Fluctuation Theorems for a Quantum Channel

Abstract

We establish the general framework of quantum fluctuation theorems by finding the symmetry between the forward and backward transitions of any given quantum channel. The Petz recovery map is adopted as the reverse quantum channel, and the notion of entropy production in thermodynamics is extended to the quantum regime. Our result shows that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer in an arbitrary non-equilibrium quantum process. We introduce a complex-valued entropy production to fully understand the relation between the forward and backward processes through the quantum channel. We find the physical meaning of the imaginary part of entropy production to witness the broken symmetry of the quantum channel. We also show that the imaginary part plays a crucial role in deriving the second law from the quantum fluctuation theorem. The dissipation and fluctuation of various quantum resources including quantum free energy, asymmetry and entanglement can be coherently understood in our unified framework. Our fluctuation theorem can be applied to a wide range of physical systems and dynamics to query the reversibility of a quantum state for the given quantum processing channel involving coherence and entanglement.

Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements

“…The evaluation of the relevant work originated by using a coherent modulation of the system Hamiltonian has been the subject of intense investigation [18][19][20][21][22][23][24]. A special focus was devoted to the study of heat and entropy production, obeying of the second law of thermodynamics, the interaction with one or more external bodies, and/or the inclusion of an observer [25][26][27][28][29][30][31][32][33][34].…”

Quantum Fluctuation Theorems beyond Two-Point Measurements

“…They additionally complement recent attempts to obtain fully quantum fluctuation theorems for mechanically driven systems [21][22][23][24] (see also Refs. [25,26]). In the following, we first derive a detailed quantum fluctuation theorem for the ratio of the probability of a conditional local trajectory of the system and its reverse.…”

Decomposable coherence and quantum fluctuation relations

“…However we find that the quantum fluctuation theorem admits an infinite series of corrections to the Crooks form in powers of β. The higher order terms account for the coherent properties of the initial pure states of the system and gives an operational perspective on related works [23,26]. There exists another natural decomposition of the fluctuation theorem in terms of 'average change in energy' and 'average change in coherence' with the caveat that the average coherence must be carefully interpreted.…”

“…It would be of interest to obtain an independent operational analysis of this transformation (perhaps as a coherent form of a maximum entropy principle). We do not expand any more on this here, but instead refer the reader to [26] and [23] for more discussion.…”

“…Firstly, it leads to a notion of decomposable and non-decomposable coherence and has an immediate description in terms of recent frameworks for quantum coherence [8-11, 17, 18]. We then extend these considerations in noisy quantum environments and show that this notion of a coherent work process connects naturally with fluctuation relations [19][20][21][22][23][24][25][26]. The coherent fluctuation theorems we develop go beyond traditional relations, such as the Crooks relation, and allow the analysis of quantum coherence in a noisy thermal environment.…”