Quantum Principles and Mathematical Computability

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algorithm” for one of the insoluble problems of mathematics, the Hilbert's tenth and equivalently the Turing halting problem. The key element of this algorithm is the computability and measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values.     

Quantum Algorithm for Hilbert's Tenth Problem

We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain Hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church—Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.     

Uncertainty-induced quantum nonlocality

Based on the skew information, we present a quantity, uncertainty-induced quantum nonlocality (UIN) to measure the quantum correlation. It can be considered as the updated version of the original measurement-induced nonlocality (MIN) preserving the good computability but eliminating the non-contractivity problem. For 2×d$2\times d$-dimensional state, it is shown that UIN can be given by a closed form. In addition, we also investigate the maximal uncertainty-induced nonlocality.     

On “Decisions and Revisions Which a Minute Will Reverse”: Consciousness,The Unconscious and Mathematical Modeling of Thinking

This article considers a partly philosophical question: What are the ontological and epistemological reasons for using quantum-like models or theories (models and theories based on the mathematical formalism of quantum theory) vs. classical-like ones (based on the mathematics of classical physics), in considering human thinking and decision making? This question is only partly philosophical because it also concerns the scientific understanding of the phenomena considered by the theories that use mathematical models of either type, just as in physics itself, where this question also arises as a physical question. This is because this question is in effect: What are the physical reasons for using, even if not requiring, these types of theories in considering quantum phenomena, which these theories predict fully in accord with the experiment? This is clearly also a physical, rather than only philosophical, question and so is, accordingly, the question of whether one needs classical-like or quantum-like theories or both (just as in physics we use both classical and quantum theories) in considering human thinking in psychology and related fields, such as decision science. It comes as no surprise that many of these reasons are parallel to those that are responsible for the use of QM and QFT in the case of quantum phenomena. Still, the corresponding situations should be understood and justified in terms of the phenomena considered, phenomena defined by human thinking, because there are important differences between these phenomena and quantum phenomena, which this article aims to address. In order to do so, this article will first consider quantum phenomena and quantum theory, before turning to human thinking and decision making, in addressing which it will also discuss two recent quantum-like approaches to human thinking, that by M. G. D’Ariano and F. Faggin and that by A. Khrennikov. Both approaches are ontological in the sense of offering representations, different in character in each approach, of human thinking by the formalism of quantum theory. Whether such a representation, as opposed to only predicting the outcomes of relevant experiments, is possible either in quantum theory or in quantum-like theories of human thinking is one of the questions addressed in this article. The philosophical position adopted in it is that it may not be possible to make this assumption, which, however, is not the same as saying that it is impossible. I designate this view as the reality-without-realism, RWR, view and in considering strictly mental processes as the ideality-without-idealism, IWI, view, in the second case in part following, but also moving beyond, I. Kant’s philosophy.     

Quantum Transparency of Barriers and Reflection from Wells for Clusters of Identical Particles

A method for solving the problem of quantum transmission through potential barriers or potential wells for a compound system consisting of several identical particles coupled via pair oscillator-type potentials in the oscillator representation of the symmetrized coordinates is considered. The efficiency of the proposed approach, algorithms and programs is demonstrated by the examples of calculation of complex energy values and analysis of metastable states of compound systems of two, three, and four identical particles on a straight line, which lead to the effect of quantum transparency of the potential barriers or quantum reflection from the wells.     

Extracting dynamical equations from experimental data is NP hard

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70?years).     

Study of a quantum scattering process by means of entropic measures

In this Letter, a scattering process of quantum particles through a potential barrier is considered. The statistical complexity and the Fisher–Shannon information are calculated for this problem. The behaviour of these entropy-information measures as a function of the energy of the incident particles is compared with the behaviour of a physical magnitude, the reflection coefficient in the barrier. We find that these statistical magnitudes present their minimum values in the same situations in which the reflection coefficient is null. These are the situations where the total transmission through the barrier is achieved, the transparency points, a typical phenomenon due to the quantum nature of the system.     

A Generalized Geometric Measurement of Quantum Discord:Exact Treatment

A generalization of the geometric measure of quantum discord is introduced in this article, based on Hellinger distance. Our definition has virtues of computability and independence of local measurement. In addition it also does not suffer from the recently raised critiques about quantum discord. The exact result can be obtained for bipartite pure states with arbitrary levels, which is completely determined by the Schmidt decomposition. For bipartite mixed states the exact result can also be found for a special case. Furthermore the generalization into multipartite case is direct. It is shown that it can be evaluated exactly when the measured state is invariant under permutation or translation. In addition the detection of quantum phase transition is also discussed for Lipkin–Meshkov–Glick and Dicke model.     

Quantum Attacks on Sum of Even–Mansour Construction with Linear Key Schedules

Shinagawa and Iwata are considered quantum security for the sum of Even–Mansour (SoEM) construction and provided quantum key recovery attacks by Simon’s algorithm and Grover’s algorithm. Furthermore, quantum key recovery attacks are also presented for natural generalizations of SoEM. For some variants of SoEM, they found that their quantum attacks are not obvious and left it as an open problem to discuss the security of such constructions. This paper focuses on this open problem and presents a positive response. We provide quantum key recovery attacks against such constructions by quantum algorithms. For natural generalizations of SoEM with linear key schedules, we also present similar quantum key recovery attacks by quantum algorithms (Simon’s algorithm, Grover’s algorithm, and Grover-meet-Simon algorithm).     

Quantum Theory and the Category of Complex Numbers

The problem of developing a formalism of quantum theory, which is both consistent with the reality of the measurements and with the invariance properties of relativistic theories, is considered. A solution is found by using a real formulation of quantum mechanics, such that there exists an interpretation of the real properties of a physical system at all times. It is demonstrated also that several concepts in quantum field theory can be recast in a real formalism. PACS: 03.65.Ca; 11.55.Ds.     

A New Book of Numbers: On the Precise Definition of Quantum Variables and the Relationships between Mathematics and Physics in Quantum Theory

Following Asher Peres’s observation that, as in classical physics, in quantum theory, too, a given physical object considered “has a precise position and a precise momentum,” this article examines the question of the definition of quantum variables, and then the new type (as against classical physics) of relationships between mathematics and physics in quantum theory. The article argues that the possibility of the precise definition and determination of quantum variables depends on the particular nature of these relationships.     

The locality problem in quantum measurements

The locality problem of quantum measurements is considered in the framework of the algebraic approach. It is shown that contrary to the currently widespread opinion one can reconcile the mathematical formalism of the quantum theory with the assumption of the existence of a local physical reality determining the results of local measurements. The key quantum experiments: double-slit experiment on electron scattering, Wheeler’s delayed-choice experiment, the Einstein-Podolsky-Rosen paradox, and quantum teleportation are discussed from the locality-problem point of view. A clear physical interpretation for these experiments, which does not contradict the classical ideas, is given.     

Logical quantization of topos theory

Traditionally set theory lies at the hub of all mathematics in the sense that every branch of mathematics, ranging from algebraic geometry to functional analysis, is to be considered as developed within some formal system of set theory. Recently topos theory, which is a natural generalization of set theory, has provided an alternative foundation of mathematics, not to say the foundation of mathematics. With these considerations in mind, we quantize topos theory logically along the lines of our previous papers. The paper culminates in the quantum treatment of classifying toposes.     

On Models of Nonlinear Evolution Paths in Adiabatic Quantum Algorithms

In this paper,we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state.If the overlap between the initial state and final state of the quantum system is not equal to zero,both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding "complexity".But when the initial state has a zero overlap with the solution state in the problem,the second model leads to an infinite time complexity of the algorithm for whatever interpolating functions being applied while the first one can still provide a constant running time.However,inspired by a related reference,a variant of the first model can be constructed which also fails for the problem when the overlap is exactly equal to zero if we want to make up the "intrinsic" fault of the second model - an increase in energy.Two concrete theorems are given to serve as explanations why neither of these two models can improve the usual adiabatic evolution algorithms for the phenomenon above.These just tell us what should be noted when using certain nonlinear evolution paths in adiabatic quantum algorithms for some special kind of problems.     

Inference in quantum physics

In quantum physics all experimental information is discrete and stochastic. But the values of physical quantities are considered to depict definite properties of the physical world. Thus physical quantities should be identified with mathematical variables which are derived from the experimental data, but which exhibit as little randomness as possible. We look for such variables in two examples by investigating how it is possible to arrive at a value of a physical quantity from intrinsically stochastic data. With the aid of standard probability calculus and elementary information theory, we are necessarily led to the quantum theoretical phases and state vectors as the first candidates for physical quantities.     

Simulation of n-qubit quantum systems: A computer-algebraic approach

During the last decade, the field of quantum computation has attracted a lot of interest and motivated many theoretical and experimental studies of n-qubit quantum systems. But apart from the promise of more efficient quantum algorithms, these investigations also revealed a number of obstacles which still have to be overcome in practice. In this context, the use of simulation programs has proved to be an appropriate method. In order to facilitate the simulation of n-qubit quantum systems, we present the Feynman software program to provide the necessary tools to define and to deal with quantum registers as well as the operators acting on them. Using an interactive design within the framework of the computer algebra system Maple, we hope that the Feynman software program will be useful not only for teaching the basic elements of quantum computing but also for studying their physical realization in the future. The text was submitted by the authors in English.     

Exponential gain in quantum computing of quantum chaos and localization

We present a quantum algorithm which simulates the quantum kicked rotator model exponentially faster than classical algorithms. This shows that important physical problems of quantum chaos, localization, and Anderson transition can be modeled efficiently on a quantum computer. We also show that a similar algorithm simulates efficiently classical chaos in certain area-preserving maps.     

Fiber Bundles in Non-relativistic Quantum Mechanics

The problem of describing a quantum mechanical system with symmetry by a fiber bundle is considered. The quantization of a fiber bundle is introduced. Fiber bundles for the Kepler problem and the rotator are constructed. The fiber bundle concept provides a new model for a physical system: it provides us with a model for an elementary particle with extension having integral values of spin.