We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.

中文翻译：

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无需纯化的纯化复杂性

我们通过将密度矩阵空间上的 Bures 度量或量子 Fisher 信息度量 (QFIM) 作为复杂性度量，将纯态复杂性的 Fubini-Study 方法推广到通用量子态。由于 Uhlmann 定理，我们表明混合状态复杂度完全等于 Fubini-Study 度量为纯化状态测量的纯化复杂度，但没有明确应用任何纯化。我们还发现在任何保留痕量的量子操作下，纯化复杂性没有增加。我们还以混合高斯状态为例，以明确说明我们对纯化复杂性的结论。