Zero-error quantum information and operator theory: emerging links

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，…，则每条边是一对（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的应用。20世纪50年代，香农意识到它们可以成功地应用于信息论。如果一个符号，比如i，通过通信通道发送，那么由于在传输过程中可能发生的错误，它可能会被接收方与另一个符号，比如j混淆。符号对（i,j）可以以这种方式混淆，形成图的边集，称为通道的混淆图。香农定义了一个渐近参数，称为信道的零错误容量（或等价地，对应的混淆图），它测量了信息可以通过零错误信道发送的程度。给定一个有n个顶点的图G，在20世纪80年代，Pauslen， Power和Smith研究了空间M_n的线性子空间S(G)，它包含所有n × n矩阵，当（i,j）不是G的边时，这些矩阵包含0个i，j元素。它在共轭转置运算下是封闭的，并且包含完全识别底层图g的单位矩阵。算子系统在分析中有着辉煌的历史和大量的应用，从而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用所描述的方式从图中获得，这是我们建议中使用的方法的关键点。目前，人们正在努力构建量子计算机。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：它背后的数学工具使用空间M_n及其无限维度的推广，其中交换规则ab = ba通常不成立。非交换性在量子通道（即用于传输量子信息的通道）的研究中具有突出的特点。最近，Duan， Severini和Winter将量子信道的混淆图定义为M_n中的某个算子系统，并证明了M_n中的每个算子系统都是以这种方式产生的。因此，在M_n非交换图中调用算子系统是很自然的。算子系统S(G)类可以用一种简单而优雅的方式被识别为所有非交换图类的一个自然子类。引入了量子零错误能力，但仍有许多重要的问题没有解决。本研究项目的目的是利用算子理论的方法来研究量子零误差能力——算子理论是算子系统研究的一般分支。我们计划获得新的估计参数的量子版本称为Lovasz数量的图表(一种数量,提供了一个简单可计算的零点误差能力),研究其他参数如non-commutative彩色数字,和奠定基础的“non-commutative图理论”,在基本操作与传递等古典图形补充,可以进行子图和同态图像成功运营商系统的上下文中。我们计划解决一些关于引入参数的渐近版本的问题，这些问题有望阐明图论和量子信息中的开放问题和猜想。

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.