Operator algebras and quantum information: 1) Connes implies Tsirelson and 2) Robust self-testing (hot topic)

We give a simple proof of Connes implies Tsirelson, and further advertise a hot topic in quantum information: optimal states and robust self-testing. We showcase here how operator algebraic techniques can be quite fruitful.

For this we begin with by recalling quantum strategies in the context of non-local games, and their description in terms of the state space on the full group algebra of certain free groups. With this description at hand, we then directly obtain the main result via an elementary lifting result by Kim, Paulsen and Schafhauser:

The Connes embedding problem implies the synchronous Tsirelson conjecture.

As such the entire proof is elementary, and bypasses all versions of Kirchberg's QWEP conjecture and thelike, as well as any reformulation such as in terms of the micro state conjecture. Moreover, it should be (likely) easier to construct minimal nonlocal games as counterexamples for the synchronous Tsirelson conjecture (which is equivalent to the full Tsirelson conjecture but in a non-trivial way) and so also nonamenable traces for above groups, in other words non-Connes embeddable operator algebras.

After this we continue (as much as time permits) with an advertisement for a topic in quantum information (a hot one):

Device-independent certification of quantum states – more precisely ROBUST SELF-TESTING – 

which has tremendous importance for the coming era of practical quantum computingand we showcase how operator algebraic techniques can be quite fruitful here.More precisely, we illustrate these techniques on the following two prominent classes of nonlocal games:

1) The tilted CHSH game.

We showcase here how to compute the quantum value using operator algebraic techniques, and how to use the same to derive uniqueness for entire optimal states, including all higher moments as opposed to correlations defined on two-moments only, where the latter compares to traditional self-testing.Moreover, we report in this example on previously unknown phase transitions on the uniqueness of optimal states when varying the parameters for the tilted CHSH game.

2) The Mermin–Peres magic square and magic pentagram game. As before, we also note here uniqueness of optimal states, which in these two examples is a basically familiar result.

The first part is based on preprint: https://arxiv.org/abs/2209.07940 

The second part on self-testing (and further robust self-testing) is based on https://arxiv.org/abs/2210.03716 and upcoming joint work with Azin Shahiri.

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