Tomography from collective measurements

We discuss the tomography of $N$-qubit states using collective measurements. The method is exact for symmetric states, whereas for not completely symmetric states the information accessible can be arranged as a mixture of irreducible SU(2) blocks. For the fully symmetric sector, the reconstruction protocol can be reduced to projections onto a canonically chosen set of pure states.


Introduction
Continuous-variable tomography has been exhaustively explored, from both theoretical and experimental viewpoints [1]. However, the corresponding problem for discrete systems stands as challenge [2]. If we look at the example of N qubits, which will be our thread in this paper, one has to make at least 2 N + 1 measurements in different bases before to determine the state of a priori unknown system [3][4][5][6]. With such an exponential scaling, it is clear that only few-qubit states can be reconstructed in a reasonable time [7,8].
In the same spirit of simplicity, one may be tempted to examine the case when one can extract only partial information from the system under consideration. This happens, e.g., in large multipartite systems, wherein addressing individual particles turns out to be a formidable task. Bose-Einstein condensates constitute an archetype of this situation: only collective spin observables can be efficiently measured through detection of the spontaneous emission correlation functions [21,22].
By assessing collective spin operators, one can only access the SU(2) invariant subspaces appearing in the decomposition of the N-qubit density matrix. The problem of partial state tomography appears thus analogous to that of permutationally invariant states.
In the present work, we show that one can obtain an explicit partial reconstruction for the N-qubit density matrix in terms of average values of correlation functions of approximately N 3 collective spin operators. In other words, we propose to arrange O(N 3 ) experimental data points inside SU(2) invariant subspaces. As an illustration, we analyze the fidelity of the reconstructed states for 2 and 3 qubits. In addition, we demonstrate that when the state belongs to the fully symmetric (Dicke) subspace, the tomographic measurements reduce to rank-one positive operator valued measurements (POVMs), and we find the corresponding operational expansion. As a bonus, we introduce a new type of discrete special functions that might find further applications in the analysis of N-qubit systems.
The paper is organized as follows. In Section 2 we briefly recall the principal aspects of discrete phase-space distribution functions and of the standard tomographic scheme. In Section 3 we provide explicit expressions for the permutationally invariant tomography for a N-qubit system, whereas in Section 4 an alternative scheme for fully symmetric states is presented. Finally, Section 5 summarizes our main results.

Tomography from collective measurements
The representation (5) requires measuring the POVM (6) with 2 2N elements. This provides a minimal complete tomography, but it is extremely demanding for N ≫ 1.
As heralded in the Introduction, to circumvent this problem we restrict ourselves to collective measurements. The information acquired from such measurements does not allow to obtain complete information about the state of the system: operators that are invariant under particle permutations (collective operators) "see" only irreducible subspaces appearing in the tensor decomposition of SU(2) ⊗N . Nonetheless, this still provides nontrivial information.
Symmetric operatorsÂ sym on N qubits are those invariant with respect to particle permutations:Â whereΠ i j is the unitary operator that swaps particles i and j. The crucial observation for what follows is that these operators possess a peculiar property: their symbols P A sym (α, β ) depend exclusively on the Hamming weights [37] of α, β , and their binary sum α + β ; that is, with Therefore, the whole information about any symmetric measurement Â sym = Tr(ρÂ sym ) is conveniently conveyed in the projectedQ-function [38,39], since, as it immediately follows from (5), the index k running in steps of two: where is a normalization factor fixed by the number of binary tuples λ , µ with Hamming weights h(λ ), h(µ), h(λ + µ) = (m, n, k).
By replacing Q ρ (α, β ) by Q lifted ρ (α, β ) in the reconstruction (5), we get where the symmetric operators∆ (±1) (m, n, k) can be jotted down aŝ Here,F mnk stands for the orthonormal set of operators (see Appendix A for details) and are discrete functions, whose properties are explored in Appendix B.1. Observe that (18) is independent of the choice of α and β : any other choice α ′ and β ′ is related by permutations, which can be applied to γ and δ as well.
Because of the properties ofF mnk , the reconstruction (15) can be reduced tô which is an explicit function of N+3 3 expectation values of collective operators. Note, in passing, that the operators (17) can be always expanded in terms of collective spin operators. For instance, by direct inspection one gets that in the simplest cases n = 0 or m = 0,F mnk are diagonal in the computational basis: j . By construction,ρ rec is nonzero only inside SU(2) invariant blocks, and coincides with the true density matrix in the fully symmetric subspace. In all the other blocks, ρ rec differs from the true value and, in particular, the irreducible subspaces of the same dimension are indistinguishable inρ rec .

Symmetric overcomplete tomography: canonical projection
The outstanding case of fully symmetric (Dicke) states [42] deserves special attention as they are widely used in numerous applications (see, e.g., [43][44][45]) and, in addition, they are efficiently generated in the laboratory [46][47][48][49]. For Dicke states, the reconstruction (19) is exact, but requires O(N 3 ) measurements of collective operators, while the density matrix contains at most N 2 + 2N independent parameters. Obviously, not all such collective measurements are independent. This redundancy can be fixed by representing the reconstructed density matrix via rank-one projectors.
For a fully symmetric density matrix it follows from (5) that sym (α, β ) is a symmetric function and actually it is a rank-one tensor∆ The unnormalized states |Ψ h(α),h(β ),h(α+β ) have the following expansion in the Dicke basis The operators (31) form an informationally complete POVM ∑ m,n,k where |Ψ mnk = N −1 mnk |Ψ mnk and For a given (N + 1)-dimensional Dicke subspace, there are only N different normalization factors N mnk . In terms of the projection of∆ (1) sym (α, β ), in Appendix C.1 we arrive at the compact resultρ where p mnk = Ψ mnk |ρ sym |Ψ mnk and whereÂ mnk are also given in Appendix C.1.
In this protocol, the total number of projections (31) required for reconstruction of symmetric states is N+3 3 = (N + 1)(N + 2)(N + 3)/6. However, it immediately follows from (35) that the probabilities p mnk are not linearly independent as they satisfy the conditions where These restrictions can be represented in a matrix form where p is the N+3 3 -dimensional probability vector andΩ is an appropriately arranged matrix (38). We have numerically found that the rank of the matrix (Ω − 1 1) is N(N 2 − 1)/6. Then, taking into account that the probabilities also satisfy the normalization condition Tr(ρ sym ) = 1, we obtain that only N 2 + 2N projections are needed for the reconstruction of fully symmetric states.

Concluding remarks
In short, we have proposed a tomographic protocol based on measuring N+3 3 = (N + 1)(N + 2)(N + 3)/6 expectation values of collective operators. The advantage of the present approach with respect to previously discussed (and experimentally verified) methods is given by the explicit expressions (19) for the reconstructed density matrix from experimental data. In addition, we have shown that restricting ourselves to fully symmetric states, the tomographic protocol is reduced to projections from an overcomplete set of pure states (32), which still allows to obtain an explicit reconstruction expression (35). Such a set of states has been worked out from the first principles of state reconstruction in an 2 N -dimensional Hilbert space.
The operatorsF mnk can be expressed in terms of a special discrete function. Taking into account the action of the monomialsẐ αXβ on the computational basis states {|κ : κ ∈ Z N 2 }, we immediately obtain for the matrix elements with The function f mk h(δ ),h(γ),h(γ + δ ) will be further analyzed below. By taking into account that Tr which shows the orthogonality used in the paper.

B Special functions
In this Appendix we discuss some relevant properties of the functions used in the derivation of our results.

B.1 Function g mnk
The discrete function (18) can be represented in the integral form where we have used the following representation of the Kronecker delta-function The above integrals can be easily computed, leading to a quite cumbersome expression in terms of finite sums: They satisfy the following dual orthogonality relations

B.2 Function f mk
Following a similar procedure, we can represent the function (42) in the integral form, Computing the integral (49) and rearranging the corresponding sums of binomial coefficients we obtain where 2 F 1 is the Hypergeometric function. It is worth noting that f mk h(δ ),h(γ),h(γ + δ ) can be obtained by a reduction from g mnk h(δ ),h(γ),h(γ + δ ) .

B.3 Function ψ ℓ
The function ψ ℓ is defined as and it can be recast as which leads to the following expression in terms of finite sums (54)

C Canonical projection
In this Section we find projections of the kernels∆ (±1) (α,β ) onto the Dicke subspace.

C.2 Projection of monomialsΠ symẐαXβΠsym
It follows immediately from (40) that the matrix elements of the monomialẐ αXβ in the Dicke basis (55) have the form where the function f ℓℓ ′ is defined in (42).