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Symmetric discrete coherent states for n qubits

We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions.


Introduction
Coherent states (CS), first introduced by Schrödinger [1], are of paramount significance for modern physical theories, as they are quantum states that follow classical trajectories. In quantum optics, CS were popularized by Glauber [2,3] for the description of correlation properties of a single-mode radiation field, where the Weyl-Heisenberg group emerges as a hallmark of noncommutativity [4].
Although generalizations along several directions have been considered (see references [5][6][7] for reviews), nowadays it seems indisputable that the pioneer work of Perelomov [8] paved the way to extend the notion of CS to quantum systems with dynamical symmetry group G. In this approach, CS appear as orbits of a certain fiducial state under a unitary irreducible representation of G acting in the corresponding Hilbert space.
This fiducial state is chosen to have maximal isotropy subgroup H, which ultimately leads to "maximal classicality". Under these circumstances, the displacement operators, transforming CS among themselves, are labeled by points in the manifold M = G/H. For many physical models, M can be equipped with an irreducible symplectic structure [9][10][11][12], so it can be considered as the phase space of a classical dynamical system. In other words, there is a one-to-one correspondence between CS and points of the classical phase space.
The crucial point of this construction is that the Hilbert space is irreducible under the action of G. This is especially clear for the symmetric representations of the unitary groups SU(n), when the classical phase space is a (2n − 2)-dimensional sphere [13]. For the discrete counterparts, even in the physically (but not mathematically!) simple n-qubit case, the symmetric subspace is only a very small portion of the whole 2 n -dimensional Hilbert space. Nonetheless, one can define a natural set of CS, constructed in a similar way as in the continuous case: acting on a fiducial state with discrete displacements; i.e., unitary operators labeled by elements of two discrete sets [14]. These two sets can be organized in a discrete 2 n × 2 n grid, on which a specific discrete geometry (including symplectic operations) can be introduced, so that such a grid turns out to be a bona fide discrete phase space [15][16][17][18][19][20][21][22][23].
Although the points of the discrete phase space label again CS, there is still an essential difference with the continuous case: the choice of the fiducial state. For continuous symmetry groups, the standard choice corresponds to an extreme state of the representation, such as the vacuum or the lowest/highest weight state. In the discrete case, the nature of the unitary displacements prevents such a simple notion and different possibilities have been discussed so far [24,25].
In this paper, we take the fiducial state as a spin CS and impose that its associated Q function fulfills reasonable symmetry conditions. This not only solves the problem, but allows us to use the system of CS to impose a partial order in phase space, which helps to recognize states pictured as distributions. Finally, we briefly speculate about detecting quantum correlations through the sum of squared Q functions.

Discrete phase space
A qubit is a two-dimensional quantum system, with Hilbert space isomorphic to C 2 . It is customary to choose two normalized orthogonal states, {|0 , |1 }, as a computational basis. The unitary operators generate the Pauli group P 1 under matrix multiplication [26]. For n qubits, the Hilbert space is the tensor product C 2 ⊗ · · · ⊗ C 2 = C 2 n . A compact way of labeling both states and elements of the corresponding Pauli group P n consists in using the finite field F 2 n [27]. This can be considered as a linear space spanned by an abstract basis {θ 1 , . . . , θ n }, so that given a field element α (henceforth, field elements will be denoted by Greek letters) the expansion allows us the identification α ⇔ (a 1 , . . . , a n ). Moreover, the basis can be chosen to be orthonormal with respect to the trace operation (the self-dual basis); that is, where tr(α) = α + α 2 + . . . + α 2 n−1 , which actually maps F 2 n → Z 2 . In this way, we associate each qubit with a particular element of the self-dual basis: qubit i ⇔ θ i . The generalized Pauli group P n is generated now by the operators Here the additive characters χ are defined as χ(α) = exp[iπ tr(α)] and |λ is an orthonormal basis in the Hilbert space of the system. Operationally, the elements of the basis can be labeled by powers of a primitive element (i.e., a root of the primitive polynomial), and read {|0 , |σ , . . . , |σ 2 n −1 = 1 }. One can verify that which is the discrete counterpart of the Weyl-Heisenberg algebra for continuous variables [4]. The operators (2.4) can be factorized into tensor products of powers of single-particle Pauli operators. This factorization can be carried out by mapping each element of F 2 n onto an ordered set of natural numbers according to where a i = tr(αθ i ) and b i = tr(β θ i ) are the corresponding expansion coefficients for α and β in the self-dual basis. Moreover, they are related through the finite Fourier transform [28] so that X µ = F Z µ F . We next recall [20] that the grid defining the phase space for n qubits can be appropriately labeled by the discrete points (α, β ), which are precisely the indices of the operators Z α and X β : α is the "horizontal" axis and β the "vertical" one. In this grid one can introduce the concept of straight lines (also called rays), which possess the same properties as in the continuous case. It is worth noting that the monomials labeled by points of the same ray, {(α, µα)} commute with each other, so that one can establish a correspondence between eigenstates of such commuting sets and states (actually bases) in the Hilbert space [20].
Following our program, we introduce the set of displacements where Φ(α, β ) is a phase required to avoid plugging extra factors when acting with D. One can immediately check that so they are unitary and Hermitian. They also constitute a complete trace-orthonormal set These operators act multiplicatively on the monomials (2.4), thus shifting phase-space points according to which justifies their designation. Finally, for later purposes, we touch on a pair of symplectic operations (z-and xrotations) that transform rays into rays according to The symbol ∝ indicates equality except for a phase. Both P µ and Q ν are unitary operators, with [P µ , X ν ] = [Q ν , Z µ ] = 0, and can be written as where |λ are the eigenstates of Z α and | λ of X β . The coefficients c λ ,ν fulfill the recurrence relation whose explicit solution can be found in reference [29] and it is unimportant for the rest of the paper.
We associate an eigenstate |ψ 0 of Z α with the horizontal axis and immediately obtain that the state associated with the ray β = µα is P µ |ψ 0 , while the vertical axis is associated with F |ψ 0 [30] . Any other straight line, parallel to a given ray, but crossing the axis β at the point β = ξ , corresponds to the state X ξ P µ |ψ 0 .
Using phase-space coordinates these zand x-rotations can be interpreted as It is clear that these two transformations are conjugate each other.

Discrete coherent states for n qubits
According to the conventional approach, we define discrete CS |α, β , labeled by phase-space points (α, β ), as the displacements of the fiducial state |Ψ 0 [19]: The state |Ψ 0 can be chosen in several ways [24]. Here, for reasons that will be apparent soon, we take |Ψ 0 as a product of identical qubit states: with ξ = e iϕ tan ϑ /2. The basis {|k, n : k = 0, . . . , n} are the Dicke states Here, {P k } denotes the complete set of all the possible qubit permutations.
In field notation, the state (3.4) can be compactly expressed as where the function h(κ) counts the number of nonzero coefficients k j in the expansion of κ in the field basis (see the appendix for a brief account of its properties). Finally, notice that one might think in imposing that the states |ξ are eigenstates of the Fourier operator [31,32] (much as the vacum is for continuous variables). Since F 2 = 1 1, this is tantamount to which leads to two possible candidates ξ ± = ± √ 2 − 1 and all the qubits pointing in the same direction. However, we prefer to follow an alternative route to fix the possible values of ξ .

P-function
Let us look for an expansion of the density matrix of the form which is the analogous to the Glauber-Sudarshan P-function for continuous variables [33,34]. It is not difficult to check that the function P(α, β ) may be recast as where Tr (with capital T to distinguish it from tr, which is the trace in the field) stands for the ordinary trace operation in Hilbert space. The kernel ∆(α, β ) reads This clearly shows that the P-function is nonsingular only when ξ |D(γ, δ )|ξ −1 exists. Since the state |ξ is factorized into single-qubit states, we have where g i , d i are the expansion coefficients of γ and δ in the self-dual basis and Using equation (3.6) one can find This obviously rules out some values of ξ for which P is singular.
As an illustrative example, consider the case of a single qubit. Then, one finds that where now a, b ∈ Z 2 . This function is singular when ξ is real, imaginary, and when |ξ | 2 = 1, i.e. on the equator of the Bloch sphere. In particular, the eigenstates of the discrete Fourier transform, when ξ ± = √ 2 ± 1, do not lead to the faithful expansion on CS for qubits, contrary to what one could expect.

Q-function
In our search for determining the values of ξ , we next look at the Q-function, defined in complete analogy with its continuous counterpart, namely C2 .
To conclude, we observe that, for a single qubit, the P-function (3.14) for the value ξ in equation (3.17) (at ϑ = π/4), becomes which is the most uniform possible. Note, in passing, that this maximum uniformity could be also adopted as a reasonable criterion to fix the value of ξ . The associated unit vector clearly reflects this uniformity
The shape of the Q-function presents a hump localized at the origin. Due to the covariance under displacements, Q |γ,δ (α, β ) = Q |ξ (α + γ, β + δ ), the ordering should be applied to the pairs (α + γ, β + δ ), but not to (α, β ) itself. In this sense, one cannot properly say that Q |γ,δ (α, β ) has a hump located at (γ, δ ) if we keep the previously established order. Nonetheless, it is clear that due to the functional form of Q |γ,δ (α, β ) the elements of the field can be easily rearranged (using the summation table) in a such way that the corresponding hump becomes centered at (γ, δ ) and has a symmetric form. In figure 2 we plot the Q-function for the -qubit CS |θ 10 , θ 10 = D(θ 10 , θ 10 )|ξ according to such a prescription.
It is also worth observing that the Q-function of an arbitrary state |Ψ can be written down as i.e. as an smearing of the P-function. In particular, the order established by the h-function helps to visualize the superpositions of several discrete CS as spatially separated humps in phase space. In figure 3 we plot the Q-function for a superposition of two CS for 5 qubits, |Ψ ∝ (|ξ + D(θ 31 , θ 31 )|ξ ), and the ordering is the same as for Q |ξ . We can clearly observe two humps with a residual symmetry.
Symmetric discrete coherent states for n qubits  As a final remark we may note that the distribution still can be re-ordered in a more symmetric form just distributing the points with the same value of h(α) on both sides of the principal peak. Although the number of such points is not always even, for a large number of qubits the distribution corresponding to the CS is practically symmetric, as it can be seen from figure 4, where we plot the Q-function for the fiducial state |ξ corresponding to 8 qubits.
It is worth comparing the form of the Q-function for the n-qubit CS (3.1) in the limit n 1 with the CS resulting from taking as the fiducial state an eigenstate of the discrete Fourier transform [31] . In the latter case the Q-function tends to a Gaussian shape [19] , while in our approach it has a step form modulated by a decreasing function f (k), which along the axes Z (β = 0), X (α = 0) and Y (α = β ) has an exponential form: where θ (k) is the Heaviside step function.

Detecting correlations in n-qubit systems
By construction, the discrete CS are factorized states, so the qubits therein do not exhibit correlations. For symmetric states, the correlations are frequently measured using the concept of spin squeezing [35][36][37][38][39][40], comparing the fluctuations of some definite operator with the standard quantum limit, given by the spin CS. Nevertheless, for nonsymmetric states similar criteria do not work well [41] for the choice of the measured operators becomes nontrivial. To study correlations in nonsymmetric n-qubit states we apply a criteria proposed in reference [42] to quantify polarization fluctuations. According to this approach, we compute the sum of squares of the Q-function: quantum correlations make such a sum lesser than the corresponding one for a CS. In fact, for the fiducial state |ξ we have To check the method, we consider a simple way to induce correlations between qubits: the application if XOR p,q gates, where the pair (p, q) indicates the qubits on which the operator is applied, namely XOR p,q |a 1 , . . . , a p , . . . , a q , . . . , a n = |a 1 , . . . , a p , . . . , a q + a p , . . . , a n . which is smaller than (5.1). In the same vein, the application of k XOR gates between different particles (i.e., now |Ψ = XOR p k ,q k . . . XOR p 1 ,q 1 D(µ, ν)|ξ , with p 1 = q 1 = . . . = p k = q k ) keeps decreasing the sum: Similarly, the application of sequences of XOR gates to the fiducial state also leads to decreasing values of the ∑ Q 2 (α, β ). This effect can be clearly seen in figure 5, where the Q-function for the state XOR 1,2 |ξ is plotted. One can observe that the heights of the Q(α, β ) are smaller, so that the distribution initially localized at the origin sparse over a substantial part of the phase space.

Conclusions
We have developed a method for constructing discrete CS from the symmetry conditions for the Q-function of the fiducial state. This has allowed us to order the points in the discrete phase space. Besides, we have applied a criterion for the detection of quantum correlations to the discrete case and have shown that it can be useful for n-qubit systems.