Processing correlated quantum systems in the one-shot setting

George, Ian

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https://hdl.handle.net/2142/127442 Copy

Description

Title

Processing correlated quantum systems in the one-shot setting

Author(s)

George, Ian

Issue Date

2024-10-02

Director of Research (if dissertation) or Advisor (if thesis)

Chitambar, Eric

Doctoral Committee Chair(s)

Chitambar, Eric

Committee Member(s)

• Goldschmidt, Elizabeth
• Leditzky, Felix
• Varshney, Lav R
• Winter, Andreas

Department of Study

Electrical & Computer Eng

Discipline

Electrical & Computer Engr

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Quantum information theory, information theory, quantum information science

Abstract

The novel ways in which correlations may be encoded non-locally in quantum mechanics have led to the promise of new technologies. This, in turn, has led to the first attempts at the development of quantum networks. Nonetheless, our understanding of quantum correlations and the limits of how we may process them over a quantum network remains nascent. This thesis studies the limits of processing quantum correlations between spatially separated parties in various settings and the mathematical tools necessary to determine such limits. To do so, this thesis studies three general problems in quantum network theory. The first problem is the minimum resources to prepare a quantum correlation between non-communicative receivers, which may be viewed as a star network. This study is separated into three pieces. The first piece is a near-tight one-shot quantum generalization of a problem introduced by Wyner, distributed source simulation. Establishing our one-shot bounds builds on recent work on soft-covering without smoothing and relies on establishing a variety of properties for one-shot entropic quantities, as well as improving our understanding of the smooth entropy framework in quantum network information theory. As distributed source simulation cannot be achieved to arbitrary error for sufficiently entangled states, the second piece characterizes the minimal error of converting one entangled state to another using zero communication. This relies on establishing new properties of the fidelity measure. The third piece establishes an entanglement-assisted version of the distributed source simulation task that can be achieved to arbitrary error for arbitrary quantum systems. This result builds on recent work on multipartite state splitting and establishing new properties of one-shot multipartite max mutual information quantities. The second problem is the extraction of perfect classical correlation from a multipartite quantum source where the receivers do not communicate. This may be seen as the reverse process of distributed source simulation. We establish that this is asymptotically possible in quantum mechanics for the first time and establish the asymptotic rate for classical-quantum states. The technical barrier in establishing the asymptotic rate is establishing a limit on how much noiseless correlation can be extracted without communication. Our major technical contribution is establishing such a limit by generalizing a technical lemma of Gács and Körner to classical-quantum states. The final problem is distributed state discrimination of quantum product states. This is motivated by its relevance to both understanding non-locality in quantum mechanics and developing quantum cryptographic schemes. Our study again breaks into three key pieces. First, we study the framework of local operations and simultaneous classical or quantum communication (LOSCC and LOSQC respectively) for product states, which is relevant to quantum cryptography. We establish the first separation between quantum and classical communication in the simultaneous communication model using only product state inputs. We also establish a new uncertainty relation that implies error bounds for discriminating product states using LOSQC. Second, we study state discrimination under local operations and classical communication (LOCC). We develop a new framework for zero-error state discrimination under LOCC. Our framework provides a simple proof of orthogonal product states that cannot be distinguished using LOCC--- a property introduced by Bennett et al. Finally, we study the thermodynamics of zero-error distributed state discrimination in the framework of Faist et al. This allows us to resolve an open problem from the aforementioned paper of Bennett et al.

Graduation Semester

2024-12

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/127442

Copyright and License Information

Copyright 2024 Ian George

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