This thesis explores foundational aspects of quantum information theory and quantum cryptography. First, we investigate quantum correlations in interactive settings, including the CHSH and graph isomorphism games. We aim to distinguish quantum correlations from non-signaling correlations by leveraging the principle of communication complexity. To this end, we employ techniques such as distributed computation, majority-function-based distillation protocols, the algebraic and geometric properties of nonlocal box wirings, and variations of some graph properties such as isomorphism, transitivity, and equitable partitions. This inquiry advances our understanding of non-physical correlations. Second, we address a key open problem in cryptography: the feasibility of unclonable encryption. We aim to construct an encryption scheme that prevents two distant parties from simultaneously obtaining information about a shared encrypted message. We introduce a candidate for unclonable encryption in the plain model, i.e., without assumptions, in working towards an unconditional proof. Our protocol is based on Clifford algebra, utilizing complex Hermitian unitary matrices that anti-commute. For small key sizes, we rigorously prove security using sum-of-squares methods, while for larger key sizes, we provide strong numerical evidence via the NPA hierarchy.
This thesis explores foundational aspects of quantum information theory and quantum cryptography. First, we investigate quantum correlations in interactive settings, including the CHSH and graph isomorphism games. We aim to distinguish quantum correlations from non-signaling correlations by leveraging the principle of communication complexity. To this end, we employ techniques such as distributed computation, majority-function-based distillation protocols, the algebraic and geometric properties of nonlocal box wirings, and variations of some graph properties such as isomorphism, transitivity, and equitable partitions. This inquiry advances our understanding of non-physical correlations. Second, we address a key open problem in cryptography: the feasibility of unclonable encryption. We aim to construct an encryption scheme that prevents two distant parties from simultaneously obtaining information about a shared encrypted message. We introduce a candidate for unclonable encryption in the plain model, i.e., without assumptions, in working towards an unconditional proof. Our protocol is based on Clifford algebra, utilizing complex Hermitian unitary matrices that anti-commute. For small key sizes, we rigorously prove security using sum-of-squares methods, while for larger key sizes, we provide strong numerical evidence via the NPA hierarchy.
This thesis explores foundational aspects of quantum information theory and quantum cryptography. First, we investigate quantum correlations in interactive settings, including the CHSH and graph isomorphism games. We aim to distinguish quantum correlations from non-signaling correlations by leveraging the principle of communication complexity. To this end, we employ techniques such as distributed computation, majority-function-based distillation protocols, the algebraic and geometric properties of nonlocal box wirings, and variations of some graph properties such as isomorphism, transitivity, and equitable partitions. This inquiry advances our understanding of non-physical correlations. Second, we address a key open problem in cryptography: the feasibility of unclonable encryption. We aim to construct an encryption scheme that prevents two distant parties from simultaneously obtaining information about a shared encrypted message. We introduce a candidate for unclonable encryption in the plain model, i.e. without assumptions, in working towards an unconditional proof. Our protocol is based on Clifford algebra, utilizing complex Hermitian unitary matrices that anti-commute. For small key sizes, we rigorously prove security using sum-of-squares methods, while for larger key sizes, we provide strong numerical evidence via the NPA hierarchy.
In quantum information, nonlocal games are particularly useful for differentiating classical, quantum, and non-signalling correlations. An example of differentiation is given by the principle of no-collapse of communication complexity, which is often interpreted as necessary for a feasible physical theory. It is satisfied by quantum correlations but violated by some non-signalling ones. In this work, we investigate this principle in the context of three nonlocal games related to graph theory, starting from the well-known graph isomorphism and graph coloring games, and introducing a new game, the vertex distance game, with a parameter D∈N, that generalizes the former two to some extent. For these three games, we prove that perfect non-signalling strategies collapse communication complexity under favorable conditions. We also define a refinement of fractional isomorphism of graphs, namely D-fractional isomorphisms, and we show that this characterizes perfect non-signalling strategies for the vertex distance game. Surprisingly, we observe that non-signalling strategies provide a finer distinction for the new game compared to classical and quantum strategies since the parameter D is visible only in the non-signalling setting.
Historically, quantum mechanics has relied on the assumption of non-communicating parties, enforced via space-like separation, to witness the gap between classical and quantum descriptions. This thesis explores the boundaries of the standard Bell non-local scenario, investigating how to preserve this gap when communication is allowed. In particular, this work focuses on restricted communication, multipartite settings, and operator-algebraic formulations. Key contributions of this work include: • Dimensional Constraints: Establishing mathematical maps from standard Bell scenarios to prepare-and-measure scenarios with dimensionally bounded channels, proving that quantum bounds and self-testing properties are preserved. • Cryptographic Constraints: Analyzing compiled non-local games using quantum homomorphic encryption. The research proves asymptotic quantum soundness for all multipartite compiled games and introduces a quantitative hierarchy, similar to the NPA hierarchy, to characterize sequential correlations. • Operational Equivalences: Demonstrating that correlations arising from arbitrary compositions of non-signalling assemblages in multipartite communication schemes always reduce to standard spatial quantum Bell correlations. • Operator-Algebraic Tools: Introducing a novel chain rule for the Radon-Nikodym derivative to dilate chains of completely-positive maps, alongside the construction of a universal C*-algebra for sequential measurements. Overall, the manuscript unifies concepts from quantum foundations, complexity theory, and operator algebras to provide a rigorous mathematical framework for characterizing and certifying multipartite sequential quantum correlations.
Quantum computers were initially proposed to surpass the limitations of classical computers in simulating physical systems. Soon after, the field of quantum computing advanced through landmark discoveries of quantum algorithms, with some conjectured to provide exponential advantages over classical counterparts. These include Shor’s factoring algorithm, which has profound implications in cryptography. Additionally, moderate yet significant polynomial separations have further increased interest in the field. However, despite this progress, unconditional proofs confirming quantum superiority on the theoretical side remain elusive. At the same time, large, noise-resilient quantum computations on the experimental side, which could challenge the extended Church-Turing thesis by potentially breaking classical computational barriers, have yet to be achieved. Meanwhile, our understanding of the differences between classical and quantum mechanics has deepened. We now possess rigorous theoretical tools, such as Bell inequalities and nonlocal games, to describe behaviors unique to these theories. Moreover, the main loopholes for observing non-classical correlations have been closed experimentally, showing that correlations in the physical world exceed those predicted by classical mechanics. This thesis investigates computational complexity differences between quantum and classical models by studying shallow-depth circuits. The primary focus lies in identifying, elucidating, and proving computational quantum advantages. Additionally, the analysis delves into the bijection between these circuits and foundational non-local game constructions, exploring how quantum-classical deviations in multi-party correlation settings translate to computational advantages. A key question addressed is how to synthesize constant-depth quantum circuits for various problems to leverage quantum advantages. To this end, circuit synthesis techniques tailored to measurement-based quantum computation are developed, enabling simultaneously optimal quantum strategies for certain nonlocal games based on a particular bijection between this model and XOR non-local games. Building on this work, we confront a pivotal challenge: If a candidate quantum circuit potentially provides a computational advantage, how can we prove that no classical circuit surpasses it? This work presents new unconditional quantum-classical separations within shallow-depth settings, distinguishing for the first time constant-depth quantum circuits with one- and two-qubit gates from biased polynomial threshold circuits, which incorporate neural-network-type functionalities. These separations extend to prime-dimensional quantum systems, and we conjecture to have identified Razborov-Smolensky-type problems uniquely solvable by specific prime dimensions. The thesis also differentiates constant-depth quantum circuits with multi-qubit Toffoli gates from their classical counterparts, emphasizing the power of parallel quantum computation. Finally, verifying shallow-depth quantum computations is challenging due to conjectured exponential separations from classical computations, limiting correctness verification through simulation. While specific quantum algorithms within BQP have been addressed, we explore whether a quantum computer can reliably execute any instance from a fixed class of quantum circuits. Moving beyond heuristic-based solutions, we propose a benchmarking protocol that certifies, with exponentially high confidence, the execution of arbitrary circuits of specified dimensions, widths, and depths.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
The emergence of quantum computing has provided new paradigms for cryptography. On the one hand, it poses significant new threats to existing classically cryptographic systems, requiring the community to define new security models that capture what a quantum adversary can do. On the other hand, it gives us new tools to design cryptographic protocols, with weaker assumptions than in the classical world, or even protocols that are impossible classically. In this survey, we first give an overview of new security definitions for classical cryptography, considering quantum adversaries who can either only use local quantum computation (post-quantum security), or even send quantum messages and in particular have access to oracle in superposition (quantum security). We explore these new notions through the examples of commitments, zero-knowledge proofs, encryption, and signatures. Then, we present what is arguably the most famous application of quantum cryptography: quantum key distribution (QKD) protocols that take advantage of unique properties of quantum mechanics to provide secure communication unconditionally. We also explore cryptography beyond QKD, focusing on unclonable cryptography: a family of cryptographic functionalities, built with quantum states, and designed to be resistant to counterfeit by leveraging the “no-cloning” theorem. We examine in particular quantum money, but also the recent notions of unclonable encryption and copy-protection, including related variants. By presenting a comprehensive survey of these topics, this paper aims to provide a thorough understanding of the current landscape and future potential of quantum cryptography.