Two-Site Quantum Random Walk

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure $\mu_n$ on the space of $n$-paths, and the $\mu_n$ in turn induce a quantum measure $\mu$ on the cylinder sets within the space $\Omega$ of untruncated paths. Although $\mu$ cannot be extended to a continuous quantum measure on the full $\sigma$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $\Omega$ in a systematic way. We begin an investigation of this problem by showing that $\mu$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $\Omega$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.Comment: 28 page

The structure of classical extensions of quantum probability theory

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed

Two quantum Simpson's paradoxes

The so-called Simpson's "paradox", or Yule-Simpson (YS) effect, occurs in classical statistics when the correlations that are present among different sets of samples are reversed if the sets are combined together, thus ignoring one or more lurking variables. Here we illustrate the occurrence of two analogue effects in quantum measurements. The first, which we term quantum-classical YS effect, may occur with quantum limited measurements and with lurking variables coming from the mixing of states, whereas the second, here referred to as quantum-quantum YS effect, may take place when coherent superpositions of quantum states are allowed. By analyzing quantum measurements on low dimensional systems (qubits and qutrits), we show that the two effects may occur independently, and that the quantum-quantum YS effect is more likely to occur than the corresponding quantum-classical one. We also found that there exist classes of superposition states for which the quantum-classical YS effect cannot occur for any measurement and, at the same time, the quantum-quantum YS effect takes place in a consistent fraction of the possible measurement settings. The occurrence of the effect in the presence of partial coherence is discussed as well as its possible implications for quantum hypothesis testing.Comment: published versio

On classical models of spin

The reason for recalling this old paper is the ongoing discussion on the attempts of circumventing certain assumptions leading to the Bell theorem (Hess-Philipp, Accardi). If I correctly understand the intentions of these Authors, the idea is to make use of the following logical loophole inherent in the proof of the Bell theorem: Probabilities of counterfactual events A and A' do not have to coincide with actually measured probabilities if measurements of A and A' disturb each other, or for any other fundamental reason cannot be performed simulaneously. It is generally believed that in the context of classical probability theory (i.e. realistic hidden variables) probabilities of counterfactual events can be identified with those of actually measured events. In the paper I give an explicit counterexample to this belief. The "first variation" on the Aerts model shows that counterfactual and actual problems formulated for the same classical system may be unrelated. In the model the first probability does not violate any classical inequality whereas the second does. Pecularity of the Bell inequality is that on the basis of an in principle unobservable probability one derives probabilities of jointly measurable random variables, the fact additionally obscuring the logical meaning of the construction. The existence of the loophole does not change the fact that I was not able to construct a local model violating the inequality with all the other loopholes eliminated.Comment: published as Found. Phys. Lett. 3 (1992) 24

Optimal State Discrimination in General Probabilistic Theories

We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a {\it Helstrom family of ensembles}) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in 2-level quantum systems and in a probabilistic model with square-state space to reproduce e.g., the optimal success probabilities for binary state discrimination and $N$ numbers of symmetric quantum states. The existences of families of ensembles in binary cases are shown both in classical and quantum theories in any generic cases.Comment: 9 pages, 6 figure

Prime Factorization in the Duality Computer

We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in classical computing. All these algorithms are polynomial in the input size.Comment: 4 page

A quantum logical and geometrical approach to the study of improper mixtures

We study improper mixtures from a quantum logical and geometrical point of view. Taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right, we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we use the convex set of states in order to construct a new lattice whose atoms are all physical states: pure states and improper mixtures. This is done in order to overcome one of the problems which appear in the standard quantum logical formalism, namely, that for a subsystem of a larger system in an entangled state, the conjunction of all actual properties of the subsystem does not yield its actual state. In fact, its state is an improper mixture and cannot be represented in the von Neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems. The new lattice also contains all propositions of the von Neumann lattice. We argue that this extension expresses in an algebraic form the fact that -alike the classical case- quantum interactions produce non trivial correlations between the systems. Finally, we study the maps which can be defined between the extended lattice of a compound system and the lattices of its subsystems.Comment: submitted to the Journal of Mathematical Physic

Extending scientific computing system with structural quantum programming capabilities

We present a basic high-level structures used for developing quantum programming languages. The presented structures are commonly used in many existing quantum programming languages and we use quantum pseudo-code based on QCL quantum programming language to describe them. We also present the implementation of introduced structures in GNU Octave language for scientific computing. Procedures used in the implementation are available as a package quantum-octave, providing a library of functions, which facilitates the simulation of quantum computing. This package allows also to incorporate high-level programming concepts into the simulation in GNU Octave and Matlab. As such it connects features unique for high-level quantum programming languages, with the full palette of efficient computational routines commonly available in modern scientific computing systems. To present the major features of the described package we provide the implementation of selected quantum algorithms. We also show how quantum errors can be taken into account during the simulation of quantum algorithms using quantum-octave package. This is possible thanks to the ability to operate on density matrices

Quantum-like Representation of Extensive Form Games: Wine Testing Game

We consider an application of the mathematical formalism of quantum mechanics (QM) outside physics, namely, to game theory. We present a simple game between macroscopic players, say Alice and Bob (or in a more complex form - Alice, Bob and Cecilia), which can be represented in the quantum-like (QL) way -- by using a complex probability amplitude (game's ``wave function'') and noncommutative operators. The crucial point is that games under consideration are so called extensive form games. Here the order of actions of players is important, such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information which is available to e.g. Bob about the previous action of Alice. In general one could not construct a classical probability space underlying a QL-game. This can happen even in a QL-game with two players. In a QL-game with three players Bell's inequality can be violated. The most natural probabilistic description is given by so called contextual probability theory completed by the frequency definition of probability