Beyond tracial states in robust self-testing and constructing derivations for quantum Markov semigroups

Abstract

This dissertation contributes to three areas of research in quantum information theory. 

1.    Cipriani and Sauvageot have shown that the generator of the L² implementation of a tracially symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a complete bimodule. Here, we study the existence of (twisted) derivations corresponding to quantum Markov semigroups which satisfy symmetry properties with respect to a non-tracial state. We show that this construction of a derivation can in general not be generalised to quantum Markov semigroups that are symmetric with respect to a non-tracial state. Next, we prove that the generator of the L² implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a twisted derivation with values in a complete bimodule, extending a subsequently released result by Wirth for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markovian operators to symmetric Markovian operators and is essential to obtain the required inner product on the complete bimodule. This transformation was discovered by adapting the approach in the first part to the twisted derivations introduced by Wirth.

     2.    Robust self-testing in non-local games allows a classical referee to certify that two untrustworthy players are able to perform a specific quantum strategy up to high precision. Proving robust self-testing results becomes significantly easier when one restricts the allowed strategies to symmetric projective maximally entangled (PME) strategies, which allow natural descriptions in terms of tracial von Neumann algebras. This has been exploited in the celebrated MIP*=RE paper and related articles to prove robust self-testing results for synchronous games when restricting to PME strategies. However, the PME assumptions are not physical, so these results need to be upgraded to make them physically relevant. In this chapter, we do just that: we prove that any perfect synchronous game which is a robust self-test when restricted to PME strategies, is in fact a robust self-test for all strategies. We then apply our result to the Quantum Low Degree Test to find an efficient n-qubit test.

3.    Families of expander graphs were first constructed by Margulis from discrete groups with property (T). Within the framework of quantum information theory, several authors have generalised the notion of an expander graph to the setting of quantum channels. In this chapter, we use discrete quantum groups with property (T) to construct quantum expanders in two ways. The first approach obtains a quantum expander family by constructing the requisite quantum channels directly from finite-dimensional irreducible unitary representations, extending earlier work of Harrow using groups. The second approach directly generalises Margulis' original construction and is based on a quantum analogue of a Schreier graph using the theory of coideals. To obtain examples of quantum expanders, we apply our machinery to discrete quantum groups with property (T) coming from compact bicrossed products.