This is a sequel to the papers (quant-ph/9910063) and (quant-ph/0004102). The aim of this paper is to give mathematical foundations to Holonomic Quantum Computation (Computer) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093). In 2-qubit case we give an explicit form to non-abelian Berry connection of quantum computational bundle which is associated with Holonomic Quantum Computation, on some parameter space. We also suggest a possibility that not only usual holonomy but also higher-dimensional holonomies must be used to prove a universality of our Holonomic Quantum Computation.
We make a brief review of (optical) Holonomic Quantum Computer (or Computation) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093), and give a mathematical reinforcement to their works.
We investigate the influence of random errors in external control parameters on the stability of holonomic quantum computation in the case of arbitrary loops and adiabatic connections. A simple expression is obtained for the case of small random uncorrelated errors. Due to universality of mathematical description our results are valid for any physical system which can be described in terms of holonomies. Theoretical results are confirmed by numerical simulations.
In this review the foundations of Geometric Quantization are explained and discussed. In particular, we want to clarify the mathematical aspects related to the geometrical structures involved in this theory: complex line bundles, hermitian connections, real and complex polarizations, metalinear bundles and bundles of densities and half-forms. In addition, we justify all the steps followed in the geometric quantization programme, from the standpoint definition to the structures which are successively introduced.
The Weltanschauung emerging from quantum theory clashes profoundly with our classical concepts. Quantum characteristics like superposition, entanglement, wave-particle duality, nonlocality, contextuality are difficult to reconcile with our everyday intuition. In this article I survey some aspects of quantum foundations and discuss intriguing connections with the foundations of mathematics.
In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$, a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.
The purpose of this paper is to survey some topics on mathematical foundations of quantum information developed mainly by the present author and co-workers for the last three decades. The topics include an axiomatic construction of quantum measurement theory based on completely positive map-valued measures, a universally valid new formulation of the uncertainty principle for error and disturbance in quantum measurements, the Wigner-Araki-Yanase limit of quantum measurements, the accuracy limit of quantum computing based on conservation laws, and a quantum interpretation based on quantum set th…
Donostia International Physics Center（多诺斯蒂亚国际物理中心）；Ikerbasque Foundation for Science（伊凯巴斯克科学基金会）；Institute of Physics, Johannes Gutenberg University（约翰内斯·古腾堡大学物理系）；Faculty of Environment and Information Sciences, Yokohama National University（横滨国立大学环境与信息科学学院）
Matrix syntax is a formal model of syntactic relations in language. The purpose of this paper is to explain its mathematical foundations, for an audience with some formal background. We make an axiomatic presentation, motivating each axiom on linguistic and practical grounds. The resulting mathematical structure resembles some aspects of quantum mechanics. Matrix syntax allows us to describe a number of language phenomena that are otherwise very difficult to explain, such as linguistic chains, and is arguably a more economical theory of language than most of the theories proposed in the contex…
It is demonstrated that in gate-based quantum computing architectures quantum walk is a natural mathematical description of quantum gates. It originates from field-matter interaction driving the system, but is not attached to specific qubit designs and can be formulated for very general field-matter interactions. It is shown that, most generally, gates are described by a set of coined quantum walks. Rotating wave and resonant approximations for field-matter interaction simplify the walks, factorizing the coin, and leading to pure continuous time quantum walk description. The walks reside on a …
A theory of quantum gravity is expected to change profoundly our understanding of black holes. Quantum theory has already shown, as a first approximation, that a black hole slowly evaporates. Non-perturbative quantum gravity also predicts that such an object could metamorphose into a white hole, its time-reverse. In this thesis, we study the foundations of such a scenario and propose a mathematical model that includes the phenomenon of evaporation.
One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information processing with classical resources? This book will introduce the reader to the mathematical framework required to answ…
We briefly review the advanced mathematical language of fiber bundle structures and how they can be used to classify two-level quantum systems based on the analysis of the topological properties of their sets of state vectors. The topological classes of quantum electronic states and matter phases are characterized by topological invariants, which can be defined geometrically as the integral of differential forms on the base manifold of the fiber bundle structure. Specifically, we demonstrate that for one-dimensional systems described by the Su-Schrieffer-Heeger (SSH) model, the set of state ve…
Quantum annealing is a generic name of quantum algorithms to use quantum-mechanical fluctuations to search for the solution of optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundation 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 Schroedinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior o…
Mathematical tools related to coherence theory and classical-quantum equivalence, due to Wigner and Glauber, are essential to modern, practical and empirical understanding of electromagnetics in areas like quantum optics and nanoelectronics. This paper specifies how an extension of these same tools (especially Glauber's "Q" mapping) can be applied to strong nuclear forces as well, and provides a "bottom-up" approach to axiomatic unification of physics, grounded in empirical reality (dice included). The Q hypothesis also has implications for quantum measurement and quantum information technolog…
We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically and exactly, state by state, by means of periodic orbit expansions. We prove analytically that the periodic orbit series exist and converge to the correct spectral eigenvalues. We investigate the convergence properties of the periodic orbit series and prove rigorously that both conditionally convergent and absolutely convergent cases can be found. We compare t…
We provide here a general mathematical framework to model attitudes towards ambiguity which uses the formalism of quantum theory as a ``purely mathematical formalism, detached from any physical interpretation''. We show that the quantum-theoretic framework enables modelling of the "Ellsberg paradox", but it also successfully applies to more concrete human decision-making (DM) tests involving financial, managerial and medical decisions. In particular, we elaborate a mathematical representation of various empirical studies which reveal that attitudes of managers towards uncertainty shift from "a…
When a quantum system is driven adiabatically through a parametric cycle in a degenerate Hilbert space, the state would acquire a non-Abelian geometric phase, which is stable and forms the foundation for holonomic quantum computation (HQC). However, in the adiabatic limit, the environmental decoherence becomes a significant source of errors. Recently, various non-adiabatic HQC schemes have been proposed, but all at the price of increased sensitivity to control errors. Alternatively, there exist theoretical proposals for speeding up HQC by the technique of "shortcut to adiabaticity" (STA), but …
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entan…
Quasi-set theory provides us a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the usual statistics (Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac) into the scope of quasi-set theory. We also show that, in order to derive Maxwell-Boltzmann statistics, it is not necessary to assume that the particles are distinguishable. In other words, Maxwell-Boltzmann statistics is possible even in an ensamble of indistinguishable particles, at least from the theoretical point of view. The main goal of this paper is to p…
For a quantum theory that includes exponentially decaying states and Breit-Wigner resonances, which are related to each other by the lifetime-width relation $τ=\frac{\hbar}Γ$, where $τ$ is the lifetime of the decaying state and $Γ$ the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz-Rigged Hilbert Space $Φ\subset\mathcal{H}\subsetΦ^\times$ of the Dirac formalism. One has to distinguish between prepared states, using a space $Φ_-\subset\mat hcal{H}$, and detected observables, using a space $Φ_+\subset\mathcal{H}$, where $-(+)$ refers to analyticity of the …
