Zero-error quantum information and operator theory: emerging links

零错误量子信息和算子理论：新兴链接

• 批准号: EP/K032763/1
• 负责人: Ivan Todorov
• 金额: $ 4.19万
• 依托单位: Queen's University Belfast
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK032763%2F1
• 关键词: Zero; error; quantum; information; operator

Graphs are among the simplest mathematical objects - by definition, a graph is a set of points called vertices, and a set of edges, each edge connecting a pair of given vertices. If the vertices are labelled by the numbers 1,2,...,n, then each edge is a pair (i,j), where i and j are between 1 and n. Graphs have a large number of applications in computer science, engineering and mathematics itself. It was Shannon in the 1950's who realised that they can be used successfully in the theory of information. If a symbol, say i, is sent through a communication channel, then due to an error that may occur during this transfer, it may be confused by the receiver with another symbol, say j. The pairs (i,j) of symbols that can be confused in such way form the set of edges of a graph, called the confusability graph of the channel. Shannon defined an asymptotic parameter, called the zero-error capacity of the channel (or, equivalently, of the corresponding confusability graph), which measures the extent to which information can be sent through the channel with zero error. Given a graph G on n vertices, in the 1980's, Pauslen, Power and Smith studied the linear subspace S(G) of the space M_n of all n by n matrices consisting of those elements that have zero i,j-entry when (i,j) is not an edge of G. The space S(G) is an operator system - that is, it is closed under the operation of conjugate transpose and contains the identity matrix - which fully identifies the underlying graph G. Operator systems have an illustrious history and a large number of applications within Analysis, and the aforementioned paper thus opened up an analytical avenue for Graph Theory. We note that not all operator systems in M_n can be obtained from graphs in the described way, and this is a crucial point for the approach used in our proposal. At present, there are ongoing efforts for the construction of quantum computers. The theoretical science that lies behind these efforts is Quantum Information Theory, a part of the general field of quantum physics. The main feature of quantum, as opposed to classical, physics, is non-commutativity: the mathematical tools behind it use the space M_n and its infinite dimensional generalisations where the commutation rule ab = ba does not hold in general. Non-commutativity features prominently in the study of quantum channels, that is, channels used to transfer quantum information. Recently, Duan, Severini and Winter defined the confusability graph of a quantum channel as a certain operator system in M_n, and showed that every operator system in M_n arises in this way. It is thus natural to call operator systems in M_n non-commutative graphs. The class of operator systems S(G) can be identified in an easy and elegant way as a natural subclass of the class of all non-commutative graphs. Quantum zero-error capacities were introduced, but a number of important questions were left open. The aim of the present research project is to study of quantum zero-error capacities using methods from Operator Theory - the general branch where the study of operator systems belongs. We plan to obtain new estimates on the quantum version of a parameter known as Lovasz number of a graph (a quantity that provides an easier computable bound for the zero-error capacity), study other parameters such as non-commutative chromatic numbers, and lay the foundations of a "non-commutative graph theory", where basic operations with classical graphs such as passing to a complement, a subgraph and a homomorphic image can be carried out successfully in the context of operator systems. We plan to address a number of questions regarding asymptotic versions of the introduced parameters that are expected to shed light on open problems and conjectures in Graph Theory and Quantum Information.

图是最简单的数学对象之一 - 根据定义，图是一组称为顶点的点和一组边，每条边连接一对给定的顶点。如果顶点用数字 1,2,...,n 标记，则每条边都是一对 (i,j)，其中 i 和 j 介于 1 和 n 之间。图在计算机科学、工程和数学本身中有大量应用。香农在 20 世纪 50 年代意识到它们可以成功地应用于信息理论。如果通过通信信道发送一个符号（例如 i），则由于传输期间可能发生的错误，接收器可能会将其与另一个符号（例如 j）混淆。以这种方式可以混淆的符号对 (i,j) 形成图的边集，称为通道的可混淆性图。香农定义了一个渐近参数，称为通道（或相应的可混淆性图）的零错误容量，它衡量信息通过通道以零错误发送的程度。给定 n 个顶点上的图 G，在 20 世纪 80 年代，Pauslen、Power 和 Smith 研究了所有 n × n 矩阵的空间 M_n 的线性子空间 S(G)，该矩阵由那些具有零 i,j 条目的元素组成，当 (i ,j) 不是 G 的边。空间 S(G) 是一个算子系统 - 即它在共轭转置运算下是封闭的，并且包含单位矩阵 - 完全标识底层图 G。算子系统在分析领域有着辉煌的历史和大量的应用，上述论文为图论开辟了一条分析途径。我们注意到，并非 M_n 中的所有算子系统都可以通过所描述的方式从图中获得，这是我们提案中使用的方法的关键点。目前，量子计算机的建设正在进行中。这些努力背后的理论科学是量子信息论，它是量子物理学一般领域的一部分。与经典物理学相反，量子的主要特征是非交换性：其背后的数学工具使用空间 M_n 及其无限维概括，其中交换规则 ab = ba 一般不成立。非交换性在量子通道（即用于传输量子信息的通道）研究中具有突出的特点。最近，Duan、Severini和Winter将量子通道的混淆图定义为M_n中的某个算子系统，并表明M_n中的每个算子系统都是这样产生的。因此，在 M_n 个非交换图中调用算子系统是很自然的。算子系统 S(G) 的类可以以简单而优雅的方式标识为所有非交换图类的自然子类。引入了量子零误差能力，但许多重要问题仍然悬而未决。本研究项目的目的是使用算子理论（算子系统研究所属的一般分支）的方法来研究量子零误差能力。我们计划获得对被称为图的洛瓦斯数（为零误差容量提供更容易计算的界限的数量）的参数的量子版本的新估计，研究其他参数，例如非交换色数，并奠定“非交换图论”的基础，其中经典图的基本运算，例如传递到补集、子图和同态图像，可以在算子系统的上下文中成功执行。我们计划解决一些有关所引入参数的渐近版本的问题，这些问题有望揭示图论和量子信息中的开放问题和猜想。

项目成果

QUANTUM CHROMATIC NUMBERS VIA OPERATOR SYSTEMS

通过运算系统计算量子色数

• DOI: 10.1093/qmath/hav004
• 发表时间: 2013-11-26
• 期刊: Quarterly Journal of Mathematics
• 影响因子: 0.7
• 作者: V. Paulsen;I. Todorov
• 通讯作者: I. Todorov

Estimating quantum chromatic numbers

估计量子色数

• DOI: 10.1016/j.jfa.2016.01.010
• 发表时间: 2014-07-25
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: V. Paulsen;S. Severini;D. Stahlke;I. Todorov;A. Winter
• 通讯作者: A. Winter