We provide an overview of a canonical formalism that describes mixed quantum-classical systems in terms of statistical ensembles on configuration space, and discuss applications to measurement theory. It is shown that the formalism allows a general description of the measurement of a quantum system by a classical apparatus without running into inconsistencies. An example of classical and quantum particles interacting gravitationally is also given.
We introduce a stability criterion for quantum statistical ensembles describing macroscopic systems. An ensemble is called "stable" when a small number of local measurements cannot significantly modify the probability distribution of the total energy of the system. We apply this criterion to lattices of spins-1/2, thereby showing that the canonical ensemble is nearly stable, whereas statistical ensembles with much broader energy distributions are not stable. In the context of the foundations of quantum statistical physics, this result justifies the use of statistical ensembles with narrow ener…
In this work, we present a general non-equilibrium ensemble formalism based on the subdynamic equation (SKE). The constructing procedure is to use a similarity transformation between Gibbsian ensemble formalism and the non-equilibrium ensemble formalism. The obtained density distribution is a projected one that can represent essence part of (irreversible) evolution of the density distribution, by which a generalized reduced density distribution for the quantum canonical ensembles is studied and applications in Cayley tree and spin network are discussed.
We show insurmountable contradictions which arise if statistical ensembles are considered a consequence of the influence of the environment of the physical systems. We regard the multiplicity of states with a definite energy value as a result of internal probabilistic processes in the macrosystem; these internal probabilistic processes are not taken into account by quantum mechanics. On a simple example it is demonstrated, that canonical distribution is not independent and equal to microcanonical one, but is a result of averaging by microcanonical ensemble.
We investigate different measures of stability of quantum statistical ensembles with respect to local measurements. We call a quantum statistical ensemble "stable" if a small number of local measurements cannot significantly modify the total-energy distribution representing the ensemble. First, we numerically calculate the evolution of the stability measure introduced in our previous work [Phys. Rev. E 94, 062106 (2016)] for an ensemble representing a mixture of two canonical ensembles with very different temperatures in a periodic chain of interacting spins-1/2. Second, we propose other possi…
General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, the…
We point out that Fratini and Hayrapetyan ignored the randomness of mixing, a basic request to prepare a statistical ensemble rather than an ensemble in general. Their analysis is irrelevant for standard statistical ensembles, their conclusions about the limitations of standard theory become unjustified.
How an Equi-ensemble Description Systematically Outperforms the Weighted-ensemble Variational Quantum Eigensolver
ICGM, Université de Montpellier, CNRS, ENSCM（蒙彼利埃大学，法国国家科学研究中心，国立高等化学学校）；Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università dell’Aquila（阿奎拉大学）
We compare pseudopure state ensemble implementations, quantified by their initial polarization and ensemble size, of Grover's search algorithm to probabilistic classical sequential search algorithms in terms of their success and failure probabilities. We propose a criterion for quantifying the resources used by the ensemble implementation via the aggregate number of oracle invocations across the entire ensemble and use this as a basis for comparison with classical search algorithms. We determine bounds for a critical polarization such that the ensemble algorithm succeeds with a greater probabi…
This proposal investigates the photon-statistics of light emitted by a statistical ensemble of cold atoms excited by the near-field of an optical nanofiber tip. Dipole-dipole interactions of atoms at such short distance from each other suppress the simultaneous emission of more than one photon and lead to antibunching of photons. We consider a mean atom number on the order of one and deal with a poissonian mixture of one and two atoms including dipole-dipole interactions and collective decay. Time tracks of the atomic states are simulated in quantum Monte Carlo simulations from which the $g^{(…
The main purpose of this tutorial is to elucidate in details what should be meant by ensemble of states in quantum mechanics, and to properly address the problem of discriminating, exactly or approximately, two different ensembles. To this aim, we review the notion and the definition of quantum ensemble as well as its relationships with the concept of statistical operator in quantum mechanics. We point out the implicit assumptions contained in introducing a correspondence between quantum ensembles and the corresponding single-particle statistical operator, and discuss some issues arising when …
Maximum-entropy ensembles are key primitives in statistical mechanics from which thermodynamic properties can be derived. Over the decades, several approaches have been put forward in order to justify from minimal assumptions the use of these ensembles in statistical descriptions. However, there is still no full consensus on the precise reasoning justifying the use of such ensembles. In this work, we provide a new approach to derive maximum-entropy ensembles taking a strictly operational perspective. We investigate the set of possible transitions that a system can undergo together with an envi…
We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of $N$ spins arranged on a $d$-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to $O(N^{1/(d+1)})$ spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory.
Many effects in the interaction between atoms and a cavity that are usually de-scribed in quantum mechanical terms (cavity quantum electrodynamics, cavity QED) can be understood and quantitatively analyzed within a classical framework. We adopt such a classical picture of a radiating dipole oscillator to derive explicit ex- pressions for the coupling of single atoms and atomic ensembles to Gaussian modes in free space and in an optical resonator. The cooperativity parameter of cavity QED is shown to play a central role, and is given a geometrical interpretation. The classical analysis yields t…
Considering the recently established arbitrariness the Schroedinger equation has to be interpreted as an equation of motion for a statistical ensemble of particles. The statistical qualities of individual particles derive from the unknown intrinsic energy components, they depend on the physical environment by way of external potentials. Due to these statistical qualities and wave function normalization, non-locality is inherent to the fundamental relations of Planck, de Broglie and Schroedinger. A local formulation of these statements is introduced and briefly assessed, the modified and local …
All ensembles of statistical mechanics are equivalent in the sense that they give the equivalent thermodynamic functions in the thermodynamic limit. However, when investigating microscopic structures in the first-order phase transition region, one must choose an appropriate statistical ensemble. The appropriate choice is particularly important when one investigates finite systems, for which even the equivalence of ensembles does not hold. We propose a class of statistical ensembles, which always give the correct equilibrium state even in the first-order phase transition region. We derive vario…
Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according to the Bures measure. The eigenvalues of the random density matrices are analyzed: we derive the eigenvalue distribution for the Bures ensemble which is shown to be broader then the quarter--circle distribution characteristic of the Hilbert--Schmidt ensemble. For measures induced by partial tracing over the environment we compute exactly the two-point eigenvalue correlation function.
We present an alternative formalism of quantum mechanics tailored to statistical ensemble in phase space. The purpose of our work is to show that it is possible to establish an alternative autonomous formalism of quantum mechanics in phase space using statistical methodology. The adopted perspective leads to obtaining within the framework of its theory the master quantum-mechanical equation without recourse to the other formalisms of quantum mechanics, and gives the idea of operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space. We…
We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by a combination of a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.
We show that two chosen ensembles of spin states, which are differently prepared but are described by the same density matrix in quantum mechanics, do not fully share the same measurable characteristics. One characteristic on which they differ is shown to be the variance of the spin along a given direction. We conclude that the statistical description of an ensemble of states as given by its density matrix, although sufficient in many cases, should be considered incomplete, as it does not fully describe the measurable characteristics of the ensemble. A discussion a posteriori on the problem is…
A system in equilibrium does not evolve -- time independence is its telltale characteristic. However, in Newtonian physics the microstate of an individual system (a point in its phase space) evolves incessantly in accord with its equations of motion. Ensembles were introduced in XIX century to bridge that chasm between continuous motion of phase space points in Newtonian dynamics and stasis of thermodynamics: While states of individual classical systems inevitably evolve, a phase space distribution of such states -- an ensemble -- can be time-independent. I show that entanglement (e.g., with t…
It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, that is that its integral is one and that the marginal properties are satisfied. However this is in general not true. We introduce a class of quantum states for which this property is satisfied, these states are dubbed "Feichtinger states" because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980's by H. Feichtinger. The properties of these states are studied, which gives us the opportuni…
