The second law of thermodynamics states that work cannot be extracted from thermal equilibrium, whose quantum formulation is known as complete passivity; a state is called completely passive if work cannot be extracted from any number of copies of the state by any unitary operations.It has been established that a quantum state is completely passive if and only if it is a Gibbs ensemble.In physically plausible setups, however, the class of possible operations is often restricted by fundamental constraints such as symmetries imposed on the system.In the present work, we investigate the concept of complete passivity under symmetry constraints.Specifically, we prove that a quantum state is completely passive under a symmetry constraint described by a connected compact Lie group, if and only if it is a generalized Gibbs ensemble including conserved charges associated with the symmetry.Remarkably, our result applies to noncommutative symmetry such as SU(2) symmetry, suggesting an unconventional extension of the notion of generalized Gibbs ensemble.Furthermore, we consider the setup where a quantum work storage is explicitly included, and prove that the characterization of complete passivity remains unchanged.Our result extends the notion of thermal equilibrium to systems protected by symmetries, and would lead to flexible design principles of quantum heat engines and batteries.Moreover, our approach serves as a foundation of the resource theory of thermodynamics in the presence of symmetries.
The second law of thermodynamics states that work cannot be extracted from thermal equilibrium, whose quantum formulation is known as complete passivity; a state is called completely passive if work cannot be extracted from any number of copies of the state by any unitary operations.It has been established that a quantum state is completely passive if and only if it is a Gibbs ensemble.In physically plausible setups, however, the class of possible operations is often restricted by fundamental constraints such as symmetries imposed on the system.In the present work, we investigate the concept of complete passivity under symmetry constraints.Specifically, we prove that a quantum state is completely passive under a symmetry constraint described by a connected compact Lie group, if and only if it is a generalized Gibbs ensemble including conserved charges associated with the symmetry.Remarkably, our result applies to noncommutative symmetry such as SU(2) symmetry, suggesting an unconventional extension of the notion of generalized Gibbs ensemble.Furthermore, we consider the setup where a quantum work storage is explicitly included, and prove that the characterization of complete passivity remains unchanged.Our result extends the notion of thermal equilibrium to systems protected by symmetries, and would lead to flexible design principles of quantum heat engines and batteries.Moreover, our approach serves as a foundation of the resource theory of thermodynamics in the presence of symmetries.
The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.
The central purpose of this thesis is to study the familiar processes of quantum information theory from a thermodynamic point of view. Chapter 1 introduces equilibrium thermodynamics and its laws. The laws are then motivated in a statistical mechanics approach which explicitly employs the language of quantum information theory. Chapter 2 is dedicated to the concept of work extraction, in particular in non- equilibrium scenarios. In this context a generalised notion of equilibrium called passivity is introduced. The chapter’s final section gives an introduction and an overview of various approaches to work extraction ranging from classical to explicitly quantum scenarios. Chapter 3 considers the question of powerful unitary operations. Focussing on the case of qubits (or rather wits, short for ‘work bits’) the chapter’s first result is an explicit protocol for maximally powerful driving for any given initial state. Building onto this, the chapter’s second main result is a proof that the evolution time (and hence the power of the corresponding work deposition process) can be decreased by a factor of 1/N for an array of N wits if global driving is permitted (rather than individual driving in parallel). Lastly, the thermodynamics of completely-positive and trace-preserving (CPTP) maps is considered in Chapter 4. It is shown that the energy change during such a process can be split into three parts: a work-like equilibrium contribution, a heat-like equilibrium contribution, and a third genuinely non-equilibrium energy difference. All three terms obtain their meaning when considering how much work can be extracted from the input and output state of the CPTP process, respectively. Furthermore, the identification of work and heat in this manner complies with a recently published version of the second law, which applies to the same class of processes. This is demonstrated by considering unital and thermal maps.
The grand canonical ensemble lies at the core of quantum and classical statistical mechanics. A small system thermalizes to this ensemble while exchanging heat and particles with a bath. A quantum system may exchange quantities represented by operators that fail to commute. Whether such a system thermalizes and what form the thermal state has are questions about truly quantum thermodynamics. Here we investigate this thermal state from three perspectives. First, we introduce an approximate microcanonical ensemble. If this ensemble characterizes the system-and-bath composite, tracing out the bath yields the system's thermal state. This state is expected to be the equilibrium point, we argue, of typical dynamics. Finally, we define a resource-theory model for thermodynamic exchanges of noncommuting observables. Complete passivity-the inability to extract work from equilibrium states-implies the thermal state's form, too. Our work opens new avenues into equilibrium in the presence of quantum noncommutation.
Thermodynamics and information have intricate interrelations. Often thermodynamics is considered to be the logical premise to justify that information is physical - through Landauer's principle -, thereby also linking information and thermodynamics. This approach towards information has been instrumental to understand thermodynamics of logical and physical processes, both in the classical and quantum domain. In the present work, we formulate thermodynamics as an exclusive consequence of information conservation. The framework can be applied to the most general situations, beyond the traditional assumptions in thermodynamics: we allow systems and thermal baths to be quantum, of arbitrary sizes and even possessing inter-system correlations.Here, systems and baths are not treated differently, rather both are considered on an equal footing. This leads us to introduce a ''temperature''-independent formulation of thermodynamics. We rely on the fact that, for a fixed amount of information, measured by the von Neumann entropy, any system can be transformed to a state with the same entropy that possesses minimal energy. This state, known as a completely passive state, acquires Boltzmann-Gibbs canonical form with an intrinsic temperature. We introduce the notions of bound and free energy and use them to quantify heat and work, respectively. Guided by the principle of information conservation, we develop universal notions of equilibrium, heat and work, Landauer's principle and universal fundamental laws of thermodynamics. We demonstrate that the maximum efficiency of a quantum engine with a finite bath is in general lower than that of an ideal Carnot engine. We introduce a resource theoretic framework for our intrinsic temperature based thermodynamics, within which we address the problem of work extraction and state transformations. Finally, the framework is extended to multiple conserved quantities.
Thermodynamics and information have intricate interrelations. Often thermodynamics is considered to be the logical premise to justify that information is physical - through Landauer's principle -, thereby also linking information and thermodynamics. This approach towards information has been instrumental to understand thermodynamics of logical and physical processes, both in the classical and quantum domain. In the present work, we formulate thermodynamics as an exclusive consequence of information conservation. The framework can be applied to the most general situations, beyond the traditional assumptions in thermodynamics: we allow systems and thermal baths to be quantum, of arbitrary sizes and even possessing inter-system correlations.Here, systems and baths are not treated differently, rather both are considered on an equal footing. This leads us to introduce a ''temperature''-independent formulation of thermodynamics. We rely on the fact that, for a fixed amount of information, measured by the von Neumann entropy, any system can be transformed to a state with the same entropy that possesses minimal energy. This state, known as a completely passive state, acquires Boltzmann-Gibbs canonical form with an intrinsic temperature . We introduce the notions of bound and free energy and use them to quantify heat and work, respectively. Guided by the principle of information conservation, we develop universal notions of equilibrium, heat and work, Landauer's principle and universal fundamental laws of thermodynamics. We demonstrate that the maximum efficiency of a quantum engine with a finite bath is in general lower than that of an ideal Carnot engine. We introduce a resource theoretic framework for our intrinsic temperature based thermodynamics, within which we address the problem of work extraction and state transformations. Finally, the framework is extended to multiple conserved quantities.
The minimal-coupling quantum heat engine is a thermal machine consisting of an explicit energy storage system, heat baths, and a working body, which alternatively couples to subsystems through discrete strokes --- energy-conserving two-body quantum operations. Within this paradigm, we present a general framework of quantum thermodynamics, where a work extraction process is fundamentally limited by a flow of non-passive energy (ergotropy), while energy dissipation is expressed through a flow of passive energy. It turns out that small dimensionality of the working body and a restriction only to two-body operations make the engine fundamentally irreversible. Our main result is finding the optimal efficiency and work production per cycle within the whole class of irreversible minimal-coupling engines composed of three strokes and with the two-level working body, where we take into account all possible quantum correlations between the working body and the battery. One of the key new tools is the introduced ``control-marginal state" --- one which acts only on a working body Hilbert space, but encapsulates all features regarding work extraction of the total working body-battery system. In addition, we propose a generalization of the many-stroke engine, and we analyze efficiency vs extracted work trade-offs, as well as work fluctuations after many cycles of the running of the engine.