FET: Small: Establishing an efficient framework for non-Gaussian states engineering and optical implementation of measurements

量子光学技术（计算、传感、通信）基本上是基于量子态的产生及其检测。例如，量子计算需要可靠地生成所谓的非高斯状态，以便不被经典计算机模拟，而传感和通信任务需要光学实现测量，这些测量可能具有抽象的数学描述，例如非高斯状态的投影。因此，非高斯性是一种资源，类似于纠缠，是量子信息的基石。该项目旨在开发一种系统的程序来理解非高斯性的资源效率生产，并提供建设性的方法来生成对大量量子信息任务有用的任何所需的非高斯态。该项目还旨在找到系统的方法，将任何给定的量子光学测量分解为一组已知光学实现的量子操作。该项目将实现的结果预计将产生双重影响：量子光子技术的进步，对国防和安全系统产生直接影响，以及通过为量子光学的基本效应提供新颖的数学工具来推进基础物理学。该项目支持的研究生将有机会获得高斯以外的连续变量量子信息系统的知识，重点是非高斯性和纠缠措施。本项目的基本成果将对未来量子互联网上的通信产生深远的影响，对发现用于光通信和传感中实现量子有限光子状态识别的接收器结构，对光子量子处理器在优化问题中的各种近期应用的实现，以及用于评估振动谱的分子模拟在药物发现中的应用产生深远的影响。也适用于特殊用途的应用，如基于簇态的全光子量子中继器，用于长距离纠缠分布。众所周知，高斯态和操作——可以用压缩态、激光和线性光学来实现——辅以单一的非高斯操作，是“通用的”，从某种意义上说，在量子物理定律允许的一组光学模式上的任何变换和测量，都可以用从上述集合中选择的元素来实现。单一的非高斯操作可以是非高斯的单一操作，如三相门。但它也可以是一种非高斯测量，如光子解析（PNR）检测，即在Fock态上的投影，这在早期和最近的工作中已被证明是状态工程的可靠选择。实际上，可以通过在多模Fock基上投射高斯态的模子集来产生非高斯态。然而，一个系统的方法来理解潜在的现象，和一个整洁的数学描述是缺乏的。这个项目的广泛目标是双重的。首先是设计一个数学工具，特别是一个非高斯度量，以一种对状态工程有用的方式量身定制。当所设想的两个不同量子态的测量值相等时，这两个态必须等于一个幺正高斯算子。这将允许结论性的结果，如果产生的非高斯状态确实是期望的目标状态（直到高斯单一操作，原则上可以使用已知的光子组件实现），量子电路应该能够产生。早期关于非高斯测度的工作不具备上述性质。与此同时，早期的状态工程研究着重于将保真度作为量子态紧密度的度量。对于成功的国家工程来说，忠实度接近一个是必要的，但不是充分的标准。最后，上述工具将使我们能够将简单的非高斯状态（例如Fock状态）“缝合”在给定应用程序中所需的更大的、一般纠缠的非高斯目标状态。最终，本项目旨在设计一种系统的方法来可靠地构建任何期望的非高斯状态。我们的结果将对量子光学的数学结构有更深入的了解，同时为光子量子技术的可扩展实现铺平道路。我们的第二个目标是找到可以使用易于制备的高斯态、高斯运算和PNR探测器实现任何量子光学测量（POVM）的配方。部分量子态上的POVM以一定的概率将一组可能状态的初始状态转换为另一种状态。利用这一特性，我们将重新编写测量（例如，真空或非真空测量-与构建用于激光通信的量子最佳接收器相关）作为输入状态（由待测量状态与辅助高斯状态组成），该状态经历高斯幺正演化，并最终部分投影到Fock状态上，从而给出给定POVM规定的适当状态集和概率。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。

Quantum optical technologies (computing, sensing, communications) are fundamentally based on the generation of quantum states and the detection thereof. For example, quantum computing requires reliable generation of the so-called non-Gaussian states in order not to be simulable by a classical computer, while sensing and communications tasks require optical implementation of measurements which might have an abstract mathematical description, e.g., projection on non-Gaussian states. Therefore, non-Gaussianity is a resource which, similar to entanglement, is a cornerstone of quantum information. This project aims to develop a systematic procedure for understanding resource-efficient production of non-Gaussianity and to provide constructive ways of generating any desired non-Gaussian state useful to a plethora of quantum informational tasks. The project also aims to find systematic ways of decomposing any given quantum optical measurement into a set of quantum operations which have known optical implementations. The results that will be enabled by this project are anticipated to have a two-fold impact: advancement of quantum photonic technologies with immediate impact on defense and security systems, and the advancement of basic physics by providing novel mathematical tools for fundamental effects in quantum optics. Graduate students supported by this project will have the opportunity to gain knowledge on continuous variable quantum information systems beyond the Gaussian regime, with an emphasis on non-Gaussianity and entanglement measures. The fundamental results from this project will have far reaching impacts on communications over the future quantum internet, on discovering receiver structures for attaining quantum-limited photonic state discrimination for applications in optical communications and sensing, on the realizability of various near-term applications of photonic quantum processors to problems in optimization, and molecular simulations for evaluating vibronic spectra with applications in drug discovery, and also to special-purpose applications such as all-photonic quantum repeaters based on cluster states for entanglement distribution over long-distances.It is known that Gaussian states and operations -- realizable using squeezed states, lasers and linear optics -- complemented with a single non-Gaussian operation, is "universal", in a sense that any transformation and measurement on a set of optical modes allowed by the laws of quantum physics, can be realized using elements chosen from this aforesaid set. The single non-Gaussian operation could be a non-Gaussian unitary such as a cubic-phase gate. But it can also be a non-Gaussian measurement such as photon resolving (PNR) detection, i.e., projection on Fock states, which have been proven a reliable choice for state engineering in early and recent works. Indeed, one can produce a non-Gaussian state by projecting a subset of a Gaussian state’s modes on the multi-mode Fock basis. However, a systematic way to understand the underlying phenomena, and a neat mathematical description is missing. The broad goals of this project are twofold. First is to devise a mathematical tool, specifically a non-Gaussianity measure, tailored in a fashion that will be useful to state engineering. When the envisioned measure is equal for two different quantum states, the two states must be equal up to a unitary Gaussian operator. This will allow for conclusive results as to if a produced non-Gaussian state is indeed the desired target state (up to a Gaussian unitary operation which is in principle implementable using known photonic components) that a quantum circuit is supposed to be able to produce. Earlier works on non-Gaussianity measures do not possess said property. At the same time earlier works on state engineering focus on fidelity as the measure of closeness of quantum states. Fidelity being close to one is a necessary, but not sufficient criterion for successful state engineering. Finally, the aforesaid tools will enable us to “stitch” together, simple non-Gaussian states, e.g., Fock states, into larger, general entangled non-Gaussian target states desired in a given application. Ultimately, this project aims to devise a systematic way of constructing any desired non-Gaussian state reliably. Our results will be of fundamental interest in a deeper understanding of the mathematical structures of quantum optics, while paving the way to scalable realizations of photonic quantum technologies. Our second goal is to find recipes with which any quantum optical measurement (POVM) can be realized using easy-to-prepare Gaussian states, Gaussian operations and PNR detectors. A POVM on a part of quantum state transforms the initial state into another state from a set of possible states with some probability. Leveraging this property, we will re-write a measurement (e.g., the vacuum-or-not measurement—which has relevance to constructing quantum optimal receivers for laser communications) as an input state (consisting of the state to be measured tensored with ancillary Gaussian states), which undergoes Gaussian unitary evolution, and finally partially projected on Fock states in a manner that gives the proper set of states and probabilities prescribed by the given POVM.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.