Practical and efficient experimental characterization of multiqubit stabilizer states

Vast developments in quantum technology have enabled the preparation of quantum states with more than a dozen entangled qubits. The full characterization of such systems demands distinct constructions depending on their specific type and the purpose of their use. Here we present a method that scales linearly with the number of qubits for characterizing stabilizer states. Our approach allows simultaneous extraction of information about the fidelity, the entanglement, and the nonlocality of the state and thus is of high practical relevance. We demonstrate the efficient applicability of our method by performing an experimental characterization of a photonic four-qubit cluster state and three- and four-qubit Greenberger-Horne-Zeilinger states. Our scheme can be directly extended to larger-scale quantum information tasks.

Figure 1

Critical IDs are minimal sets of mutually commuting $N$-qubit observables that relate to specific multiqubit states. Each row [e.g., light yellow (light gray) circle in (a)] is a different joint observable, where the implied tensor product symbols are omitted for compactness, while each column [e.g. dark red (dark gray) circle in (a)] corresponds to a different qubit. When each single-qubit Pauli observable appears an even number of times in each column of a negative ID, the set enables us to prove the GHZ theorem. The tables represent (a) a whole negative ID related to the three-qubit GHZ state, (b) a whole negative ID related to the four-qubit linear cluster state, and (c) a partial positive ID related to the four-qubit GHZ state.

Figure 2

A femtosecond-pulsed UV-laser beam passes twice through a $\beta$-barium borate (BBO) crystal, producing pairs of polarization-entangled photons. The photons are emitted in forward and backward directions and are recombined on polarizing beam splitters (PBSs). Walk-off effects are compensated using HWPs and half-thick BBOs. Additional HWPs set the entangled pairs to a selected Bell state. By postselecting fourfold coincidence events we obtain the desired cluster state or GHZ state. Polarization analysis is implemented with motorized tomographic optic components.

Figure 3

(a) Measured expectation values for the $\mathbf{ID}{5}_w^4$ (on the left) and results of the maximization of ${\gamma }_{\mathcal{C}}$ for different four-qubit entangled states (on the right). and are reached by exchanging the order of qubits in the linear cluster state. In the dashed box we report the experimental result of the ID-Bell parameter. (b) Measured expectation values for the $\mathbf{ID}{4}_w^3$ (on the left) and results of the maximization of ${\gamma }_{\mathcal{C}}$ for three-qubit entangled states (on the right). In the dashed box we report the experimental result of the ID-Bell parameter.

Figure 4

Comparison of fidelities obtained with different methods for the four-qubit linear cluster, four-qubit GHZ state, and three-qubit GHZ state. The QST (red/first bar for every state) and SG (blue/second bar) approaches scale exponentially, while the ID (yellow/third bar), GoSG (green/fourth bar), and Wit (purple/fifth bar) approaches scale linearly with the number of qubits. Within the error bars the IDs set lower bounds, in agreement with the QST results. The SG fidelities tend to overestimate the QST ones. The GoSG and Wit bounds, like the IDs, are consistent with the rest of the methods. Note that $F_{GoSG}<0.5$ for the four-qubit linear cluster, so it is not sufficient to certify that the state can violate a Bell-type inequality. The error bars derive from Poissonian statistics and thus correspond to a lower limit.

Figure 5

Different types of cluster states.

Figure 6

Reconstructed density matrix (real part) of the four-qubit cluster state ($F_{QST}=0.629\pm 0.007$). The imaginary part is not shown since its components are below $0.05$.

Figure 7

All eight equivalent $\text{ID}{5}_w^4$ whose joint eigenstate is $|C_{\mathrm{lin}}\rangle$.

Figure 8

Reconstructed density matrix (real part) of the three-qubit GHZ state ($F_{QST}=0.672\pm 0.015$). The imaginary part has components below $0.07$ and is not shown.

Figure 9

Reconstructed density matrix (real part) of the four-qubit GHZ state ($F_{QST}=0.701\pm 0.008$). The imaginary part is not shown since its components are below $0.03$.