The triangle wave versus the cosine: How classical systems can optimally approximate EPR-B correlations

The famous singlet correlations of a composite quantum system consisting of two spatially separated components exhibit notable features of two kinds. The first kind consists of striking certainty relations: perfect correlation and perfect anti-correlation in certain settings. The second kind consists of a number of symmetries, in particular, invariance under rotation, as well as invariance under exchange of components, parity, or chirality. In this note, I investigate the class of correlation functions that can be generated by classical composite physical systems when we restrict attention to systems which reproduce the certainty relations exactly, and for which the rotational invariance of the correlation function is the manifestation of rotational invariance of the underlying classical physics. I call such correlation functions classical EPR-B correlations. It turns out that the other three (binary) symmetries can then be obtained "for free": they are exhibited by the correlation function, and can be imposed on the underlying physics by adding an underlying randomisation level. We end up with a simple probabilistic description of all possible classical EPR-B correlations in terms of a "spinning coloured disk" model, and a research programme: describe these functions in a concise analytic way. We survey open problems, and we show that the widespread idea that "quantum correlations are more extreme than classical physics allows" is at best highly inaccurate, through giving a concrete example of a classical correlation which satisfies all the symmetries and all the certainty relations and which exceeds the quantum correlations over a whole range of settingsComment: This version, arXiv:1312.6403v.6, as accepted by "Entropy" 27 February 202

Correlations and fluctuations of matrix elements and cross sections

The fluctuations and correlations of matrix elements of cross sections are investigated in open systems that are chaotic in the classical limit. The form of the correlation functions is discussed within a statistical analysis and tested in calculations for a damped quantum kicked rotator. We briefly comment on the modifications expected for systems with slowly decaying correlations, a typical feature in mixed phase spaces.Comment: 7 pages, 7 figure

Coarse Grainings and Irreversibility in Quantum Field Theory

In this paper we are interested in the studying coarse-graining in field theories using the language of quantum open systems. Motivated by the ideas of Calzetta and Hu on correlation histories we employ the Zwanzig projection technique to obtain evolution equations for relevant observables in self-interacting scalar field theories. Our coarse-graining operation consists in concentrating solely on the evolution of the correlation functions of degree less than $n$, a treatment which corresponds to the familiar from statistical mechanics truncation of the BBKGY hierarchy at the n-th level. We derive the equations governing the evolution of mean field and two-point functions thus identifying the terms corresponding to dissipation and noise. We discuss possible applications of our formalism, the emergence of classical behaviour and the connection to the decoherent histories framework.Comment: 25 pages, Late

Approximating Time-Dependent Quantum Statistical Properties

Computing quantum dynamics in condensed matter systems is an open challenge due to the exponential scaling of exact algorithms with the number of degrees of freedom. Current methods try to reduce the cost of the calculation using classical dynamics as the key ingredient of approximations of the quantum time evolution. Two main approaches exist, quantum classical and semi-classical, but they suffer from various difficulties, in particular when trying to go beyond the classical approximation. It may then be useful to reconsider the problem focusing on statistical time-dependent averages rather than directly on the dynamics. In this paper, we discuss a recently developed scheme for calculating symmetrized correlation functions. In this scheme, the full (complex time) evolution is broken into segments alternating thermal and real-time propagation, and the latter is reduced to classical dynamics via a linearization approximation. Increasing the number of segments systematically improves the result with respect to full classical dynamics, but at a cost which is still prohibitive. If only one segment is considered, a cumulant expansion can be used to obtain a computationally efficient algorithm, which has proven accurate for condensed phase systems in moderately quantum regimes. This scheme is summarized in the second part of the paper. We conclude by outlining how the cumulant expansion formally provides a way to improve convergence also for more than one segment. Future work will focus on testing the numerical performance of this extension and, more importantly, on investigating the limit for the number of segments that goes to infinity of the approximate expression for the symmetrized correlation function to assess formally its convergence to the exact result

Cross-correlations and Entanglement in Cavity QED

Every quantum system subjected to measurements is an open quantum system. The cavity QED system is elegant in that it probes the interaction between two quantum systems, the atom and the field, while its loss mechanisms are well understood and can be externally monitored. The study of cross-correlations in cavity QED is important for understanding how entanglement evolves in open quantum systems. As quantum information science grows we need to learn more about entanglement and how it can be quantified and measured. Correlation functions have been used to compare an electromagnetic field (intensity) of one mode with the electromagnetic field (intensity) of the same mode at a later time or different spatial location. In quantum optics, correlation functions have been calculated and measured to probe the nonclassical field that results from the interaction of a single mode of the electromagnetic field and an ensemble of two-level atoms (the canonical cavity QED system). This field can exhibit antibunching, squeezing, and can violate inequalities required for a classical field. Entanglement in the steady state of a cavity QED system cannot be measured directly with traditional correlation functions (Hanbury-Brown and Twiss type experiments). Cross-correlations, however, interrogate directly both modes of the entangled pair, the transmitted (cavity) and the fluorescent (atom) intensities, and can act as an entanglement witness. This thesis presents the implementation of a cross-correlation measurement in a cavity QED system. The work has required the construction of an apparatus that incorporates laser cooling and trapping with quantum optics to carefully control both the external (center of mass motion) and internal (atomic state) degrees of freedom of a collection of atoms that interact with a single mode of a high finesse Fabry-Perot cavity. We examine theoretically and experimentally a new intensity cross-correlation function which probes the evolution of the cavity field conditioned on the detection of a fluorescent photon from an atom in the cavity. The results open the possibility to generalize the dynamics of entanglement as a physical resource necessary for the nascent quantum information science

Quantum chaos and the emergence of statistical physics

The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results

Łamanie nierówności Leggetta-Garga w otwartych układach kwantowych

Pełny tekst artykułu nr 3, dołączonego do rozprawy, dostępny jest lokalnie w sieci bibliotek Uniwersytetu Śląskiego: http://www.bc.us.edu.pl/publication/16367This dissertation concerns an analysis of obtained theoretical values of temporal correlation functions in open quantum systems in the context of Leggett-Garg inequalities. The violation of these inequalities indicates that a system reveals non-classical correlations. A special case of the temporal correlations, analysed in this work, are used to tests of macrorealism, likewise as an indicator of the “quantumness” of a system or in order to perform the quantum information protocols. Despite of the well grounded results on the temporal quantum correlations in isolated systems, the open systems are still barely explored. The main motivation to study this subject is a possibility to obtain better models of real physical systems and to develop new methods to control the amount of the non-classical correlations. The main research objective is to establish an influence on the amount of the non-classical correlations in the measured subsystem by a coupling with the environment. In this work, four distinct physical models of open quantum systems are presented. In the first one there is revealed a violation of Leggett-Garg inequality in the system weakly coupled to thermal environment where especially is discussed the process of decoherence and dissipation. The main result is an observation that, under some conditions, the violation of the inequality is independent of environment properties like temperature. The second model concerns an analysis of temporal quantum correlations in the systems that dynamics is governed by the angular momentum operators and driven by classical white noise. In this case strict analytical results reveal an exponential dumping of the non-classical correlations as well as a property that such dumping can be less effective for systems with larger state space. The last but not least two models for which is calculated the Leggett-Garg correlator are the systems which interact with the environment due to the spin-spin coupling. In this context, a physical model of atoms from the first group of the periodic table and a model of quantum-classical hybrids, for which is discussed the semi-classical approach, is proposed. In both examples it is proven that for “more macroscopic” systems as well as for “classical environments”, it is possible to observe higher violation of the Leggett-Garg inequality

Stability of Local Quantum Dissipative Systems

This is the author accepted manuscript. The final version is available from Springer at http://link.springer.com/article/10.1007%2Fs00220-015-2355-3.Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are potentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time which scales logarithmically with the system size. The proof relies on Lieb-Robinson bounds, which describe a finite group velocity for propagation of information in local systems. As a main example, we prove that classical Glauber dynamics is stable under local perturbations, including perturbations in the transition rates which may not preserve detailed balance