Bell's theorem, a landmark result in the foundations of physics, establishes that quantum mechanics is a non-local theory. It asserts, in particular, that two spatially separated, but entangled, quantum systems can be correlated in a way that cannot be mimicked by classical systems. A direct operational consequence of Bell's theorem is the existence of statistical tests which can detect the presence of entanglement. Remarkably, certain correlations not only witness entanglement, but they give quantitative bounds on the minimum dimension of quantum systems attaining them. In this work, we show that there exists a correlation which is not attainable by quantum systems of any arbitrary finite dimension, but is attained exclusively by infinite-dimensional quantum systems (such as infinite-level systems arising from quantum harmonic oscillators). This answers the long-standing open question about the existence of a finite correlation witnessing infinite entanglement.
Bell's theorem, a landmark result in the foundations of physics, establishes that quantum mechanics is a non-local theory. It asserts, in particular, that two spatially separated, but entangled, quantum systems can be correlated in a way that cannot be mimicked by classical systems. A direct operational consequence of Bell's theorem is the existence of statistical tests which can detect the presence of entanglement. Remarkably, certain correlations not only witness entanglement, but they give quantitative bounds on the minimum dimension of quantum systems attaining them. In this work, we show that there exists a correlation which is not attainable by quantum systems of any arbitrary finite dimension, but is attained exclusively by infinite-dimensional quantum systems (such as infinite-level systems arising from quantum harmonic oscillators). This answers the long-standing open question about the existence of a finite correlation witnessing infinite entanglement.
Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether Cq=Cqs). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows for either infinite answer sets (and finite question sets) or infinite question sets (and finite answer sets), there exist correlations that are achievable using an infinite-dimensional quantum strategy, but not a finite-dimensional one. For the former case, our proof exploits a recent result [Nat. Comm. 8, 15485 (2017)], which shows self-testing of any pure bipartite entangled state of arbitrary local dimension d, using question sets of size 3 and 4 and answer sets of size d. For the latter case, a key step in our proof is to show a novel self-test, inspired by [Nat. Comm. 8, 15485 (2017)], of all bipartite entangled states of any local dimension d, using question sets of size O(d), and answer sets of size 4 and 3 respectively.
A key goal in the foundations of quantum mechanics is to identify operational constraints characterizing physical theories. Bell inequalities do so for classical probability theory, while Tsirelson's bound provides a first step for quantum mechanics. Here, we address the dual question: Can one certify that all correlations predicted by quantum theory are actually realizable? In this work, we construct a finite number of two-body correlations such that the only probabilistic theory that (1) realizes them, and (2) does so in a way that is stable under iterated teleportation, is quantum theory. For n quantum systems of dimension d each, Condition (1) can be verified using poly(d,n) measurement settings. Condition (2) may be understood as a hierarchy of tests, one for each number N of teleportation steps. There is thus a sense in which finite-dimensional quantum theory can be self-tested. In particular, one can certify the existence of Bell inequality violations larger than any that have been directly observed.
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.
Determining the relationship between quantum correlation sets is a long-standing open problem. The most well-studied part of the hierarchy is captured by the chain of inclusions Cq⊆Cqs⊊Cqa⊆Cqc. The separation Cqs=Cqa, showing that the set of quantum spatial correlations is not closed, was proven in breakthrough work by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)]. Resolving the question of Cqa=Cqc would resolve the Connes Embedding Conjecture and would represent major progress in the mathematical field of operator algebras. In this work, we resolve the ambiguity in the first inclusion, showing that Cq=Cqs. We provide an explicit construction of a correlation that can be attained on a tensor product of infinite-dimensional Hilbert spaces but not finite-dimensional ones. This property is also conjectured to be possessed by any correlation which maximally violates the I3322 inequality.
Recent works in foundations of quantum (field) theory and relativistic quantum information try to better grasp the interplay between the structure of quantum correlations and the constraints imposed by causality on physical operations. Chapter 1 is dedicated to the study of the conceptual implications of quantum nonlocality, a concept that subsumes that of entanglement in a certain way. We detail the recent information-theoretic approaches to understanding the structure of nonlocal correlations, and the issues the latter raise concerning the ability of local observers to isolate a system from its environment. Chapter 2 reviews in what sense imposing Poincare invariance affects entanglement detection and quantification procedures. This invariance ultimately forces a description of all quantum systems within the framework of quantum field theory, which leads to the impossibility of localized finite-energy states and to the divergence of all entanglement measures for local observers. We provide a solution to these two problems by showing that there exists a generic equivalence between a finite spatial resolution of the measurement apparatus and the exclusion of high-energy degrees of freedom from the definition of the observed system. This equivalence allows for an epistemic interpretation of the standard quantum formalism describing nonrelativistic localized systems and their correlations, hence a clarification of the origin of the finite measures of entanglement between such systems. Chapter 3 presents a recent theoretical framework that predicts the existence of correlations with indefinite causal order. In analogy to the information-theoretic approaches to nonlocal correlations, we introduce some principles that constrain the structure of such correlations, which is a first step toward a clear understanding of their physical origin.
A long-standing open question is to characterize the classical communication cost of simulating the correlations arising from measurements on finite-dimensional quantum systems. We consider this question in the prepare-and-measure scenario. First we construct a classical simulation model involving a finite average number of bits of one-way communication that simulates exactly the correlations arising from arbitrary d-dimensional quantum states and arbitrary measurements. Second, we show that no classical simulation protocol involving a finite average message dimension can exist for d≥4, hence strengthening a recent result of de Gois, Santos and Vieira. Our first result relies on a more general simulation theorem which can be applied to certain continuous variable (i.e. infinite-dimensional) quantum systems, and even beyond quantum theory, e.g. to generalized probabilistic theories. In particular, we show that multimode Gaussian states of bounded average energy and general Gaussian measurements admit an exact classical simulation with a finite expected number of bits. Overall, our results show that the classical simulation cost of quantum communication is essentially as high as it could be.