量子操作的瓶颈维度
The bottleneck dimension of quantum operations
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中文总结 AI 辅助
本文针对可编程量子设备因噪声导致的相干性问题，提出与基无关的维度瓶颈框架，定义三种不等价的量子维度概念并确定其层级，推导噪声通用量子处理器的白噪声阈值，揭示保持相干性随维度增大要求更高的规律。
中文摘要 AI 辅助
可编程量子设备名义上作用于希尔伯特空间，其维度随组成部分数量呈指数增长，但噪声的存在使得这些设备不太可能在整个庞大的希尔伯特空间中保持相干性。那么，应将这种不完善设备关联的有效相干量子维度是多少？为回答该问题，本文引入一种与基无关的操作框架，该框架对可编程变换施加维度瓶颈，具体而言，我们探究其处理的量子信息能被压缩到何种程度。将这一想法形式化后，我们确定了三种不等价的概念，即d-可压缩性、d-可模拟性和d-可嵌入性，它们因施加瓶颈所用的因果结构不同而不同，且形成严格的层级关系。该框架统一了几个现有概念：量子测量的联合可测量性与可模拟性，以及态系综的绝对维度，均可作为特例被推导出来。我们以互补基下的噪声量子比特测量为例阐释该层级关系，并确定n维所有噪声酉信道（即噪声通用量子处理器）成为d-可压缩、d-可模拟和d-可嵌入的白噪声阈值。这些阈值证实了预期：随着名义维度n增大，在整个希尔伯特空间中保持相干性的要求越来越高。
英文摘要
Programmable quantum devices nominally act on a Hilbert space whose dimension grows exponentially with the number of constituents, but the presence of noise makes it unlikely that they remain coherent across all of this immense Hilbert space. Then, what is the effective coherent quantum dimension that should be associated with such imperfect devices? To answer this question we here introduce an operational basis-independent framework which imposes a dimension bottleneck on the programmable transformations. Concretely we ask how strongly the quantum information they process can be compressed. Formalizing this idea we identify three inequivalent notions, termed $d$-compressibility, $d$-simulability and $d$-embeddability, which differ in the causal structure used to impose the bottleneck and form a strict hierarchy. The framework unifies several existing notions: joint measurability and simulability of quantum measurements, and the absolute dimensionality of state ensembles, are recovered as special cases. We illustrate the hierarchy with noisy qubit measurements in complementary bases, and we determine the white-noise thresholds at which the set of all noisy unitary channels in dimension $n$, a noisy universal quantum processor, becomes $d$-compressible, $d$-simulable and $d$-embeddable. The thresholds confirm the expectation -- maintaining coherence across the full Hilbert space becomes increasingly demanding as the nominal dimension $n$ increases.
