Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0304142

Algorithm of Reduction

N. K. Solovarov

Published Apr 22, 2003• 1 citation
arXivHTMLPDF

Abstract

We show that the von Neumann's algorithm of reduction (i.e. the algorithm of calculating the density matrix of the observable subsystem from the density matrix of the closed quantum system) corresponds to the special approximation at which the unobservable subsystem is supposed to be in the steady state of minimum information (infinite temperature). We formulate the generalized algorithm of reduction that includes as limiting cases the von Neumann's reduction and the self-congruent correlated reduction most corresponding to the quantum nondemolition measurement. We demonstrate the correlation in dynamics of subsystems with exactly soluble models of quantum optics: 1) about the dynamics of a pair of interacting two-level atoms, and 2) about the dynamics of a two-level atom interacting with a single-mode resonant field.

DOI
10.48550/arxiv.quant-ph/0304142

Related papers

Correlation or decoherence? Quantum beats of atomic inversion in a resonant coherent field

N. K. Solovarov

It is shown that the standard reduction procedure (i.e., the calculation of the density matrix of the observable subsystem from the density matrix of a closed quantum system) bringing about decoherence corresponds to the limiting approximation, where the unobservable subsystem is assumed to be in the stationary state with minimum information (infinite temperature). An approximate set of interrelation (correlation) equations for the density matrices of the subsystems is derived. It is shown that the correlation of atom and field can be manifested as the inversion beats of a two-level atom in the known experimental scheme of resonator QED. Experimental observation of such beats would indicate that the observable subsystem (atom) generally conserves information about quantum coherence of the unobservable subsystem (field).

Journal of Experimental and Theoretical Physics Letters•Sep 1, 2003•1 citation

Relaxation of interacting open quantum systems

Vladislav Yu. Shishkov, E. S. Andrianov, Александр Александрович Пухов +2 more

Abstract We consider the transition from the description of a closed quantum system consisting of an open quantum system and a reservoir to the description of the open quantum system alone by eliminating the reservoir degrees of freedom by averaging over them. An approach based on the Lindblad master equation for the density matrix is used. A general scheme is developed for deriving the Lindblad superoperator that emerges after averaging the von Neumann equation over the reservoir degrees of freedom. This scheme is illustrated with the cases of radiation of a two-level atom into free space and the dynamics of the transition of a two-level atom from a pure state to a mixed state due to interaction with a dephasing reservoir. Special attention is paid to the open system consisting of several subsystems each of which independently interacts with the reservoir. In the case of noninteracting subsystems, the density matrix is a tensor product of the subsystem density matrices, and the Lindblad superoperator of the system is a sum of Lindblad superoperators of those subsystems. The interaction between the subsystems results not only in the emergence of the corresponding term in the Hamiltonian of the combined system but also in the nonadditivity of the Lindblad superoperators. This is often overlooked in modern literature, possibly leading, as is shown in this methodological note, to serious errors; for example, the second law of thermodynamics could be violated.

Physics-Uspekhi•Apr 20, 2019•31 citations

The von Neumann model of measurement in quantum mechanics

Pier A. Mello

We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called "Wigner's formula" emerges in the strong-coupling limit, while "Kirkwood's quasi-probability distribution" is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.

arXiv:1311.7649•AIP conference proceedings•Jan 1, 2014•12 citations

The externally-driven nonequilibrium dicke model

V. M. Bastidas

In this thesis, we study the nonequilibrium dynamics of the time dependent single mode Dicke model in the thermodynamic limit in the context of the Liouville-von Neumann canonical method. We report an exact solution for this model under a non-equilibrium situation, generated by time dependent control parameters. We compute the correlations: in the atom-field ground state by means of the density operator for the pure state of the universe, and the corresponding reduced density operator for the atoms when the field is in resonance with the atomic ensemble. We obtain a closed relationship between the correlations in the ground state, the localisation of the atom¿s Wigner function in phase space, and the localisation of the wave function of the universe. As a measure of bipartite entanglement, we calculate the linear entropy from which we study the atom-field entanglement dynamics. In particular, we find a strong relationship between the stability of the auxiliary dynamical parameters, the reported entanglement, and the dynamics of certain physical observables. Furthermore, we use the Liouville-von Neumann technique to study the influence of quantum and thermal fluctuations on the dynamics of the normal and the superradiant phase. In addition, we calculate the density matrix for the quantum mechanical evolution of an initial thermal equilibrium state in the case of an instantaneous atom-field coupling and time independent ω and ω0.

•Jan 1, 2009•0 citations

Entropy reduction of quantum measurements

Максим Евгеньевич Широков

It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.

arXiv:1011.3127•Journal of Mathematical Physics•May 1, 2011•26 citations

Generalised Equilibrium Reduced-Density Operator

Zbigniew Jacyna‐Onyszkiewicz

Equilibrium reduced-density operators for subsystems of the universe are derived from the generalised Schrödinger variational principle, additionally assuming that the values of von Neumann entropy for the subsystems are fixed and higher than zero.The obtained reduced-density operator may be useful for description of the properties of an arbitrary system (macroscopic, mesoscopic, nanoscopic, or microscopic) in the equilibrium mixed state.

Acta Physica Polonica A•May 1, 2018•0 citations
We show that the von Neumann's algorithm of reduction (i.e. the algorithm of calculating the density matrix of the observable subsystem from the density matrix of the closed quantum system) corresponds to the special approximation at which the unobservable subsystem is supposed to be in the steady state of minimum information (infinite temperature). We formulate the generalized algorithm of reduction that includes as limiting cases the von Neumann's reduction and the self-congruent correlated reduction most corresponding to the quantum nondemolition measurement. We demonstrate the correlation in dynamics of subsystems with exactly soluble models of quantum optics: 1) about the dynamics of a pair of interacting two-level atoms, and 2) about the dynamics of a two-level atom interacting with a single-mode resonant field.
It is shown that the standard reduction procedure (i.e., the calculation of the density matrix of the observable subsystem from the density matrix of a closed quantum system) bringing about decoherence corresponds to the limiting approximation, where the unobservable subsystem is assumed to be in the stationary state with minimum information (infinite temperature). An approximate set of interrelation (correlation) equations for the density matrices of the subsystems is derived. It is shown that the correlation of atom and field can be manifested as the inversion beats of a two-level atom in the known experimental scheme of resonator QED. Experimental observation of such beats would indicate that the observable subsystem (atom) generally conserves information about quantum coherence of the unobservable subsystem (field).
Abstract We consider the transition from the description of a closed quantum system consisting of an open quantum system and a reservoir to the description of the open quantum system alone by eliminating the reservoir degrees of freedom by averaging over them. An approach based on the Lindblad master equation for the density matrix is used. A general scheme is developed for deriving the Lindblad superoperator that emerges after averaging the von Neumann equation over the reservoir degrees of freedom. This scheme is illustrated with the cases of radiation of a two-level atom into free space and the dynamics of the transition of a two-level atom from a pure state to a mixed state due to interaction with a dephasing reservoir. Special attention is paid to the open system consisting of several subsystems each of which independently interacts with the reservoir. In the case of noninteracting subsystems, the density matrix is a tensor product of the subsystem density matrices, and the Lindblad superoperator of the system is a sum of Lindblad superoperators of those subsystems. The interaction between the subsystems results not only in the emergence of the corresponding term in the Hamiltonian of the combined system but also in the nonadditivity of the Lindblad superoperators. This is often overlooked in modern literature, possibly leading, as is shown in this methodological note, to serious errors; for example, the second law of thermodynamics could be violated.
We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called "Wigner's formula" emerges in the strong-coupling limit, while "Kirkwood's quasi-probability distribution" is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.
In this thesis, we study the nonequilibrium dynamics of the time dependent single mode Dicke model in the thermodynamic limit in the context of the Liouville-von Neumann canonical method. We report an exact solution for this model under a non-equilibrium situation, generated by time dependent control parameters. We compute the correlations: in the atom-field ground state by means of the density operator for the pure state of the universe, and the corresponding reduced density operator for the atoms when the field is in resonance with the atomic ensemble. We obtain a closed relationship between the correlations in the ground state, the localisation of the atom¿s Wigner function in phase space, and the localisation of the wave function of the universe. As a measure of bipartite entanglement, we calculate the linear entropy from which we study the atom-field entanglement dynamics. In particular, we find a strong relationship between the stability of the auxiliary dynamical parameters, the reported entanglement, and the dynamics of certain physical observables. Furthermore, we use the Liouville-von Neumann technique to study the influence of quantum and thermal fluctuations on the dynamics of the normal and the superradiant phase. In addition, we calculate the density matrix for the quantum mechanical evolution of an initial thermal equilibrium state in the case of an instantaneous atom-field coupling and time independent ω and ω0.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.