Experimental Comparison of Efficient Tomography Schemes for a Six-Qubit State

Quantum state tomography suffers from the measurement effort increasing exponentially with the number of qubits. Here, we demonstrate permutationally invariant tomography for which, contrary to conventional tomography, all resources scale polynomially with the number of qubits both in terms of the measurement effort as well as the computational power needed to process and store the recorded data. We demonstrate the benefits of combining permutationally invariant tomography with compressed sensing by studying the influence of the pump power on the noise present in a six-qubit symmetric Dicke state, a case where full tomography is possible only for very high pump powers.

Quantum state tomography suffers from the measurement effort increasing exponentially with the number of qubits. Here, we demonstrate permutationally invariant tomography for which, contrary to conventional tomography, all resources scale polynomially with the number of qubits both in terms of the measurement effort as well as the computational power needed to process and store the recorded data. We evaluate permutationally invariant tomography by comparing it to full tomography for six-photon states obtained from spontaneous parametric down-conversion. We show that their results are compatible within the statistical errors. For low rank states, we further optimize both schemes using compressed sensing. We demonstrate the benefits of combining permutationally invariant tomography with compressed sensing by studying the influence of the pump power on the noise present in a six-qubit symmetric Dicke state, a case where full tomography is possible only for very high pump powers. Introduction.-The number of controllable qubits in quantum experiments is steadily growing [1][2][3][4]. Yet, to fully characterize a multi-qubit state via quantum state tomography (QST), the measurement effort scales exponentially with the number of qubits. Moreover, the amount of data to be saved and the resources to process them scale exponentially, too. Thus, the limit of conventional QST will soon be reached. The question arises how much information about a quantum state can be inferred without all the measurements a full QST would require. Protocols have been proposed which require significantly fewer measurement settings if one has additional knowledge about a state, e.g., that it is of low rank, a matrix product state or a permutationally invariant (PI) state [5][6][7][8][9][10][11][12][13]. Some of these approaches only require a polynomially increasing number of measurements. Still, it must be remembered that the post-processing of the data must also be scalable.
The goal of this paper is twofold: First, we implement and compare different schemes for quantum state tomography in a six-photon experiment. In detail, we perform the largest full QST of a photon experiment so far, and additionally apply PI tomography and tomography based on compressed sensing (CS), which both require less effort. Our comparison shows that PI and CS tomography or even a combination of both faithfully represent the properties of the quantum state in the experiment with clearly reduced effort and resources. Second, we use PI and CS tomography to characterize six-photon quantum states in a regime where full state tomography is not fea-sible. More precisely, we determine the relation between noise due to higher-order emissions and the laser power and characterize the entanglement properties of the respective states.
Scalable scheme for measurements.-Let us first consider the measurement effort needed for tomography. For full QST, each N -qubit state is associated with a normalized non-negative Hermitian matrix with 4 N − 1 real free parameters. Since all free parameters have to be determined, any scheme suitable to fully analyze an arbitrary state, as, e.g., the standard Pauli tomography scheme, suffers from an exponentially increasing measurement effort [14][15][16]. PI states in contrast are described by only N +3 N − 1 = O(N 3 ) free parameters. Tomography in the PI subspace can be performed by measuring (global) operators of the form A ⊗N i with A i = n i σ, i.e., measurements of the polarization along the same direction n i for every photon [17]. Here, | n i | = 1 and σ = (σ x , σ y , σ z ) with Pauli operators σ i (i = x, y, z). Each single measurement setting A ⊗N i delivers N expectation values of the operators where the summation is over all permutations Π k and i refers to the eigenbasis of A i . This reduces the number of necessary settings to D N = N +2 . Note, if one allows global entangled measurements this number can be further reduced [18] with the symmetric subspace is bounded from below by the expectation value of the projector to the symmetric subspace P (N ) s which can be estimated with these three measurements only [17,19].
Scalable representation of states and operators.-While the above approach clearly reduces the experimental effort required for a tomographically complete set of data, it offers the possibility to efficiently store and process the measured data. Describing states in the PI subspace enables an efficient representation with only polynomial scaling of the storage space and processing time [20,21].
Consider the angular momentum basis states |j, j z , α for the N -qubit Hilbert space, with J 2 |j, j z , α = j(j + 1)|j, j z , α , J z |j, j z , α = j z |j, j z , α , where the total spin numbers are restricted to be j = j min , j min +1, ..., N 2 starting from j min = 0 for N even and j min = 1 2 for N odd, while j z = − N 2 , − N 2 + 1, ..., N 2 . Here, α = 1, 2, ..., d j is a label to remove the degeneracy (of degree d j [22]) of the eigenstates of J 2 and J 2 z . In this basis, according to standard angular momentum theory, PI states can be written in a simple block diagonal form with j being the density operators of the spin-j subspace and p j a probability distribution. Hence, it is sufficient to consider only the N 2 blocks j = p j j /d j (of which each has a multiplicity of d j , see Fig. 1) with the largest block -the symmetric subspace -being of dimension (N +1)×(N +1) and multiplicity d N 2 = 1. Consequently, a PI state can be stored efficiently.
Even if the state to be analyzed is not PI, as long as the observable to be measured is PI one can hugely benefit from the scheme, thanks to the fact that a similarly scalable decomposition can be found for any PI operator O, i.e., O = j 1 1 dj ⊗ O j . Together with Eq. (1) this yields an efficient way to calculate the expectation values O = Tr( O) = j p j Tr( j O j ) also for non-PI states. Note, while in the regular case the trace has to be taken over the product of two 2 N -dimensional matrices, now we only have about N 2 terms with traces of at most (N + 1)-dimensional matrices. Again, the effort reduces from exponential to polynomial. For the six-qubit case (j ∈ j min = 0, 1, 2, N 2 = 3) this means that the state to be analyzed as well as each measurement operator can be described by only four Hermitian matrices of size 7×7, 5 × 5, 3 × 3 and 1 × 1, respectively, reducing the number of parameters from 4 6 − 1 = 4095 to 9 6 − 1 = 83 only. Data analysis starts with the counts c n i observed measuring M n i , and the frequencies f n i = c n i / k c k i , respectively. Solving the system of linear equations f n i ≈ M n i = Tr( M n i ) for the free parameters of usually results in a non-positive and thus unphysical density matrix ( 0) due to statistical errors and misalignment. Here, typically a maximum likelihood (ML) fitting algorithm is used to find the physical state that optimally agrees with the measured data [14,23,24]. We use a convex optimization algorithm [20,25] which guarantees a unique minimum and fast convergence. The performance of our algorithm is illustrated best by the fact that a 20 qubit PI state can be reconstructed in less than 10 minutes on a standard desktop computer.
State reconstruction of low rank states and compressed sensing.-As shown recently, low rank states, i.e. states with only few non-zero eigenvalues, enable state reconstruction even if the underlying set of data obtained from random Pauli measurements is tomographically incomplete [7]. There, the measurement effort to analyze a state of rank r with r2 N free parameters scales like O(r2 N log 2 N ) -clearly achieving optimal scaling up to a log factor. Despite of the still exponential scaling, the squareroot improvement can be considerable for practical pump powers. Since in many cases the state to be experimentally prepared is at the same time PI and of low rank we propose to combine the two methods whenever possible [24,26].
Experimental full state tomography.-In the following, using experimental data, various QST schemes are compared, in particular concerning the number of settings necessary to obtain (almost) full knowledge about the state. For this purpose, we perform, for the first time, full QST of a six-photon state. At very high pump power (8.4 W) of the down conversion source we collect data for the complete set of Pauli measurement settings. PI tomography is performed to test it against full QST and to analyze states emitted for lower pump powers. For both strategies we also perform data analysis for tomographically incomplete data based on CS.
The six-photon state observed in this work is the symmetric Dicke state |D . In general, symmetric Dicke states are defined as where |H/V i denotes horizontal/vertical polarization in the i th mode and the P i represent all the distinct permutations. In order to experimentally observe |D we distribute an equal number of H and V polarized photons over six output modes and apply conditional detection (more details on the setup are given in the Supplemental Material [24] and [29,30]). The setup uses cavity enhanced spontaneous parametric down-conversion (SPDC) [31] with special care taken to further reduce losses of all components and to optimize the yield of |D Data are recorded at a pump power of 8.40 ± 0.56 W over 4 minutes for each of the 3 6 = 729 Pauli settings. The six-photon count rate was 58 events per minute on average, leading to a count statistic of about 230 events per basis setting within a total measurement time of approximately 50 hours [32]. The reconstructed density matrix can be seen in Fig. 2(a). Due to the emission characteristics of SPDC, the commonly used fidelity [33] is not a good measure of the quality of the tomography here. Instead, Table I lists the overlap with all the various Dicke states. The sum reaches high values proving that the state is close to the symmetric subspace.
Evidently, the experimental state is a mixture of mainly |D and thus CS might be used beneficially. The question arises, how many settings are required for CS for a faithful reconstruction of the density matrix. We chose random subsets of up to 300 settings from the 729 settings for full tomography. Fig. 2(d) gives the probability distribution of the fidelity of the reconstructed matrix for a bin size of 0.01 with respect to the results of full tomography. While for a low number of settings (< 10) the results are randomly spread out, the overlap is on average ≥ 0.800 already for   PI tomography should be clearly more efficient. To test whether it is applicable, we first determined the lower bound for the projection of the state onto the symmetric subspace, i.e., the largest block in Fig. 1, P (6) s from the settings σ ⊗6 x , σ ⊗6 y and σ ⊗6 z by analyzing all photons under ±45 • , right/left-circular and H/V polarization. We found P (6) s ≥ 0.922 ± 0.055 indicating that it is legitimate to use PI tomography which for 6 qubits only requires 25 more settings [24]. Under the same experimen-tal conditions as before and four minutes of data collection per setting we performed the experiment within two hours only. The density matrix PI obtained is shown in Fig. 2(b) with its symmetric subspace representation shown in Fig. 3(a). The fidelities with the symmetric Dicke states for PI tomography can be found again in Table I } n+1,n+1 = 1). The overlap between the reconstructed states using either full or PI tomography is 0.922 which is equivalent to the fidelity of 0.923 between full tomography and its PI part. Clearly, PI tomography rapidly and precisely determines the PI component of the state.
PI tomography with CS.-To speed up analysis even further, based on subsets of the data used for PI tomography, we derived the density matrix PI,CS , see Fig. 3(b). Here, the fidelity averaged over a series of different samples is above 0.950 for 16 or more settings, as shown in Fig. 3(c). Again, both methods are compatible within one standard deviation. In summary, our results prove that PI tomography (with CS) enables fast and precise state reconstruction with minimal experimental and computational effort.  at different ultra violet (UV) pump powers for PI tomography and CS in the PI subspace from 12 settings. The error bars were determined by nonparametric bootstrapping [27,28]. (b) The influence of the pump power on the higher-order noise expressed via the noise q and the asymmetry parameter λ.
Application to noise analysis.-As the count rates for six-photon states depend on the cube of the pump power, full QST is not possible for lower pump power within reasonable measurement times and thus does not allow us to analyze the particular features of multiphoton states obtained form SPDC. As SPDC is a spontaneous process, with certain probability there are also cases where 8 photons have been emitted but where only 6 have been detected leading to an admixture of the states D (2) 6 and D (4) 6 . Ideally, the amplitude of the two admixtures should be the same but due to polarization dependent coupling efficiencies of H and V photons [34,35] this is not the case. Therefore, we extended the noise model [36] to better specify the experimental , the noise q and the asymmetry parameter λ. Both q and λ can be determined from the fidelities to the Dicke states (see also the Supplemental Material [24]). At 8.4 W noise parameters of q = 0.807 ± 0.013 and λ = 0.234 ± 0.015 were obtained from full tomography, which agree well with the results of PI tomography (q = 0.867 ± 0.041 and λ = 0.273 ± 0.059). After convincing ourselves that (CS) PI tomography is in excellent agreement with full QST, we now can perform six-qubit tomography also for low pump powers.
We performed PI analysis at 3.7 W, 5.1 W, 6.4 W, and 8.6 W [see Fig. 4(a)] with sampling times of 67 hours, 32 hours, 18 hours, 15 hours, and average counts per setting of 340, 390, 510, and 610, respectively. PI tomography shows an increase of the noise parameter q from 0.677 ± 0.029 for 3.7 W to 0.872 ± 0.023 for 8.6 W due to the increasing probability of 8 photon emission for high pump power [37]. Note, the ratio between 6 photon detection from 8 photon emission relative to detection from 6 photon emission is given by q/(1 − q), i.e., for a pump power of 8.6 W we obtain six-fold detection events with 90% probability from eight photon emissions of which two photons were lost. Although fluctuating, the asymmetry parameter λ does not show significant dependence on the pump power and lies in the interval [0.136 ± 0.042, 0.200 ± 0.053] for PI tomography (within [0.101 ± 0.116, 0.190 ± 0.071] for CS in the PI subspace). This confirms that the difference in the coupling efficiency of H and V does not change also for high pump powers [see Fig. 4(b)]. The fidelity between the ML fits and the noise model noise exp (p, λ) is > 0.925 for all pump levels and for CS in the PI subspace it is > 0.897. The high values indicate that our noise model adequately describes the experimental results.
Entanglement witnesses with respect to symmetric states are PI operators and thus can be determined efficiently. Here, we use this fact to test whether, in spite of the higher-order noise, the reconstructed states still exhibit genuine six-partite entanglement. We use the witness W = 0.420 · 1 1 − 0.700|D 6 |, where an expectation value W < 0 rules out any biseparability [24]. For the corresponding pump powers from 3.7 W to 8.6 W, we observed −0.088 ± 0.006, −0.078 ± 0.006, −0.075 ± 0.006 and −0.048 ± 0.005 [37]. Clearly, due to the high probability of D (3) 6 states in the higher-order noise the entanglement is maintained also for high pump powers [38].
Conclusions.-We compared standard quantum state tomography with the significantly more efficient permutationally invariant tomography and with compressed sensing also in the permutationally invariant subspace. For this purpose we used data of the symmetric Dicke state |D obtained from spontaneous parametric down conversion. All methods give compatible results within their statistical errors. The number of measurement settings was gradually reduced from 729 for full tomography, to 270 for compressed sensing, 28 for permutationally invariant tomography, and only 16 for compressed sensing in the permutationally invariant subspace giving in total a reduction of about a factor of 50 without significantly changing the quantities specifying the state. We applied this highly efficient state reconstruction scheme to study the dependence of higher-order noise on the pump power, clearly demonstrating the benefits of efficient state reconstruction for the analysis of multi-qubit states required for future quantum computation and quantum simulation applications.

Supplemental Material
The setup The photon source is based on a femtosecond enhancement cavity in the UV with a 1 mm thick β-barium borate (BBO) crystal cut for type II phase matching placed inside [31] (Fig. S1). In order to compensate for walk off effects a half-wave plate (HWP) and a second BBO crystal of 0.5 mm are applied. Spatial filtering is achieved by coupling the photons into a single mode fiber (SM) and an interference filter (IF) (∆λ = 3 nm) enables spectral filtering. Distributing the photons into six spatial modes is realized by 3 beam splitters with a splitting ratio of 50:50 (BS 1 , BS 3 , BS 4 ) and two beam splitters with a ratio of 66:33 (BS 2 , BS 4 ). Yttrium-vanadate (YVO 4 ) crystals are used to compensate for unwanted phase shifts. State analysis is realized by half-wave and quarter-wave plates (QWP) and polarizing beam splitters (PBS). The photons are detected by fiber-coupled single photon counting modules connected to a FPGA-based coincidence logic.
In Fig. S1 (lower right corner) a visualization of the measurement directions on the Bloch sphere is depicted. Each point (a x , a y , a z ) on the sphere corresponds to a measurement operator of the form a x σ x +a y σ y +a z σ z . In order to perform PI tomography for 6 qubits 28 operators have to be measured.

State reconstruction
The target function to be minimized is the logarithmic likelihood which is given by k,s n k,s Nmax log(p k,s ) where n k,s labels the number of counts for the outcome k when measuring setting s with the corresponding probability p k,s for the guessˆ . In order to take into account slightly different total count numbers per setting, the n k,s have to be divided by the maximum count number observed in one setting N max = max(N s ).
For CS exactly the same target function has to be minimized with the only difference that the underlying set of measurement data is tomographically incomplete. For a description, see text.

Convergence of CS in the PI subspace
As described in the main text, we performed PI tomography together with CS in the PI subspace at different UV pump powers. In order to investigate the convergence of CS, series of different samples were randomly chosen from the full set of measurements. For all pump powers, the average fidelity with respect to all PI settings is above 0.950 as soon as the number of settings is ≥ 12 (out of 28), see Fig. S2.

Noise model
As already explained in the main part of this paper, SPDC is a spontaneous process and therefore with a certain probability 8 photons are emitted from the source. The loss of two of these 8 photons in the linear optical setup and subsequent detection leads to an admixture of the states D (2) 6 and D (4) 6 for the case that either two H or two V polarized photons are not detected, respectively. However, in the case that one H and one V polarized photon remain undetected a considerable amount of this higher-order noise consists of the target state D (3) 6 thus preserving genuine multipartite entanglement even at high UV pump powers. The probabilities of the respective states to occur can be deduced from simple combinatorics, see Fig. S3. From this simple noise model, an Figure S2. Probability to observe a certain fidelity for arbitrarily chosen tomographically incomplete sets of settings in comparison with PI tomography from 28 settings at different pump levels. As soon as the number of settings surpasses 12, the state is almost perfectly determined, i.e., the overlap with respect to the states reconstructed from all settings ≥ 0.950. (S2) would be expected. However, this is not observed experimentally since the emission angles of down-conversion photons are polarization dependent [34,35] leading to an asymmetry in the coupling into the single mode fiber used. Therefore, the noisemodel was extended by the asymmetry parameter λ. Both q and λ can be deduced form the fidelities F with respect to the Dicke states |D (S3)