Decomposable coherence and quantum fluctuation relations

Abstract

In Newtonian mechanics, any closed-system dynamics of a composite system in a microstate will leave all its individual subsystems in distinct microstates, however this fails dramatically in quantum mechanics due to the existence of quantum entanglement. Here we introduce the notion of a `coherent work process', and show that it is the direct extension of a work process in classical mechanics into quantum theory. This leads to the notion of `decomposable' and `non-decomposable' quantum coherence and gives a new perspective on recent results in the theory of asymmetry as well as early analysis in the theory of classical random variables. Within the context of recent fluctuation relations, originally framed in terms of quantum channels, we show that coherent work processes play the same role as their classical counterparts, and so provide a simple physical primitive for quantum coherence in such systems. We also introduce a pure state effective potential as a tool with which to analyze the coherent component of these fluctuation relations, and which leads to a notion of temperature-dependent mean coherence, provides connections with multi-partite entanglement, and gives a hierarchy of quantum corrections to the classical Crooks relation in powers of inverse temperature.

Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations

“…Furthermore, the fluctuation relations give way to an interpretation involving coherent work states, a generalisation of Newtonian work for fully quantum dynamics. It was proved that the energetic and coherent properties of the coherent work is totally captured in this fluctuation setting [ 41 ].…”

“…Our starting point is a global “fully quantum fluctuation theorem” from [ 29 ], a more general relation than that explicated in [ 32 , 40 , 41 ], which can be used to derive a whole family of quantum fluctuation relations. A defining property of quantum systems is their ability to reside in superpositions of states belonging to different energy eigenspaces, a property often referred to simply as coherence.…”

“…While often not explicitly modelled, the dynamics of the energy supply can contribute non-trivially to the evolution of the driven system. Thus, to enable a more careful analysis of the energy and coherence changes of the system, we consider an inclusive (this is in contrast to the exclusionary picture of the original Crooks and Jarzynski equalities) approach [ 29 , 30 , 31 , 40 , 41 , 62 ], which introduces a battery and assumes the system (S) and battery (B) evolve together under a time independent Hamiltonian .…”

“…For the uncorrelated initial states and measurement operators related by the mapping , the global fluctuation relation [ 29 , 40 , 41 ] can be written as in terms of the quantum generalisation of the change in free energy, as well as a quantum generalisation of the work supplied by the battery. The function is an effective potential that specifies the relevant energy value within the fluctuation theorem context.…”

“…Conversely, for a projector onto an energy eigenstate, the effective potential, , is the corresponding energy from which we regain the classical work term using a two point projective measurement scheme. More generally, when restricting to projective measurement operators, the function is a cumulant generating function in the parameter that captures the statistical properties of measurements of H on [ 41 ].…”