Shelton Jacinto | Searcharxiv

arXiv2023

A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator $ρ_S$. 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 $ρ_S$. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from $ρ_S$. 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-grainied space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to maximum entanglement between system and 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 w.r.t the purity of $ρ_S$. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

quant-ph↗

arXiv2021

Canonical Quantum Coarse-Graining and Surfaces of Ignorance

In this paper we introduce a canonical quantum coarse-graining and use negentropy to connect ignorance as measured by quantum information entropy and ignorance related to quantum coarse-graining. For our procedure, macro-states are the set of purifications $\{|\barΓ^ρ\rangle\}$ associated with density operator $ρ$ and micro-states are elements of $\{|\barΓ^ρ\rangle\}$. Unlike other quantum coarse-graining procedures, ours always gives a well-defined unique coarse-graining of phase space. Our coarse-graining is also unique in that the volumes of phase space associated with macro-states are computed from differential manifolds whose metric components are constructed from the Lie group symmetries that generate $\{|\barΓ^ρ\rangle\}$. We call these manifolds surfaces of ignorance, and their volumes quantify the lack of information in $ρ$ as measured by quantum information entropies. To show that these volumes behave like information entropies, we compare them to the von Neumann and linear entropies for states whose symmetries are given by $SO(3)$, $SU(2)$, and $SO(N)$. We also show that our procedure reproduces features of Boltzmann's original coarse-graining by showing that the majority of phase space consists of states near or at equilibrium. As a consequence of this coarse-graining, it is shown that an inherent flag variety structure underlies composite Hilbert spaces.

quant-ph↗