Operator Algebras and Quantum Information

这项资助将研究算子代数和量子信息理论之间有很强联系的问题。我们将研究量子实验的随机性可能产生的概率密度。目前，有几种数学模型来描述这些概率密度，并且不知道它们是否都产生相同的密度矩阵集。这些模型中的两个是否给出相同的密度集，现在已知等价于算子代数中的一个重要的公开问题，Connes嵌入猜想。** 许多没有获胜策略的两人游戏，当一个人被限制使用经典生成的随机性时，当随机性来自纠缠量子实验时，可以有获胜策略。我们将研究这些量子概率密度的所有模型是否一致的问题，试图确定这些不同模型的获胜策略是否相同。我们研究的许多博弈都源于图论，并且具有这样的性质：如果一个人被限制在要求一个获胜的经典策略，那么当且仅当博弈参数与通常的图参数相同时，这才可能发生。 但是，当人们允许这些获胜的量子策略时，这些参数的值可以小得多。例如，图的着色数是所需的最小颜色数，使得每个顶点可以被分配一种颜色，并且没有两个相邻的顶点接收相同的颜色。在图形着色游戏中，两名玩家试图通过回答一系列问题来说服裁判，他们使用一定数量的颜色进行着色。 这些问题的设计是为了让他们永远不需要泄露实际的颜色。如果参与者使用经典概率密度，那么他们可以使用并总是获胜的最少颜色数就是着色数，但是如果他们使用纠缠量子实验来产生他们的答案，那么他们可以设计一种策略，总是给出正确的答案，但使用的颜色要少得多。 这个最小值我们称之为量子色数。 但这个整数可能取决于我们使用的量子力学模型。以类似的方式，我们能够为许多其他图形参数生成新的量子值。 试图计算这些值总是归结为算子代数中的问题，对此我们没有明确的答案，也没有算法。 其中一些问题被称为NP难问题。这些问题都可以归结为算符系统的张量积问题，而对这些张量积和算符系统的研究是我们长期的研究课题。算符系统和完全正映射是量子信息理论的基础，我们将继续追求这些对象的理论和应用。

This grant will study problems that have a strong link between operator algebras and quantum information theory. We will study the probability densities that can arise from the randomness of quantum experiments. Currently, there are several mathematical models for what these probability densities are and it is not known if they all yield the same sets of densities matrices. Whether or not two of these models give the same sets of densities is now known to be equivalent to one of the important open problems in operator algebras, the Connes embedding conjecture. **Many two-person games which have no winning strategy, when one is restricted to use classically generated randomness, can have winning strategies when the randomness arises from entangled quantum experiments. We will study the problem of whether or not all models for these quantum probability densities agree by trying to determine if the games that have winning strategies for these different models are the same or not.**Many games that we study arise from graph theory and have the property that if one is restricted to asking for a winning classical strategy, then that can happen if and only if the game parameters are the same as the usual graph parameters. But when one allows these winning quantum strategies the values of these parameters can be much smaller. For example, the coloring number of a graph is the smallest number of colors needed so that each vertex can be assigned a color with no two adjacent vertices receiving the same color. In the graph coloring game, two players try to convince a referee that they have a coloring, using a certain number of colors, by answering a series of questions. The questions are designed so that they never need to divulge the actual coloring. If the players use classical probability densities, then the fewest number of colors that they can use and always win is exactly the coloring number, but if they use an entangled quantum experiment to produce their answers, then they can design a strategy that always gives correct answers but uses far fewer colors. This smallest value we call the quantum chromatic number. But this integer could depend on which model of quantum mechanics that we use. In a similar manner, we are able to generate new quantum values for many other graph parameters. Trying to compute these values always reduces to a problem in operator algebras, for which we have no clear answers and no algorithms. Some of these problems are known to belong to a family of problems called the NP-hard problems.** These problems can all be interpreted as questions about tensor products of operator systems, and the study of these tensor products and operator systems is our longer term research project. Operator systems and completely positive maps are fundamental to quantum information theory and we will continue to pursue both the theory and the applications of these objects.**