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Science in China Series A-Mathematics (in Chinese)

量子效应的下确界问题

启慧 李, 鸿科 杜, 春源 邓

Published Mar 1, 2006• 0 citations
Publisher

Abstract

Hilbert空间H上的一个量子效应A是满足0≤A≤I的正算子, 简称为效应A, 用E(H)表示所有效应构成的集合. Hilbert空间上两个效应的下确界问题是确定在何种条件下效应A和B∈E(H)的下确界A∧B存在.本文用算子谱理论的方法, 给出了两个效应A,B∈E(H)存在下确界A∧B的充分和必要的条件, 从而完全解决了该问题.

DOI
10.1360/za2006-36-3-320

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•Jan 1, 2012•0 citations
Hilbert空间H上的一个量子效应A是满足0≤A≤I的正算子, 简称为效应A, 用E(H)表示所有效应构成的集合. Hilbert空间上两个效应的下确界问题是确定在何种条件下效应A和B∈E(H)的下确界A∧B存在.本文用算子谱理论的方法, 给出了两个效应A,B∈E(H)存在下确界A∧B的充分和必要的条件, 从而完全解决了该问题.
A quantum effect is an operator AAA on a complex Hilbert space HHH that satisfies 0≤A≤I0\leq A\leq I0≤A≤I, E(H){\cal E} (H)E(H) is the set of all quantum effects on HHH. In 2001, Professor Gudder and Nagy studied the sequential product A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 of A,B∈E(H)A, B\in {\cal E}(H)A,B∈E(H). In 2005, Professor Gudder asked: Is A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 the only sequential product on E(H){\cal E} (H)E(H)? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on E(H){\cal E} (H)E(H); secondly, we present a general method for constructing sequential products on E(H){\cal E} (H)E(H); finally, we study some properties of the sequential products constructed by the method
The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P(H)⊆E(H). Under the natural order, E(H) becomes a partially ordered set that is not a lattice if dimH⩾2. A physically significant and useful characterization of the pairs A,B∊E(H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A∊E(H), P∊P(H) and give an explicit expression for A∧P. We also give a characterization of when A∧(I−A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem.
Let HHH be a finite-dimensional Hilbert space, dim⁡H≥2\dim H \ge 2dimH≥2. We prove that every continuous coexistency preserving map on the effect algebra E(H)E(H)E(H) is either a standard automorphism of E(H)E(H)E(H), or a standard automorphism of E(H)E(H)E(H) composed with the orthocomplementation. We present examples showing the optimality of the result.
不同於一般電腦的數位位元資訊只有兩種可能數值0與1，量子電腦利用的是量子位元，係一個二維希爾伯特(Hilbert)空間中的單位向量，其表示方法為0與1的線性疊加態。量子模擬利用量子物理的基本線性疊加原理，來得到更有效率的方法處理計算科學的問題。本論文討論兩種量子演算法，量子隨機行走法則與量子退火最佳化演算法，在本論文中分成兩大部分，在第一部分中，我們研究在各種圖像中的量子隨機行走法則。研究隨機漫步有助於我們了解各種自然界中的隨機過程，如擴散作用與布朗運動。隨機漫步也已經被運用在許多的電腦演算法中，如搜尋演算法或者最佳化演算法。量子的隨機漫步係建立於量子力學的波函數，也是古典隨機漫步原理的延伸。但古典與量子隨機漫步卻有著非常不同的特性，比如量子隨機漫步傳播的速度大於古典的隨機漫步，且量子隨機漫步並不會像古典隨機漫步一樣會趨向穩定態。量子隨機漫步的時間演進係一由么正(unitary)算符所規範的么正過程，根據定義的不同，我們區分離散時間與連續時間兩種量子隨機漫步。在本論文中，我們研究與比較古典與量子的隨機漫步，分析在圖像以及無序環境上的模擬行走。第二部分，我們利用量子退火式演算解最佳化問題。退火係一材料從高溫控制降溫速度使其保持平衡態最後達成完美結晶結構的物理過程。與傳統的退火演算法利用熱擾動的方法不同，量子退火演算法利用的是量子擾動，使系統在其各種可能的解之間穿隧(tunneling)，以有效的達到最佳解。在本論文中，我們利用建立於路徑積分的蒙地卡羅(Monte Carlo)量子退火演算法，找出自旋玻璃的基態能量。我們以離散的虛數時間方法進行標準的量子蒙地卡羅以及連續的虛數時間方法進行路徑積分的蒙地卡羅，將這兩種量子方法與退火演算法結果的做分析比較。