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

Abstract

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ${\rho }_S$. 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 ${\rho }_S$. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ${\rho }_S$. 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 ${\rho }_S$. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

arXiv comment12 pages, 10 figures. arXiv admin note: text overlap with arXiv:2111.07836

References in corpus (20)

• On the volume of the set of mixed entangled states
• Quantum thermalization through entanglement in an isolated many-body system
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Canonical Typicality
• The Physics of Maxwell's demon and information
• Aspects of generic entanglement
• On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems
• The microscopic origin of thermodynamic entropy in isolated systems
• Weak and strong typicality in quantum systems
• Thermodynamic entropy of a many body energy eigenstate
• Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems
• Quantum coarse-grained entropy and thermalization in closed systems
• Macroscopic and Microscopic Thermal Equilibrium
• Foundations of statistical mechanics from symmetries of entanglement
• Emerging dynamics arising from coarse-grained quantum systems
• Quantum Coarse-Graining, Symmetries and Reducibility of Dynamics
• Macro-to-micro quantum mapping and the emergence of nonlinearity
• Spin-entanglement wave in a coarse-grained optical lattice
• Fuzzy measurements and coarse graining in quantum many-body systems
• Decay of quantumness in a measurement process: Action of a coarse-graining channel