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arXiv 1209.5233quant-phmath-phmath.MPmath.OA
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量子通道的谱矩与熵刚性：一个三模光子见证

Spectral moments and entropy rigidity of quantum channels: a three-mode photonic witness

Foundation Science Education Center, Hangzhou City University（杭州城市学院基础科学教育中心）；School of Mathematical Sciences, Hangzhou Dianzi University（杭州电子科技大学数学系）

Laizhen Luo, Lin Zhang

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22 pages
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AI总结：

本研究通过三模光子实验，利用谱矩和熵刚性区分量子通道，证明恒定输出纯度与第三谱矩可确定主化保持通道，熵相等条件更严格，并构造了qutrit输入验证退极化噪声下的熵分裂。

摘要：

主化（majorization）和冯·诺依曼熵（von Neumann entropy）提供了关于量子光学态中谱无序的相关但不等效的描述。我们研究了保持这些描述的通道，并提出了一个三模光子测试来区分它们的差异。对于输入和输出维度相等的通道，我们证明了在单一简单谱态（simple-spectrum state）的酉轨道（unitary orbit）上，恒定的输出纯度和第三谱矩决定了保持主化的通道形式，而无需假设该通道是酉的。对于量子比特（qubits），仅输出纯度就足够了。保持熵相等则更具限制性：在维度$d\geqslant3$中，即使在最大混合态附近的局部条件，也仅允许替换通道（replacement channels）和酉通道（unitary channels）。量子比特允许额外的退极化族（depolarizing family），但这一例外在引入一个空闲辅助量子比特（idle auxiliary qubit）的扩展下消失。我们构造了精确熵匹配的qutrit输入，其输出熵在非平凡退极化噪声下以三次阶分裂。提出的单光子实现使用三个光学模式、随机Weyl变换以及粒子数或谱矩测量。还获得了制备失配、读出不确定性和与输入无关的损耗的解析表达式。结果将二次和三次谱信息与量子通道结构的实验可解释测试联系起来。

英文摘要：

Majorization and von Neumann entropy provide related but inequivalent descriptions of spectral disorder in quantum optical states. We investigate the channels preserving these descriptions and propose a three-mode photonic test of their difference. For channels with equal input and output dimensions, we show that constant output purity and third spectral moment on the unitary orbit of one simple-spectrum state determine the majorization-preserving channel forms, without assuming unitality.For qubits, output purity alone is sufficient. Preservation of entropy equality is substantially more restrictive: in dimension $d\geqslant3$, even a local condition near the maximally mixed state permits only replacement and unitary channels. Qubits admit an additional depolarizing family, but this exception disappears under extension by an idle auxiliary qubit. We construct exactly entropy-matched qutrit inputs whose output entropies split at cubic order under nontrivial depolarizing noise. A proposed single-photon implementation uses three optical modes, randomized Weyl transformations and population or spectral-moment measurements.Explicit expressions are also obtained for preparation mismatch,readout uncertainty and input-independent loss. The results connect quadratic and cubic spectral information with experimentally interpretable tests of quantum-channel structure.

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