Symmetric Informationally Complete Measurements and Quantum Computation

到目前为止，大部分量子信息科学都是根据量子理论的教科书表示来发展的，这主要相当于向量和复数矩阵的数学语言。但这种语言可能隐藏的东西和它揭示的东西一样多。某些量子基础方法暗示了这一点，例如量子贝叶斯主义，其中寻找用概率（非负实数）表示理论的替代方法，完成了大部分概念上的繁重工作。这是因为这些表示更接近于从决策理论的角度分析经典和量子之间的区别。这表明，这种性质的表征可能对发展量子信息技术也至关重要，例如，通过提供如何使用传统计算资源最好地模拟量子系统的基准和证明量子霸权。开发一种有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，这将对跨学科产生影响，数论是物理课程的一个陌生的数学领域。近年来，许多研究都集中在这些表示中最对称的可能性上——那些基于“对称信息完整量子测量”或sic的表示——因为它们在惊人数量的应用中简化了功能和最优性：从最优量子态断层扫描到纠缠探测器、新颖的密钥分发方案、设备无关随机数生成中的组件、维度观察等等。然而，这些表示带来了两个问题。首先，尚不清楚量子系统是否总能满足SIC存在的条件（尽管目前已知SIC存在于至少264个维度中，并且强烈认为存在于所有其他维度中）。其次，当存在的条件能够被满足时，除了定义它们的全局对称性外，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物理学和代数数论之间的联系，特别是希尔伯特的第12个问题，来弥补后者，如果不是前者的话。本质上，我们需要的是开发一个特殊（超越）函数的工具包，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审视一些现象，并建立一个开源代码库，以促进更广泛的应用研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.