Wigner function and Bell’s inequalities for even and odd coherent states

The Wigner function and the symplectic tomogram of an entangled quantum state, which is a superposition of the photon’s coherent states (even and odd coherent states), is studied. Photon statistics and violation of Bell’s inequality for the photon state are discussed.     

Bell-type inequalities in classical probability theory

A review of the tomographic probability representation for qudit sates is presented. Properties of related stochastic matrices are considered. Tomograms of two qubits and three qubits are used to study the Bell-type inequalities. The Bell-type inequalities in the standard classical probability theory are discussed. Joint probability distributions of classical systems with several random variables and their properties in the case of factorized distribution functions are considered.     

Violation of Bell’s inequality in neutral kaons system

We show by general considerations that it is not possible to test violation of the existing versions of Bell’s inequality in entangled neutral kaons system using experimentally accessible thin regenerators. We point out the loophole in the recent argument (A Bramon and M Nowakowski, Phys. Rev. Lett. 83, 1 (1999)) that claimed such a test to be possible.     

The effect of recommended role models in prisoner’s dilemma game

Previous studies concerning the prisoner’s dilemma game on graphs conventionally assume that individuals select role models from their replacement graphs at random. We propose a extended prisoner’s dilemma game model to study the impact of recommended role models on the evolution of cooperation in a homogeneous population. Individuals are endowed with the capacity to recommend the ones they imitated in the past to their neighbors for strategy updating. Numerical simulations show that cooperation can be improved significantly when recommendation is allowed. Our results might be helpful in explaining the widespread cooperation in the real world.     

Proposal to Test a Transient Deviation from Quantum Mechanics’ Predictions for Bell’s Experiment

Quantum mechanics predicts correlations between measurements performed in distant regions of a spatially spread entangled state to be higher than allowed by intuitive concepts of Locality and Realism. These high correlations forbid the use of nonlinear operators of evolution (which would be desirable for several reasons), for they may allow faster-than-light signaling. As a way out of this situation, it has been hypothesized that the high quantum correlations develop only after a time longer than L/c has elapsed (where L is the spread of the entangled state and c is the velocity of light). In shorter times, correlations compatible with Locality and Realism would be observed instead. A simple hidden variables model following this hypothesis is described. It is based on a modified Wheeler–Feynman theory of radiation. This hypothesis has not been disproved by any of the experiments performed to date. A test achievable with accessible means is proposed and described. It involves a pulsed source of entangled states and stroboscopic record of particle detection during the pulses. Data recorded in similar but incomplete optical experiments are analyzed, and found consistent with the proposed model. However, it is not claimed, in any sense, that the hypothesis has been validated. On the contrary, it is stressed that a complete, specific test is absolutely needed.     

Randomness, evenness, and Rényi’s index

This paper links together the notion of entropy and the notion of inequality indices—the former is applied in Statistical Physics to measure randomness, and the latter is applied in Economics to measure evenness. We explore the profound similarities between these diametric notions, construct a mathematical transformation between them, and show how randomness can be used to measure evenness-and vice versa. In particular, we devise and study Rényi’s index—a randomness-based measure of evenness with special properties. Rényi’s index is established as an effectual gauge of statistical heterogeneity in the context of general probability laws defined on the positive half-line.     

Comments on Shimony’s “An Analysis of Stapp’s ‘A Bell-Type Theorem without Hidden Variables’ ”

The hidden-variable theorems of Bell and followers depend upon an assumption, namely the hidden-variable assumption, that conflicts with the precepts of quantum philosophy. Hence from an orthodox quantum perspective those theorems entail no faster-than-light transfer of information. They merely reinforce the ban on hidden variables. The need for some sort of faster-than-light information transfer can be shown by using counterfactuals instead of hidden variables. Shimony’s criticism of that argument fails to take into account the distinction between no-faster-than-light connection in one direction and that same condition in both directions. The argument can be cleanly formulated within the framework of a fixed past, open future interpretation of quantum theory, which neatly accommodates the critical assumptions that the experimenters are free to choose which experiments they will perform. The assumptions are compatible with the Tomonaga–Schwinger formulation of quantum field theory, and hence with orthodox quantum precepts, and with the relativistic requirement that no prediction pertaining to an outcome in one region can depend upon a free choice made in a region spacelike-separated from the first.     

On the naturalness of Einstein’s equation

We compute all 2-covariant tensors naturally constructed from a semi-Riemannian metric which are divergence-free and have weight greater than −2.     

Impact of conformity on the evolution of cooperation in the prisoner’s dilemma game

We present an evolutionary model of the prisoner’s dilemma game taking into account two cognitive mechanisms: (1) payoff-biased strategy transmission and (2) a conformity mechanism involving a tendency to copy the most frequent nearby strategy in a group. Moreover, for two types of conformity, a minority mechanism and a majority rule, a dual process holds whereby the types differ in both the factors that give rise to them and the effects they have. By contrast, a signal process suggests that differences between the two forms of influence are primarily of degree and that fundamental processes are at work in both. We explore the model using both well-mixed and spatially structured populations. When the temptation to defect is low and both conformism and local interactions are present, the system can reach high levels of cooperation or even a full cooperation state. Furthermore, we find a stronger effect of conformity and a higher level of cooperation among the population regardless of the group size. This indicates that conformity follows a signal process. However, when the temptation to defect is rather large, results for the minority influence change non-monotonically with conformism cohesion. This is remarkably different from the results under majority rule, which is considered as support for the dual process.     

Spatial contraction of the Poincaré group and Maxwell’s equations in the electric limit

The contraction of the Poincaré group with respect to the space translations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincaré group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIRs) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non-relativistic limit of Maxwell’s equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.     

Solving the Einstein-Podolsky-Rosen puzzle: the origin of non-locality in Aspect-type experiments

So far no mechanism is known, which could connect the two measurementsin an Aspect-type experiment. Here, we suggest such a mechanism, basedon the phase of a photon’s field during propagation. We showthat two polarization measurements are correlated, even if no signalpasses from one point of measurement to the other. The non-local connectionof a photon pair is the result of its origin at a common source, wherethe two fields acquire a well defined phase difference. Therefore,it is not actually a non-local effect in any conventional sense. Weexpect that the model and the detailed analysis it allows will havea major impact on quantum cryptography and quantum computation.     

Proof of absence of spooky action at a distance in quantum correlations

I prove that there is no spooky action at a distance and nonlocal state-reduction during measurements on quantum entangled systems. The prediction of quantum theory as well as experimental results are in conflict with the concept of nonlocal state-reduction, as conclusively shown here under very general assumptions. This has far-reaching implications in the interpretation of quantum mechanics in general, and demands a radical change in its present interpretation of measurements on entangled multiparticle systems. Motivated by these results we re-examine Bell’s theorem for correlations of entangled systems and find that the correlation function used by Bell fails to incorporate phase correlations at source. It is the use of such an unphysical correlation function, and not failure of locality, that leads to the Bell’s inequalities.     

Parrondo’s paradox and complementary Parrondo processes

Parrondo’s Paradox has gained a fair amount of attention due to it being counter-intuitive. Given two stochastic processes, both of which are losing in nature, it is possible to have an overall net increase in capital by periodically or randomly alternating between the two processes. In this paper, we analyze the paradox with a different approach, in which we start with one process and seek to derive its complementary process. We will also state the conditions required for this to occur. Possible applications of our results include the development of future models based on the paradox.     

The paradox of group behaviors based on Parrondo’s games

We assume a multi-agent model based on Parrondo’s games. The model consists of game A between individuals and game B. In game A, two behavioral patterns are defined: competition and inaction. A controlled alternation strategy of behavioral pattern that gives a single player the highest return is proposed when game A+B is played randomly. Interesting phenomena can be found in collective games where a large number of individuals choose the behavioral pattern by voting. When game B is the capital-dependent version, the outcome can be better for the players to vote randomly than to vote according to their own capital. An explanation of such counter-intuitive phenomena is given by noting that selfish voting prevents the competition behavior of game A that is essential for the total capital to grow. However, if game B is the history-dependent version, this counter-intuitive phenomenon will not happen. The reason is selfish voting results in the competition behavior of game A, and finally it produces the winning results.     

Evolutionary prisoner’s dilemma game in flocks

We investigate an evolutionary prisoner’s dilemma game among self-driven agents, where collective motion of biological flocks is imitated through averaging directions of neighbors. Depending on the temptation to defect and the velocity at which agents move, we find that cooperation can not only be maintained in such a system but there exists an optimal size of interaction neighborhood, which can induce the maximum cooperation level. When compared with the case that all agents do not move, cooperation can even be enhanced by the mobility of individuals, provided that the velocity and the size of neighborhood are not too large. Besides, we find that the system exhibits aggregation behavior, and cooperators may coexist with defectors at equilibrium.     

Hamilton’s principle and associated variational principles in polar thermopiezoelectricity

We express Hamilton’s principle for a regular region of thermopiezoelectric polar materials. First, we obtain a four-field variational principle which leads, as its Euler-Lagrange equations, to the divergence equations and the associated natural boundary conditions only. Next, we adjoin the rest of the fundamental equations into the variational principle through an involutory transformation. Thus, we formulate a differential type of unified variational principles operating on all the field variables. The unified variational principle is extended for the region with a fixed internal surface of discontinuity and for a curvilinear laminated region as well. The principles derived in invariant form are expressible in a system of particular coordinate system most appropriate to the geometry of the regions. They are indicated to recover some of earlier principles as special cases.     

Quantum computation in algebraic number theory: Hallgren’s efficient quantum algorithm for solving Pell’s equation

Pell’s equation is x²−dy²=1, where d is a square-free integer and we seek positive integer solutions x,y>0. Let (x₀,y₀) be the smallest solution (i.e., having smallest ). Lagrange showed that every solution can easily be constructed from A so given d it suffices to compute A. It is known that A can be exponentially large in d so just to write down A we need exponential time in the input size . Hence we introduce the regulator R=lnA and ask for the value of R to n decimal places. The best known classical algorithm has sub-exponential running time . Hallgren’s quantum algorithm gives the result in polynomial time with probability . The idea of the algorithm falls into two parts: using the formalism of algebraic number theory we convert the problem of solving Pell’s equation into the problem of determining R as the period of a function on the real numbers. Then we generalise the quantum Fourier transform period finding algorithm to work in this situation of an irrational period on the (not finitely generated) abelian group of real numbers. This paper is intended to be accessible to a reader having no prior acquaintance with algebraic number theory; we give a self-contained account of all the necessary concepts and we give elementary proofs of all the results needed. Then we go on to describe Hallgren’s generalisation of the quantum period finding algorithm, which provides the efficient computational solution of Pell’s equation in the above sense.     

Cautious strategy update promotes cooperation in spatial prisoner’s dilemma game

In the realistic world, individual cautiousness can be often involved or observed when a rational individual makes a decision. However, it remains unclear how such individual cautiousness influences the evolution of cooperative behavior. To this end, we consider a Fermi strategy updating rule, where each individual is assigned a cautiousness index that controls its learning activity, and then study the evolution of cooperation in the spatial prisoner’s dilemma game. Interestingly, it is found that cooperation can be significantly promoted when individuals’ cautiousness is considered. In particular, there exists an optimal range of the degree of cautiousness resulting in the highest cooperation level. The remarkable promotion of cooperation, as well as the emerging phase transition is explained by configurational analysis. The sensitivity of cooperation to initial states with different fractions of cooperators is also discussed. The result illustrates that high densities of cooperators can be established at small initial fractions of cooperators. The detailed mechanism for such phenomenon is explained by analyzing the typical initial configurations.