Simultaneous certification of entangled states and measurements in bounded dimensional semiquantum games

Certification of quantum systems and operations is a central task in quantum information processing. Most current schemes rely on a tomography with fully characterized devices, while this may not be met in real experiments. Device characterizations can be removed with device-independent tests, but it is technically challenging at the moment. In this paper, we investigate the problem of certifying entangled states and measurements via semiquantum games, a type of nonlocal quantum games with well characterized quantum inputs, balancing practicality and device independence. We first design a specific bounded-dimensional semiquantum game, with which any pure entangled state and Bell state measurement operators can be simultaneously certified. Afterward via a duality treatment of state and measurement, we interpret the dual form of this game as a source-independent bounded-dimensional entanglement swapping protocol and show the whole process, including any entangled projector and Bell states, can be certified with this protocol. In particular, our results do not require a complete Bell state measurement, which is beneficial for experiments and practical use.

Figure 1

The semiquantum game. The systems are denoted by their corresponding Hilbert spaces. In this game, from the referee's perspective, well-characterized states $\left\{{\psi }_x^{A_0}\right\},\left\{{\psi }_y^{B_0}\right\}$ are sent to Alice and Bob, respectively, where each of the two players measures the input quantum state received and her or his own party of ${\rho }^{AB}$ jointly. The joint measurements are expressed as POVMs $\left\{M_a^{A_{0}A}\right\},\left\{M_b^{B{B}_0}\right\}$

Figure 2

The uncharacteristic entanglement swapping protocol. In this protocol, two uncharacterized states ${\rho }^{A_{0}A}\in \mathcal{D}({\mathcal{H}}_{A_0}\otimes {\mathcal{H}}_A),{\rho }^{B{B}_0}\in \mathcal{D}({\mathcal{H}}_B\otimes {\mathcal{H}}_{B_0})$ are prepared. No information is known about the two states except for that they are both a pair of qubits. A so-called Bell state measurement is performed on the systems ${\mathcal{H}}_A,{\mathcal{H}}_B$. One can use well-characterized local measurements $\left\{N_a^{A_0}\right\},\left\{N_b^{B_0}\right\}$ on systems $A_0,B_0$ to certify the results.