Symmetric Informationally Complete Measurements and Quantum Computation

到目前为止，量子信息科学的大部分发展都是在量子理论的教科书表述中进行的，主要是复数的向量和矩阵的数学语言。 但这种语言可能隐藏了它所揭示的东西。 某些量子基础方法暗示了这一点，如量子贝叶斯主义，其中找到替代方法来表示理论的概率（非负的真实的数）做了大部分的概念繁重。 这是因为这些表示更接近于分析经典和量子在决策理论方面的区别的目标。 这表明，这种性质的表示对开发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，并证明量子优越性。 在这些方面开发一种有效的量子信息处理语言是该项目的目标，由于它在量子力学和数论之间建立了联系，这将对物理课程产生跨学科的影响。近年来，许多研究都集中在最对称的可能，这种表示-那些基于“对称信息完整的量子测量”或SIC-为他们的简化权力和最优性在一个惊人的应用数量：从最佳的量子状态断层扫描，纠缠探测器，新的密钥分配方案，组件在设备无关的随机数生成，尺寸见证，等等。 然而，这些表示的承诺伴随着两个问题。 首先，不知道qudit系统是否总是满足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）。 第二，当存在条件可以满足时，除了定义它们的整体对称性，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SIC和代数数论之间的联系，特别是希尔伯特的第12个问题，来解决后者，如果不是前者的话。 从本质上讲，所需要的是开发一个特殊（先验）功能的工具包，使表征更易于管理。 有了这个工具包，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。这个奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.