Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful.
We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including:
- establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras;
- formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics;
- investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations;
- developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.
Contextuality in quantum physics provides a key resource for quantum information and computation. The topological approach in [Abramsky and Brandenburger, New J. Phys., 2011, Abramsky et al., CSL 2015, 2015] characterizes contextuality as "global inconsistency" coupled with "local consistency", revealing it to be a phenomenon also found in many other fields. This has yielded a logical method of detecting and proving the "global inconsistency" part of contextuality. Our goal is to capture the other, "local consistency" part, which requires a novel approach to logic that is sensitive to the topo…
In this work we present a generalization of the recently developed Hardy-like logical proof of contextuality and of the so-called KCBS contextuality inequality for any qudit of dimension greater than three. Our approach uses compatibility graphs that can only be satisfied by qudits. We find a construction for states and measurements that satisfy these graphs and demonstrate both logical and inequality based contextuality for qudits. Interestingly, the quantum violation of the inequality is constant as dimension increases. We also discuss the issue of imprecision in experimental implementations…
While quantum computers are expected to yield considerable advantages over classical devices, the precise features of quantum theory enabling these advantages remain unclear. Contextuality--the denial of a notion of classical physical reality--has emerged as a promising hypothesis. Magic states are quantum resources critical for practically achieving universal quantum computation. They exhibit the standard form of contextuality that is known to enable probabilistic advantages in a variety of computational and communicational tasks. Strong contextuality is an extremal form of contextuality desc…
We unify the resource-theoretic and the cohomological perspective on quantum contextuality. At the center of this unification stands the notion of the contextual fraction. For both symmetry and parity based contextuality proofs, we establish cohomological invariants which are witnesses of state-dependent contextuality. We provide two results invoking the contextual fraction, namely (i) refinements of logical contextuality inequalities, and (ii) upper bounds on the classical cost of Boolean function evaluation, given the contextual fraction of the corresponding measurement-based quantum computa…
Many seemingly paradoxical effects are known in the predictions for outcomes of measurements made on pre- and post-selected quantum systems. A class of such effects, which we call ``logical pre- and post-selection paradoxes'', bear a striking resemblance to proofs of the Bell-Kochen-Specker theorem, which suggests that they demonstrate the contextuality of quantum mechanics. Despite the apparent similarity, we show that such effects can occur in noncontextual hidden variable theories, provided measurements are allowed to disturb the values of the hidden variables.
If a quantum system is prepared and later post-selected in certain states, "paradoxical" predictions for intermediate measurements can be obtained. This is the case both when the intermediate measurement is strong, i.e. a projective measurement with Luders-von Neumann update rule, or with weak measurements where they show up in anomalous weak values. Leifer and Spekkens [quant-ph/0412178] identified a striking class of such paradoxes, known as logical pre- and post-selection paradoxes, and showed that they are indirectly connected with contextuality. By analysing the measurement-disturbance re…
We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with high probability are contextual. The class of contextual MBQCs includes an example which is of practical interest and has a super-polynomial speedup over the best known classical algorithm, namely the quantum algorithm that solves the Discrete Log problem.
A review is made of the field of contextuality in quantum mechanics. We study the historical emergence of the concept from philosophical and logical issues. We present and compare the main theoretical frameworks that have been derived. Finally, we focus on the complex task of establishing experimental tests of contextuality. Throughout this work, we try to show that the conceptualisation of contextuality has progressed through different complementary perspectives, before summoning them together to analyse the signification of contextuality experiments. Doing so, we argue that contextuality eme…
In the paper, a value assignment for projection operators relating to a quantum system is equated with assignment of truth-values to the propositions associated with these operators. In consequence, the Kochen-Specker theorem (its localized variant, to be exact) can be treated as the statement that a logic of those projection operators does not obey the principle of bivalence. This implies that such a logic has a gappy (partial) semantics or many-valued semantics.
Contextuality is a key feature of quantum mechanics. We present the sheaf-theoretic approach to contextuality introduced by Abramsky and Brandenburger, and show how it covers a range of logical and physical phenomena "at the borders of paradox".
Contextuality and nonlocality are non-classical properties exhibited by quantum statistics whose implications profoundly impact both foundations and applications of quantum theory. In this paper we provide some insights into logical contextuality and inequality-free proofs. The former can be understood as the possibility version of contextuality, while the latter refers to proofs of quantum contextuality/nonlocality that are not based on violations of some noncontextuality (or Bell) inequality. The present work aims to build a bridge between these two concepts from what we call possibilistic p…
The purpose of this note is to clarify the logical relationship between joint measurability and contextuality for quantum observables in view of recent developments [1-4].
Aliakbarzadeh, Kitto, and Bruza (arXiv:1909.13048v1) criticize the Contextuality-by-Default (CbD) theory in three ways: (1) they claim an internal contradiction within the theory, that consists in both denying and accepting the equality of stochastically unrelated random variables; (2) they find CbD deficient because when one confines one's attention to consistently connected systems (those with no disturbance/signaling) one can dispense with the double-indexation of random variables; and (3) they see no relationship between double-indexation and ontic states. This commentary shows that (re$\:…
A theory is universal contextual if its prediction cannot be reproduced by an ontological model satisfying both preparation and measurement noncontextuality assumptions. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely li…
Emergence and Recovery of (logical) Kochen-Specker Contextuality via Hamilton Extension
S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国立中心）；Indian Institute of Technology Jodhpur（焦浦印度理工学院）；Indian Statistical Institute Kolkata（加尔各答印度统计研究所）
Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory and any putative underlying model of it satisfying the constraint of KS-noncontextuality. A logical proof of the KS theorem is one that relies only on the compatibility relations amongst a set of projectors (a KS set) to witness this inconsistency. These compatibility relations can be represented by a hypergraph, referred to as a contextuality scenario. Here we consider contextuality scenarios that we term KS-uncolourable, e.g., those which appear in logical proofs of the KS theorem. We introduce a hypergraph framewor…
Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We g…
Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that…
The notion of contextuality, which emerges from a theorem established by Simon Kochen and Ernst Specker (1960-1967) and by John Bell (1964-1966), is certainly one of the most fundamental aspects of quantum weirdness. If it is a questioning on scholastic philosophy and a study of contrafactual logic that led Specker to his demonstration with Kochen, it was a criticism of von Neumann's "proof" that led John Bell to the result. A misinterpretation of this famous "proof" will lead them to diametrically opposite conclusions. Over the last decades, remarkable theoretical progresses have been made on…
We discuss quantum non-locality and contextuality, emphasising logical and structural aspects. We also show how the same mathematical structures arise in various areas of classical computation.
The finite set of subsystems of a finite quantum system with variables in ${\mathbb Z}(n)$, is studied as a Heyting algebra. The physical meaning of the logical connectives is discussed. It is shown that disjunction of subsystems is more general concept than superposition. Consequently the quantum probabilities related to projectors in the subsystems, are incompatible with associativity of the join in the Heyting algebra, unless if the variables belong to the same chain. This leads to contextuality, which in the present formalism has as contexts, the chains in the Heyting algebra. Logical Bell…
Contextuality - the obstruction to describing quantum mechanics in a classical statistical way - has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate non-linear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality - the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the spe…
In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum c…
We discuss quantum non-locality and contextuality using the notion of transition sets. This approach provides a way to obtain a direct logical contradiction with locality/non-contextuality in the EPRB gedanken experiment as well as a clear graphical illustration of what violations of Bell inequalities quantify. In particular, we show graphically how these violations are related to measures of non-local transition sets. We also introduce a new form of contextuality, {\em measurement ordering contextuality}, i.e. there exists commuting operators $\hat{\mathcal{A}}$ and $\hat{\mathcal{B}}$ such t…
We establish a strong link between two apparently unrelated topics: the study of conflicting information in the formal framework of valuation algebras, and the phenomena of non-locality and contextuality. In particular, we show that these peculiar features of quantum theory are mathematically equivalent to a general notion of \emph{disagreement} between information sources. This result vastly generalises previously observed connections between contextuality, relational databases, constraint satisfaction problems, and logical paradoxes, and gives further proof that contextual behaviour is not a…
Homomorphisms between relational structures play a central role in finite model theory, constraint satisfaction and database theory. A central theme in quantum computation is to show how quantum resources can be used to gain advantage in information processing tasks. In particular, non-local games have been used to exhibit quantum advantage in boolean constraint satisfaction, and to obtain quantum versions of graph invariants such as the chromatic number. We show how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad o…
In Itamar Pitowsky's reading of the Gleason and the Kochen-Specker theorems, in particular, his Logical Indeterminacy Principle, the emphasis is on the value indefiniteness of observables which are not within the preparation context. This is in stark contrast to the prevalent term {\em contextuality} used by many researchers in informal, heuristic yet omni-realistic and potentially misleading ways. This paper discusses both concepts and argues in favor of value indefiniteness in all but a continuum of contexts intertwining in the vector representing a single pure (prepared) state. Even more re…
Results of measurements give legitimacy to a physical theory. What if acquiring these results in the first place necessitates what the same theory considers to be an interaction? In this note, we assume that theories account for interactions so that they are empirically traceable, and that observations necessarily go with such an interaction with the observed system. We investigate consequences of this assumption: The unfolding language game, inspired by "quantum logic," leads to a class of contextual and probabilistic theories. Contextuality becomes a means to render interactions, thus also m…
Although `quantum contextuality' is one of the most fundamental non-classical feature, its generic role in information processing and computation is an open quest. In this article, we present a family of distributed computing tasks pertaining to every logical proof of Kochen-Specker (KS) contextuality in two different one-way communication scenarios: (I) communication of bounded dimensional system, (II) communication of unbounded dimensional system while keeping certain information oblivious, namely, oblivious communication (OC). As the later remains largely unexplored, we introduce a general …
We show that the established physics of spin valves together with the recently discovered giant spin-Hall effect could be used to construct Read and Write units that can be integrated into a single spin switch with input-output isolation, gain and fan-out similar to CMOS inverters, but with the information stored in nanomagnets making it non-volatile. Such spin switches could be interconnected, with no external amplification, just with passive circuit elements, to perform logic operations. Moreover, since the digitization and storage occurs naturally in the magnets, the voltages can be used to…
Quantum Hamiltonian Computing is a recent approach that uses quantum systems, in particular a single molecule, to perform computational tasks. Within this approach, we present explicit methods to construct logic gates using two different designs, where the logical outputs are encoded either at fixed energy and spatial positioning of the quantum states, or at different energies. We use these results to construct quantum Boolean adders involving a minimal number of quantum states with the two designs. We also establish a matrix algebra giving an analogy between classical Boolean logic gates and …
The concept of complementarity in combination with a non-Boolean calculus of propositions refers to a pivotal feature of quantum systems which has long been regarded as a key to their distinction from classical systems. But a non-Boolean logic of complementary features may also apply to classical systems, if their states and observables are defined by partitions of a classical state space. If these partitions do not satisfy certain stability criteria, complementary observables and non-Boolean propositional lattices may be the consequence. This is especially the case for non-generating partitio…
We give an overview and conceptual discussion of some of our results on contextuality and non-locality. We focus in particular on connections with the work of Itamar Pitowsky on correlation polytopes, Bell inequalities, and Boole's "conditions of possible experience".
Tohoku University（东北大学）；Portland State University（波特兰州立大学）
It has been experimentally proven that realizing universal quantum gates using higher-radices logic is practically and technologically possible. We developed a Parallel Genetic Algorithm that synthesizes Boolean reversible circuits realized with a variety of quantum gates on qudits with various radices. In order to allow synthesizing circuits of medium sizes in the higher radix quantum space we performed the experiments using a GPU accelerated Genetic Algorithm. Using the accelerated GA we compare heuristic improvements to the mutation process based on cost minimization, on the adaptive cost o…
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not…
In this paper we show that there is a direct correspondence between quantum Boolean operations and certain forms of classical (non-quantum) logic known as Reed-Muller expansions. This allows us to readily convert Boolean circuits into their quantum equivalents. A direct result of this is that the problem of synthesis and optimization of quantum Boolean logic can be tackled within the field of Reed-Muller logic.
We consider orthomodular posets endowed with a symmetric difference. We call them ODPs. Expressed in the quantum logic language, we consider quantum logics with an XOR-type connective. We study three classes of "almost Boolean" ODPs, two of them defined by requiring rather specific behaviour of infima and the third by a Boolean-like behaviour of Frink ideals. We establish a (rather surprising) inclusion between the three classes, shadding thus light on their intrinsic properties. (More details can be found in the Introduction that follows.) Let us only note that the orthomodular posets pursued…
We report first steps towards elucidating the relationship between contextuality, measurement-based quantum computation (MBQC) and the non-classical logic of a topos associated with the computation. We show that, in a class of MBQC, classical universality requires non-classical logic, which is 'consumed' during the course of the computation, thereby pinpointing another potential quantum computational resource.
Since the enlightening proofs of quantum contextuality first established by Kochen and Specker, and also by Bell, various simplified proofs have been constructed to exclude the non-contextual hidden variable theory of our nature at the microscopic scale. The conflict between the non-contextual hidden variable theory and quantum mechanics is commonly revealed by Kochen-Specker (KS) sets of yes-no tests, represented by projectors (or rays), via either logical contradictions or noncontextuality inequalities in a state-(in)dependent manner. Here we first propose a systematic and programmable const…
We investigate dynamical properties of a quantum generalization of classical reversible Boolean networks. The state of each node is encoded as a single qubit, and classical Boolean logic operations are supplemented by controlled bit-flip and Hadamard operations. We consider synchronous updating schemes in which each qubit is updated at each step based on stored values of the qubits from the previous step. We investigate the periodic or quasiperiodic behavior of quantum networks, and we analyze the propagation of single site perturbations through the quantum networks with input degree one. A no…
The fundamental algebraic concepts of quantum mechanics, as expressed by many authors, are reviewed and translated into the framework of the relatively new non-distributive system of Boolean fractions (also called conditional events or conditional propositions). This system of ordered pairs (A|B) of events A, B, can express all of the non-Boolean aspects of quantum logic without having to resort to a more abstract formulation like Hilbert space. Such notions as orthogonality, superposition, simultaneous verifiability, compatibility, orthoalgebras, orthocomplementation, modularity, and the Sasa…
