An Axiomatization for Quantum Processes to Unifying Quantum and Classical Computing

We establish an axiomatization for quantum processes, which is a quantum generalization of process algebra ACP (Algebra of Communicating Processes). We use the framework of a quantum process configuration 〈p, ϱ〉, but we treat it as two relative independent part: the structural part p and the quantum part ϱ, because the establishment of a sound and complete theory is dependent on the structural properties of the structural part p. We let the quantum part ϱ be the outcomes of execution of p to examine and observe the function of the basic theory of quantum mechanics. We establish not only a strong bisimilarity for quantum processes, but also a weak bisimilarity to model the silent step and abstract internal computations in quantum processes. The relationship between quantum bisimilarity and classical bisimilarity is established, which makes an axiomatization of quantum processes possible. An axiomatization for quantum processes called qACP is designed, which involves not only quantum information, but also classical information and unifies quantum computing and classical computing. qACP can be used easily and widely for verification of most quantum communication protocols.

     

Memory Effects in Quantum Dynamics Modelled by Quantum Renewal Processes

Simple, controllable models play an important role in learning how to manipulate and control quantum resources. We focus here on quantum non-Markovianity and model the evolution of open quantum systems by quantum renewal processes. This class of quantum dynamics provides us with a phenomenological approach to characterise dynamics with a variety of non-Markovian behaviours, here described in terms of the trace distance between two reduced states. By adopting a trajectory picture for the open quantum system evolution, we analyse how non-Markovianity is influenced by the constituents defining the quantum renewal process, namely the time-continuous part of the dynamics, the type of jumps and the waiting time distributions. We focus not only on the mere value of the non-Markovianity measure, but also on how different features of the trace distance evolution are altered, including times and number of revivals.     

Effective Interactions Due to Quantum Fluctuations

A class of quantum lattice models is considered, with Hamiltonians consisting of a classical (diagonal) part and a small off-diagonal part (e.g. hopping terms). In some cases when the classical part has an infinite degeneracy of ground states, the quantum perturbation may stabilize some of them. The mechanism of this stabilization stems from effective potential created by the quantum perturbation. Conditions are found when this strategy can be rigorously controlled and the low temperature phase diagram of the full quantum model can be proven to be a small deformation of the zero temperature phase diagram of the classical part with the effective potential added. As illustrations we discuss the asymmetric Hubbard model and the Bose–Hubbard model. Received: 28 April 1998 / Accepted: 19 March 1999     

Dispersive Quantum Systems

A dispersive quantum system is a quantum system which is both isolated and non-time reversal invariant. This article presents precise definitions for those concepts and also a characterization of dispersive quantum systems within the class of completely positive Markovian quantum systems in finite dimension (through a homogeneous linear equation for the non-Hamiltonian part of the system’s Liouvillian). To set the framework, the basic features of quantum mechanics are reviewed focusing on time evolution and also on the theory of completely positive Markovian quantum systems, including Kossakowski–Lindblad’s standard form for Liouvillians. After those general considerations, a simple two-dimensional example is presented and then applied to describe the neutrino oscillation, with the introduction of a new “dispersive parameter.”     

Quantum Pseudo-Telepathy

Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with what we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required. Pseudo-telepathy is a surprising application of quantum information processing to communication complexity. Thanks to entanglement, perhaps the most nonclassical manifestation of quantum mechanics, two or more quantum players can accomplish a distributed task with no need for communication whatsoever, which would be an impossible feat for classical players. After a detailed overview of the principle and purpose of pseudo-telepathy, we present a survey of recent and not-so-recent work on the subject. In particular, we describe and analyse all the pseudo-telepathy games currently known to the authors. Supported in Part by Canada’s Natural Sciences and Engineering Research Council (NSERC), the Canada Research Chair programme and the Canadian Institute for Advanced Research (CIAR). Supported in part by a scholarship from Canada’s NSERC. Supported in part by Canada’s NSERC Québec’s Fonds de recherche sur la nature et les technologies (FQRNT), the CIAR and the Mathematics of Information Technology and Complex Systems Network (MITACS).     

Quantum photonic network on chip

We provide an overview of quantum photonic network on chip. We begin from the discussion of the pros and cons of several material platforms for engineering quantum photonic chips. Then we introduce and analyze the basic building blocks and functional units of quantum photonic integrated circuits. In the main part of this review, we focus on the generation and manipulation of quantum states of light on chip and are particularly interested in some applications of advanced integrated circuits with different functionalities for quantum information processing, including quantum communication, quantum computing, and quantum simulation. We emphasize that developing fully integrated quantum photonic chip which contains sources of quantum light, integrate circuits, modulators, quantum storage, and detectors are promising approaches for future quantum photonic technologies. Recent achievements in the large scale photonic chips for linear optical computing are also included. Finally, we illustrate the challenges toward high performance quantum information processing devices and conclude with promising perspectives in this field.     

Statistics of Local Value in Quantum Mechanics

Given a quantum mechanical observable and a state, one can construct a classical observable, that is, a real function on the configuration space, such that it is the optimal estimate of the quantum observable, in the sense of minimum variance. This optimal estimate turns out to be the quantum mechanical local value, which arises from several contexts such as de Broglie–Bohm's casual approach to quantum mechanics, instantaneous frequency in time–frequency analysis, Nelson's quantum fluctuations formalism, and phase-space approach to quantum mechanics. Accordingly, any observable can be decomposed into a local value part and a quantum fluctuation part, which are independent, both geometrically and statistically. Furthermore, the current density in quantum mechanics, the osmotic velocity in stochastic mechanics, and the Fisher information in classical statistical inference, arise naturally in connection with local value. In particular, Heisenberg uncertainty principle can be quantified more precisely by virtue of local value.     

Dynamics of Quantum Fisher Information in Homodyne-Mediated Feedback Control

We investigate the quantum Fisher information(QFI) dynamics of a dissipative two-level system in homodynemediated quantum feedback control. The analytical results demonstrate that the maximum values and stable values of the QFI can be greatly enhanced via feedback control. The quantum feedback plays a more evident role in the improvement of classical Fisher information. The classical part can reach a high stable value, while the quantum part eventually decays to zero whatever the feedback parameter is.     

Coherence and Entanglement Dynamics in Training Variational Quantum Perceptron

In quantum computation, what contributes supremacy of quantum computation? One of the candidates is known to be a quantum coherence because it is a resource used in the various quantum algorithms. We reveal that quantum coherence contributes to the training of variational quantum perceptron proposed by Y. Du et al., arXiv:1809.06056 (2018). In detail, we show that in the first part of the training of the variational quantum perceptron, the quantum coherence of the total system is concentrated in the index register and in the second part, the Grover algorithm consumes the quantum coherence in the index register. This implies that the quantum coherence distribution and the quantum coherence depletion are required in the training of variational quantum perceptron. In addition, we investigate the behavior of entanglement during the training of variational quantum perceptron. We show that the bipartite concurrence between feature and index register decreases since Grover operation is only performed on the index register. Also, we reveal that the concurrence between the two qubits of index register increases as the variational quantum perceptron is trained.     

Quantum corrections for Boltzmann equation

We present the lowest order quantum correction to the semiclassical Boltzmann distribution function, and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation, and the quantum Wigner distribution function is expanded in powers of Planck constant, too. The negative quantum correlation in the Wigner distribution function which is just the quantum correction terms is naturally singled out, thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework. Supported by the National Natural Science Foundation of China (Grant No. 10404037) and the Scientific Research Fund of GUCAS (Grant No. 055101BM03)     

Quantum Capacity under Adversarial Quantum Noise: Arbitrarily Varying Quantum Channels

We investigate entanglement transmission over an unknown channel in the presence of a third party (called the adversary), which is enabled to choose the channel from a given set of memoryless but non-stationary channels without informing the legitimate sender and receiver about the particular choice that he made. This channel model is called an arbitrarily varying quantum channel (AVQC). We derive a quantum version of Ahlswede’s dichotomy for classical arbitrarily varying channels. This includes a regularized formula for the common randomness-assisted capacity for entanglement transmission of an AVQC. Quite surprisingly and in contrast to the classical analog of the problem involving the maximal and average error probability, we find that the capacity for entanglement transmission of an AVQC always equals its strong subspace transmission capacity. These results are accompanied by different notions of symmetrizability (zero-capacity conditions) as well as by conditions for an AVQC to have a capacity described by a single-letter formula. In the final part of the paper the capacity of the erasure-AVQC is computed and some light shed on the connection between AVQCs and zero-error capacities. Additionally, we show by entirely elementary and operational arguments motivated by the theory of AVQCs that the quantum, classical, and entanglement-assisted zero-error capacities of quantum channels are generically zero and are discontinuous at every positivity point.     

Quantum cosmology and stationary states

A model for quantum gravity, in which the conformal part of the metric is quantized using the path integral formalism, is presented. Einstein's equations can be suitably modified to take into account the effects of quantum conformal fluctuations. A closed Friedman model can be described in terms of well-defined stationary states. The “ground state” sets a lower bound (at Planck length) to the scale factor preventing the collapse. A possible explanation for matter creation and quantum nature of matter is suggested.     

Quantum teleportation of optical images with frequency conversion

A new version of quantum teleportation of optical images in continuous variables is considered. The new scheme is based on quantum entanglement in continuous variables between light fields of different frequencies and allows wavelength conversion between input and teleported images. The protocol of quantum teleportation with frequency conversion can be used as a combined part of light/matter interface in quantum parallel data processing and in a parallel quantum memory.     

Quantum Riemann surfaces III. the exceptional cases

We discuss quantum deformations of Riemann surfaces whose fundamental groups are Abelian (the exceptional Riemann surfaces). We prove uniformization theorems, state deformation estimates, and study the dependence of quantum Riemann surfaces on the deformation parameter.Supported in part by NSF grant DMS-9206936.Supported in part by DOE grant DE-FG02-88ER25065.     

Quantum graphs: Applications to quantum chaos and universal spectral statistics

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.     

Quantum Phenomena in a Classical Model

This work is part of a program which has the aim to investigate which phenomena can be explained by nonlinear effects in classical mechanics and which ones require the new axioms of quantum mechanics. In this paper, we construct a nonlinear field equation which admits soliton solutions. These solitons exibit a dynamics which is similar to that of quantum particles.     

θ Statistics and Quantum Group

General theory of quantum statistics with internal degree of freedom in two dimensions is studied. Because of the topological nontrivial properties in the configuration space of indistinguishable particles, the intrinsic part of many-body wave function is associated with the representation of quantum group, which describes θ statistics. The statistical potential and the effective Hamiltonian of general two-dimensional system can be derived from the Kohno connection.     

Quantum coherent operators: A generalization of coherent states

We introduce a technique to compare different, but related, quantum systems, thereby generalizing the way that coherent states are used to compare quantum systems to classical systems in semiclassical analysis. We then use this technique to estimate the dependence of the free energy of the quantum Heisenberg model on the spin value, and to estimate the relation between the ferromagnetic and antiferromagnetic free energies.Work supported in part by the U.S. National Science Foundation grant PHY-9019433.Work supported in part by the U.S. National Science Foundation grant DMS-9002416.     

Stress Tensor Fluctuations and Passive Quantum Gravity

The quantum fluctuations of the stress tensor of a quantum field are discussed, as are the resulting space-time metric fluctuations. Passive quantum gravity is an approximation in which gravity is not directly quantized, but fluctuations of the space-time geometry are driven by stress tensor fluctuations. We discuss a decomposition of the stress tensor correlation function into three parts, and consider the physical implications of each part. The operational significance of metric fluctuations and the possible limits of validity of semiclassical gravity are discussed.