q-analog qudit Dicke states

Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their d-level generalization. We define here q-deformed qudit Dicke states using the quantum algebra suq(d) . We show that these states can be compactly expressed as a weighted sum over permutations with q-factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a q-dependence does not change the circuit gate count.

We consider in this paper a q-analog of qudit Dicke states, which we call q-qudit Dicke states.In principle, there are many possible one-parameter (q) deformations of qudit Dicke states, which reduce to (1.3) in the limit q → 1.We consider here a particular deformation based on Quantum Groups, which has a number of attractive features, as we will see.The qubit case (d = 2) has been considered in [34].Our main result is a formula (3.12) that generalizes (1.3) for q-qudit Dicke states, namely where inv(w) denotes the inversion number and J( ⃗ k) the maximum inversion number, see Sections 2 and 3 for more details.We use this result to compute the entanglement entropy of q-qudit Dicke states (4.7).The inversion number is an important permutation statistic in Combinatorics [33]; it is interesting to see that it and the related q-combinatorial identities (3.17) and (4.6) appear in Quantum Information.We expect that q-qudit Dicke states will find applications similar to those of their undeformed (q = 1) counterparts, but with the advantage of having available a free parameter q as an additional degree of freedom.This paper is organized as follows.In Section 2 we briefly review su q (d), the q-deformation of the su(d) algebra.In Section 3 we use the generators of su q (d) to define q-qudit Dicke states, and we note their key properties.In particular, we note a recursion relation for q-qudit Dicke states (3.8) (proved in Appendix A), which we use to derive the result (1.5).In Section 4 we compute the bipartite entanglement entropy of q-qudit Dicke states, using the Schmidt decomposition (4.2) that is proved with the help of (1.5) in Appendix B. In Section 5 we take advantage of the recursive nature of the q-qudit Dicke states to formulate an efficient deterministic method of preparing these states on a quantum computer.In Section 6 we briefly discuss our results, and note some possible directions for further investigation.Mathematica and Qiskit codes supporting these findings are provided as Supplementary Material.
2 Review of su q (d) We briefly review here the su q (d) algebra with q > 0, following [35,36], which we will use in the following section to define q-qudit Dicke states.
For a single qudit, the generators of the quantum algebra su q (d) = U q (su(d)) are the same as for the classical (undeformed) algebra.In particular, we define the Cartan generators and the Chevalley generators where e ij are elementary d × d matrices with matrix elements (e ij ) ab = δ ai δ bj .Other generators can be obtained by taking commutators.
For n qudits, living in the vector space (C d ) ⊗n , the corresponding generators are given by the coproducts [35,36] ±(1) i occur at the jth location, and we sum j over all n positions.It is important to note for the next section that X ±(n) i may be defined recursively The algebra includes the commutator relations ) where we use the bracket notation For more details about su q (d), see e.g.[35,36].

q-qudit Dicke states
We define here q-qudit Dicke states, and note some of their key properties, including the recursion formula (3.8), the sum formula (3.12) and its extension to complex values of q (3.19), and the duality symmetry (3.22).
The single-qudit computational basis states are given as usual by the d-dimensional vectors The single-qudit operators defined in (2.1), (2.2) perform the following mappings on these basis states:

Definition and recursive property
We use the su q (d) generators (2.3) and (2.4) to define the q-qudit Dicke states for fixed q > 0 and ⃗ k by the "operator formula"1 where and the q-factorial is defined for non-negative integers n by Indeed, for a single qudit, it is easy to see that the nested commutator [X where the basis states are defined in (3.1).Hence, for q → 1, the formula (3.4) can be seen to give the qudit Dicke state (1.3). 2he q-qudit Dicke state (3.4) satisfies the important recursion with the initial conditions |D 1 q (ŝ)⟩ = |s⟩ as given in (3.7),where ŝ is the sth unit vector in d dimensions ŝ = (0 ↑ 0 , . . ., 0, 1 In the sum over s in (3.8), we implicitly skip over s = i if k i = 0, since |D n−1 q ( ⃗ k − î)⟩ is then not defined.A detailed proof of this recursion is given in Appendix A; the main idea is to exploit the recursive nature of X −(n) i in (2.5).As we shall see in Section 5, it is due to the recursive nature of the q-qudit Dicke states that these states can be efficiently generated on a quantum computer.

Sum formula
The q-qudit Dicke states can be written explicitly in terms of permutations, generalizing the q = 1 result (1.3).To do so, we need to borrow a notion from Combinatorics (see, e.g.[33]).Every permutation w of M ( ⃗ k) can be regarded as a "word" with n "letters" w i in the alphabet {0, 1, . . ., d − 1}, i.e. w = w 1 w 2 . . .w n .We refer to the permutation in weakly increasing order, satisfying w i ≤ w j for all i < j, e.g.0112, as the identity permutation e( ⃗ k).The inversion number of a permutation w, written inv(w), represents the minimum number of adjacent transpositions it takes to go from the identity permutation to w.The inversion number of w can be efficiently computed by where z j equals the number of letters w i > w j satisfying i < j; that is z j is the number of letters to the left of w j larger than w j .For example, for w = 1201, we have and hence inv(w) = 3.
We let J( ⃗ k) denote the maximum inversion number of all permutations of M ( ⃗ k); clearly this corresponds to the permutation in reverse-order of the identity permutation e( ⃗ k) (i.e.weakly decreasing).Thus where the term k i k j with i < j represents the k j j's to the left of the k i i's in the reverse of the identity.
We claim that the q-qudit Dicke states (3.4) may be expressed by the "sum formula" where we sum over all permutations w of M ( ⃗ k).For example, the q-qudit Dicke state corresponding to (1.4) with ⃗ k = (1, 2, 1) is To prove that the sum formula (3.12) is valid, all we need to do is show that this formula satisfies the recursion in (3.8) along with the same initial conditions.Clearly, (3.12) has the same initial conditions |D 1 q (ŝ)⟩ = |s⟩.
Note that any permutation w that ends with the letter s can be expressed as The inversion numbers for w and w s are related by where the sum denotes the number of adjacent transpositions it takes to move the right-most s in the identity permutation |e( ⃗ k)⟩ n to the nth position, obtaining |e( ⃗ The sum ∑ r≠s k r represents the number of non-s letters that cross over the right-most s in e( ⃗ k) when reversing the identity permutation e( ⃗ k).Substituting J( ⃗ k), inv(w), and |w⟩ in terms of J( ⃗ k− ŝ), inv(w s ), and |w s ⟩⊗|s⟩ in (3.12), using the fact and summing over all possible s, we see that (3.12) satisfies recursion (3.8).That the factor with inv(w) is present in front of |w⟩ in (3.12) is not entirely surprising, as the proof of the recursion of the q-qudit Dicke states (see Appendix A) makes use of counting interchanges of operators that each produces some q-factor.
The sum formula (3.12) makes clear the orthogonality of any pair of q-qudit Dicke states with different ⃗ k's.The identity 3 confirms that the q-qudit Dicke states (3.12) are normalized to unity, and thus are orthonormal.This identity also shows that the q-multinomial can be expressed as a Laurent polynomial in q with non-negative integer coefficients cf. the denominator in (3.13), which is a nontrivial generalization of the fact [n] = q n−1 + q n−3 + ⋯ + q 3−n + q 1−n for positive integers n.

Complex q
We have so far assumed that the deformation parameter q is positive q > 0. Let us now briefly consider a complex deformation parameter q = qe iα , with q = |q| > 0 and α real.It is not obvious how to define corresponding q-qudit Dicke states; indeed, replacing q by q in (3.4) could lead to complications if q is a root of unity, i.e. q p = 1 for some positive integer p, since then [p] = 0. We therefore define q-qudit Dicke states instead by the recursion (3.8) with q replaced by q, i.e.
where the brackets are given by (2.8) with q = |q| > 0, which is well-defined (after choosing the square-root branch cut) even if q is a root of unity.It follows that 3 This identity has a long and interesting history recounted in [33].In Eq. (1.68) of [33], this identity is written in the form where the boldface denotes the q-multinomial using a definition of q-deformation that is different from (2.8), namely (x) . Substituting this into (3.16) and letting q = q −2 leads to our form (3.17).
where again the brackets are given by (2.8) with q = |q| > 0.
As an example, let us consider the case d = 2 , n = 3 , ⃗ k = (2, 1) with q = e 2πi/3 .The brackets are to be computed with the absolute value of q; that is, with q = |q| = 1.Therefore, and the q-qudit Dicke state (3.19) is given by Note that defining the q-qudit Dicke states by (3.18) ensures that they are properly normalized and q † is given by the complex conjugate of q.The special case q = −1, which corresponds to antisymmetric states, has been considered in the literature, see e.g.[25,27,31,37] and references therein.

Duality symmetry
The sum formula (3.19) can be used to show that the q-qudit Dicke states have the "duality" symmetry where rev denotes the reverse of ⃗ k, and C is the d × d antidiagonal "charge conjugation" matrix which performs the mapping on the single-qudit basis states (3.1).This is a generalization of the well-known duality property of qubit Dicke states |D n (n − l, l)⟩ = X ⊗n |D n (l, n − l)⟩.

Entanglement entropy
Quantum entanglement is a key resource of Quantum Information.An important measure of quantum entanglement is the bipartite entanglement entropy, see e.g. the reviews [38,39].For qubit Dicke states (d = 2 , q = 1), the bipartite entanglement entropy was computed in [10][11][12]; this result was generalized for qudit Dicke states (general d, but q = 1) in [26,30].Moreover, the generalization to q-qubit Dicke states (d = 2 , q ≠ 1) was done in [34].We extend the entanglement entropy computation here to the general case of q-qudit Dicke states, with q > 0; in fact, for the case of complex q where the states are given by (3.19), the entropy has a similar form as the case q = |q|.
In order to compute the Von Neumann bipartite entanglement entropy, we begin by finding the Schmidt decomposition of the q-qudit Dicke states.We recall that the d-tuple ⃗ k consists of non-negative integers k i satisfying k is then referred to as a weak d-composition of n [33], so that we can associate all weak compositions with q-qudit Dicke states.For example, ⃗ k = (1, 2, 1) is a weak 3-composition of 4. We wish to partition the state with n qudits into two parts, with l and n − l qudits, for a positive integer l < n.

Thus, given ⃗
k and l, we consider d-tuples ⃗ a that are weak d-compositions of l satisfying a i ≤ k i .We let A l ( ⃗ k) denote the set of all such ⃗ a; explicitly, For example, for ⃗ k = (1, 2, 1) and l = 2, we have With these conditions, we have essentially 'cut' ⃗ k into ⃗ a and ⃗ k − ⃗ a.We can then associate the q-qudit Dicke states |D l q (⃗ a)⟩ and The Schmidt decomposition of the q-qudit Dicke states for fixed l can be written as with A proof of this decomposition is given in Appendix B. We note that √ λ l q ( ⃗ k, ⃗ a) can be understood as Clebsch-Gordon coefficients relating symmetric representations of su q (d).In the limit as q → 1, λ l q ( ⃗ k, ⃗ a) reduces to the multivariate hypergeometric distribution [30], which has a clear combinatorial meaning: it represents the ratio of the number of permutations w of M ( ⃗ k) that contain in the first l positions the letters i with multiplicity a i and in the last n − l positions the letters i with multiplicity k i − a i to the number of all permutations w.
| be the density matrix of a pure q-qudit Dicke state, which in light of the Schmidt decomposition (4.2) can be expressed as The reduced density matrix ρ l q ( ⃗ k) is obtained by tracing over the last n − l qudits, which, using the orthonormality of the q-qudit Dicke states, forces implying ρ l q ( ⃗ k) is diagonal in the q-qudit Dicke state basis, with eigenvalues λ l q ( ⃗ k, ⃗ a).The eigenvalues of a density matrix are non-negative and sum to unity, implying which we recognize as the q-Vandermonde identity for multinomials. 4The Von Neumann bipartite entanglement entropy is given by where λ l q ( ⃗ k, ⃗ a) is given by (4.3).For the general case of complex q, where the states are given by (3.19), the eigenvalues of ρ l q ( ⃗ k) are given by |λ l q ( ⃗ k, ⃗ a)|, which simplifies to λ l q ( ⃗ k, ⃗ a) with q = |q|, see (4.3).It follows that the entanglement entropy for complex q is given by (4.7) with q = |q|.
We also see from Fig. 1a (blue vs red) that states whose ⃗ k's have similar proportions (in this example, k 0 ≥ k 1 ≥ k 2 ) have EE curves (EE vs. l) with similar shape.On the contrary, we see from Fig. 1b

State preparation
The problem of preparing a general quantum state on a quantum computer [41,42] (i.e., by acting with unitary transformations on a simple reference state) is interesting but difficult.For the case of qubit Dicke states (d = 2 , q = 1), an efficient deterministic algorithm has been developed, see [15][16][17][18] and references therein.A generalization to the case of qudits (general d , q = 1) was recently given in [32].We consider here briefly the problem of preparing q-qudit Dicke states, and find that adding a q-dependence adds little difficulty to the algorithm.We begin by laying out the general approach for arbitrary d, and then work out in detail a suitable circuit for the case of qubits (d = 2).Finally, we show that this circuit can be simplified by removing certain gates.

General d
Similarly to [15,32], we begin by looking for a unitary operator U n (independent of ⃗ k) that generates |D n q ( ⃗ k)⟩ for all ⃗ k by acting on the identity permutation |e( ⃗ k)⟩ , for all ⃗ k, where q = qe iα with q = |q| > 0 and α real.The recursive nature of the q-qudit Dicke states (3.18) indicates that U n can be constructed recursively, as the q-qudit Dicke states on the LHS and RHS of (5.2) can be constructed by applying U n and U n−1 to the states |e( ⃗ k)⟩ and |e( ⃗ k − ŝ)⟩ (for all s), respectively.This motivates finding an operator W n (independent of ⃗ k) that performs the mapping for all ⃗ k.In terms of this operator W n , we can clearly see the recursion (5.4) Using the initial condition U 1 = I, we can telescope the recursion (5.4) into a product of W m operators where the product goes from left to right with increasing m.The problem therefore reduces to constructing quantum circuits for the W m operators.

d = 2
As an example, we consider the simplest case, namely d = 2 (qubits).We associate ⃗ k = (k 0 , k 1 ) with (n − l, l), so that (5.3) reduces to finding a gate decomposition for W m (m ≤ n) that satisfies for all l = 1, 2, . . ., m − 1 (W m acts as the identity when l = 0 or l = m), where as before the brackets are given by (2.8) with q = |q| > 0. We introduce the operator I m,l acting on the lth, (l − 1)th, and 0th qubit, that performs the transformation5 and otherwise acts as identity (as long as the 0th qubit is in the state |1⟩).The corresponding circuit diagram is given by Fig. 2, with one-qubit unitary u-gates whose angles θ, ϕ, λ depend on m, l, q, α as follows: In this Section, following Qiskit conventions, we label m-qubit vector spaces from 0 to m − 1, going from right to left; and in circuit diagrams, the m vector spaces are represented by corresponding wires labeled from the top (0) to the bottom (m − 1).
Hence, a quantum circuit that performs the transformation (5.6) for all l = 1, 2, . . ., m − 1 is given by an ordered product of such operators where the product goes from right to left with increasing l.
As an explicit example with n = 5, we see from (5.5) that and the corresponding complete circuit diagram is shown in Fig. 3.This circuit can be used to prepare the 5-qubit q-Dicke state |D 5 q (5 − l, l)⟩ = U 5 |e(5 − l, l)⟩ from the initial state |e(5 − l, l)⟩ for any l ∈ {1, . . ., 4}.For the particular case l = 3, the shaded gates are redundant and can therefore be removed, as explained below.

W4 W3 W2
Figure 3: Circuit diagram for U 5 (5.12), with W 's (separated by red dashed lines) given by (5.11) in terms of I's in Fig. 2.This circuit can be used to prepare the state |D 5 q (5 − l, l)⟩ for any l ∈ {1, . . ., 4}.For the particulr case l = 3, the shaded gates can be removed.

Simplifying the operators
While the U n operator (5.5) in terms of W m 's (5.11) does generate a q-qubit Dicke state for any ⃗ k = (n − l, l), its gate count can generally be reduced.Indeed, we now proceed to prune away the redundant gates, and thereby remain with a simplified operator U n (n − l, l) in terms of corresponding simplified operators W m (n − l, l), such that U n (n − l, l) |e(n − l, l)⟩ = |D n q (n − l, l)⟩ , (5.13) which are tailored for a fixed value of l.We begin by considering how the right-most factor in (5.5), W n , acts on |e(n − l, l)⟩ for a fixed l.It is clear that we can remove n − 2 factors in the product (5.11),simplifying to W n (n − l, l) = I n,l .For example, for l = 3 in Fig. 3, we can remove gates I 5,1 , I 5,2 , I 5,4 in W 5 .
We next consider how W n−1 ⊗ I acts on I n,l |e(n − l, l)⟩.Rewriting (5.11) as we find that all the terms in the right-most product can be removed, as their controls are in qubit positions between and including 1 and l − 1, where the qubits take the value of |1⟩.For example, for l = 3 in Fig. 3, the term I 4,1 in W 4 can be removed, as the controls lie on wires 1 and 2. Similarly, the terms in the left-most product can be removed as they also never get activated.Thus, the factor W n−1 in (5.5) can be simplified to Similar analysis can be done on the general W m factors in the product (5.5),leading us to generalize [15] to q-qubit Dicke states by creating the quantum circuit for fixed l where performs the transformation (5.6) for a fixed l.Note that the presence of max/min in (5.16) is simply enforcing the requirement 1 ≤ j ≤ m − 1 in I m,j .
The number of I-gates in U n (n − l, l) is given by which satisfies I n (l) = I n (n − l), and I n (l) ∼ ln for l ≪ n.We see that there is no gate count advantage to using the dual symmetry (3.22) |D n q (n − l, l)⟩ = X ⊗n |D n 1/q (l, n − l)⟩ as the gate count is symmetric under l → n − l.However, by using the dual symmetry to restrict to l ≤ n/2, one can minimize the separation of the wires on which the I-gates act.
Considering as an example the particular case (n − l, l) = (2, 3), we see from Eqs. (5.13) and (5.15) that |D 5 q (2, 3)⟩ = U 5 (2, 3) |e(2, 3)⟩, with where the W's are given by (5.16), as shown in Fig. 4. Note that this circuit can be obtained by removing the shaded gates from the circuit in Fig. 3.An implementation in Qiskit of this and additional examples, as well as Mathematica code for verifying the results, are available as Supplementary Material.
Note that the only dependence on q is through the angles in the u(m, l) gates (5.8); hence, the gate count for general q is the same as for q = 1 [15][16][17][18].For d > 2, we expect that the q-dependence can be implemented in a similar way.

Conclusion
We have used Quantum Groups to define the notion of q-qudit Dicke states through the operator formula (3.4), and we have shown that these states can be expressed by the compact and useful sum formula (3.12).0 |1⟩ u(5, 3) Figure 4: Circuit diagram for preparing the state |D 5 q (2, 3)⟩ = U 5 (2, 3) |e(2, 3)⟩, where U 5 (2, 3) is given by (5.18) and the W's (separated by red dashed lines) are given by (5.16) in terms of I's in Fig. 2.
A key ingredient of the latter is the inversion number, which is an important permutation statistic in Combinatorics.Basic properties of these states and their bipartitions are expressed by celebrated q-combinatorial identities (3.17), (4.6).This interplay among Quantum Groups, Combinatorics and Quantum Information suggests that there may be deeper relations among these subjects.
We have seen that the recursive nature of the q-qudit Dicke states (3.8) leads to a simple algorithm for their construction on a quantum computer.We note that the Schmidt decomposition (4.2) can be regarded as a generalization of this recursion; indeed, (3.8) can be obtained by letting l = n − 1 in (4.2).It should be possible to further generalize the Schmidt decomposition of q-qudit Dicke states to multipartite decompositions, and study multipartite entanglement.It would be interesting to see if other algorithms can be obtained by similarly exploiting these generalizations of the recursion (3.8).It may also be possible to compute the n → ∞ limit of the bipartite entanglement entropy with q > 0, as has been done for the undeformed (q = 1) case in [10,11,26,30].
As noted in the Introduction, Dicke states have been exploited for numerous tasks in quantum information and computation.We expect that q-qudit Dicke states will find similar applications, but with the advantage of having available a free parameter q as an additional degree of freedom.Larger values of d will allow for more choices of ⃗ k, and therefore richer structure, see e.g.Fig. 1.As discussed in Section 5, the preparation of such states is not much more difficult than for undeformed qudit Dicke states.

A.1 New operators and their properties
Let us recall that the single-qudit operators H (1) i (2.1) and X ±(1) i (2.2) perform the mappings (3.2) and (3.3), respectively, on the computational basis states.We now define new operators on n qudits6 which, for n = 1, perform the mappings It is natural to wonder if the X (n) i operators satisfy relations analogous to (2.4) and (2.5).In fact, where "rem" denotes additional remainder terms that vanish when applied to the state Σ|0, i, i + 1, . . ., d − 1⟩, where Σ|0, i, j, . . .⟩ denotes an arbitrary sum of kets of n qudits with each ket consisting of 0's, i's, j's, etc.This may be understood by substituting (2.4) into (A.1): When the products are expanded into a sum of terms, each such term will have exactly i operators X −(1) k 's spread throughout the n locations.The n terms for which the X −(1) k 's occur at the same location together form the square-brackets term in (A.5).The rest of the terms have an interrupted sequence of lowering operators at some location (e.g. the expansion of X (n) 3 includes terms which some location equals , which is missing X ) and thus acts trivially on the state Σ|0, i, i + 1, . . ., d − 1⟩ using (3.3).
We conclude this subsection with two useful claims.