Quantum-like dynamics of decision-making

In cognitive psychology, some experiments for games were reported, and they demonstrated that real players did not use the “rational strategy” provided by classical game theory and based on the notion of the Nasch equilibrium. This psychological phenomenon was called the disjunction effect. Recently, we proposed a model of decision making which can explain this effect (“irrationality” of players) Asano et al. (2010, 2011) [23] and [24]. Our model is based on the mathematical formalism of quantum mechanics, because psychological fluctuations inducing the irrationality are formally represented as quantum fluctuations Asano et al. (2011) [55]. In this paper, we reconsider the process of quantum-like decision-making more closely and redefine it as a well-defined quantum dynamics by using the concept of lifting channel, which is an important concept in quantum information theory. We also present numerical simulation for this quantum-like mental dynamics. It is non-Markovian by its nature. Stabilization to the steady state solution (determining subjective probabilities for decision making) is based on the collective effect of mental fluctuations collected in the working memory of a decision maker.     

Entanglement,superselection rules and supersymmetric quantum mechanics

In this paper we show that the energy eigenstates of supersymmetric quantum mechanics (SUSYQM) with non-definite “fermion” number are entangled states. They are “physical states” of the model provided that observables with odd number of spin variables are allowed in the theory like it happens in the Jaynes–Cummings model. Those states generalize the so-called “spin-spring” states of the Jaynes–Cummings model which have played an important role in the study of entanglement.     

Doves and hawks in economics revisited: An evolutionary quantum game theory based analysis of financial crises

The last financial and economic crisis demonstrated the dysfunctional long-term effects of aggressive behaviour in financial markets. Yet, evolutionary game theory predicts that under the condition of strategic dependence a certain degree of aggressive behaviour remains within a given population of agents. However, as a consequence of the financial crisis, it would be desirable to change the “rules of the game” in a way that prevents the occurrence of any aggressive behaviour and thereby also the danger of market crashes. The paper picks up this aspect. Through the extension of the well-known hawk-dove game by a quantum approach, we can show that dependent on entanglement, evolutionary stable strategies also can emerge, which are not predicted by the classical evolutionary game theory and where the total economic population uses a non-aggressive quantum strategy.     

Entanglements induced by cavity interacting with two-coupled atoms

We consider a quantum optics model where the cavity interacts with two-coupled atoms. The atom-atom entanglement, atoms-cavity entanglement and the mixture for the two atoms are investigated, and discuss the effects of the initial conditions, atom-atom coupling and the mean number of photons on the entanglements and mixture. We find that atom-atom coupling plays an important role in the entanglement and mixture. Numerical results show that under some conditions the phenomena of “entanglement sudden death” and “entanglement collapse and revival” emerge.     

Relativity of Pure States Entanglement

Entanglement of any pure state of an N×N bi-partite quantum system may be characterized by the vector of coefficients arising by its Schmidt decomposition. We analyze various measures of entanglement derived from the generalized entropies of the vector of Schmidt coefficients. For N≥3 they generate different ordering in the set of pure states and for some states their ordering depends on the measure of entanglement used. This odd-looking property is acceptable, since these incomparable states cannot be transformed to each other with unit efficiency by any local operation. In analogy to special relativity the set of pure states equivalent under local unitaries has a causal structure so that at each point the set splits into three parts: the “Future,” the “Past,” and the set of noncomparable states.     

Notes on nonlocal projective measurements in relativistic systems

In quantum mechanical bipartite systems, naive extensions of von Neumann’s projective measurement to nonlocal variables can produce superluminal signals and thus violate causality. We analyze the projective quantum nondemolition state-verification in a two-spin system and see how the projection introduces nonlocality without entanglement. For the ideal measurements of “R-nonlocal” variables, we argue that causality violation can be resolved by introducing further restrictions on the post-measurement states, which makes the measurement “Q-nonlocal”. After we generalize these ideas to quantum mechanical harmonic oscillators, we look into the projective measurements of the particle number of a single mode or a wave-packet of a relativistic quantum field in Minkowski space. It turns out that the causality-violating terms in the expectation values of the local operators, generated either by the ideal measurement of the “R-nonlocal” variable or the quantum nondemolition verification of a Fock state, are all suppressed by the IR and UV cutoffs of the theory. Thus relativistic quantum field theories with such projective measurements are effectively causal.     

Classical images as quantum entanglement: An image processing analogy of the GHZ experiment

In this paper we present an optical analogy of quantum entanglement by means of classical images. As in previous works, the quantum state of two or more qbits is encoded by using the spatial modulation in amplitude and phase of an electromagnetic field. We show here that bidimensional encoding of two qbit states allows us to interpret some non local features of the joint measurement by the assumption of “astigmatic” observers with different resolving power in two orthogonal directions. As an application, we discuss the optical simulation of measuring a system characterized by multiparticle entanglement. The simulation is based on a local representation of entanglement and a classical interferometric system. In particular we show how to simulate the Greenberger-Horne Zeilinger (GHZ) argument and the experimental results which interpretation illustrates the conflict between quantum mechanics and local realism.     

Long-distance quantum communication with “polarization” maximally entangled states

We propose a scheme for long-distance quantum communication where the elementary entanglement is generated through two-photon interference and quantum swapping is performed through one-photon interference. Local “polarization” maximally entangled states of atomic ensembles are generated by absorbing a single photon from on-demand single-photon sources. This scheme is robust against phase fluctuations in the quantum channels, moreover speeds up long-distance high-fidelity entanglement generation rate.     

Two quantum dialogue protocols without information leakage

By swapping the entanglement of Bell states, we propose two quantum dialogue protocols, which have two different characters, one with the “two-step” character and the other with the “ping-pong” character. The common merits of two dialogue protocols are that the information leakage does not exist.     

Entanglement Dynamics of the Mixed Two-qubit System in Different Noisy Channels

We investigate the entanglement dynamics of the two-qubit entanglement quantum system when they transmitted through the Pauli channels and the depolarizing channel both independently or collectively. Making use of the concurrence we found that the entanglement of a kind of mixed two-qubit system defined in this paper can be preserved in the collective Pauli σ _y noise channel, but the entanglement of the other kind of mixed two-qubit system can be preserved in the collective Pauli σ _z noise channel. Meanwhile, our quantum systems will undergoing the entanglement sudden death (ESD) in collective depolarizing channel when they return to the maximally entangled Bell states. The reason is the Landblad operators in depolarizing channel are non-commuting operators and this finding is in accord with the previous study.     

Eigenvalues of collective emission in multi-slice slab configurations

We compute the eigenmodes of collective emission from multi-slice slab configurations, using the transfer matrix formalism. We elucidate within this formalism the phenomena of “Invisible Gaps” in multiple-slice configuration and of “Precocious Superradiance” in periodic structures previously observed in numerical solutions of Maxwell-Bloch equations.     

Binary choices in small and large groups: A unified model

Two different ways to model the diffusion of alternative choices within a population of individuals in the presence of social externalities are known in the literature. While Galam’s model of rumors spreading considers a majority rule for interactions in several groups, Schelling considers individuals interacting in one large group, with payoff functions that describe how collective choices influence individual preferences. We incorporate these two approaches into a unified general discrete-time dynamic model for studying individual interactions in variously sized groups. We first illustrate how the two original models can be obtained as particular cases of the more general model we propose, then we show how several other situations can be analyzed. The model we propose goes beyond a theoretical exercise as it allows modeling situations which are relevant in economic and social systems. We consider also other aspects such as the propensity to switch choices and the behavioral momentum, and show how they may affect the dynamics of the whole population.     

Entanglement and the process of measuring the position of a quantum particle

We explore the entanglement-related features exhibited by the dynamics of a composite quantum system consisting of a particle and an apparatus (here referred to as the “pointer”) that measures the position of the particle. We consider measurements of finite duration, and also the limit case of instantaneous measurements. We investigate the time evolution of the quantum entanglement between the particle and the pointer, with special emphasis on the final entanglement associated with the limit case of an impulsive interaction. We consider entanglement indicators based on the expectation values of an appropriate family of observables, and also an entanglement measure computed on particular exact analytical solutions of the particle–pointer Schrödinger equation. The general behavior exhibited by the entanglement indicators is consistent with that shown by the entanglement measure evaluated on particular analytical solutions of the Schrödinger equation. In the limit of instantaneous measurements the system’s entanglement dynamics corresponds to that of an ideal quantum measurement process. On the contrary, we show that the entanglement evolution corresponding to measurements of finite duration departs in important ways from the behavior associated with ideal measurements. In particular, highly localized initial states of the particle lead to highly entangled final states of the particle–pointer system. This indicates that the above mentioned initial states, in spite of having an arbitrarily small position uncertainty, are not left unchanged by a finite-duration position measurement process.     

Observing quantum trajectories: From Mott’s problem to quantum Zeno effect and back

The experimental results of Kocsis et al., Mahler et al. and the proposed experiments of Morley et al. show that it is possible to construct “trajectories” in interference regions in a two-slit interferometer. These results call for a theoretical re-appraisal of the notion of a “quantum trajectory” first introduced by Dirac and in the present paper we re-examine this notion from the Bohm perspective based on Hamiltonian flows. In particular, we examine the short-time propagator and the role that the quantum potential plays in determining the form of these trajectories. These trajectories differ from those produced in a typical particle tracker and the key to this difference lies in the active suppression of the quantum potential necessary to produce Mott-type trajectories. We show, using a rigorous mathematical argument, how the active suppression of this potential arises. Finally we discuss in detail how this suppression also accounts for the quantum Zeno effect.     

Quantitative complementarity relations in bipartite systems: Entanglement as a physical reality

We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a “triality” relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine quantum aspect of bipartite systems. A generalized duality relation, that also governs possible violations of the Bell’s inequality, arises between single- and bipartite properties.     

Sudden death of entanglement: Classical noise effects

When a composite quantum state interacts with its surroundings, both quantum coherence of individual particles and quantum entanglement will decay. We have shown that under vacuum noise, i.e., during spontaneous emission, two-qubit entanglement may terminate abruptly in a finite time [T. Yu, J.H. Eberly, Phys. Rev. Lett. 93 (2004) 140404], a phenomenon termed entanglement sudden death (ESD). An open issue is the behavior of mixed-state entanglement under the influence of classical noise. In this paper we investigate entanglement sudden death as it arises from the influence of classical phase noise on two qubits that are initially entangled but have no further mutual interaction.     

Collective states in highly symmetric atomic configurations and single-photon traps

We study correlated states in circular and linear-chain configurations of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behavior is mediated by exchange of transverse photons between the atoms. For a circular atomic configuration containing N atoms, the collective energy eigenstates can be determined by group-theoretical means making use of the fact that the configuration possesses a cyclic symmetry group Z _N . For these circular configurations, the carrier spaces of the various irreducible representations of the symmetry group are at most two-dimensional, so that the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. As a consequence, the radiationless energy eigenstates carry a Z _N quantum number p = 0, 1, …, N, which is analogous to the angular momentum quantum number l = 0, 1, … carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s states are the only electronic wave functions that can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing Z _N quantum number p. When a central atom is present, the p = 0 state splits into two, showing level crossing at certain radii; in the regions between these radii, damped oscillations between two “ extreme” p = 0 states occur, where the excitation occupies either the outer atoms or the central atom only. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of λ/2 emerges both in the linear-chain and in the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jumplike “critical” behavior for next-neighbor distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configurations exhibit exponential photon trapping once the next-neighbor distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for single-photon traps.     

Hydrodynamics and phases of flocks

We review the past decade’s theoretical and experimental studies of flocking: the collective, coherent motion of large numbers of self-propelled “particles” (usually, but not always, living organisms). Like equilibrium condensed matter systems, flocks exhibit distinct “phases” which can be classified by their symmetries. Indeed, the phases that have been theoretically studied to date each have exactly the same symmetry as some equilibrium phase (e.g., ferromagnets, liquid crystals). This analogy with equilibrium phases of matter continues in that all flocks in the same phase, regardless of their constituents, have the same “hydrodynamic”—that is, long-length scale and long-time behavior, just as, e.g., all equilibrium fluids are described by the Navier-Stokes equations. Flocks are nonetheless very different from equilibrium systems, due to the intrinsically nonequilibrium self-propulsion of the constituent “organisms.” This difference between flocks and equilibrium systems is most dramatically manifested in the ability of the simplest phase of a flock, in which all the organisms are, on average moving in the same direction (we call this a “ferromagnetic” flock; we also use the terms “vector-ordered” and “polar-ordered” for this situation) to exist even in two dimensions (i.e., creatures moving on a plane), in defiance of the well-known Mermin-Wagner theorem of equilibrium statistical mechanics, which states that a continuous symmetry (in this case, rotation invariance, or the ability of the flock to fly in any direction) can not be spontaneously broken in a two-dimensional system with only short-ranged interactions. The “nematic” phase of flocks, in which all the creatures move preferentially, or are simply oriented preferentially, along the same axis, but with equal probability of moving in either direction, also differs dramatically from its equilibrium counterpart (in this case, nematic liquid crystals). Specifically, it shows enormous number fluctuations, which actually grow with the number of organisms faster than the “law of large numbers” obeyed by virtually all other known systems. As for equilibrium systems, the hydrodynamic behavior of any phase of flocks is radically modified by additional conservation laws. One such law is conservation of momentum of the background fluid through which many flocks move, which gives rise to the “hydrodynamic backflow” induced by the motion of a large flock through a fluid. We review the theoretical work on the effect of such background hydrodynamics on three phases of flocks—the ferromagnetic and nematic phases described above, and the disordered phase in which there is no order in the motion of the organisms. The most surprising prediction in this case is that “ferromagnetic” motion is always unstable for low Reynolds-number suspensions. Experiments appear to have seen this instability, but a quantitative comparison is awaited. We conclude by suggesting further theoretical and experimental work to be done.     

Entanglement propagation and typicality of measurements in the quantum Kac ring

We study the time evolution of entanglement in a new quantum version of the Kac ring, where two spin chains become dynamically entangled by quantum gates, which are used instead of the classical markers. The features of the entanglement evolution are best understood by using knowledge about the behavior of an ensemble of classical Kac rings. For instance, the recurrence time of the quantum many-body system is twice the length of the chain and “thermalization” only occurs on time scales much smaller than the dimension of the Hilbert space. The model thus elucidates the relation between the results of measurements in quantum and classical systems: While in classical systems repeated measurements are performed over an ensemble of systems, the corresponding result is obtained by measuring the same quantum system prepared in an appropriate superposition repeatedly.