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Quantum Correlations, Certifying Quantum Devices, and the Quest for Infinite Entanglement

Citation

Coladangelo, Andrea Wei (2020) Quantum Correlations, Certifying Quantum Devices, and the Quest for Infinite Entanglement. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VQC0-ZK80. https://resolver.caltech.edu/CaltechTHESIS:01162020-013448014

Abstract

Quantum information has the potential to disrupt the present computational landscape. Much of this potential rests on the existence of efficient quantum algorithms for classically intractable problems and of quantum cryptographic protocols for tasks that are provably impossible to realize classically. At the heart of many quantum advantages is one of the most counterintuitive features of quantum mechanics, known as entanglement. The central motivating question of this thesis is the following: if quantum devices will perform tasks that are beyond the reach of classical devices, can we hope to certify that they are performing these tasks correctly? Bell's theorem, a landmark result in physics, provides a partial answer to this question: it asserts that measurements on spatially isolated, but entangled, particles can result in outcomes that are correlated in a way that cannot be explained by any local hidden variable theory (such as Newtonian physics). A direct operational consequence of this theorem is that one can devise a statistical test to certify the presence of entanglement (and hence of genuine quantumness). Remarkably, nature allows us to take this certification one step further: in some cases, the correlation of measurement outcomes is sufficient to single out a unique quantum setup compatible with this correlation. This phenomenon is often referred to as self-testing, and is the central topic of this thesis.

In the first part of this thesis, we review the basic terminology and results in the theory of self-testing. We then explore a concrete application to the problem of verifiably delegating a quantum computation. Our main technical contribution is a test that robustly certifies products of single-qubit Clifford measurements on many EPR pairs. We employ this test to obtain a protocol which allows a classical user to verifiably delegate her quantum computation to two spatially isolated quantum servers. The overall complexity of our protocol is near-optimal, requiring resources that scale almost linearly in the size of the circuit being delegated.

In the second part of this thesis, the driving question is the following: what is the class of quantum states and measurements that can be certified through self-testing? Does self-testing only apply to a few special cases, like EPR pairs or copies of EPR pairs, or are these instances of a more general phenomenon? One of the main results of this thesis is that we settle this question for the case of bipartite states. We show the existence of a self-testing correlation for any pure bipartite entangled state of any finite local dimension. We then move on to explore the multipartite case, and we show that a significantly larger class of states can be self-tested than was previously known. This includes all multipartite partially entangled GHZ states, and more generally all multipartite qudit states which admit a Schmidt decomposition.

In the final part of this thesis, we explore connections of the theory of self-testing to basic questions about entanglement and quantum correlation sets. In particular, we set out to understand the expressive power of infinite-dimensional quantum systems. We consider two questions: can spatially isolated quantum systems of infinite dimension produce correlations that are unattainable by finite-dimensional systems? Does there exist a correlation that cannot be attained exactly by spatially isolated quantum systems (not even infinite-dimensional ones), but can be approximated arbitrarily well by a sequence of finite or infinite-dimensional systems? The first question was posed by Tsirelson in 1993, and its answer has been elusive. One of the main results of this thesis is a resolution of this question. The second question is better known as the "non-closure of the set of quantum correlations", and was answered affirmatively in a breakthrough of Slofstra. We give a new elementary proof of this result which leverages one of our self-testing results and a phenomenon known as embezzlement.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Quantum information, quantum correlations, entanglement
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Computing and Mathematical Sciences
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Vidick, Thomas G.
Thesis Committee:
  • Preskill, John P. (chair)
  • Chen, Xie
  • Brandao, Fernando
  • Vidick, Thomas Georges
Defense Date:18 November 2019
Non-Caltech Author Email:andrea.coladangelo (AT) gmail.com
Record Number:CaltechTHESIS:01162020-013448014
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:01162020-013448014
DOI:10.7907/VQC0-ZK80
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/1709.09267arXivAdapted for Chapter 3.
https://www.doi.org/10.1007/978-3-030-17659-4_9DOIAdapted for Chapter 4.
https://doi.org/10.1038/ncomms15485DOIAdapted for Chapter 5.
https://doi.org/10.1103/PhysRevA.98.052115DOIAdapted for Chapter 5.
https://doi.org/10.1088/1367-2630/aad89bDOIAdapted for Chapter 5.
https://arxiv.org/abs/1804.05116arXivAdapted for Chapter 6.
https://arxiv.org/abs/1904.02350arXivAdapted for Chapter 6.
http://www.rintonpress.com/xxqic17/qic-17-910/0831-0865.pdfPublisherAdapted for Appendix A.
ORCID:
AuthorORCID
Coladangelo, Andrea Wei0000-0002-6773-2711
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:13624
Collection:CaltechTHESIS
Deposited By: Andrea Coladangelo
Deposited On:04 Feb 2020 20:31
Last Modified:12 Jun 2020 18:04

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