Negativity and contextuality are equivalent notions of nonclassicality

Two notions of nonclassicality that have been investigated intensively are: (i) negativity, that is, the need to posit negative values when representing quantum states by quasiprobability distributions such as the Wigner representation, and (ii) contextuality, that is, the impossibility of a noncontextual hidden variable model of quantum theory. Although both of these notions were meant to characterize the conditions under which a classical explanation cannot be provided, we demonstrate that they prove inadequate to the task and we argue for a particular way of generalizing and revising them. With the refined version of each in hand, it becomes apparent that they are in fact one and the same. We also demonstrate the impossibility of noncontextuality or non-negativity in quantum theory with a novel proof that is symmetric in its treatment of measurements and preparations.     

The Status of Determinism in Proofs of the Impossibility of a Noncontextual Model of Quantum Theory

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine’s theorem do not challenge this conclusion.     

Logical foundation of quantum mechanics

The subject of this article is the reconstruction of quantum mechanics on the basis of a formal language of quantum mechanical propositions. During recent years, research in the foundations of the language of science has given rise to adialogic semantics that is adequate in the case of a formal language for quantum physics. The system ofsequential logic which is comprised by the language is more general than classical logic; it includes the classical system as a special case. Although the system of sequential logic can be founded without reference to the empirical content of quantum physical propositions, it establishes an essential part of the structure of the mathematical formalism used in quantum mechanics. It is the purpose of this paper to demonstrate the connection between the formal language of quantum physics and its representation by mathematical structures in a self-contained way.     

Optical scheme to demonstrate state-independent quantum contextuality

The contradiction between classical and quantum physics can be identified through quantum contextuality, which does not need composite systems or spacelike separation. Contextuality is proven either by a logical contradiction between the noncontextuality hidden variable predictions and those of quantum mechanics or by the violation of noncontextual inequality. We propose an experimental scheme of state-independent contextual inequality derived from the Mermin proof of the Kochen-Specker (KS) theorem in eight-dimensional Hilbert space, which could be observed either in an individual system or in a composite system. We also show how to resolve the compatibility problems. Our scheme can be implemented in optical systems with current experiment techniques.     

Contextuality and the probability representation of quantum states

We consider the notions of contextuality and noncontextuality within the framework of the probability representation of quantum states. We present an example of qutrit states and violation of the noncontextuality inequalities using the spin tomogram and tomographic symbols of the observables.     

On Noncontextual,Non-Kolmogorovian Hidden Variable Theories

One implication of Bell’s theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (Br J Philos Sci 66(4): 905–927, 2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.     

A compact program code for simulations of quantum algorithms in classical computers

A general quantum simulation language on a classical computer provides the opportunity to compare an experiential result from the development of quantum computers with mathematical theory. The intention of this research is to develop a program language that is able to make simulations of all quantum algorithms in same framework. This study examines the simulation of quantum algorithms on a classical computer with a symbolic programming language. We use the language Mathematica to make simulations of well-known quantum algorithms. The program code implemented on a classical computer will be a straight connection between the mathematical formulation of quantum mechanics and computational methods. This gives us an uncomplicated and clear language for the implementations of algorithms. The computational language includes essential formulations such as quantum state, superposition and quantum operator. This symbolic programming language provides a universal framework for examining the existing as well as future quantum algorithms. This study contributes with an implementation of a quantum algorithm in a program code where the substance is applicable in other simulations of quantum algorithms.     

Teaching a laser beam to go straight

In classical physics a beam of light propagates in a perfectly straight line and this means that we can measure small displacements with unlimited accuracy. However, this is not correct for real laser beams when we take the quantum properties of light into account. Spatial measurements will be limited by quantum noise, similar to the limitations for optical communication and sensing. Here we derive the spatial quantum noise limit and show how to measure it. Next we demonstrate that we can use specially prepared light with quantum correlations, so-called squeezed light, to improve spatial measurements to below this quantum limit. In this way we prepare a beam which goes in a straighter line than the output of any conventional laser.     

Quasiclassicality and Quantum Measurement

We address two problems arising in the quantum measurement process: A rigorous definition of quasiclassical systems and its implications for the observed collapse of the wave function. For a mathematical definition of quasiclassical systems, we recall the structure of models for the classical world. They describe the dynamics of some simultaneously measurable quantities, thereby ignoring many properties of the modeled real world phenomena, especially all quantum mechanical ones. In this article, we define a quasiclassical system as a quantum system which allows such a simplified modelling. By classifying such quasiclassical systems, it is shown that they naturally correspond to classical systems in the usual sense. By describing quantum measurements with the aid of quasiclassical systems, we then observe an effect that is similar to decoherence: While the latter implies that off-diagonal entries of the density matrix vanish, in the former they correspond to the parts of the system that are not modeled and thus can be ignored. Especially, they do not influence any measurements of the properties contained in the classical model. Mathematically, this allows to treat the output of a quantum measurement as a classical probability distribution. Finally, we discuss some implications of this definition of quasiclassicality on the interpretation of quantum mechanics.     

Toward an architecture for quantum programming

It is becoming increasingly clear that, if a useful device for quantum computation will ever be built, it will be embodied by a classical computing machine with control over a truly quantum subsystem, this apparatus performing a mixture of classical and quantum computation. This paper investigates a possible approach to the problem of programming such machines: a template high level quantum language is presented which complements a generic general purpose classical language with a set of quantum primitives. The underlying scheme involves a run-time environment which calculates the byte-code for the quantum operations and pipes it to a quantum device controller or to a simulator. This language can compactly express existing quantum algorithms and reduce them to sequences of elementary operations; it also easily lends itself to automatic, hardware independent, circuit simplification. A publicly available preliminary implementation of the proposed ideas has been realised using the language.Received: 25 June 2002, Published online: 30 July 2003PACS: 03.67.Lx Quantum computation     

Connecting Quantum Contextuality and Genuine Multipartite Nonlocality with the Quantumness Witness

The Clauser-Horne-Shimony-Holt-type noncontextuality inequality and the Svetlichny inequality are derived from the Alicki-van Ryn quantumness witness.Thus connections between quantumness and quantum contextuality,and between quantumness and genuine multipartite nonlocality are established.     

Fractional Fourier transform and geometric quantization

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the “fractional Fourier transform” provides a simple example of this construction. As an application of this technique we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schrödinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.     

Quantum Uncertainties and Holism Seem to Render Irrelevant Qudit-Semantics

We consider a semantics based on the peculiar holistic features of the quantum formalism. Any formula of the language gives rise to a quantum circuit that transforms the density operator associated to the formula into the density operator associated to the atomic subformulas in a reversible way. The procedure goes from the whole to the parts against the compositionality-principle and gives rise to a semantic characterization for a new form of quantum logic that has been called “Łukasiewicz quantum computational logic”. It is interesting to compare the logic based on qubit-semantics with that on qudit-semantics. Having in mind the relationships between classical logic and Łukasiewicz-many valued logics, one could expect that the former is stronger than the fragment of the latter. However, this is not the case. From an intuitive point of view, this can be explained by recalling that the former is a very weak form of logic. Many important logical arguments, which are valid either in Birkhoff and von Neumann’s quantum logic or in classical logic, are generally violated.     

Complementary classical fidelities as an efficient criterion for the evaluation of experimentally realized quantum operations

It is shown that a good estimate of the fidelity of an experimentally realized quantum process can be obtained by measuring the outputs for only two complementary sets of input states. The number of measurements required to test a quantum network operation is therefore only twice as high as the number of measurements required to test a corresponding classical system.     

Quantum Secure Direct Communication with W State

A new theoretical scheme for quantum secure direct communication is proposed, where four-qubit symmetric W state functions as quantum channel. It is shown that two legitimate users can directly transmit the secret messages by using Bell-basis measurements and classical communication. The scheme is completely secure if the quantum channel is perfect. Even if the quantum channel is unsecured, it is still possible for two users to perform their secure communication. One bit secret message can be transmitted by sending a bit classical information.