算子空间、量子概率与量子信息的交叉研究

The remarkable interplay between operator spaces and quantum probability has started to emerge in the last years and shown to be very impressive and promising. But applications of these theories to quantum information began to appear only recently. Noncommutative Lp-spaces are at the intersection of these areas and play a crucial role in recent mathematical research motivated by concepts and problems from quantum physics. It would be appropriate to investigate and develop these links in depth, in particular through the applications of noncommutative martingale and matricial inequalities. These inequalities of quantum probabilistic nature have applications to theoretical and mathematical physics. The originality of this project lies in the systematic exploration of these interactions to build up a coherent program that goes far beyond the current state of the art. We will exploit our original and novel approaches to these fields where noncommutative Lp-spaces have been rarely or have not been explored so far. We firmly believe that the research in these domains is at a crossroad, and that this is the right time to investigate them through the insight of operator spaces and noncommutative Lp-spaces. This project will initiate new studies across a wide spectrum of fields at the frontiers in noncommutative analysis...This project will follow three directions. The first one is devoted to the study of operator spaces by using quantum probabilistic models. It deals with Grothendieck's program which requires notably the concrete realisation of operator spaces. Another aspect concerns the study of fundamental examples of completely bounded maps such as Fourier or Schur multipliers and their applications to approximation properties. The second direction goes conversely and treats the exploration of quantum probability and noncommutative harmonic analysis by operator spaces. The principal and ambitious objective here is the creation of an analytic theory of quantum stochastic integration which now seems possible thanks to the noncommutative martingale inequalities. This analytic theory of quantum stochastic integration should open a large avenue of applications exactly like Ito's integral in the classical probability theory. Markov dilations and harmonic analysis on quantum Markov semigroups will be part of our objectives in this second direction too. The third one deals with applications of operator spaces and quantum probability to quantum information. We intend, in particular to study various Bell type inequalities and quantum connections. The different notions of entropy or capacity and their additivity problem will be another important aspect of this part. Finally, exploiting the full potential of the interplay between the three topics should allow us to solve fundamental open problems in these areas.

算子空间与量子概率的交叉研究近年来十分活跃，它们正被有效地应用于其他领域如量子信息。本项目的研究内容为：1）用量子概率的方法研究算子空间的问题。将通过算子空间的具体实例研究Grothendieck纲领；研究完全有界Fourier-Schur乘子及其在逼近性质中的应用。2）用算子空间的理论和方法研究量子概率与非交换调和分析，主要目标是用非交换鞅不等式理论的最新成果建立量子随机积分的Lp理论，它将为量子概率开辟广阔的应用领域；研究量子Markov半群上的调和分析。3）算子空间和量子概率在量子信息中的应用。将用算子空间与量子概率的最新成果和方法研究各种Bell型不等式与量子关联的问题；研究量子熵、量子容度及其可加性问题。我们坚信量子概率和量子信息的交叉研究正处在一个关键的发展时期，算子空间和非交换Lp空间理论在这些领域的应用正当其时，本项目必将在非交换分析中开辟一个充满前景的交叉研究方向。

该重点项目的研究内容主要分为方面：1）用量子概率的方法研究算子空间与非交换调和分析的问题。2）用算子空间的理论和方法研究量子概率。3）算子空间和量子概率在量子信息中的应用。算子空间和量子概率之间的相互作用已显现出重要的理论意义, 而量子概率和量子信息的交叉研究正处在一个关键的发展时期, 本项目的研究开辟了一个充满前景的交叉研究方向。.我们在多个研究方向上取得了重要进展, 特别是在量子环面上的调和分析方面, 我们建立了量子环面上的Sobolev空间、Besov空间和Triebel空间理论，给出了这些空间的等价刻画, 证明了Besov空间和Sobolev空间的嵌入定理和插值定理等, 揭示了量子环面上函数空间之间的内在联系，刻画了这些空间上的完全有界Fourier乘子, 这部分工作还被应用于研究非交换几何, 第一次得到在真正非交换情形下研究Alain Connes的量子可微性。相关工作表明量子环面上的调和分析在非交换几何和非交换偏微分方程中的应用前景广阔。.在量子信息理论方面, 我们综合应用量子信息理论的技巧和算子代数方法得到了该距离的两个下界, 由此构造了对AQBC的一些具体反例。为此, 我们还建立了Grothendieck不等式与AQBC之间的联系。 .在非交换分析方面，我们证明了对所有p>0情形的Davis分解和原子分解，给出了相应的范数同时控制结果, 刻画了小指标非交换鞅Hardy空间的对偶空间，解决了非交换鞅不等式的一些遗留问题；这部分结果将对Lp中量子随机积分的分析理论的建立起到至关重要的作用。.我们还在超有限II1型因子方面，在多变量函数空间上的复合算子理论方面, 在函数空间上算子理论方面, 以及在在向量值调和分析及其应用方面取得了重要的进展, 取得了非常丰富的研究成果。