Abstract
Passive states, i.e., those states from which no work can be extracted via unitary operations, play an important role in the foundations and applications of quantum thermodynamics. They generalize the familiar Gibbs thermal states, which are the sole passive states being stable under tensor product. Here, we introduce a partial order on the set of passive states that captures the idea of a passive state being virtually cooler than another one. This partial order, which we build by defining the notion of relative passivity, offers a fine-grained comparison between passive states based on virtual temperatures (just like thermal states are compared based on their temperatures). We then characterize the quantum operations that are closed on the set of virtually cooler states with respect to some fixed input and output passive states. Viewing the activity, i.e., non-passivity, of a state as a resource, our main result is then a necessary and sufficient condition on the transformation of a class of pure active states under these relative passivity-preserving operations. This condition gives a quantum thermodynamical meaning to the majorization relation on the set of non-increasing vectors due to Hoffman. The maximum extractable work under relative passivity-preserving operations is then shown to be equal to the ergotropy of these pure active states. Finally, we are able to fully characterize passivity-preserving operations in the simpler case of qubit systems, and hence to derive a state interconversion condition under passivity-preserving qubit operations. The prospect of this work is a general resource-theoretical framework for the extractable work via quantum operations going beyond thermal operations.
Highlights
A major focus in the area of thermodynamics is work extraction
We have introduced a partial order on the set of passive states which generalizes the natural ordering of thermal states in terms of temperature
We show that, if used as a refrigerator, a virtually cooler state can cool down an external qubit system to a further extent as compared with the cooling effected by this other state
Summary
A major focus in the area of thermodynamics is work extraction. The laws of thermodynamics, which are expressed as a set of phenomenological rules, govern the transformations of states leading to work extraction (see, e.g., Ref. [1]). After characterizing the convex set of virtually cooler states than a given passive state, we consider the class of quantum operations that preserve this partial order. For transformations of pure qubit states under strictly incoherent RPPOs, the necessary and sufficient condition based on Hoffman majorization remains true for passivity-preserving operations. The latter operations, which we denote as PPOs, are those that map the set of passive states into itself. We provide the necessary and sufficient condition for the transformation of a specific class of active pure states under RPPOs, which makes the connection with Hoffman majorization and gives it a thermodynamical meaning. In Appendix E, we briefly touch upon the problem of defining the notion of extractable work under quantum channels beyond the thermal case and provide an example of work extraction under a RPPO for a qubit system
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have