Zwitters: Particles between quantum and classical

We describe both quantum particles and classical particles in terms of a classical statistical ensemble, with a probability distribution in phase space. By use of a wave function in phase space both can be treated in the same quantum formalism. Quantum particles are characterized by a specific choice of observables and time evolution of the probability density. Then interference and tunneling are found within classical statistics. Zwitters are (effective) one-particle states for which the time evolution interpolates between quantum and classical particles. Experimental bounds on a small parameter can test quantum mechanics.     

Physical uniformities on the state space of nonrelativisitic quantum mechanics

Uniformities describing the distinguishability of states and of observables are discussed in the context of general statistical theories and are shown to be related to distinguished subspaces of continuous observables and states, respectively. The usual formalism of quantum mechanics contains no such physical uniformity for states. Using recently developed tools of quantum harmonic analysis, a natural one-to-one correspondence between continuous subspaces of nonrelativistic quantum and classical mechanics is established, thus exhibiting a close interrelation between physical uniformities for quantum states and compactifications of phase space. General properties of the completions of the quantum state space with respect to these uniformities are discussed.     

玻色-爱因斯坦凝聚系统的量子Fisher信息与混沌

量子Fisher信息作为经典Fisher信息的自然推广,与量子信息中的纠缠判断具有密切联系.在表现为典型量子混沌特征的受击两分量玻色-爱因斯坦凝聚系统中,研究了与经典相空间对应的纠缠和量子Fisher信息动力学性质. 结果表明,初次撞击后的系统量子态是纠缠的,与初态所处相空间中的混乱程度无关.而量子Fisher信息的动力学演化对系统初态非常敏感,当初态处于混沌区域时,量子Fisher信息值比初态处于规则区域时大.利用这种较好的量子-经典对应关系,得到量子Fisher信息可以刻画量子混沌的结论. 关键词： 量子Fisher信息 玻色-爱因斯坦凝聚 量子混沌 量子-经典对应     

Quantum potential and the spatial distribution of observable values

Quantum potential theory points to the possibility and usefulness of assigning values of the momentum observable at each point on the configuration space. We give a formulation of such an idea for all observables together with an analysis of the meaning of such local values within quantum mechanics. The formulation is easily extended to obtain generalised phase space distributions.     

Strategies for Positive Partial Transpose (PPT) States in Quantum Metrologies with Noise

Quantum metrology overcomes standard precision limits and has the potential to play a key role in quantum sensing. Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurements. Conventional bounds to the measurement precision such as the shot noise limit are not as fundamental as the Heisenberg limits, and can be beaten with quantum strategies that employ ‘quantum tricks’ such as squeezing and entanglement. Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered to be too weakly entangled for applications. Since no pure entanglement can be distilled from them, they are also called bound entangled states. We provide strategies, using which multipartite quantum states that have a positive partial transpose with respect to all bi-partitions of the particles can still outperform separable states in linear interferometers.     

Amplitude phase-space model for quantum mechanics

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrödinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.     

Geometrization of quantum mechanics

Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are near the vacuum. In particular the origin of complex structure is described.     

A causal quantum theory in phase space

The causal theory for the coherent state representation of quantum mechanics is derived. The general conditions for the classical limit are given and it is shown that phase space classical mechanics can be obtained as a limit even for stationary states, in contrast to the de Broglie-Bohm quantum theory of motion.     

Classical Mechanics Is not the ħ, → 0 Limit of Quantum Mechanics

Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied. It is shown that the sets have a common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vice versa, there exist tomograms of quantum states which are not admissible in classical mechanics. The role of different transformations of reference frames in the phase space of classical and quantum systems (scaling and rotation) determining the admissibility of tomograms as well as the role of quantum uncertainty relations are elucidated. The union of all admissible tomograms of both quantum and classical states is discussed in the context of interaction of quantum and classical systems. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively. The role of the semigroup of scaling transforms of the Planck's constant is discussed.     

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.     

Statistics of Wave Functions for a Point Scatterer on the Torus

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

Epistemic Entanglement due to Non-generating Partitions of Classical Dynamical Systems

Quantum entanglement relies on the fact that pure quantum states are dispersive and often inseparable. Since pure classical states are dispersion-free they are always separable and cannot be entangled. However, entanglement is possible for epistemic, dispersive classical states. We show how such epistemic entanglement arises for epistemic states of classical dynamical systems based on phase space partitions that are not generating. We compute epistemically entangled states for two coupled harmonic oscillators.     

Classical formulation of quantum mechanics

The concept of quantum state is given in terms of classical probability for position in squeezed and rotated classical reference frames in phase space. Stationary states and energy levels of the quantum system are obtained in a classical formulation of quantum mechanics. The positive probability density of the harmonic oscillator position is obtained by solving a new eigenvalue equation of standard quantum mechanics instead of the Schrödinger equation. The orthogonality and completeness relations are found for the eigendistributions.     

Quantum dynamics and decoherence in generalized Coleman-Hepp model and boson detector model

Decoherence process in quantum mechanical entangled spin states is formulated and solved on the basis of a generalized Coleman-Hepp model and a boson detector model. These models are exactly solved to give reduced density matrices and the von Neumann entropy. Detailed studies are done on environmental fluctuations which cause decoherence in the correlated entangled states: a single detector model and two detectors model are examined with the use of analytic solutions and numerical evaluations.Received: 13 May 2004, Published online: 10 August 2004PACS: 03.65.-w Quantum mechanics - 05.30.-d Quantum statistical mechanics     

Deterministic entanglement of photons in two superconducting microwave resonators

Quantum entanglement, one of the defining features of quantum mechanics, has been demonstrated in a variety of nonlinear spinlike systems. Quantum entanglement in linear systems has proven significantly more challenging, as the intrinsic energy level degeneracy associated with linearity makes quantum control more difficult. Here we demonstrate the quantum entanglement of photon states in two independent linear microwave resonators, creating N-photon NOON states (entangled states |N0> + |0N>) as a benchmark demonstration. We use a superconducting quantum circuit that includes Josephson qubits to control and measure the two resonators, and we completely characterize the entangled states with bipartite Wigner tomography. These results demonstrate a significant advance in the quantum control of linear resonators in superconducting circuits.     

Quantum Mechanics is Incomplete but it is Consistent with Locality

Quantum mechanics is seen to be incomplete not because it cannot explain the correlations that characterize entanglement without invoking either non-locality or realism, both of which, despite special relativity or no-go theorems, are at least conceivable. Quantum mechanics is incomplete, in a perhaps broader than hidden variable sense, because it fails to address within its theoretical structure the question of how even a single particle, by being in a given quantum state, causes the frequency distribution of measurement values specified by the state. This incompleteness of quantum mechanics as it is currently conceived is both fundamental and indefeasible. Failure to address the question of how the states of entangled particles are given effect to yield the correlations they specify is simply a particular albeit attention arresting instance of this incompleteness. But if that is so then quantum mechanics cannot be held to be inconsistent with locality.     

Hamiltonian Formulation of Statistical Ensembles and Mixed States of Quantum and Hybrid Systems

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems.