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和检测。例如，量子计算需要可靠地产生所谓的非高斯态，以便不被经典计算机模拟，而传感和通信任务需要测量的光学实现，该测量可能具有抽象的数学描述，例如，在非高斯态上的投影。因此，非高斯性是一种资源，类似于纠缠，它是量子信息的基石。这个项目的目的是开发一个系统的程序来理解资源高效的非高斯性的产生，并提供建设性的方法来产生任何所需的非高斯态，对大量的量子信息任务有用。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具有已知光学实现的量子操作。该项目的成果预计将产生双重影响：量子光子技术的进步将对国防和安全系统产生直接影响，并通过为量子光学中的基本效应提供新的数学工具来促进基础物理学的进步。这个项目支持的研究生将有机会获得超越高斯区的连续变量量子信息系统的知识，重点是非高斯性和纠缠测量。这个项目的基本结果将对未来量子互联网上的通信产生深远的影响，对于发现接收器结构以实现光通信和传感中的量子受限光子状态识别，对于光子量子处理器的各种近期应用在优化问题上的实现，对于评估振动光谱的分子模拟和在药物发现中的应用，以及对于特殊用途的应用，例如用于远距离纠缠分布的基于簇态的全光子量子中继器。众所周知，高斯态和使用压缩态、激光和线性光学可实现的运算--辅以单个非高斯运算--是“通用的”，在某种意义上，可以使用从上述集合中选择的元素来实现对量子物理定律所允许的一组光学模式的任何变换和测量。单个非高斯运算可以是诸如三次相位门的非高斯么正运算。但它也可以是非高斯测量，如光子分辨(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。事实上，人们可以通过在多模Fock基础上投影高斯态的模子集来产生非高斯态。然而，缺乏一种系统的方法来理解潜在的现象，以及一种简洁的数学描述。这个项目的总体目标有两个。首先是设计一种数学工具，特别是一种非高斯度量，其量身定做的方式将对国家工程有用。对于两个不同的量子态，当所设想的测量相等时，这两个态必须等于一个酉高斯算符。这将允许关于所产生的非高斯态是否确实是量子电路应该能够产生的期望的目标态(直到原则上可以使用已知的光子组件来实现的高斯么正运算)的决定性结果。早期关于非高斯主义衡量标准的著作并不拥有上述财产。同时，早期的状态工程工作关注的是保真度，作为衡量量子态封闭性的标准。保真度接近1是成功进行国家工程的必要标准，但不是充分标准。最后，上述工具将使我们能够将简单的非高斯态(例如，Fock态)缝合在一起，形成给定应用中所需的更大的、一般的纠缠非高斯目标态。最终，这个项目的目标是设计一种系统的方法来可靠地构建任何所需的非高斯态。我们的结果将对更深入地理解量子光学的数学结构，同时为光子量子技术的可扩展实现铺平道路，这将是我们的根本兴趣。我们的第二个目标是找到使用易于制备的高斯态、高斯运算和PNR探测器来实现任何量子光学测量(POVM)的配方。量子态的一部分上的POVM以一定的概率将初始状态从一组可能的状态转换为另一种状态。利用这一性质，我们将重写测量(例如，真空或非测量-与构建激光通信的量子最佳接收器相关)作为输入状态(由张量的状态和辅助的高斯态组成)，它经历高斯么正演化，并最终部分投影到Fock态上，给出给定POVM规定的适当状态和概率集。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。

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.