Abstract
Self-testing protocols are methods to determine the presence of shared entangled states in a device independent scenario, where no assumptions on the measurements involved in the protocol are made. A particular type of self-testing protocol, called parallel self-testing, can certify the presence of copies of a state, however such protocols typically suffer from the problem of requiring a number of measurements that increases with respect to the number of copies one aims to certify. Here we propose a procedure to transform single-copy self-testing protocols into a procedure that certifies the tensor product of an arbitrary number of (not necessarily equal) quantum states, without increasing the number of parties or measurement choices. Moreover, we prove that self-testing protocols that certify a state and rank-one measurements can always be parallelized to certify many copies of the state. Our results suggest a method to achieve device-independent unbounded randomness expansion with high-dimensional quantum states.
Highlights
Bell nonlocality describes measurement correlations which are rigidly incompatible with the notion of local determinism [Bel64, BCP+14]
Bell nonlocality plays a crucial role in socalled device-independent protocols
The simplest example of such phenomenon is the maximal violation of the Clauser-Horn-ShimonyHolt (Bell) inequality [CHSH69], which can be used to self-test the maximally entangled pair of qubits and mutually unbiased local measurements [PR92, Tsi93, SW87]
Summary
Bell expressions imply that the outputs corresponding to different i-s are mutually independent. The correlations obtained by performing the experiment R = {|φ+ , Ma|x, Nb|y} in the parallel scheme of Fig. 1c self-test the reference experiment {|φ+ ⊗n, M⊗a|nx, N⊗b|yn} This means that an unbounded amount of entanglement can be certified in a device-independent manner with the minimum number of local measurements possible. The correlations {p(a, b|x, y)} obtained by performing the reference experiments n times in the parallel scheme of Fig. 1c self-test the state i |ψi The proof of this is almost identical to the proof of Theorem 1, instead in (4) the Bell expressions Ii are different for each i; see Appendix C for more details.
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