The mixedness of a quantum state is usually seen as an adversary to topological quantization of observables. For example, exact quantization of the charge transported in a so-called Thouless adiabatic pump is lifted at any finite temperature in symmetry-protected topological insulators. Here, we show that certain directly observable many-body correlators preserve the integrity of topological invariants for mixed Gaussian quantum states in one dimension. Our approach relies on the expectation value of the many-body momentum-translation operator, and leads to a physical observable --- the ``ensemble geometric phase'' (EGP) --- which represents a bona fide geometric phase for mixed quantum states, in the thermodynamic limit. In cyclic protocols, the EGP provides a topologically quantized observable which detects encircled spectral singularities (``purity-gap'' closing points) of density matrices. While we identify the many-body nature of the EGP as a key ingredient, we propose a conceptually simple, interferometric setup to directly measure the latter in experiments with mesoscopic ensembles of ultracold atoms.

中文翻译：

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探索密度矩阵的拓扑

量子态的混合通常被视为可观测对象拓扑量化的对手。例如，在任何受对称保护的拓扑绝缘子中，在任何有限的温度下，对所谓的Thouless绝热泵中传输的电荷的精确定量都可以提高。在这里，我们表明某些可直接观察到的多体相关器为一维混合高斯量子态保留了拓扑不变量的完整性。我们的方法依赖于多体动量平移算符的期望值，并导致物理可观察到的-``整体几何相''（EGP）-代表了混合量子的真实几何相状态，处于热力学极限。在循环协议中 EGP提供了一种拓扑量化的可观察值，可检测密度矩阵的环绕光谱奇点（``纯度间隙''闭合点）。虽然我们将EGP的多体性质确定为关键成分，但我们提出了一种概念上简单的干涉测量设置，可以在使用超冷原子的介观集合体进行的实验中直接测量后者。