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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lieb-Robinson bounds ： ウィキペディア英語版 Lieb-Robinson bounds The Lieb-Robinson bound is a theoretical upper limit on the speed at which information can propagate in non-relativistic quantum systems. It demonstrates that information cannot travel instantaneously in quantum theory, even when the relativity limits of the speed of light are ignored. In the study of quantum systems such as quantum optics, quantum information theory, atomic physics, and condensed matter physics, it is important to know that there is a finite speed with which information can propagate. The theory of relativity shows that no information, or anything else for that matter, can travel faster than the speed of light. When non-relativistic mechanics is considered, however, (Newton's equations of motion or Schrödinger's equation of quantum mechanics) it had been thought that there is then no limitation to the speed of propagation of information. This is not so for certain kinds of quantum systems of atoms arranged in a lattice, often called quantum spin systems. This is important conceptually and practically, because it means that, for short periods of time, distant parts of a system act independently. The surprising existence of such a finite limit to the speed of propagation, up to exponentially small error terms, was discovered mathematically in 1972.〔E. Lieb, D. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251–257, (1972)〕 It turns the locality properties of physical systems into the existence of an upper bound for this speed. This bound is known as the Lieb-Robinson bound and the speed is known as the Lieb-Robinson velocity. The velocity is not universal, because it depends on the details of the system under consideration, but, for each system, there is a finite velocity. One of the practical applications of Lieb-Robinson bounds is quantum computing. Current proposals to construct quantum computers built out of atomic-like units mostly rely on the existence of this finite speed of propagation to protect against too rapid dispersal of information. Review articles can be found in the following references, for example,〔B. Nachtergaele, R. Sims, Much Ado About Something: Why Lieb-Robinson bounds are useful, IAMP News Bulletin, October 2010, 22–29, (2010)〕〔M. Kliesch, C. Gogolin, J. Eisert, Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems, arXiv:1306.0716, (2013)〕〔M. B. Hastings, Locality in quantum systems, arXiv:1008.5137〕 == Set up == To define the bound, it is necessary to first describe basic facts about quantum mechanical systems composed of several units, each with a finite dimensional Hilbert space. Lieb-Robinson bounds are considered on a $d$-dimensional lattice ($d\; =\; 1,\; 2$ or $3$) such as the square lattice $\backslash Gamma\; =\; \backslash mathbb^d$. A Hilbert space of states $\backslash mathcal\_x$ is associated with each point $x\backslash in\backslash Gamma$. The dimension of this space is finite, but this was generalized in 2008 (see below). This is called ''quantum spin system''. For every finite subset of the lattice, $X\; \backslash subset\backslash Gamma$, the associated Hilbert space is given by the tensor product :$\backslash mathcal\_X=\backslash otimes\_\; \backslash mathcal\_x$. $\backslash mathcal\_X$ is a subspace of $\backslash mathcal\_Y$ if $X\; \backslash subset\; Y$. An observable $A$ supported on (i.e., depends only on) a finite set $X\backslash subset\; \backslash Gamma$ is a linear operator on the Hilbert space $\backslash mathcal\_X$. When $\backslash mathcal\_x$ is finite dimensional choose a finite basis of operators that span the set of linear operators on $\backslash mathcal\_x$. Then any observable on $\backslash mathcal\_x$ can be written as a sum of basis operators on $\backslash mathcal\_x$. The Hamiltonian of the system is described by an interaction $\backslash Phi(\backslash cdot)$. The ''interaction'' is a function from the finite sets $X\backslash subset\backslash Gamma$ to self-adjoint observables $\backslash Phi(X)$ supported in $X$. The interaction is assumed to be finite range (meaning that $\backslash Phi(X)=0$ if the size of $X$ exceeds a certain prescribed size) and translation invariant. These requirements were lifted later, see:〔M. Hastings, T. Koma, Spectral Gap and Exponential Decay of Correlations, Commun. Math. Phys. 256, 781, (2006)〕〔 Although translation invariance is usually assumed, it is not necessary to do so. It is enough to assume that the interaction is bounded above and below on its domain. Thus, the bound is quite robust in the sense that it is tolerant of changes of the Hamiltonian. A finite range ''is'' essential, however. An interaction is said to be of finite range if there is a finite number $R$ such that for any set $X$ with diameter greater than $R$ the interaction is zero, i.e., $\backslash Phi(X)=0$. The Hamiltonian of the system with interaction $\backslash Phi$ is defined formally by: :$H\_\backslash Phi=\backslash sum\_\backslash Phi(X)$. The laws of quantum mechanics say that corresponding to every physically observable quantity there is a self-adjoint operator $A$. For every observable $A$ with a finite support the Hamiltonian defines a continuous one-parameter group $\backslash tau\_t$ of transformations of the observables $\backslash tau\_t$ given by :$\backslash tau\_t(A)=e^Ae^.$ Here, $t$ has the physical meaning of time. (Technically speaking, this time evolution is defined by a power-series expansion that is known to be a norm-convergent series $\backslash tau\_t(A)=A+it()+\backslash frac\; =\; AB-BA$ is called the commutator of the operators $A$ and $B$, while the symbol $\backslash |\; O\; \backslash |$ denotes the norm, or size, of an operator $O$. It is very important to note that the bound has nothing to do with the ''state'' of the quantum system, but depends only on the Hamiltoninan governing the dynamics. Once this operator bound is established it necessarily carries over to any state of the system. A positive constant $c$ depends on the norms of the observables $A$ and $B$, the sizes of the supports $X$ and $Y$, the interaction, the lattice structure and the dimension of the Hilbert space $\backslash mathcal\_x$. A positive constant $v$ depends on the interaction and the lattice structure only. The number $a>0$ can be chosen at will provided $d(X,Y)/v|t|$ is chosen sufficiently large. In other words, the further out one goes on the light cone, $d(X,Y)-v|t|$, the sharper the exponential decay rate is. (In later works authors tended to regard $a$ as a fixed constant.) The constant $v$ is called the group velocity or Lieb-Robinson velocity. The bound () is presented slightly differently from the equation in the original paper.〔 This more explicit form () can be seen from the proof of the bound 〔 Lieb-Robinson bound shows that for times the norm on the right-hand side is exponentially small. This is the exponentially small error mentioned above. The reason for considering the commutator on the left-hand side of the Lieb–Robinson bounds is the following: The commutator between observables $A$ and $B$ is zero if their supports are disjoint. The converse is also true: if observable $A$ is such that its commutator with any observable $B$ supported outside some set $X$ is zero, then $A$ has a support inside set $X$. This statement is also approximately true in the following sense:〔S. Bachmann, S. Michalakis, B. Nachtergaele, R. Sims, Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems, Commun. Math. Phys. 309, 835–871, (2012)〕 suppose that there exists some $\backslash epsilon\; >\; 0$ such that $\backslash |(B)\backslash |\; \backslash leq\; \backslash epsilon\; \backslash |B\backslash |$ for some observable $A$ and any observable $B$ that is supported outside the set $X$. Then there exists an observable $A(\backslash epsilon)$ with support inside set $X$ that approximates an observable $A$, i.e. $\backslash |A\; -\; A(\backslash epsilon)\backslash |\; \backslash leq\; \backslash epsilon$. Thus, Lieb-Robinson bounds say that the time evolution of an observable $A$ with support in a set $X$ is supported (up to exponentially small errors) in a $\backslash delta$-neighborhood of set $X$, where $\backslash delta\; >\; v|t|$ with $v$ being the Lieb-Robinson velocity. Outside this set there is no influence of $A$. In other words, this bounds assert that the speed of propagation of perturbations in quantum spin systems is bounded. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lieb-Robinson bounds」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.