A resource theoretic approach to work and voltage

Abstract

Resource theories offer a versatile and mathematically powerful framework that can be used to identify and quantify resources under a wide range of conditions including physical limitations and mathematical convenience. In this dissertation, resource theoretic approach has been used in both quantum and classical settings that are practically relevant to ascertain the usefulness of a given state/source for certain practically relevant tasks. A particular emphasis is given to energy, a resource of paramount importance. In the quantum setting, quantum thermodynamics has seen increasing interest in recent years. In majority of the existing works, the results have been obtained for systems in contact with large heat baths. With enhancements in technological capabilities, it is now possible to control and manipulate single-particle quantum systems. In particular, this raises a fundamental question of how these single-particle quantum systems behave thermodynamically without heat baths. In this setting, active states--states that are useful for work extraction under time-dependent perturbations--emerge as a resource. The first part of this dissertation focuses on characterising activity of a given state by establishing a resource theory, where the free operations correspond to the operational scenario of an experimenter manipulating a quantum system by employing energy-preserving operations with some probability or resetting the input state to some non-active state. State transformations under these operations turn out to have simple conditions. We contrast these operations with an alternative set of operations called the passivisation covariant operations. These operations are covariant with respect to activity-breaking channels, that is, quantum processes that transform every quantum state into a passive state. Lastly, a broad class of resource quantifiers is introduced. In the classical scenario, energy is a fundamental resource. This has led to the development of techniques to harvest energy from a variety of such as the voltage generated by converting human motion into electricity sources. These voltage sources are typically tiny and transient, and often exhibit an element of randomness. In addition, they include dissipation, which can be modelled as an internal resistance. The second part of this dissertation investigates how to quantify the usefulness of a given transient voltage source. Using the systematic approach of resource theory, several candidate measures quantifying the usefulness of a given transient voltage source are proposed, establishing an inter-convertibility hierarchy between such transient voltage sources. At the bottom of the hierarchy are resistors at the ambient temperature, while sources with low internal resistance and high internal voltages rank at the top. Three potential quantifiers that respect the hierarchy are provided. Among the three, one is particularly useful as it captures the amount of ``$\unitdc$" the source contains, meaning the source has an internal resistance of 1$\Omega$ and produces 1V DC in 1s. Further, it also has an interesting operational interpretation regarding the number of $\unitdc$ that can be distilled from the given voltage source or the number of sources that are to be combined to create a given source.