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.

Figure 1

Every PI state can be decomposed into a block diagonal form. Exemplarily shown is the combination of $d_j$ block matrices $\widetilde{{\varrho }_j}$ which are all identical.

Figure 2

ML reconstruction of the state $|D_6^{(3)}⟩$ obtained from (a) full and (b) PI tomography and (c) CS with 270 settings performed at a pump power of 8.4 W. The respective fidelities are 0.604, 0.590, and 0.615 with mutual overlaps of $F({\varrho }_{\text{full}},{\varrho }_{\mathrm{PI}})=0.922$, $F({\varrho }_{\text{full}},{\varrho }_{\mathrm{CS}})=0.950$, and $F({\varrho }_{\mathrm{PI}},{\varrho }_{\mathrm{CS}})=0.908$. (d) Probability to obtain a certain fidelity for CS with a certain number of randomly chosen settings in comparison with full tomography.

Figure 3

Symmetric subspaces ($j=3$) obtained with (a) PI tomography and (b) CS in the PI subspace with 16 settings. The central bars can be associated with the target state $|D_6^{(3)}⟩$ and the small bars next to it with $|D_6^{(2)}⟩$ and $|D_6^{(4)}⟩$ originating from higher-order noise. (c) Probability to observe a certain fidelity for arbitrarily chosen tomographically incomplete sets of settings in comparison with PI tomography from 28 settings. For 16 settings, the overlap is $\ge 0.950$ on average.

Figure 4

(a) Observed fidelities with the states $|D_6^{(2)}⟩$, $|D_6^{(3)}⟩$, and $|D_6^{(4)}⟩$ at different ultraviolet (UV) pump powers for PI tomography and CS in the PI subspace from 12 settings. The error bars were determined by nonparametric bootstrapping [23]. (b) The influence of the pump power on the higher-order noise expressed via the noise $q$ and the asymmetry parameter $\lambda$ (upper part) and the phase estimation sensitivity expressed via the quantum Fisher information (QFI) (lower part).