Parallel Self-Testing of the GHZ State with a Proof by Diagrams

Abstract

Quantum self-testing addresses the following question: is it possible to verify the existence of a multipartite state even when one's measurement devices are completely untrusted? This problem has seen abundant activity in the last few years, particularly with the advent of parallel self-testing (i.e., testing several copies of a state at once), which has applications not only to quantum cryptography but also quantum computing. In this work we give the first error-tolerant parallel self-test in a threeparty (rather than two-party) scenario, by showing that an arbitrary number of copies of the GHZ state can be self-tested. In order to handle the additional complexity of a three-party setting, we use a diagrammatic proof based on categorical quantum mechanics, rather than a typical symbolic proof. The diagrammatic approach allows for manipulations of the complicated tensor networks that arise in the proof, and gives a demonstration of the importance of picture-languages in quantum information.

Self-testing of quantum systems: a review

“…In section 5.3 we saw many ways to self-test n EPR pairs by using parallel repetition of CHSH or Magic Square game. Up to date, the only parallel self-test of some multipartite state is shown in [BKM19]. The authors use diagramatic proofs based on categorical quantum mechanics [CK17], to prove that parallel repetition of the GHZ game robustly self-tests n copies of the GHZ state.…”

Certified quantum measurement of Majorana fermions

“…The equivalence is up to a local isometry, because the measurement statistics are unaffected by a local change of basis and by the existence of an auxiliary subsystem on which the measurements act trivially. The notion of self-testing was formalized by Mayers and Yao [9], and since then, self-testing protocols for many other states and measurement scenarios [18][19][20][21][22][23] have been discovered. Such protocols are often called device-independent because they rely only on the statistics of measurement outcomes, and not on any physical assumptions about the measurement apparatus.…”

Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension

“…To prove Proposition 2.4, we need the following lemma to combine operator relations with respect to the same shared state. Lemma 2.5 (Lemma 7 of [4]). Let |ψ ∈ H be a quantum state.…”