Self-testing of two-qutrit systems

Introduction. The rapid development of quantum technologies creates un urgent need to design efficient and robust certification methods that would allow the end user of a quantum device to verify whether it operates in a... [ view full abstract ]

Introduction. The rapid development of quantum technologies creates un urgent need to design efficient and robust certification methods that would allow the end user of a quantum device to verify whether it operates in a quantum way and cannot be mimicked by purely classical methods. One of such methods, formulated in the cryptographic context, is self-testing [1]. It allows for certification of entangled quantum states in quantum devices together with measurements made on it only from the observed non-local correlations.

Results. We derive a self-testing statement for qutrit systems which, unlike the previous approaches [2,3], does not in any way rely on self-testing results for qubit states. To this aim, we first introduce a general class of Bell inequalities in a d-outcome scenario with d being a prime number, which is a modification of the CHSH-d Bell inequality [4], and prove that it is maximally violated by the maximally entangled state of two qudits. We then prove an exact self-testing statement for the case of d=3: up to local isometries, the quantum state shared by Alice and Bob is the maximally entangled state of two qutrits, whereas the three measurements performed by each player are mutually unbiased bases. Thus, our results give rise to a device-independent certification method of the two-qutrit maximally entangled state and measurements corresponding to mutually unbiased bases. 

Methods. The key method used to derive our results is the so-called sum of squares decomposition of the so-called shifted Bell operator, which allows to determine the maximal quantum violation of a Bell inequality. Importantly, such a decomposition implies certain conditions on a quantum state and measurements violating maximally our Bell inequality. By solving them we obtain the aforementioned self-testing statement for two-qutrit entangled quantum systems.

Discussion. Unlike the previous approaches, we provide the first, to the best of our knowledge, self-testing statement of a composite quantum system of local dimension larger than two that exploits a genuinely d-outcome Bell inequality. It would be certainly of a great interest to generalize our results to any prime local dimension.

References:

[1] D. Mayers, and A. Yao, Proc. 39th Ann. Symp. on Foundations of Computer Science (FOCS), 503 (1998).

[2] T. H. Yang and M. Navascues, Phys. Rev. A 87, 050102(R) (2013).

[3] A. Coladangelo, K. T. Goh, V. Scarani, Nat. Comm. 8, 15485 (2017).

[4] H. Buhrman and S. Massar, Phys. Rev. A 72, 052103 (2005).