The classical methods used by recursion theory and formal logic to block paradoxes do not work in quantum information theory. Since quantum information can exist as a coherent superposition of the classical ``yes'' and ``no'' states, certain tasks which are not conceivable in the classical setting can be performed in the quantum setting. Classical logical inconsistencies do not arise, since there exist fixed point states of the diagonalization operator. In particular, closed timelike curves need not be eliminated in the quantum setting, since they would not lead to any paradoxical outcome controllability. Quantum information theory can also be subjected to the treatment of inconsistent information in databases and expert systems. It is suggested that any two pieces of contradicting information are stored and processed as coherent superposition. In order to be tractable, this strategy requires quantum computation.
GHZ paradoxes are presented for all even numbers of qubits from four up. They are obtained from proofs of the Kochen-Specker (KS) theorem by showing how the assumption of noncontextuality can be justified on the basis of locality. The nature of the entangled states involved in our paradoxes is discussed. Some multiqubit proofs of the KS theorem are also presented in the form of diagrams from which they are visually obvious. The implications of our results are discussed.
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.
As a serious attempt for constructing a new foundation for describing micro-incidents from a local causal standpoint, I explained before that each micro-entity can be assumed to be composed of a probability field joined to a particle (called a particle-field or PF system) which is endowed with energy, having an objective character. In this essay, some odd quantum phenomena including the tunneling effect, the measurement problem and the EPR theorem are reexamined in detail from the standpoint of the new theory.
This paper explores the consequences of denying the "emptiness of paths not taken," EPNT, premise of Bernstein, Greenberger, Horne, and Zeilinger (BGHZ) in their paper titled, Bell theorem without inequalities.[ ] Carrying out the negation of EPNT leads to the concept of a "shadow stream." Streams are essentially particle implementations of the paths in Feynman path-integrals, resulting in a simple and consistent extension of the standard axioms for quantum mechanics. The construct provides elegant resolutions of single- and multi-particle interference paradoxes. Moreover, combining the argume…
Recently, a geometric embedding of the classical space and classical phase space of an n-particle system into the space of states of the system was constructed and shown to be physically meaningful. Namely, the Newtonian dynamics of the particles was recovered from the Schroedinger dynamics by constraining the state of the system to the classical phase space submanifold of the space of states. A series of theorems related to the embedding and the Schroedinger evolution with a random Hamiltonian was proven and shown to be applicable to the process of measurement in classical and quantum mechani…
This thesis reports progress in the analysis of causality and multi-agent logical paradoxes in quantum and post-quantum theories. These research areas are highly relevant for the foundations of physics as well as the development of quantum technologies. In the first part, focussing on causality, we develop techniques for using generalised entropies to analyse distinctions between classical and non-classical causal structures. We derive new properties of Tsallis entropies of systems that follow from the relevant causal structure, and apply these to obtain new necessary constraints for classical…
We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice $BW_n$ ($n=2^m$). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs $v-l$ involving $v$ rays and $l$ $2n$-dimensional bases of $n$-qubits. Specifically, we look a…
Using the newly introduced general ordering theorem (GOT) by Shähandeh and Bazrafkan, we derive and generalize some quantum optical identities and give their applications.
Device-Independent (DI) security is the best form of quantum cryptography, providing information-theoretic security based on the very laws of nature. In its highest form, security is guaranteed against adversaries limited only by the no-superluminal signalling rule of relativity. The task of randomness amplification, to generate secure fully uniform bits starting from weakly random seeds, is of both cryptographic and foundational interest, being important for the generation of cryptographically secure random numbers as well as bringing deep connections to the existence of free-will. DI no-sign…
We present a generalized Greenberger-Horne-Zeilinger (GHZ) theorem, which involves more than two local measurement settings for some parties, and cannot be reduced to one with less settings. Our results hold for an odd number of parties. We use a set of observables, which are incompatible but share a common eigenstate, here a GHZ state. Such observables are called concurrent. The idea is illustrated with an example of a three-qutrit system and then generalized to systems of higher dimensions, and more parties. The GHZ paradoxes can lead to, e.g., secret sharing protocols.
The principles and methods of the Conformal Quantum Geometrodynamics (CQG) based on the Weyl's differential geometry are presented. The theory applied to the case of the relativistic single quantum spin 1/2 leads a novel and unconventional derivation of Dirac's equation. The further extension of the theory to the case of two spins 1/2 in EPR entangled state and to the related violation of Bell's inequalities leads, by an exact albeit non relativistic analysis, to an insightful resolution of all paradoxes implied by quantum nonlocality.
The structural connections between the Kochen-Specker (KS) theorem, pre- and post-selection (PPS) paradoxes, and anomalous weak values are explored in detail. All PPS paradoxes, such as the 3-box paradox, the Quantum Cheshire Cat, and the Quantum Pigeonhole principle, construct a particular type of ontological model that assigns an eigenvalue to each observable (independent of context) of a system such that these assignments are consistent with the PPS. It is shown that such an ontological model must be explicitly contextual in the sense of the KS theorem, or otherwise implies either a restric…
There exist limits of self-inspection due to self-referential paradoxes, incompleteness and fixed point theorems. As quantum mechanics dictates the exchange of discrete quanta, measurements and self-inspection of quantized systems are fundamentally limited.
A formula for the commutator of tensor product matrices is used to shows that, for qubits, compatibility of quantum multiparty observables almost never implies local compatibility at each site and to predict when this happens/does not happen in a concise manner. In particular, it is shown that two ``fully nontrivial'' $n$-qubit observables are compatible locally and globally if and only if they are equal up to sign. In addition, the formula gives insight into the construction of new paradoxes of the type of the Kochen-Specker Theorem, which can then be easily rephrased into proposals for new n…
We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, eith…
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…
When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen-Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextu…
We introduce Prove-It, a Python-based general-purpose interactive theorem-proving assistant designed with the goal of making formal theorem proving as easy and natural as informal theorem proving (with moderate training). Prove-It uses a highly-flexible Jupyter notebook-based user interface that documents interactions and proof steps using LaTeX. We review Prove-It's highly expressive representation of expressions, judgments, theorems, and proofs; demonstrate the system by constructing a traditional proof-by-contradiction that $\sqrt{2}\notin\mathbb{Q}$; and discuss how the system avoids incon…
The possibility of a fundamental consistency between the basic quantum principles and reduction (so-called wave function reduction) is reexamined. The mathematical description of an organized macroscopic device is constructed explicitly as a convenient tool for this investigation. Previous no-go theorems excluding consistency are disproved on this ground, because their assumptions neglected the occurrence of organization in a real measuring apparatus. A scheme for deriving reduction from quantum mechanics is also proposed as a conjecture.
A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, these conditions determine that the holonomy group of the two-qubit interaction model contains $SU(2) \times SU(2)$. In particular, a universal two-qubit logic gate is attainable for this model.
The entangled quantum states play a key role in quantum information. The association of the quantum state vector with each individual physical system in an attributive way is a source of many false paradoxes and inconsistencies. The paradoxes are avoided if the purely statistical interpretation (SI) of the quantum state vector is adopted. According the SI the quantum theory (QT) does not provide any deterministic prediction for any individual experimental result obtained for a free physical system, for a trapped ion or for a quantum dot. In this article it is shown that if the SI is used then,…
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…
Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws fo…
In previous articles we presented a simple set of axioms named Contexts, Systems and Modalities (CSM), where the structure of quantum mechanics appears as a result of the interplay between the quantized number of modalities accessible to a quantum system, and the continuum of contexts that are required to define these modalities. In the present article we discuss further how to obtain (or rather infer) Born's rule within this framework. Our approach is compared with other former and recent derivations, and its strong links with Gleason's theorem are particularly emphasized.
Busch's theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleason's theorem. Here we show that a further generalisation is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Ou…
Quantum contextuality is one of the fundamental notions in quantum mechanics. Proofs of the Kochen-Specker theorem and noncontextuality inequalities are two means for revealing the contextuality phenomenon in quantum mechanics. It has been found that some proofs of the Kochen-Specker theorem, such as those based on rays, can be converted to a state-independent noncontextuality inequality, but it remains open whether it is true in general, i.e., whether any proof of the Kochen-Specker theorem can always be converted to a noncontextuality inequality. In this paper, we address this issue. We prov…
This text is an introduction to an operational outlook on Bell inequalities, which has been very fruitful in the past few years. It has lead to the recognition that Bell tests have their own place in applied quantum technologies, because they quantify non-classicality in a device-independent way, that is, without any need to describe the degrees of freedom under study and the measurements that are performed. At the more fundamental level, the same device-independent outlook has allowed the falsification of several other alternative models that could hope to reproduce the observed statistics wh…
