Work extractable from quantum systems can be viewed as a resource to perform some tasks. Here, we try to relate the work extractable to some resource theory. To do this, we consider completely passive states and passive states as free states and, by identifying the free operations, formulate resource theories where the maximum work extractable is a monotone. In particular, we study how the free operations act on the free states, showing how a sort of temperature plays a pivotal role in these resource theories.
Work extractable from quantum systems can be viewed as a resource to perform some tasks. Here, we try to relate the work extractable to some resource theory. To do this, we consider completely passive states and passive states as free states and, by identifying the free operations, formulate resource theories where the maximum work extractable is a monotone. In particular, we study how the free operations act on the free states, showing how a sort of temperature plays a pivotal role in these resource theories.
Resource theories are a set of tools, coming from the field of quantum information theory, that find applications in the study of several physical scenarios. These theories describe the physical world from the perspective of an agent, who acts over a system to modify its quantum state, while having at disposal a limited set of operations. A noticeable example of a physical theory which has recently been described with these tools is quantum thermodynamics, consisting in the study of thermodynamic phenomena at the nano-scale. In the standard approach to resource theories, it is usually the case that the constraints over the set of available operations single out a unique resource. In this thesis, we extend the resource theoretic framework to include situations where multiple resources can be identified, and we apply our findings to the study of quantum thermodynamics, to gain a better understanding of quantities like work and heat in the microscopic regime. We introduce a mathematical framework to study resource theories with multiple resources, and we explore under which conditions these multi-resource theories are reversible. Furthermore, we investigate the interconversion of resources, i.e., in which situations it is possible to exchange between two different kinds of resources. We then apply this formalism to quantum thermodynamics, where the two resources under consideration are energy and entropy. This multi-resource theory allows us to explore thermodynamics when the system under examination is closed or coupled with a thermal environment with a finite size. In addition to our work on multi-resource theories, we study the states of equilibrium of closed systems, known as passive states, and analyse under which circumstances it is possible to extract energy from these states. We show that passivity is energetically unstable, and that, even for closed systems, the only stable states are those with a well-defined temperature.
Entanglement quantification aims to assess the value of quantum states for quantum information processing tasks. A closely related problem is state convertibility, asking whether two remote parties can convert a shared quantum state into another one without exchanging quantum particles. Here, we explore this connection for quantum entanglement and for general quantum resource theories. For any quantum resource theory which contains resource-free pure states, we show that there does not exist a finite set of resource monotones which completely determines all state transformations. We discuss how these limitations can be surpassed, if discontinuous or infinite sets of monotones are considered, or by using quantum catalysis. We also discuss the structure of theories which are described by a single resource monotone and show equivalence with totally ordered resource theories. These are theories where a free transformation exists for any pair of quantum states. We show that totally ordered theories allow for free transformations between all pure states. For single-qubit systems, we provide a full characterization of state transformations for any totally ordered resource theory.
Resource theory is a widely-applicable framework for analyzing the physical resources required for given tasks, such as computation, communication, and energy extraction. In this paper, we propose a general scheme for studying and understanding resource theories based on resource-destroying maps, which leave resource-free states unchanged but erase the resource stored in all other states. In particular, we introduce a group of simple conditions for the class of resource-free operations, which play a key role in the general framework of resource theories. Our theory reveals fundamental connections among free states, free operations and resource measures. To illustrate the properties of resource destruction, we explicitly discuss our theory in the contexts of coherence and discord.
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., nonpassivity, 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 nonincreasing 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.
Quantum thermodynamics can be cast as a resource theory by considering free access to a heat bath, thereby viewing the Gibbs state at a fixed temperature as a free state and hence any other state as a resource. Here, we consider a multipartite scenario where several parties attempt at extracting work locally, each having access to a local heat bath (possibly with a different temperature), assisted with an energy-preserving global unitary. As a specific model, we analyze a collection of harmonic oscillators or a multimode bosonic system. Focusing on the Gaussian paradigm, we construct a reasonable resource theory of local activity for a multimode bosonic system, where we identify as free any state that is obtained from a product of thermal states (possibly at different temperatures) acted upon by any linear-optics (passive Gaussian) transformation. The associated free operations are then all linear-optics transformations supplemented with tensoring and partial tracing. We show that the local Gaussian extractable work (if each party applies a Gaussian unitary, assisted with linear optics) is zero if and only if the covariance matrix of the system is that of a free state. Further, we develop a resource theory of local Gaussian extractable work, defined as the difference between the trace and symplectic trace of the covariance matrix of the system. We prove that it is a resource monotone that cannot increase under free operations. We also provide examples illustrating the distillation of local activity and local Gaussian extractable work.
Resource theory is a widely applicable framework for analyzing the physical resources required for given tasks, such as computation, communication, and energy extraction. In this Letter, we propose a general scheme for analyzing resource theories based on resource destroying maps, which leave resource-free states unchanged but erase the resource stored in all other states. We introduce a group of general conditions that determine whether a quantum operation exhibits typical resource-free properties in relation to a given resource destroying map. Our theory reveals fundamental connections among basic elements of resource theories, in particular, free states, free operations, and resource measures. In particular, we define a class of simple resource measures that can be calculated without optimization, and that are monotone nonincreasing under operations that commute with the resource destroying map. We apply our theory to the resources of coherence and quantum correlations (e.g., discord), two prominent features of nonclassicality.