Geometric Quantum Thermodynamics

Abstract

Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We develop both microcanonical and canonical ensembles, introducing continuous mixed states as distributions on the manifold of quantum states. We call out the experimental consequences for a gas of qudits. We define quantum heat and work in an intrinsic way, including single-trajectory work, and reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the First and Second Laws of Thermodynamics and Jarzynki's Fluctuation Theorem. The result is a more transparent physics, than conventionally available, in which the mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned.

Quantum and classical ergotropy from relative entropies

“…In a second part of the analysis, we turn to a unified framework, namely geometric quantum mechanics. Exploiting this approach [18][19][20], we define the geometric relative entropy. With this it becomes particularly trans-parent to characterize the one-time measurement approach to quantum work [21][22][23][24].…”

“…Only very recently, Anza and Crutchfield [18][19][20] recognized that for such thermodynamic considerations socalled geometric quantum mechanics [47][48][49] is a uniquely suited paradigm. In standard quantum theory, a quantum state is described by a density operator ρ, which can be expanded in many different decompositions of pure states.…”

“…We proceed by briefly outlining the main notions of geometric quantum mechanics. For a more complete exposition we refer to the literature [18][19][20][47][48][49]. In the geometric approach, a pure quantum state |ψ is described as a point in a complex projective space V d ≡ CP d−1 [48], where d is the dimension of the Hilbert space [50].…”

“…Remarkably, it is not hard to see that ̺ B is a representation of the geometric canonical ensemble as proposed by Anza and Crutchfield [18,20]. The geometric canonical ensemble is defined as the geometric state that maximizes the corresponding Shannon entropy under the usual boundary conditions [53].…”

“…The geometric canonical ensemble is defined as the geometric state that maximizes the corresponding Shannon entropy under the usual boundary conditions [53]. Specifically, we have [18,20].…”