Abstract: Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrodinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrodinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.

Abstract: We study the statistics of the work done on a quantum critical system by quenching a control parameter in the Hamiltonian. We elucidate the relation between the probability distribution of the work and the Loschmidt echo, a quantity emerging usually in the context of dephasing. Using this connection we characterize the statistics of the work done on a quantum Ising chain by quenching locally or globally the transverse field. We show that for local quenches starting at criticality the probability distribution of the work displays an interesting edge singularity.

Abstract: We describe a quantum algorithm that solves combinatorial optimization problems by quantum simulation of a classical simulated annealing process. Our algorithm exploits quantum walks and the quantum Zeno effect induced by evolution randomization. It requires order $1/\sqrt{\ensuremath{\delta}}$ steps to find an optimal solution with bounded error probability, where $\ensuremath{\delta}$ is the minimum spectral gap of the stochastic matrices used in the classical annealing process. This is a quadratic improvement over the order $1/\ensuremath{\delta}$ steps required by the latter.

Abstract: The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.

Abstract: Several finite-dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite-dimensional quantum systems. Moreover, we show that any quasi-probability representation is equivalent to a frame representation and then prove that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus.

Abstract: A quantum state is nonclassical if its Glauber-Sudarshan $P$ function fails to be interpreted as a probability density. This quantity is often highly singular, so that its reconstruction is a demanding task. Here we present the experimental determination of a well-behaved $P$ function showing negativities for a single-photon-added thermal state. This is a direct visualization of the original definition of nonclassicality. The method can be useful under conditions for which many other signatures of nonclassicality would not persist.

Abstract: Quantum field theory has been a great success for physics, but it is difficult for mathematicians to learn because it is mathematically incomplete. Folland, who is a mathematician, has spent considerable time digesting the physical theory and sorting out the mathematical issues in it. Fortunately for mathematicians, Folland is a gifted expositor. The purpose of this book is to present the elements of quantum field theory, with the goal of understanding the behavior of elementary particles rather than building formal mathematical structures, in a form that will be comprehensible to mathematicians. Rigorous definitions and arguments are presented as far as they are available, but the text proceeds on a more informal level when necessary, with due care in identifying the difficulties. The book begins with a review of classical physics and quantum mechanics, then proceeds through the construction of free quantum fields to the perturbation-theoretic development of interacting field theory and renormalization theory, with emphasis on quantum electrodynamics. The final two chapters present the functional integral approach and the elements of gauge field theory, including the Salam-Weinberg model of electromagnetic and weak interactions.

Abstract: Time-continuous measurement in quantum mechanics has long been an open theoretical issue because of the peculiarity of single quantum measurement itself. The Markovian theory emerged 20 years ago [1–3] from foundational considerations. The requests in quantum optics (and elsewhere) triggered another, partly independent, line of progress with expanding applications [4]. So far the Markovian theory of continuous measurement has become completely understood while the general non-Markovian one has remained an open issue even conceptionally. Markovian time-continuous quantum measurement the

Abstract: Ontological theories of quantum mechanics describe a single system by means of well-defined classical variables and attribute the quantum uncertainties to our ignorance about the underlying reality represented by these variables. We consider the general class of ontological theories describing a quantum system by a set of variables with Markovian (either deterministic or stochastic) evolution. We provide proof that the number of continuous variables cannot be smaller than $2N\ensuremath{-}2$, $N$ being the Hilbert-space dimension. Thus, any ontological Markovian theory of quantum mechanics requires a number of variables which grows exponentially with the physical size. This result is relevant also in the framework of quantum Monte Carlo methods.

Abstract: The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the so-called quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems. In contrast with the approaches found in the literature, we do not assume that the system is continuously monitored. Instead, our starting point is a relatively realistic unitary dynamics of the full system

Abstract: Our aim in this paper is to take quite seriously Heinz Post’s claim that the non-individuality and the indiscernibility of quantum objects should be introduced right at the start, and not made a posteriori by introducing symmetry conditions. Using a different mathematical framework, namely, quasi-set theory, we avoid working within a label-tensor-product-vector-space-formalism, to use Redhead and Teller’s words, and get a more intuitive way of dealing with the formalism of quantum mechanics, although the underlying logic should be modified. We build a vector space with inner product, the Q-space, using the non-classical part of quasi-set theory, to deal with indistinguishable elements. Vectors in Q-space refer only to occupation numbers and permutation operators act as the identity operator on them, reflecting in the formalism the fact of unobservability of permutations. Thus, this paper can be regarded as a tentative to follow and enlarge Heinsenberg’s suggestion that new phenomena require the formation of a new “closed” (that is, axiomatic) theory, coping also with the physical theory’s underlying logic and mathematics.

Abstract: Chemistry began as a discipline to document materials that restore health, as pharmacy is today. During the 16th to 18th centuries, we have learned that material consists of compounds that are combinations and variation of only about 90 chemical elements, each with a unique atomic weight. The variation of their macroscopic properties as a function of the atomic weight is very interesting. For example, lithium, sodium, potassium and cesium react with water vigorously, and their reactivity increases as their atomic weights increase. This discovery led to their arrangement in a periodic table in the 19th century.

Abstract: Motivated by several classic decision-theoretic paradoxes, and by analogies with the paradoxes which in physics motivated the development of quantum mechanics, we introduce a projective generalization of expected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting decision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium.

Abstract: The paper is devoted to the study of nonlinear stochastic Schrodinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born--Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator.

Abstract: We give a lower bound on the probability of error in quantum state discrimination. The bound is a weighted sum of the pairwise fidelities of the states to be distinguished.

Abstract: We study the multi-parameter quantum groups defined by the generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category O.W e present two explicit descriptions here: as a Hopf 2-cocycle deformation, and as the multi-parameter quantum shuffle realization of the positive part.

Abstract: The hypothesis of quantum nonequilibrium at the big bang is shown to have observable consequences. For a scalar field on expanding space, we show that relaxation to quantum equilibrium (in de Broglie-Bohm theory) is suppressed for field modes whose quantum time evolution satisfies a certain inequality, resulting in a 'freezing' of early quantum nonequilibrium for these particular modes. For an early radiation-dominated expansion, the inequality implies a corresponding physical wavelength that is larger than the (instantaneous) Hubble radius. These results make it possible, for the first time, to make quantitative predictions for nonequilibrium deviations from quantum theory, in the context of specific cosmological models. We discuss some possible consequences: corrections to inflationary predictions for the cosmic microwave background, non-inflationary super-Hubble field correlations, and relic nonequilibrium particles.

Abstract: This Perspective discusses the role that quantum information plays in determining the quantum-mechanical aspects of matter. Beginning with the entwined concepts of information and entropy, the article discusses how quantum information theory can supply us with novel concepts and techniques for understanding how matter behaves at the most microscopic of levels.

Abstract: It is shown that in the complex trajectory representation of quantum mechanics, the Born's Psi^{\star}\Psi probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

Abstract: Quantum solitons are discovered with the help of generalized quantum hydrodynamics. The solitons have the character of the stable quantum objects in the self consistent electric field. Particularly these effects can be considered as explanation of the existence of lightning balls. The delivered theory demonstrates the great possibilities of the generalized quantum hydrodynamics in investigation of the quantum solitons. The theory leads to solitons as typical formations in the generalized quantum hydrodynamics.

Abstract: In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach and successively apply it to quantum hydrodynamic models and quantum diffusion models. The results of numerical simulations show that these models capture well the various features of quantum transport.

Abstract: "Quantum trajectories" are solutions of stochastic differential equations of non-usual type. Such equations are called "Belavkin" or "Stochastic Schr\"odinger Equations" and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called "Quantum Repeated Measurements". In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics.

Abstract: The first famous t.hought experiment of Einstein gives rise to his theories of relativity. the bedrock of modern astrophysics and cosmology. His second famous thought experiment begin..c; the investigation iuto the foundations of quantum mechanics. It leads to a paradox. inspiring \'arious 'no-go' theorems pro....en by Bell, Kochen. and Spe~ker. Physi~ists and philosophers worldwide become increasingly dissatisfied with the probabilistic complemen­ tarity interpretation (Born-Bohr) and eventually offer their own accounts of the theory. By the end of the 20th century. two alternative approaches stand out as the best candidates: Both the hidden \'ariables int.erpretation (de Broglie-Bohm) and the many worlds interpre­ tation (Everett-De\Yitt) give compelling descriptions of what the true nature of quantum reality could be. In this paper, a chronological O\'erview of all these events is given, followed by a philosophical analysis of the three aforementioned interpretations. Ultimately. it is con­ cluded that the many worlds interpretation should be adopted as the best understanding of the formalism of quantum mechanics and. therefore, should be used in the multiversity textbooks.

Abstract: Quantum information theory has given rise to a renewed interest in, and a new perspective on, the old issue of understanding the ways in which quantum mechanics differs from classical mechanics. The task of distinguishing between quantum and classical theory is facilitated by neutral frameworks that embrace both classical and quantum theory. In this paper, I discuss two approaches to this endeavour, the algebraic approach, and the convex set approach, with an eye to the strengths of each, and the relations between the two. I end with a discussion of one particular model, the toy theory devised by Rob Spekkens, which, with minor modifications, fits neatly within the convex sets framework, and which displays in an elegant manner some of the similarities and differences between classical and quantum theories. The conclusion suggested by this investigation is that Schrodinger was right to find the essential difference between classical and quantum theory in their handling of composite systems, though Schrodinger's contention that it is entanglement that is the distinctive feature of quantum mechanics needs to be modified.

Abstract: This paper advocates a concept of quantum information whose origins can be traced to Schumacher [1995. Quantum coding. Physical Review A 51, 2738–2747]. The concept of quantum information advocated is elaborated using an analogy to Shannon's theory provided by Schumacher coding. In particular, this paper extends Timpson's [2004. Quantum information theory and the foundations of quantum mechanics . Ph.D. dissertation, University of Oxford. Preprint, quant-ph/0412063] framework for interpreting Shannon information theory to the quantum context. Entanglement fidelity is advocated as the appropriate success criterion for the reproduction of quantum information. The relationship between the Shannon theory and quantum information theory is discussed.

Abstract: In this paper, we consider the problem of discriminating two given quantum operations. Based on the Bloch representation of a single qubit, we give an implicit expression that can be used to evaluate the exact minimum error probability of discriminating any two single-qubit quantum operations by unentangled input states. In particular, for the Pauli channels discussed in Sacchi (2005 Phys. Rev. A 71 062340), we use a more intuitive and visual method to deal with their discrimination problem. Also, we consider the condition for perfect discrimination of two quantum operations.