Symmetric Informationally Complete Measurements and Quantum Computation

到目前为止，大多数量子信息科学的发展都是在教科书上介绍量子理论，主要是复数的矢量和矩阵的数学语言。但这种语言可能隐藏了很多东西，也揭示了一些东西。某些量子基础方法暗示了这一点，比如量子贝叶斯主义，在这种方法中，找到用概率(非负实数)来表示理论的替代方法，完成了大部分概念上的繁重任务。这是因为这些表示法更接近于在决策理论方面分析经典和量子之间的区别的目标。这表明，这种性质的表示对于发展量子信息技术也可能至关重要，例如，通过提供基准，说明如何使用传统计算资源最好地模拟量子系统，并证明量子优势。以这些术语开发一种高效的量子信息处理语言是这个项目的目标，由于它在量子力学和数论之间建立了联系，它将产生跨学科的影响，数论是物理课程之外的一个数学领域。近年来，许多研究都集中在这类表示中最对称的可能表示上--那些基于“对称信息完全量子测量”或SIC的表示，因为它们的简化能力和在数量惊人的应用中的最佳性：从最优的量子状态层析成像，到纠缠检测器，新颖的密钥分发方案，设备无关随机数生成的组件，维度见证，等等。然而，这些陈述的承诺伴随着两个陷阱。首先，QUDIT系统是否总能满足SIC存在的条件尚不清楚(尽管目前已知SIC至少存在于264维，并强烈认为存在于所有其他维)。其次，当存在的条件能够满足时，除了定义它们的整体对称性之外，解总是显得异常复杂。在这个项目中，该小组计划通过利用最近发现的SICS和代数数论之间的联系，特别是希尔伯特的第12个问题，来补救后一个问题，如果不是前者的话。从本质上讲，需要的是开发一个特殊(先验)功能的工具包，通过它使表示更易于管理。有了工具包，该小组将从这个更有效的角度重新研究一些现象，并建立一个开放源代码基础，以促进更广泛的应用研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

So far, most of quantum information science has been developed in terms of the textbook presentation of quantum theory, which chiefly amounts to the mathematical language of vectors and matrices of complex numbers. But this language may hide as much as it reveals. This is hinted at by certain quantum foundational approaches such as Quantum Bayesianism where finding alternative ways to represent the theory in terms of probabilities (nonnegative real numbers) does most of the conceptual heavy lifting. This is because these representations live much closer to the goal of analyzing the distinction between classical and quantum in terms of decision theory. This suggests that representations of this nature may also be of critical importance to developing quantum information technologies, for instance by providing benchmarks for how to best simulate quantum systems using conventional computational resources and certifying quantum supremacy. Developing an efficient language for quantum information processing in these terms is the goal of this project, which will have influence across disciplines thanks to the connection it makes between quantum mechanics and number theory, an area of mathematics foreign to the physics curriculum. In recent years much research has focused on the most symmetric possible of such representations—those based on “symmetric informationally complete quantum measurements” or SICs—for their simplifying power and optimality in a surprising number of applications: From optimal quantum-state tomography, to entanglement detectors, novel key distribution schemes, components in device-independent random number generation, dimension witnessing, and more. However, the promise of these representations comes with two catches. First, it is not known whether the conditions for SIC existence can always be satisfied for qudit systems (though SICs are currently known to exist in at least 264 dimensions and strongly believed to exist in all others). Second, when the conditions for existence can be satisfied, except for the global symmetry defining them, the solutions always appear monstrously complex. In this project, the group plans to remedy the latter matter, if not the former, by exploiting recently discovered connections between SICs and algebraic number theory, particularly Hilbert’s 12th problem. Essentially what is called for is the development of a tool pack of special (transcendental) functions by which to make the representation more manageable. With the tool pack in hand, the group will reexamine a number of phenomena from this more efficient perspective and build an open-source code base to facilitate applied research more broadly.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.