Continuous QND measurements: no quantum noise

Monitoring an arbitrary observable is analyzed in the framework of Restricted-Path-Integral (RPI) theory of continuous quantum measurements. While in an usual (quantum-demolition) continuous measurement the measurement noise contains both classical and quantum parts, only the classical noise is shown to be present in a quantum nondemolition (QND) continuous measurement. As a result, no absolute restrictions exist on measurability of a QND observable and the measurement output satisfies the classical equation of motion. Monitoring the energy gives an example of a discrete-spectrum observable. Received: 7 April 1996 / Revised version: 7 August 1996     

Principles of maximally classical and maximally realistic quantum mechanics

Recently Auberson, Mahoux, Roy and Singh have proved a long standing conjecture of Roy and Singh: In 2N-dimensional phase space, a maximally realistic quantum mechanics can have quantum probabilities of no more than N+1 complete commuting cets (CCS) of observables coexisting as marginals of one positive phase space density. Here I formulate a stationary principle which gives a nonperturbative definition of a maximally classical as well as maximally realistic phase space density. I show that the maximally classical trajectories are in fact exactly classical in the simple examples of coherent states and bound states of an oscillator and Gaussian free particle states. In contrast, it is known that the de Broglie-Bohm realistic theory gives highly nonclassical trajectories.     

The Non-Negativity of Probabilities and the Collapse of State

The dynamical equation, being the combination of Schrödinger and Liouville equations, produces noncausal evolution when the initial state of interacting quantum and classical mechanical systems is as it is demanded in discussions regarding the problem of measurement. It is found that state of quantum mechanical system instantaneously collapses due to the non-negativity of probabilities.     

Conditional probabilities with a quantal and a kolmogorovian limit

We give a definition for the conditional probability that is applicable to quantum situations as well as classical ones. We show that the application of this definition to a two-dimensional probabilistic model, known as the epsilon model, allows one to evolve continuously from the quantum mechanical probabilities to the classical ones. Between the classical and the quantum mechanical, we identify a region that is neither classical nor quantum mechanical, thus emphasizing the need for a probabilistic theory that allows for a broader spectrum of probabilities.     

Operator Deformations in Quantum Measurement Theory

In this paper, we develop a rigorous observable- and symmetry generator-related framework for quantum measurement theory by applying operator deformation techniques previously used in noncommutative quantum field theory. This enables the conventional observables (represented by unbounded operators) to play a role also in the more general setting. In addition, it gives a way of explicitly calculating the so-called instrument of the measurement process.     

Constructing quantum games from symmetric non-factorizable joint probabilities

We construct quantum games from a table of non-factorizable joint probabilities, coupled with a symmetry constraint, requiring symmetrical payoffs between the players. We give the general result for a Nash equilibrium and payoff relations for a game based on non-factorizable joint probabilities, which embeds the classical game. We study a quantum version of Prisoners' Dilemma, Stag Hunt, and the Chicken game constructed from a given table of non-factorizable joint probabilities to find new outcomes in these games. We show that this approach provides a general framework for both classical and quantum games without recourse to the formalism of quantum mechanics.     

Probabilistic analysis of three-player symmetric quantum games played using the Einstein-Podolsky-Rosen-Bohm setting

This Letter extends our probabilistic framework for two-player quantum games to the multiplayer case, while giving a unified perspective for both classical and quantum games. Considering joint probabilities in the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting for three observers, we use this setting in order to play general three-player noncooperative symmetric games. We analyze how the peculiar non-factorizable joint probabilities provided by the EPR-Bohm setting can change the outcome of a game, while requiring that the quantum game attains a classical interpretation for factorizable joint probabilities. In this framework, our analysis of the three-player generalized Prisoner's Dilemma (PD) shows that the players can indeed escape from the classical outcome of the game, because of non-factorizable joint probabilities that the EPR setting can provide. This surprising result for three-player PD contrasts strikingly with our earlier result for two-player PD, played in the same framework, in which even non-factorizable joint probabilities do not result in escaping from the classical consequence of the game.     

Quantum structures due to fluctuations of the measurement situation

We analyze the meaning of the nonclassical aspects of quantum structures. We proceed by introducing a simple mechanistic macroscopic experimental situation that gives rise to quantum-like structures. We use this situation as a guiding example for our attempts to explain the origin of the nonclassical aspects of quantum structures. We see that the quantum probabilities can be introduced as a consequence of the presence of fluctuations on the experimental apparatuses, and show that the full quantum structure can be obtained in this way. We define the classical limit as the physical situation that arises when the fluctuations on the experiment apparatuses disappear. In the limit case we come to a classical structure, but in between we find structures that are neither quantum nor classical. In this sense, our approach not only gives an explanation for the nonclassical structure of quantum theory, but also makes it possible to define and study the structure describing the intermediate new situations. By investigating how the nonlocal quantum behavior disappears during the limiting process, we can explain theapparentlocality of the classical macroscopic world. We come to the conclusion that quantum structures are the ordinary structures of reality, and that our difficulties of becoming aware of this fact are due to prescientific prejudices, some of which we point out.     

Towards a Resolution of Dilemma: Nonlocality or Nonobjectivity?

This paper discusses a possible resolution of the nonobjectivity-nonlocality dilemma in quantum mechanics in the light of experimental tests of the Bell inequality for two entangled photons and a Bell-like inequality for a single neutron. My conclusion is that these experiments show that quantum mechanics is nonobjective: that is, the values of physical observables cannot be assigned to a system before measurement. Bell’s assumption of nonlocality has to be rejected as having no direct experimental confirmation, at least thus far. I also consider the relationships between nonobjectivity and contextuality. Specifically, I analyze the impact of the Kochen-Specker theorem on the problem of contextuality of quantum observables. I argue that, just as von Neumann’s “no-go” theorem, the Kochen-Specker theorem is based on assumptions that do not correspond to the real physical situation. Finally, I present a theory of measurement based on a classical, purely wave model (pre-quantum classical statistical field theory), a model that reproduces quantum probabilities. In this model continuous fields are transformed into discrete clicks of detectors. While this model is classical, it is nonobjective. In this case, nonobjectivity is the result of the dependence of experimental outcomes on the context of measurement, in accordance with Bohr’s view.     

Classical Versus Quantum Probability in Sequential Measurements

We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we consider a quantum mechanical measurement device or the presence of an environment. We then examine the same issues in alternative interpretations of quantum theory. We first show that multi-time probabilities cannot be naturally defined in terms of a frequency operator. We next prove that local hidden variable theories cannot reproduce the predictions of quantum theory for sequential measurements, even when the degrees of freedom of the measuring apparatus are taken into account. Bohmian mechanics, however, does not fall in this category. We finally examine an alternative proposal that sequential measurements can be modeled by a process that does not satisfy the Kolmogorov axioms of probability. This removes contextuality without introducing non-locality, but implies that the empirical probabilities cannot be always defined (the event frequencies do not converge). We argue that the predictions of this hypothesis are not ruled out by existing experimental results (examining in particular the “which way” experiments); they are, however, distinguishable in principle.     

Quantum structures: An attempt to explain the origin of their appearance in nature

We explain quantum structure as due to two effects: (a) a real change of state of the entity under the influence of the measurement and (b) a lack of knowledge about a deeper deterministic reality of the measurement process. We present a quantum machine, with which we can illustrate in a simple way how the quantum structure arises as a consequence of the two mentioned effects. We introduce a parameter that measures the size of the lack of knowledge of the measurement process, and by varying this parameter, we describe a continuous evolution from a quantum structure (maximal lack of knowledge) to a classical structure (zero lack of knowledge). We show that for intermediate values of we find a new type of structure that is neither quantum nor classical. We apply the model to situations of lack of knowledge about the measurement process appearing in other aspects of reality. Specifically, we investigate the quantumlike structures that appear in the situation of psychological decision processes, where the subject is influenced during the testing and forms some opinions during the testing process. Our conclusion is that in the light of this explanation, the quantum probabilities are epistemic and not ontological, which means that quantum mechanics is compatible with a determinism of the whole.     

Schemes for Remotely Preparing a Six-Particle Entangled Cluster-Type State

By constructing some useful measurement bases, we put forward two novel schemes via different entanglement resources to realize remote preparation of a six-particle entangled cluster-type state with high probabilities. It is shown that through a three-particle projective measurement and two-step two-particle projective measurement under the novel sets of mutually orthogonal basis vectors, the original state can be prepared with the probability 50 % and 100 %, respectively. And for the first scheme, the special cases of the prepared state that the success probability reaches up to 100 % are discussed by the permutation group. Compared with the previous proposal, the success probabilities of the proposed schemes are greatly improved. Furthermore, the present schemes are extended to the non-maximally entangled quantum channel, and the classical communication costs are calculated.     

Quantum particles from classical statistics

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being “classical” or “quantum” ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross‐fertilization between classical statistics and quantum physics.     

Environment-induced decoherence I. The Stern-Gerlach measurement

A dynamical model for the collapse of the wave function in a quantum measurement process is proposed by considering the interaction of a quantum system (spin -1/2) with a macroscopic quantum apparatus interacting with an environment in a dissipative manner. The dissipative interaction leads to decoherence in the superposition states of the apparatus, making its behaviour classical in the sense that the density matrix becomes diagonal with time. Since the apparatus is also interacting with the system, the probabilities of the diagonal density matrix are determined by the state vector of the system. We consider a Stern-Gerlach type model, where a spin-1/2 particle is in an inhomogeneous magnetic field, the whole set up being in contact with a large environment. Here we find that the density matrix of the combined system and apparatus becomes diagonal and the momentum of the particle becomes correlated with a spin operator, selected by the choice of the system-apparatus interaction. This allows for a measurement of spin via a momentum measurement on the particle with associated probabilities in accordance with quantum principles.     

Survival of Classical and Quantum Particles in the Presence of Traps

We present a detailed comparison of the motion of a classical and of a quantum particle in the presence of trapping sites, within the framework of continuous-time classical and quantum random walk. The main emphasis is on the qualitative differences in the temporal behavior of the survival probabilities of both kinds of particles. As a general rule, static traps are far less efficient to absorb quantum particles than classical ones. Several lattice geometries are successively considered: an infinite chain with a single trap, a finite ring with a single trap, a finite ring with several traps, and an infinite chain and a higher-dimensional lattice with a random distribution of traps with a given density. For the latter disordered systems, the classical and the quantum survival probabilities obey a stretched exponential asymptotic decay, albeit with different exponents. These results confirm earlier predictions, and the corresponding amplitudes are evaluated. In the one-dimensional geometry of the infinite chain, we obtain a full analytical prediction for the amplitude of the quantum problem, including its dependence on the trap density and strength.     

Quantum Probability’s Algebraic Origin

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.