142 research outputs found
Generators of KMS Symmetric Markov Semigroups on B(h) Symmetry and Quantum Detailed Balance
We find the structure of generators of norm continuous quantum Markov
semigroups on B(h) that are symmetric with respect to the scalar product
tr(\rho^{1/2}x\rho^{1/2}y) induced by a faithful normal invariant state
invariant state \rho and satisfy two quantum generalisations of the classical
detailed balance condition related with this non-commutative notion of
symmetry: the so-called standard detailed balance condition and the standard
detailed balance condition with an antiunitary time reversal
Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples
We solve, mainly by counterexamples, many natural questions regarding maximal
commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a
von Neumann algebra. In particular, we discuss the structure of the generators
of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra
invariant and show that there exists Markov CP-semigroups on M_d without
invariant maximal commutative subalgebras for any d>2.Comment: After the elemenitation in Version 2 of a false class of examples in
Version 1, we now provide also correct examples for unital CP-maps and Markov
semigroups on M_d for d>2 without invariant masa
Bell's Inequality Violations: Relation with de Finetti's Coherence Principle and Inferential Analysis of Experimental Data
It is often believed that de Finetti's coherence principle naturally leads, in the nite case, to the Kolmogorov's probability theory of random phenomena, which then implies Bell's inequality. Thus, not only a violation of Bell's inequality looks paradoxical in the Kolmogorovian framework, but it should violate also de Finetti's coherence principle. Firstly, we show that this is not the case: the typical theoretical violations of Bell's inequality in quantum physics are in agreement with de Finetti's coherence principle. Secondly, we look for statistical evidence of such violations: we consider the experimental data of measurements of polarization of photons, performed to verify empirically violations of Bell's inequality, and, on the basis of the estimated violation, we test the null hypothesis of Kolmogorovianity for the observed phenomenon. By standard inferential techniques we compute the p-value for the test and get a clear strong conclusion against the Kolmogorovian hypothesis
Supercritical Poincar\'e-Andronov-Hopf bifurcation in a mean field quantum laser equation
We deal with the dynamical system properties of a
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation with mean-field
Hamiltonian that models a simple laser by applying a mean field approximation
to a quantum system describing a single-mode optical cavity and a set of two
level atoms, each coupled to a reservoir. We prove that the mean field quantum
master equation has a unique regular stationary solution. In case a relevant
parameter , i.e., the cavity cooperative parameter, is less
than , we prove that any regular solution converges exponentially fast to
the equilibrium, and so the regular stationary state is a globally
asymptotically stable equilibrium solution. We obtain that a locally
exponential stable limit cycle is born at the regular stationary state as
passes through the critical value . Then, the mean-field
laser equation has a Poincar\'e-Andronov-Hopf bifurcation at of supercritical-like type. Namely, we derive rigorously, at the level of
density matrices --for the first time--, the transition from a global attractor
quantum state, where the light is not emitted, to a locally stable set of
coherent quantum states producing coherent light. Moreover, we establish the
local exponential stability of the limit cycle in case a relevant parameter is
between the first and second laser thresholds appearing in the semiclassical
laser theory. Thus, we get that the coherent laser light persists over time
under this condition. In order to prove the exponential convergence of the
quantum state, we develop a new technique for proving the exponential
convergence in open quantum systems that is based in a new variation of
constant formula. Applying our main results we find the long-time behavior of
the von Neumann entropy, the photon-number statistics, and the quantum variance
of the quadratures
Generic q-Markov semigroups and speed of convergence of q-algorithms
We study a special class of generic quantum Markov semigroups, on the algebra of all bounded operators on a Hubert space HS, arising in the stochastic limit of a generic system interacting with a boson Fock reservoir. This class depends on an orthonormal basis of HS. We obtain a new estimate for the trace distance of a state from a pure state and use this estimate to prove that, under the action of a semigroup of this class, states with finite support with respect to the given basis converge to equilibrium with a speed which is exponential, but with a polynomial correction which makes the convergence increasingly worse as the dimension of the support increases (Theorem 5.1). We interpret the semigroup as an algorithm, its initial state as input and, following Belavkin and Ohya,10 the dimension of the support of a state as a measure of complexity of the input. With this interpretation, the above results mean that the complexity of the input "slows down" the convergence of the algorithm. Even if the convergence is exponential and the slow down the polynomial, the constants involved may be such that the convergence times become unacceptable from a computational standpoint. This suggests that, in the absence of estimates of the constants involved, distinctions such as "exponentially fast" and "polynomially slow" may become meaningless from a constructive point of view. We also show that, for arbitray states, the speed of convergence to equilibrium is controlled by the rate of decoherence and the rate of purification (i.e. of concentration of the probability on a single pure state). We construct examples showing that the order of magnitude of these two decays can be quite differen
Sanov and central limit theorems for output statistics of quantum Markov chains
In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not
Homogeneous Open Quantum Random Walks on a lattice
We study Open Quantum Random Walks for which the underlying graph is a
lattice, and the generators of the walk are translation-invariant. We consider
the quantum trajectory associated with the OQRW, which is described by a
position process and a state process. We obtain a central limit theorem and a
large deviation principle for the position process, and an ergodic result for
the state process. We study in detail the case of homogeneous OQRWs on a
lattice, with internal space
Quantum theory: the role of microsystems and macrosystems
We stress the notion of statistical experiment, which is mandatory for
quantum mechanics, and recall Ludwig's foundation of quantum mechanics, which
provides the most general framework to deal with statistical experiments giving
evidence for particles. In this approach particles appear as interaction
carriers between preparation and registration apparatuses. We further briefly
point out the more modern and versatile formalism of quantum theory, stressing
the relevance of probabilistic concepts in its formulation. At last we discuss
the role of macrosystems, focusing on quantum field theory for their
description and introducing for them objective state parameters.Comment: 12 pages. For special issue of J.Phys.A, "The Quantum Universe", on
the occasion of 70th birthday of Professor Giancarlo Ghirard
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