Quantum correlations; quantum probability approach (original) (raw)
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Lectures on General Quantum Correlations and their Applications
Quantum Science and Technology, 2017
Monogamy is an intrinsic feature of quantum correlations that gives rise to several interesting quantum characteristics which are not amenable to classical explanations. The monogamy property imposes physical restrictions on unconditional sharability of quantum correlations between the different parts of a multipartite quantum system, and thus has a direct bearing on the cooperative properties of states of multiparty systems, including large many-body systems. On the contrary, a certain party can be maximally classical correlated with an arbitrary number of parties in a multiparty system. In recent years, the monogamy property of quantum correlations has been applied to understand several key aspects of quantum physics, including distribution of quantum resources, security in quantum communication, critical phenomena, and quantum biology. In this chapter, we look at some of the salient developments and applications in quantum physics that have been closely associated with the monogamy of quantum discord, and "discord-like" quantum correlation measures.
Unified View of Quantum and Classical Correlations
Physical Review Letters, 2010
We discuss the problem of separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states.
An outline of quantum probability
1 INDEX Introduction (1a.) Foundations of quantum theory (1b.) Quantum probability and the paradoxes of quantum theory (1c.) Von Neumann' s measurement theory (1d.) Contemporary measurement theory (1e.) Open systems and quantum noise (1f.) Stochastic calculus (1g.) Laws of large numbers and central limit theorems (1h.) Conditioning PART I: ALGEBRAIC PROBABILITY THEORY (2.) Algebraic probability spaces (3.) Algebraic random variables (4.) Stochastic Processes (5.) The local algebras of a stochastic process (6.) Independence (7.) Example: quantum spin systems (8.) A combinatorial lemma (9.) The Boson law of large numbers for independent random variables (10.) The central limit theorem for product maps (11.) Boson and Fermion Gaussian maps (12.) The quantum commutation relations as GNS representations (13.) The quantum commutation relations (14.) De Finetti' s theorem (15.) Conditioning: expected subalgebras (16.) Conditional amplitudes on B(H o ) (17.) Transition expectations and Markovian operators (18.) Markov chains, stationarity, ergodicity (19.) Conditional density amplitudes, potentials and invariant weights (20.) Multiplicative functionals and the discrete Feynman integral (21.) Quantum Markov chains and high temperature superconductivity models (22.) Kümmerer's Markov chains (23.) The algebraic states of Fannes, Nachtergaele and Slegers (24.) 1-dependence and the Ibragimov-Linnik conjecture (25.) 1-dependent quantum Markov chains 2 (26.) Commuting conditional density amplitudes (27.) Diagonalizable states (28.) A nonlinear chain of harmonic oscillators (29.) Generalized random walks (30.) The diffusion limit of the coherent chain (31.) Cecchini' s Markov chains PART II : STOCHASTIC CALCULUS (32.) Simple stochastic integrals (33.) Semimartingales and integrators (34.) Forward derivatives (35.) The o(dt)-notation (36.) Stochastic differential equations (37.) Meyer brackets and Ito tables (38.) The weak Itô formula (39.) The unitarity conditions (40.) The Boson Lévy theorem PART III : CONDITIONING (41.) The standard space of a von Neumann algebra (42.) The ϕ-conditional expectation
Elements of Quantum Probability
This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of quantum coin tosses are discussed, closely related to V.F.R. Jones approach to braid group representations, to spin relaxion, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability.
On entanglement of states and quantum correlations
Eprint Arxiv Math Ph 0202030, 2002
In this paper we present the novel qualities of entanglement of formation for general (so also infinite dimensional) quantum systems and we introduce the notion of coefficient of quantum correlations. Our presentation stems from rigorous description of entanglement of formation.
Quantum Correlations in Classical Statistics
Quantum correlations can be naturally formulated in a classical statistical system of infinitely many degrees of freedom. This realizes the underlying non-commutative structure in a classical statistical setting. We argue that the quantum correlations offer a more robust description with respect to the precise definition of observables.
Quantum correlations from incomplete classical statistics
2001
We formulate incomplete classical statistics for situations where the knowledge about the probability distribution outside a local region is limited. The information needed to compute expectation values of local observables can be collected in a quantum mechanical state vector, whereas further statistical information about the probability distribution outside the local region becomes irrelevant. The translation of the available information between neighboring local regions is expressed by a Hamilton operator. A quantum mechanical operator can be associated to each local observable, such that expectation values of "classical" observables can be computed by the usual quantum mechanical rules. The requirement that correlation functions should respect equivalence relations for local obeservables induces a non-commutative product in classical statistics, in complete correspondence to the quantum mechanical operator product. We also discuss the issue of interference and the complex structure of quantum mechanics within our classical statistical setting. 1
Dual quantum-correlation paradigms exhibit opposite statistical-mechanical properties
Physical Review A, 2012
We report opposite statistical mechanical behaviors of the two major paradigms in which quantum correlation measures are defined, viz., the entanglement-separability paradigm and the informationtheoretic one. We show this by considering the ergodic properties of such quantum correlation measures in transverse quantum XY spin-1 2 systems in low dimensions. While entanglement measures are ergodic in such models, the quantum correlation measures defined from an information-theoretic perspective can be nonergodic.
Journal of Physics A: Mathematical and Theoretical, 2015
Quantum correlation lies at the very heart of almost all the non-classical phenomena exhibited by quantum systems composed of more than one subsystem. In the recent days it has been pointed out that there exists quantum correlation, namely discord which is more general than entanglement. Some authors have investigated that for certain initial states the quantum correlations as well as classical correlation exhibit sudden change under simple Markovian noise. We show that, this dynamical behavior of the both types of correlations can be explained using the idea of complementary correlations introduced in [arXiv:1408.6851]. We also show that though certain class of mixed entangled states can resist the monotonic decay of quantum correlations,it is not true for all mixed states. Moreover, pure entangled states of two qubits will never exhibit such sudden change.
Quantum correlations from classically correlated states
Physica A: Statistical Mechanics and its Applications, 2014
Consider a bipartite quantum system with at least one of its two components being itself a composite system. By tracing over part of one (or both) of these two subsystems it is possible to obtain a reduced (separable) state that exhibits quantum correlations even if the original state of the full system is endowed only with classical correlations. This effect, first pointed out by Li and Luo in [PRA 78, 024303 (2008)], is of considerable interest because there is a growing body of evidence suggesting that quantum correlations in non-entangled, mixed states may constitute a useful resource to implement non trivial information related tasks. Here we conduct a systematic exploration of the aforementioned effect for particular families of states of quantum systems of low dimensionality (three qubits states). In order to assess the non-classicality of the correlations of the reduced states we use an indicator of quantum correlations based upon the state disturbances generated by the measurement of local observables. We show, for a three-qubit system, that there exists a relationship between the classical mutual information of the original classically correlated states and the maximum quantum correlation exhibited by the reduced states.