All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory} But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding However, it appeared that this new resource is
very complex and difficult to detect Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon A basic role of entanglement
witnesses in detection of entanglement is emphasized

Quantum entanglement

We show that under the influence of pure vacuum noise two entangled qubits become completely disentangled in a finite-time, and in a specific example we find the time to be given by ln((2+sqrt[2] / 2) times the usual spontaneous lifetime.

https://arxiv.org/pdf/quant-ph/0404161.pdf

Finite-time disentanglement via spontaneous emission.

We construct a general measure for the degree of non-Markovian behavior in open quantum systems. This measure is based on the trace distance which quantifies the distinguishability of quantum states. It represents a functional of the dynamical map describing the time evolution of physical states, and can be interpreted in terms of the information flow between the open system and its environment. The measure takes on nonzero values whenever there is a flow of information from the environment back to the open system, which is the key feature of non-Markovian dynamics.

Measure for the degree of non-markovian behavior of quantum processes in open systems.

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Geometry of Quantum States: Frontmatter

A new development in the dynamical behavior of elementary quantum systems is the surprising discovery that correlation between two quantum units of information called qubits can be degraded by environmental noise in a way not seen previously in studies of dissipation. This new route for dissipation attacks quantum entanglement, the essential resource for quantum information as well as the central feature in the Einstein-Podolsky-Rosen so-called paradox and in discussions of the fate of Schrodinger9s cat. The effect has been labeled ESD, which stands for early-stage disentanglement or, more frequently, entanglement sudden death. We review recent progress in studies focused on this phenomenon.

Sudden Death of Entanglement

The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. In this paper we examine master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel.

Depolarizing channel as a completely positive map with memory

The theory of zero-range potentials is investigated in an arbitrary number of dimensions. Except for the trivial one-dimensional case the zero-range potentials are described by nonlocal operators called Fermi pseudopotentials. It is shown that in odd dimensions the Fermi pseudopotentials involve a very simple regularization operator. In even dimensions with the help of dimensional regularization, explicit formulas for the Fermi pseudopotentials are derived. The Green's functions, the propagators, and the exact solutions of the Lippmann-Schwinger equations are derived in explicit forms. In odd dimensions d=1 and 3 the Fermi pseudopotentials can be applied to describe multiphoton processes of atoms and molecules with very short-range interactions. In even dimension d=2 the Fermi pseudopotential can be applied to describe tunneling from laser-driven quantum wells. Physical applications involving higher d are also possible.

Fermi pseudopotential in arbitrary dimensions.

For a leaky cavity driven by an active medium, we clarify the connection between the concepts of single-quasimode squeezing (for the intracavity field) and spectral squeezing (for the input and output fields). Our analysis is based on a rigorous construction of the Fox-Li quasimodes of an empty cavity by a superposition of the true modes of a perfect, large (infinitely large in the limit) cavity which fully subsumes the leaky cavity. To illustrate the physical importance of spectral squeezing (as defined here and as used by other authors), we consider a novel model of a microscopic system that is sensitive to the generalized quadratures of the field. We obtain a simple picture of the reasons why the quantum-noise reduction is in general different inside and outside the cavity. An important special case of our picture is that if the {ital intracavity} field is perfectly squeezed, then the {ital output} field shows no squeezing at all. Our considerations apply to a wide class of systems, examples of which are discussed in the companion paper.

Treatment of the spectrum of squeezing based on the modes of the universe. I. Theory and a physical picture.

A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium (chi(3) nonlinearity) is presented. We numerically solved this equation taking a coherent state as an initial condition. The dissipation effects are discussed. We provide examples of quantum interference, sub-Planck phase space structures, and Gaussian vs non-Gaussian dynamical evolution of the state. The results also apply to the description of a nanomechanical resonator with an intrinsic Duffing nonlinearity.

/pdf/wigner-function-evolution-of-quantum-states-in-the-presence-4ctzorgmfn.pdf

Wigner function evolution of quantum states in the presence of self-Kerr interaction

We examine stochastic maps in the context of quantum optics. Making use of the master equation, the damping basis, and the Bloch picture we calculate a nonunital, completely positive, trace-preserving map with unequal damping eigenvalues. This results in what we call the squeezed vacuum channel. A geometrical picture of the effect of stochastic noise on the set of pure state qubit density operators is provided. Finally, we study the capacity of the squeezed vacuum channel to transmit quantum information and to distribute Einstein-Podolsky-Rosen states.

Krzysztof Wódkiewicz

Papers

Depolarizing channel as a completely positive map with memory

Fermi pseudopotential in arbitrary dimensions.

Treatment of the spectrum of squeezing based on the modes of the universe. I. Theory and a physical picture.

Wigner function evolution of quantum states in the presence of self-Kerr interaction

Quantum Markov channels for qubits