On the Significance of the Quantum Mechanical Covariance Matrix

The characterization of quantum correlations, being stronger than classical, yet weaker than those appearing in non-signaling models, still poses many riddles. In this work we show that the extent of binary correlations in a general class of nonlocal theories can be characterized by the~existence of a certain covariance matrix. The set of quantum realizable two-point correlators in the~bipartite case then arises from a subtle restriction on the structure of this general covariance matrix. We also identify a class of theories whose covariance does not have neither a quantum nor an "almost quantum" origin, but which nevertheless produce the accessible two-point quantum mechanical correlators. Our approach leads to richer Bell-type inequalities in which the extent of nonlocality is intimately related to a non-additive entropic measure. In particular, it suggests that the Tsallis entropy with parameter q=1/2 is a natural operational measure of non-classicality. Moreover, when~generalizing this covariance matrix we find novel characterizations of the quantum mechanical set of correlators in multipartite scenarios. All these predictions might be experimentally validated when adding weak measurements to the conventional Bell test (without adding postselection)

Why the Quantum Must Yield to Gravity

After providing an extensive overview of the conceptual elements -- such as Einstein's `hole argument' -- that underpin Penrose's proposal for gravitationally induced quantum state reduction, the proposal is constructively criticised. Penrose has suggested a mechanism for objective reduction of quantum states with postulated collapse time T = h/E, where E is an ill-definedness in the gravitational self-energy stemming from the profound conflict between the principles of superposition and general covariance. Here it is argued that, even if Penrose's overall conceptual scheme for the breakdown of quantum mechanics is unreservedly accepted, his formula for the collapse time of superpositions reduces to T --> oo (E --> 0) in the strictly Newtonian regime, which is the domain of his proposed experiment to corroborate the effect. A suggestion is made to rectify this situation. In particular, recognising the cogency of Penrose's reasoning in the domain of full `quantum gravity', it is demonstrated that an appropriate experiment which could in principle corroborate his argued `macroscopic' breakdown of superpositions is not the one involving non-rotating mass distributions as he has suggested, but a Leggett-type SQUID or BEC experiment involving superposed mass distributions in relative rotation. The demonstration thereby brings out one of the distinctive characteristics of Penrose's scheme, rendering it empirically distinguishable from other state reduction theories involving gravity. As an aside, a new geometrical measure of gravity-induced deviation from quantum mechanics in the manner of Penrose is proposed, but now for the canonical commutation relations [Q, P] = ih.Comment: 33 pages (TeX, uses mtexsis) plus 3 figures (epsf). To appear in ``Physics Meets Philosophy at the Planck Scale'' (Cambridge University Press). Two footnotes adde

Entangling the motion of two optically trapped objects via time-modulated driving fields

We study entanglement of the motional degrees of freedom of two tethered and optically trapped microdisks inside a single cavity. By properly choosing the position of the trapped objects in the optical cavity and driving proper modes of the cavity it is possible to equip the system with linear and quadratic optomechanical couplings. We show that a parametric coupling between the fundamental vibrational modes of two tethered mircodiscs can be generated via a time modulated input laser. For a proper choice of the modulation frequency, this mechanism can drive the motion of the microdisks into an inseparable state in the long time limit via a two-mode squeezing process. We numerically confirm the performance of our scheme for current technology and briefly discuss an experimental setup which can be employed for detecting this entanglement by employing the quadratic coupling. We also comment on the perspectives for generating such entanglement between the oscillations of optically levitated nanospheres.Comment: 9 pages, 3 figure

The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue of DCDS-

Light-Front Bethe-Salpeter Equation

A three-dimensional reduction of the two-particle Bethe-Salpeter equation is proposed. The proposed reduction is in the framework of light-front dynamics. It yields auxiliary quantities for the transition matrix and the bound state. The arising effective interaction can be perturbatively expanded according to the number of particles exchanged at a given light-front time. An example suggests that the convergence of the expansion is rapid. This result is particular for light-front dynamics. The covariant results of the Bethe-Salpeter equation can be recovered from the corresponding auxiliary three-dimensional ones. The technical procedure is developed for a two-boson case; the idea for an extension to fermions is given. The technical procedure appears quite practicable, possibly allowing one to go beyond the ladder approximation for the solution of the Bethe-Salpeter equation. The relation between the three-dimensional light-front reduction of the field-theoretic Bethe-Salpeter equation and a corresponding quantum-mechanical description is discussed.Comment: 42 pages, 5 figure