing but the one-qubit states and , respectively (these qubit states are generally associated with the angular momentum states and , respectively). As a more instructive example, for , we have the Dicke states

Each vector is a symmetric (with respect to ) linear combination of the vectors of .

For fixed N, we have so that the set constitutes an orthonormal system in the space . Let us denote as the space of dimension spanned by the symmetric vectors with . Then, the set is an orthonormal basis of .

3.2. Dicke States and Representations of

The nilpotency relations in Equation (12) imply that the representation space of the generalized Weyl–Heisenberg algebra with , see Equations (13)–(16), is of dimension . The representation vectors can be determined using repeated actions of the raising operator combined with the actions of the operators and defined by relations similar to Equations (3)–(5). It can be shown that these representation vectors are Dicke states. The proof is as follows.

First, the action of the operator on the ground state of yields or equivalently

Second, the action of on gives or

From repeated application of the raising operator on the state of , we obtain

By using Equation (20), we finally get the ladder relation

Similarly, for the lowering operator , we have

Equations (21) and (22) can be rewritten as where and

Note that gives the action of the raising and lowering qubit operators on the extremal Dicke states and of .

The Dicke states are eigenstates of the operator K defined in Equation (10), see also Equation (5) in the case . Indeed, we have in agreement with the fact that K is a number operator: it counts the number of qubits of type in the Dicke state . From Equations (23)–(25), we recover the commutation relations in Equation (13). Therefore, the generalized Weyl–Heisenberg algebra , with , generated by the operators , , K, and provides an algebraic description of a qudit (d-level) system viewed as a collection of qubits. In fact, the vectors of the representation space of the algebra are the Dicke states which are symmetric superpositions of states of a multiqubit system.

3.3. Decomposition of Dicke States

Let us consider again the action of on the ground state involving N qubits . We have seen that see Equation (19). On another side, we have

This gives which can be rewritten as where the states on the right-hand side member contains qubits. Thus, we get

A comparison of Equations (26) and (27) yields

By applying the creation operator on both sides of Equation (28), and by using Equation (21), we obtain

Repeating this process k times, we end up with

Equation (29) can be simplified to give where .

For and , there are two terms in the decomposition of : one is a tensor product involving the qubit and the other a tensor product involving the qubit . The decomposition of Equation (30) of the Dicke states is trivial in the cases and . For and , the significance of Equation (30) is clear. These two particular cases correspond to a factorization of the Dicke state for N qubits into the tensor product of a Dicke state for qubits with a state for one qubit.

3.4. Dicke States and Angular Momentum States

To close this section, a link between Dicke states and angular momentum states is in order. The Lie algebra su(2) of the group SU(2) can be realized by means of the angular momentum operators , , and . The irreducible representation of su(2) can be constructed from the set of angular momentum states. We know that according to the Condon and Shortley phase convention [55]. Let us put

Therefore, the state can be denoted as since, for fixed j, then N is fixed and implies . Consequently, Equations (31)–(33) can be rewritten as to be compared with Equations (21), (22), and (25). This leads to the identification which establishes a link between the Weyl–Heisenberg algebra with and the Lie algebra su(2). The rewriting of the Dicke state , see Equation (18), in terms of the variables j and m yields where the summation over runs on the permutations of the symmetric group restricted to the identity permutation and the permutations between the states and exclusively (only the permutations leading to distinct vectors are permitted).

4. Separable Qudit States

4.1. Factorization of a Qudit

In this section, we start from a qudit (d-level state) and study on which condition such a state is separable in the direct product of qubit states.

The most general state in the space can be considered as a qudit constituted from qubits. In other words, in terms of Dicke states we have

We may ask the question: On which condition can the vector be factorized as involving a state for qubits and a state for one qubit?

The use of Equation (30) yields which can be rewritten as where

Clearly, the state is separable if there exists z in such that

Then, where

It is easy to show that Equation (36) implies

Consequently, we get the recurrence relation that admits the solution where the coefficient can be calculated from the normalization condition . This leads to up to a phase factor. Thus, the introduction of Equation (38) into Equation (34) leads to the separable state

To identify the various factors occurring in the decomposition of the separable state in Equation (39), as a tensor product, we note that the use of Equation (38) in Equation (35) gives

Hence, Equation (37) takes the form where stands for a single qubit in the Majorana representation [4] (the vector is nothing but a SU(2) coherent state for a spin as can be seen by identifying the qubits and to the spin states and , respectively). By iteration of Equation (40), we obtain with N factors .

As a résumé, we have the following result. If the qudit state given by Equation (34) is separable, then it can be written so that is completely separable into the tensor product of N identical SU(2) coherent states for a spin .

4.2. Separable States and Coherent States

Let us consider the unitary displacement operator for the i-th qubit. The action of on the i-th qubit can be calculated to be

By introducing into Equation (43), we obtain

Hence, we have where is the coherent state defined in Equation (41). This well-known result can be extended to the case of N qubits. The action of the operator , where and are given in Equation (10), on the Dicke state reads

Therefore, the separable state given by Equation (42) can be written in three different forms, namely where the last member coincides, modulo some changes of notation, with the Perelomov coherent state derived in [56] (see Formulas (122) and (123) in [56]).

5. Majorana Description

We now go back to the general case where the qudit state of is not necessarily a separable state. This state (normalized to unity) can be written in two different forms, namely, as in Equation (34) or, according to the Majorana description [4], as (see Section 7 for a discussion of the equivalence between Equations (44) and (45) in the framework of the Bargmann function associated with and the so-called Majorana stars). In Equation (45), the state (with ) is given by Equation (41) with . Furthermore, is a normalization factor and the sum over runs here over all the permutations of the symmetric group . The coefficients can be expressed in terms of the coefficients . The case where N is arbitrary is rather intricate. Therefore, for pedagogical reasons, we start with the case of qubits.

5.1. The Case

For (), on the one hand we have where the Dicke states with are

On the other hand,

Therefore, we have to compare with

This leads to

Of course, the complex numbers and are the roots of the equation

Therefore, by combining Equations (47) and (48), we end up with the quadratic equation so that and are given by for ( for ). Observe that, when the so-called concurrence C defined by (see Ref. [13]) vanishes, we have . Therefore, the state is separable.

5.2. The Case N Arbitrary

The case N arbitrary is very much involved. Equations (47) and (49) for can be generalized as follows. In the general case of N qubits, the vector of the space , normalized via , is given by Equation (44) in terms of Dicke states or by Equation (45) in the Majorana representation. The coefficients are connected to the complex numbers through where is the elementary symmetric polynomial (invariant under ) in N variables defined as and the normalization factors are given by and

Note that where stands for the permanent of the matrix of elements

Finally, for fixed , the numbers are the roots (Majorana roots) of the polynomial equation of degree N which generalizes (49).

The complete proof of Equation (56) is based on the fact that two generic qubit states and , with , are orthogonal if and only if the variables and satisfy . The state is orthogonal to the N states for . This orthogonality condition shows that the variables are indeed solutions of Equation (56).

To sum up, we have the following central result. Any vector in the space reads where the normalization factors , and (with ) can be calculated from Equations (52) and (53) and the variables are given in terms of by Equation (56). Note that Equation (53) can be rewritten as so that

Therefore, Equation (57) becomes up to a phase factor.

As a check of the last result, note that the introduction of Equation (38) into Equation (56) yields a trivial identity. Furthermore, in the particular case where the solutions of Equation (56) are identical, i.e., then Equation (58) leads to the completely separable state in Equation (42). In this particular case, from Equation (55) we have (which is the maximum value of ). Therefore, in the general case, the quantity can be used for characterizing the degree of entanglement of the state .

5.3. The Cases , 3, 4, and 5

5.3.1. Case

The state is the most general qubit (linear combination of the basic qubits and ). Of course, the notion of separability does not apply in this case.

5.3.2. Case

The general normalized qutrit vector is where the Dicke states with are cf. Equation (46). In the Majorana description, Equations (45), (54), and (59) give with where and are the roots of Equation (50) of the quadratic Equation (49).

It can be shown that where the concurrence C for a two-qubit system is defined by Equation (51). Thus, another expression for C is

The possible values of C and are

Therefore, a vanishing concurrence (which reflects the absence of entanglement) corresponds to ; in the particular case , we have that leads to the separable state . Furthermore, for (which characterizes entangled states), we have . Consequently, in the general case ( and arbitrary), constitutes an alternative to the concurrence C for measuring the degree of entanglement of the general qutrit .

It is interesting to note that can be alternatively written as where the vectors (with and ) are unit vectors in the space which serve to locate points on the Bloch sphere. Therefore, entangled states are obtained for (in this case, takes its minimal value ).

Note the following relation valid for arbitrary i and j. This relation will be useful for deriving closed-form expressions of in higher dimensional cases.

5.3.3. Case

In this case, the general state of is made of qubits. It takes the form where the Dicke states with are In the Majorana representation, we have where the states are given by Equation (41) with and the complex numbers are solutions of the polynomial equation of degree 3

The normalization factor reads with or alternatively where the components of the vectors () are given by Equation (60). From Equation (61), we get that clearly shows that

The case of complete separability corresponds to . The minimal value is obtained for entangled states.

5.3.4. Case

In this case, the variables () are solutions of the equation of degree 4

The calculation of yields or with

The minimal value can be obtained from or from any analogue equality deduced from Equation (62) by permutations of the indices .

5.3.5. Case d Arbitrary

The general case is approached in Section 5.2. For d arbitrary, it can be shown that the situation where corresponding to complete separability and to entangled states. Therefore, the parameter can serve as a measure of the entanglement of the symmetric qudit state described by qubits.

The minimal value of can be obtained when

Thus, Equation (59) can be reduced to

The condition in Equation (63) implies that

In this case, we have and the minimal value of is

The same result can be obtained equally well, due to the invariance of under permutation symmetry, from any of the following conditions instead of the condition in Equation (63).

6. Fubini–Study Metric

6.1. The Separable Case

The adequate approach to deal with the geometrical properties of a quantum state manifold is based on the derivation of the corresponding Fubini–Study metric [57]. The Fubini–Study metric is defined by the infinitesimal distance between two neighboring quantum states. This derivation is simplified by adopting the coherent states formalism. Indeed, for a single qubit coherent state this is realized in the following way. Let us define the Kähler potential as

Using Equation (41) of the coherent state , we have and the metric tensor becomes so that the Fubini–Study metric reads which coincides with the metric of the unit sphere. This provides us with a simple way to describe the 2-sphere , or equivalently the complex projective space , usually regarded as the space of states of a -spin particle.

This can be generalized to the completely separable state constructed from the tensor product of N qubit coherent states. In this case, the Kähler potential is given by

This leads to

The metric tensor g is defined via its components

Finally, the Fubini–Study line element is associated with the complex space .

In the special case where the complex variables are identical, i.e., , the state reduces to the coherent state given by Equation (42). In this case, the Fubini–Study metric takes the form which describes the unit 2-sphere, of radius , written in stereographic coordinates.

6.2. The Arbitrary Case

We now apply the just described geometrical picture to calculate the Fubini–Study metric for an arbitrary multiqubit symmetric state . Here, we define the Kähler potential through as a generalization of Equations (64) and (65). It is easy to show that

Hence, we obtain in terms of the parameter defined by Equation (59). As a result, the Kähler potential splits into two parts: one term is the Kähler potential corresponding to a completely separable state involving N qubits, cf. Equation (66), and the other term depends exclusively on the parameter which characterizes the degree of entanglement of the state . Then, the components of the corresponding metric tensor g are and the Fubini–Study line element is

In the special case where , corresponding to a completely separable state, the last equation gives back Equation (67) valid for a multiqubit separable state. This is a further indication that the parameter encodes the geometrical aspects due to the entanglement of a multiqubit symmetric state.

7. Majorana Stars and Zeros of the Bargmann Function

7.1. The Main Idea

An arbitrary normalized state of the space can be written either in terms of the Dicke states with (see Equation (44)) or in terms of the coherent states for (see Equation (45)). The variables , called Majorana stars [58], can be determined from the zeros of the Bargmann function associated with the state . In fact, denoting by the zeros of the Bargmann function , we shall show that the Majorana stars can be obtained from the Bargmann zeros via and we shall give the equation satisfied by the variables .

7.2. Determining the Bargmann Zeros

In the analytic Fock–Bargmann representation [59], an arbitrary normalized state of is represented by the Bargmann function defined by where the bra follows from the coherent state corresponding to the completely separable state in Equation (42). Thus, we have which can be decomposed as where with

In fact, the polynomial is of degree , where is the maximum value of the index k for which . Therefore, the polynomial can be factorized as where () are called the Bargmann zeros.

7.3. Expression of in Terms of the Bargmann Zeros

We now look for the expression of the state vector in terms of the Bargmann zeros (). To this end, we remark that the scalar product between the state and the coherent state (see Equation (41)) is

Thus, the polynomial can be written as

Furthermore, by noting that we extend the definition of the states (initially defined for ) by taking so that the Bargmann function takes the form

Since the representation is unique up to permutations of the , the Bargmann function can be rewritten as or alternatively as where the normalization constant is given by

Comparing Equations (68) and (71), we find that the state can be expressed as in terms of the zeros for of the Bargmann function and of their extension .

7.4. Expression of in Terms of the Majorana Stars

We note that the states with can be written in terms of the coherent states by putting

We verify that up to irrelevant phase factors. Hence, the symmetric qudit state given by Equation (72) can be expressed as in terms of the coherent states . Equation (73) is identical to Equation (45); thus, we recover Equation (45).

7.5. Equation Satisfied by the Majorana Stars

The zeros of the Bargmann function in Equation (69) satisfy . From Equation (70), we thus get or in terms of the in agreement with Equation (56).

8. Concluding Remarks

In this study, we discussed the role of a specific generalized Weyl–Heisenberg algebra in the algebraic structure of qubits and qudits. The use of this generalized Weyl–Heisenberg algebra was based on the fact that qubits are neither fermions nor bosons. Indeed, in the standard theoretical approach of quantum information, a qubit is a vector in a two-dimensional Hilbert space as for fermions and the Hilbert space of a multiqubit system has a tensor product structure similar to for bosons. In this respect, the commutation rules of the raising and lowering operators for qubits are not specified by relations of bosonic type or of fermionic type.

By using a collection of qubits, we gave a realization of the d-dimensional representation space of the generalized Weyl–Heisenbeg algebra. In particular, we demonstrated that the vectors of this representation space coincide with the Dicke states. These states are of special interest for describing multiqubit quantum systems possessing exchange symmetry. Another advantage of this algebraic description via the generalized Weyl–Heisenberg algebra concerns the separability of multiqubit states invariant under permutations. Hence, starting from the decomposition of Dicke states, we investigated the condition for the separability of symmetric qudits made of qubit states. Our results show that exchange symmetry implies that the superposition of Dicke states are globally entangled unless they are fully separable and coincide with the coherent states, in the Perelomov sense, associated with the generalized Weyl–Heisenberg algebra.

In the Majorana description of a symmetric qudit state in terms of symmetrized tensor products of qubits, we introduced a parameter connected to the permanent of the matrix characterizing the overlap between the N qubits. This parameter provides us with a quantitative measure of the entanglement for the qudit arising from N qubits. This was illustrated in the special case , for which the parameter constitutes an alternative to the Wootters concurrence C for qubits. Therefore, we propose that be called perma-concurrence as a contraction of permanent and concurrence. Other examples of were given for and 5. The results highlight the interest of the perma-concurrence for measuring the entanglement of a symmetric qudit state developed in terms of tensor products of qubit coherent states.

In Section 7, we further investigated the formalism of qubit coherent states to describe qudit states in the Fock–Hilbert space corresponding to the generalized Weyl–Heisenberg algebra. More precisely, we used the Fock–Bargmann representation for describing any symmetric qudit constructed from qubits with the help of an analytic function, the so-called Bargmann function. The zeros of the Bargmann function were related to the Majorana stars which provide an alternative way to describe Fock–Hilbert states as tensor products of qubit coherent states labeled by complex variables, namely, Majorana stars on the Bloch sphere.

Recently, new entropic and information inequalities for one qudit, which differs from a multiqubit system, have been developed [60]. Therefore, it will be a challenge to ask whether the qudit picture proposed in this paper can be adapted in terms of linear combinations of Dicke states.

To close this paper, note that it might be interesting to introduce Dicke states into the construction of the so-called mutually unbiased bases used in quantum information. This approach, feasible in view of the connection between mutually unbiased bases and angular momentum states [61], could be the object of a future work.