There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, ie, quantum simulation Quantum simulation promises to have applications in the study of many problems in, eg, condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field

Quantum Simulation

Tables of integrals

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.

An ongoing theme in quantum physics is the interaction of small quantum systems with an environment. If that environment has many degrees of freedom and is weakly coupled, it can often be reasonable to treat its decohering effect on the small system using a ``memoryless,'' or Markovian description. This Colloquium shows that for many phenomena a more refined, non-Markovian, treatment is necessary. The suite of developing theoretical tools is reviewed, with which recent progress on this problem has been based.

/pdf/colloquium-non-markovian-dynamics-in-open-quantum-systems-141503q23f.pdf

Colloquium: Non-Markovian dynamics in open quantum systems

We study the dynamics of quantum and classical correlations in the presence of nondissipative decoherence. We discover a class of initial states for which the quantum correlations, quantified by the quantum discord, are not destroyed by decoherence for times $tl\overline{t}$. In this initial time interval classical correlations decay. For $tg\overline{t}$, on the other hand, classical correlations do not change in time and only quantum correlations are lost due to the interaction with the environment. Therefore, at the transition time $\overline{t}$ the open system dynamics exhibits a sudden transition from classical to quantum decoherence regime.

/pdf/sudden-transition-between-classical-and-quantum-decoherence-amrq1n9dxf.pdf

Sudden transition between classical and quantum decoherence.

An open quantum system loses its ‘quantumness’ when information about the state leaks into its surroundings. Researchers now control this so-called decoherence in a single photon. By rotating an optical filter, the information flow between the photon and its environment can be tuned. This concept could be harnessed for future quantum technologies.

/pdf/experimental-control-of-the-transition-from-markovian-to-non-4oy1jbpsrb.pdf

Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems

Recently, a measure for the non-Markovian behavior of quantum processes in open systems has been developed, which is based on the quantification of the flow of information between the open system and its environment [Phys. Rev. Lett. 103, 210401 (2009)]. The information flow is connected to the rate of change of the trace distance between quantum states, which can be interpreted in terms of the distinguishability of these states. Here, we elaborate the mathematical details of this theory, present applications to specific physical models, and discuss further theoretical and experimental implications as well as relations to alternative approaches proposed recently.

Jyrki Piilo

Papers

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

Colloquium: Non-Markovian dynamics in open quantum systems

Sudden transition between classical and quantum decoherence.

Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems

Measure for the non-Markovianity of quantum processes