Classical and Quantum Markov Processes Associated with q-Bessel Operators

2020 ◽  
Vol 27 (01) ◽  
pp. 2050005
Author(s):  
Khadija Bessadok ◽  
Franco Fagnola ◽  
Skander Hachicha
Keyword(s):  
Hilbert Space ◽  
Markov Processes ◽  
Infinitesimal Generator ◽  
Bounded Operators ◽  
Initial State ◽  
Bessel Operators ◽  
Fundamental Properties ◽  
Invariant States ◽  
Quantum Markov Processes ◽  
Support Projection

We study the fundamental properties of classical and quantum Markov processes generated by q-Bessel operators and their extension to the algebra of all bounded operators on the Hilbert space [Formula: see text]. In particular, we find a suitable generalized Gorini–Kossakowski–Sudarshan–Lindblad representation for the infinitesimal generator of q-Bessel operator and show that both the classical and quantum Markov processes are transient for α > 0 and recurrent for α = 0. We also show that they do not admit invariant states and, moreover that the support projection of any initial state instantaneously fills the full space.

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

scholarly journals The χ2-divergence and mixing times of quantum Markov processes

2010 ◽  
Vol 51 (12) ◽  
pp. 122201 ◽  
Author(s):  
K. Temme ◽  
M. J. Kastoryano ◽  
M. B. Ruskai ◽  
M. M. Wolf ◽  
F. Verstraete
Keyword(s):  
Markov Processes ◽  
Mixing Times ◽  
Quantum Markov Processes

scholarly journals FROZEN DISCORD IN NON-MARKOVIAN DEPHASING CHANNELS

2011 ◽  
Vol 09 (03) ◽  
pp. 981-991 ◽  
Author(s):  
LAURA MAZZOLA ◽  
JYRKI PIILO ◽  
SABRINA MANISCALCO
Keyword(s):  
Hilbert Space ◽  
Colored Noise ◽  
Geometric Interpretation ◽  
Quantum Decoherence ◽  
Initial State ◽  
Memory Kernel ◽  
Single Qubit ◽  
Classical State ◽  
Classical Correlations ◽  
Quantum And Classical Correlations

We investigate the dynamics of quantum and classical correlations in a system of two qubits under local colored-noise dephasing channels. The time evolution of a single qubit interacting with its own environment is described by a memory kernel non-Markovian master equation. The memory effects of the non-Markovian reservoirs introduce new features in the dynamics of quantum and classical correlations compared to the white noise Markovian case. Depending on the geometry of the initial state, the system can exhibit frozen discord and multiple sudden transitions between classical and quantum decoherence [L. Mazzola, J. Piilo and S. Maniscalco, Phys. Rev. Lett. 104 (2010) 200401]. We provide a geometric interpretation of those phenomena in terms of the distance of the state under investigation to its closest classical state in the Hilbert space of the system.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

Composable Markov processes

1970 ◽  
Vol 7 (2) ◽  
pp. 400-410 ◽  
Author(s):  
Tore Schweder
Keyword(s):  
Social Sciences ◽  
Markov Process ◽  
Markov Processes ◽  
Infinitesimal Generator ◽  
A Priori ◽  
The Social

Many phenomena studied in the social sciences and elsewhere are complexes of more or less independent characteristics which develop simultaneously. Such phenomena may often be realistically described by time-continuous finite Markov processes. In order to define such a model which will take care of all the relevant a priori information, there ought to be a way of defining a Markov process as a vector of components representing the various characteristics constituting the phenomenon such that the dependences between the characteristics are represented by explicit requirements on the Markov process, preferably on its infinitesimal generator.

Identifying Coefficients in the Spectral Representation for First Passage Time Distributions

1987 ◽  
Vol 1 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Mark Brown ◽  
Yi-Shi Shao
Keyword(s):  
Markov Processes ◽  
Time Distribution ◽  
Infinitesimal Generator ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Spectral Approach ◽  
Generator Matrix ◽  
Eigenvalues And Eigenvectors ◽  
First Passage

The spectral approach to first passage time distributions for Markov processes requires knowledge of the eigenvalues and eigenvectors of the infinitesimal generator matrix. We demonstrate that in many cases knowledge of the eigenvalues alone is sufficient to compute the first passage time distribution.

Composable Markov processes

1970 ◽  
Vol 7 (02) ◽  
pp. 400-410 ◽  
Author(s):  
Tore Schweder
Keyword(s):  
Social Sciences ◽  
Markov Process ◽  
Markov Processes ◽  
Infinitesimal Generator ◽  
A Priori ◽  
The Social

Many phenomena studied in the social sciences and elsewhere are complexes of more or less independent characteristics which develop simultaneously. Such phenomena may often be realistically described by time-continuous finite Markov processes. In order to define such a model which will take care of all the relevant a priori information, there ought to be a way of defining a Markov process as a vector of components representing the various characteristics constituting the phenomenon such that the dependences between the characteristics are represented by explicit requirements on the Markov process, preferably on its infinitesimal generator.

Sign in / Sign up

Export Citation Format

Share Document