213 research outputs found
Beyond heat baths II: Framework for generalized thermodynamic resource theories
Thermodynamics, which describes vast systems, has been reconciled with small
scales, relevant to single-molecule experiments, in resource theories. Resource
theories have been used to model exchanges of energy and information. Recently,
particle exchanges were modeled; and an umbrella family of thermodynamic
resource theories was proposed to model diverse baths, interactions, and free
energies. This paper motivates and details the family's structure and
prospective applications. How to model electrochemical, gravitational,
magnetic, and other thermodynamic systems is explained. Szilard's engine and
Landauer's Principle are generalized, as resourcefulness is shown to be
convertible not only between information and gravitational energy, but also
among diverse degrees of freedom. Extensive variables are associated with
quantum operators that might fail to commute, introducing extra nonclassicality
into thermodynamic resource theories. An early version of this paper partially
motivated the later development of noncommutative thermalization. This
generalization expands the theories' potential for modeling realistic systems
with which small-scale statistical mechanics might be tested experimentally.Comment: Minor updates (contributions clarified, material restored from v1,
references updated). 18 pages (including 2 figures) + appendice
Symmetric Laplacians, Quantum Density Matrices and their Von-Neumann Entropy
We show that the (normalized) symmetric Laplacian of a simple graph can be
obtained from the partial trace over a pure bipartite quantum state that
resides in a bipartite Hilbert space (one part corresponding to the vertices,
the other corresponding to the edges). This suggests an interpretation of the
symmetric Laplacian's Von Neumann entropy as a measure of bipartite
entanglement present between the two parts of the state. We then study extreme
values for a connected graph's generalized R\'enyi- entropy. Specifically,
we show that
(1) the complete graph achieves maximum entropy,
(2) the -regular graph: a) achieves minimum R\'enyi- entropy among all
-regular graphs, b) is within of the minimum R\'enyi- entropy
and of the minimum Von Neumann entropy among all connected
graphs, c) achieves a Von Neumann entropy less than the star graph.
Point contrasts sharply with similar work applied to (normalized)
combinatorial Laplacians, where it has been shown that the star graph almost
always achieves minimum Von Neumann entropy. In this work we find that the star
graph achieves maximum entropy in the limit as the number of vertices grows
without bound.
Keywords: Symmetric; Laplacian; Quantum; Entropy; Bounds; R\'enyi
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