Universität zu Köln

Ligthart, Laurens ORCID: 0000-0001-7634-2259 (2024). Semidefinite Programming Techniques for the Quantum Causal Compatibility Problem. PhD thesis, Universität zu Köln.

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Abstract

Characterizing the set of correlations that can arise from performing measurements in a quantum description of Nature is a relevant, but challenging task. Such a characterization, known as the quantum causal compatibility problem, provides us with insights in the advantages of quantum theory over a classical theory, but can also show its limitations. This problem becomes particularly challenging when the quantum states and measurements are required to be compatible with a given causal structure. A causal structure dictates the causal dependencies of the parties and systems involved in the experiment. One can think of the Bell scenario as one of the simplest causal structures, in which two spatially separated parties, Alice and Bob, are assumed to perform a measurement on a shared source. In more general causal structures we might have more parties, and more sources that are independent of each other. Recently, a systematic way of analyzing the correlations in classical and quantum causal structures was proposed in the form of the inflation technique. In the inflation technique the causal dependencies, which are difficult to encode algorithmically, are relaxed to easy-to-encode symmetry constraints on a larger number of parties. For the classical case, this provides a converging hierarchy of linear programs for the causal compatibility problem. For the quantum case, it instead yields a hierarchy of increasingly restrictive semidefinite programming relaxations. It is, however, unknown whether this hierarchy is also complete. One of the main results of this thesis is to show that a modified version of the quantum inflation technique is convergent for the quantum causal compatibility problem. This modified hierarchy introduces an additional parameter, r, that restricts the Schmidt rank of the observables. For each value of r we provide a hierarchy of compatibility tests that is complete in the sense that it will detect, at some finite level, any probability distribution that is incompatible with the causal model under the Schmidt rank constraint. Such compatibility tests are formulated as non-commutative polynomial optimization problems, of which we provide a C*-algebraic description. Additionally, we develop a separate hierarchy of semidefinite programs, which we call the polarization hierarchy. It is shown that the polarization hierarchy, as well as the original quantum inflation hierarchy are complete for the causal structure known as the bilocal scenario. In the bilocal scenario there are three distant parties, Alice, Bob and Charlie, performing measurements on two independent sources: one shared by Alice and Bob and the other by Bob and Charlie. We show that a model for Bob’s algebra, which consists of two commuting subalgebras, can be constructed from the commutants of representations of Alice’s and Charlie’s algebras. This construction also gives insight into a bilocal version of Tsirelson’s problem. We show that if Alice’s and Charlie’s systems can be modeled with a finite dimensional representation, the commuting observables model and tensor product model of locality in quantum theory coincide. Our convergence results rely centrally on the fact that certain symmetries in the limit imply independence of random variables or quantum states. Such statements are collectively known as de Finetti theorems. For the specific setup considered in this thesis, namely the C*-algebraic description of quantum mechanics, a de Finetti theorem had not yet been proven beyond the special case of the minimal tensor product. Another result of the thesis is therefore the proof that a quantum de Finetti theorem also holds for general tensor products of C*-algebras. The quantum causal compatibility problem can be seen as a version of the quantum network compatibility problem – in which we ask which quantum states can be produced in a certain causal structure – where the output is assumed to be a classical state. We show that the techniques developed for the causal compatibility problem can be adapted to the more general setting of quantum networks. Furthermore, an analytic proof is given of the fact that graph states cannot be produced in bipartite quantum networks. This proof again relies on the inflation technique by linking correlations of different inflations of the network to each other. By assuming that the correlations arise from a graph state, it can be shown that the bound of a particular inequality can be violated, which leads to a contradiction. Lastly, we show that the polarization hierarchy can be used to optimize over a large class of optimization problems known as state polynomial optimization. In such problems, the goal is to optimize an objective function that is a polynomial in the expectation values of observables, under similar polynomial constraints. This allows us, for example, to optimize over covariances or non-linear Bell inequalities. We also give an alternative version of a recently developed semidefinite programming hierarchy that solves this problem, in which we incorporate the polarization trick.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Ligthart, Laurens laurensligthart@gmail.com orcid.org/0000-0001-7634-2259 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-720634
Date:	25 January 2024
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Physics > Institute for Theoretical Physics
Subjects:	Mathematics Physics
Uncontrolled Keywords:	Keywords Language quantum, causality, causal structure, C*-algebra, quantum network, semidefinite program, optimization, de Finetti, Tsirelson English
Date of oral exam:	30 November 2023
Referee:	Name Academic Title Gross, David Prof. Dr. Vallentin, Frank Prof. Dr.
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/72063