Beam splitters are routinely used for generating entanglement between modes in the optical and microwave domains, requiring input states that are not convex combinations of coherent states. This leads to the ability to generate entanglement at a beam splitter as a notion of quantumness. A similar, yet distinct, notion of quantumness is the amount of entanglement generated by two-mode squeezers (i.e., four-wave mixers). We show that squeezed-vacuum states, paradoxically, both minimize and maximize these notions of quantumness, with the crucial resolution of the paradox hinging upon the relative phases between the input states and the devices. Our notion of quantumness is intrinsically related to eigenvalue equations involving creation and annihilation operators, governed by a set of inequalities that leads to generalized cat and squeezed-vacuum states.

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压缩状态如何最大化和最小化相同的量子概念

分束器通常用于在光域和微波域中的模式之间产生纠缠，要求输入状态不是相干状态的凸组合。这导致在分束器处产生纠缠的能力作为量子性的概念。一个类似但不同的量子概念是由双模式挤压器（即四波混频器）产生的纠缠量。我们证明，自相矛盾的是，挤压真空状态既最小化又最大化了这些量子概念，而这个悖论的关键解决方案取决于输入状态和设备之间的相对相位。我们的量子概念本质上与涉及创生和湮灭算符的特征值方程相关，