Ray Shannon, Alsing Paul M, Cafaro Carlo, Jacinto H S
Air Force Research Laboratory, Rome, NY 13441, USA.
Griffiss Institute, Rome, NY 13441, USA.
Entropy (Basel). 2023 May 12;25(5):788. doi: 10.3390/e25050788.
In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρS. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.
在本文中，我们为与任意约化密度算子ρS相关的纯化流形构造了度规张量和体积。我们还定义了一种量子粗粒化（CG），以研究宏观态为纯化流形（我们称之为无知表面（SOI））且微观态为ρS的纯化的体积。在此背景下，体积充当宏观态的多重性，量化了ρS中缺失的信息量。通过使用利用SU(2)、SO(3)和SO(N)表示生成SOI的示例，我们展示了CG的两个特征：（1）从较小体积的非典型宏观态开始的系统会演化为体积更大的宏观态，直到在系统与环境变得严格更纠缠的过程中达到平衡宏观态；（2）平衡宏观态占据了粗粒化空间的绝大部分，尤其是随着总系统维度变大时。这里，平衡宏观态对应于系统与环境之间的最大纠缠。为了在所考虑的示例中证明特征（1），我们表明体积的行为类似于冯·诺依曼熵，即对于纯态它为零，对于最大混合态它最大，并且相对于ρS的纯度是一个凹函数。这两个特征对于关于热化的典型性论证和玻尔兹曼最初的CG至关重要。
