Quantum thermodynamics

Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent theories address the physical phenomena of light and matter.
In 1905, Albert Einstein argued that the requirement of consistency between thermodynamics and electromagnetism leads to the conclusion that light is quantized obtaining the relation formula_1. This paper is the dawn of quantum theory. In a few decades 
quantum theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. 
It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. 
In addition, there is a quest for the theory to be relevant for a single individual quantum system.
Dynamical view.
There is an intimate connection of quantum thermodynamics with the theory of open quantum systems.
Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics.
The main assumption is that the entire world is a large closed system, and therefore, time evolution
is governed by a unitary transformation generated by a global Hamiltonian. For the combined system
bath scenario, the global Hamiltonian can be decomposed into:
where formula_3 is the system Hamiltonian, formula_4 is the bath Hamiltonian and formula_5 is the system-bath interaction.
The state of the system is obtained from a partial trace over the combined system and bath:
formula_6.
Reduced dynamics is an equivalent description of the system dynamics utilizing only system operators.
Assuming Markov property for the dynamics the basic equation of motion for an open quantum system is the Lindblad equation (GKLS):
formula_8 is a (Hermitian) Hamiltonian part and formula_9:
is the dissipative part describing implicitly through system operators formula_11 the influence of the bath on the system.
The Markov property imposes
that the system and bath are uncorrelated at all times formula_12.
The L-GKS equation is unidirectional and leads any initial state formula_13 to a steady state solution which is an invariant of the equation of motion formula_14.
The Heisenberg picture supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator, formula_15, has the form:
where the possibility that the operator, formula_15 is explicitly time-dependent, is included.
Emergence of time derivative of first law of thermodynamics.
When formula_18 the first law of thermodynamics emerges:
where power is interpreted as formula_20
and the heat current formula_21.
Additional conditions have to be imposed on the dissipator formula_22 to be consistent with thermodynamics.
First the invariant formula_23 should become an equilibrium Gibbs state.
This implies that the dissipator formula_24 should commute with the unitary part generated by formula_25. 
In addition an equilibrium state is stationary and stable. This assumption is used to derive
the Kubo-Martin-Schwinger stability criterion for thermal equilibrium i.e. KMS state.
A unique and consistent approach is obtained by deriving the generator, formula_9, in the weak system bath
coupling limit.
In this limit, the interaction energy can be neglected. This approach represents a thermodynamic idealization: it allows energy transfer, while keeping a tensor product separation
between the system and bath, i.e., a quantum version of an isothermal partition.
Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that in
phenomenological treatments, one cannot combine arbitrary system Hamiltonians, formula_3, with a given L-GKS generator. This observation is particularly important in the context of quantum thermodynamics,
where it is tempting to study Markovian dynamics with an arbitrary control Hamiltonian. Erroneous derivations of the quantum master equation can easily lead to a violation of the laws of thermodynamics.
An external perturbation modifying the Hamiltonian of the system will also modify the heat flow. As a result, the L-GKS generator has to be renormalized. For a slow change, one can adopt the adiabatic approach and use the instantaneous system’s Hamiltonian to derive formula_9. An important class of problems in quantum thermodynamics is periodically driven systems. Periodic quantum heat engines and power-driven refrigerators fall into this class.
A reexamination of the time-dependent heat current expression using quantum transport techniques has been proposed.
A derivation of consistent dynamics beyond the weak coupling limit has been suggested.
Emergence of the second law.
The second law of thermodynamics is a statement on the irreversibility of dynamics or, the breakup of time reversal symmetry (T-symmetry). This should be consistent with the empirical direct definition: heat will flow spontaneously from a hot source to a cold sink.
From a static viewpoint, for a closed quantum system, the 2nd law of thermodynamics is a consequence of the unitary evolution. In this approach, one accounts for the entropy change before and after a change in the entire system. A dynamical viewpoint is based on local accounting for the entropy changes
in the subsystems and the entropy generated in the baths.
Entropy.
In thermodynamics, entropy is related to a concrete process. In quantum mechanics, this translates to the ability to measure and manipulate the system based on the information gathered by measurement. An example is the case of Maxwell’s demon, which has been resolved
by Leó Szilárd.
The entropy of an observable is associated with the complete projective measurement of an observable,
formula_29, where the operator formula_30 has a spectral decomposition: formula_31
where formula_32 is the projection operators of the eigenvalue formula_33.
The probability of outcome j is formula_34
The entropy associated with the observable formula_35 is the Shannon entropy with respect to the possible outcomes:
The most significant observable in thermodynamics is the energy represented by the Hamiltonian operator formula_37, and its associated energy entropy, formula_38.
John von Neumann suggested to single out the most informative observable to characterize the entropy of the system. This invariant is obtained by minimizing the entropy with respect to all possible observables. The most informative observable operator commutes with the state of the system. The
entropy of this observable is termed the Von Neumann entropy and is equal to:
As a consequence, formula_40 for all observables. At thermal equilibrium the energy entropy is equal to the von Neumann entropy: formula_41.
formula_42 is invariant to a unitary transformation changing the state. The Von Neumann entropy formula_43 is additive only for a system state that is composed of a tensor product
of its subsystems:
Clausius version of the II-law.
No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.
This statement for N-coupled heat baths in steady state becomes:
A dynamical version of the II-law can be proven, based on Spohn’s inequality
which is valid for any L-GKS generator, with a stationary state, formula_47.
Consistency with thermodynamics can be employed to verify quantum dynamical models of transport. For example, local models for networks where local L-GKS equations are connected through weak links have been shown to violate the second law of thermodynamics.
Quantum and thermodynamic adiabatic conditions and quantum friction.
Thermodynamic adiabatic processes have no entropy change. Typically, an external control modifies
the state. A quantum version of an adiabatic process can be modeled by an externally controlled time dependent
Hamiltonian formula_48. If the system is isolated, the dynamics are unitary, and therefore, formula_42 is
a constant. A quantum adiabatic process is defined by the energy entropy formula_50 being constant.
The quantum adiabatic
condition is therefore equivalent to no net change in the population of the instantaneous energy levels.
This implies that the Hamiltonian should commute with itself at different times: formula_51.
When the adiabatic conditions are not fulfilled, additional work is required to reach the final control
value. For an isolated system, this work is recoverable, since the dynamics is unitary and can be reversed. In this case, quantum friction can be suppressed using shortcuts to adiabaticity as demonstrated in the laboratory using a unitary Fermi gas in a time-dependent trap.
The coherence stored in the off-diagonal elements of the density operator carry the required information
to recover the extra energy cost and reverse the dynamics. Typically, this energy is not recoverable, due
to interaction with a bath that causes energy dephasing. The bath, in this case, acts like a measuring
apparatus of energy. This lost energy is the quantum version of friction.
Emergence of the dynamical version of the third law of thermodynamics.
There are seemingly two independent formulations of the third law of thermodynamics both originally were stated by Walther Nernst. The first formulation is known as the Nernst heat theorem, and can be phrased as:
The second formulation is dynamical, known as the "unattainability principle"
At steady state the second law of thermodynamics implies that the total entropy production is non-negative.
When the cold bath approaches the absolute zero temperature,
it is necessary to eliminate the entropy production divergence at the cold side
when formula_52, therefore
For formula_54 the fulfillment of the second law depends on the entropy production of the other baths, 
which should compensate for the negative entropy production of the cold bath. 
The first formulation of the third law modifies this restriction. 
Instead of formula_55 the third law imposes formula_56, 
guaranteeing that at absolute zero the entropy production at the cold bath is zero: formula_57. 
This requirement leads to the scaling condition of the heat current formula_58.
The second formulation, known as the unattainability principle can be rephrased as;
The dynamics of the cooling process is governed by the equation
where formula_60 is the heat capacity of the bath. Taking formula_58 and formula_62 with formula_63, we can quantify this formulation by evaluating the characteristic exponent formula_64 of the cooling process,
This equation introduces the relation between the characteristic exponents formula_64 and formula_67. When formula_68 then the bath is cooled to zero temperature in a finite time, which implies a valuation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem.
Typicality as a source of emergence of thermodynamic phenomena.
The basic idea of quantum typicality is that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. 
This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average.
Quantum ergodic theorem originated by John von Neumann is a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of termed normal typicality, i.e. the statement that, for typical large systems, every initial wave function formula_69 from an energy shell is ‘normal’: it evolves in such a way that formula_70 for most t, is macroscopically equivalent to the micro-canonical density matrix.
Resource theory.
The second law of thermodynamics can be interpreted as quantifying state transformations which are statistically unlikely so that they become effectively forbidden. The second law typically applies to systems composed of many particles interacting; Quantum thermodynamics resource theory is a formulation of thermodynamics in the regime where it can be applied to a small number of particles interacting with a heat bath. For processes which are cyclic or very close to cyclic, the second law for microscopic systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. These second laws are not only relevant for small systems, but also apply to individual macroscopic systems interacting via long-range interactions, which only satisfy the ordinary second law on average. By making precise the definition of thermal operations, the laws of thermodynamics take on a form with the first law defining the class of thermal operations, the zeroth law emerging as a unique condition ensuring the theory is nontrivial, and the remaining laws being a monotonicity property of generalised free energies.
Engineered reservoirs.
Nanoscale allows for the preparation of quantum systems in physical states without classical analogs. There, complex out-of-equilibrium scenarios may be produced by the initial preparation of either the working substance or the reservoirs of quantum particles, the latter dubbed as "engineered reservoirs". There are different forms of engineered reservoirs. Some of them involve subtle quantum coherence or correlation effects, while others rely solely on nonthermal classical probability distribution functions.
The latter are dubbed nonequilibrium incoherent reservoirs. Interesting phenomena may emerge from the use of engineered reservoirs such as efficiencies greater than the Otto limit,
violations of Clausius inequalities, or simultaneous extraction of heat and work from the reservoirs.
In general, the thermodynamics and efficiency of such systems require particular analysis. However, for the special case of NIR, the efficiency of steady-state quantum machines connected to them can be treated within a unified picture.
Further reading.
Deffner, Sebastian and Campbell, Steve. "Quantum Thermodynamics: An introduction to the thermodynamics of quantum information", (Morgan & Claypool Publishers, 2019).
F. Binder, L. A. Correa, C. Gogolin, J. Anders, G. Adesso (eds.) "Thermodynamics in the Quantum Regime. Fundamental Aspects and New Directions." (Springer 2018)
Jochen Gemmer, M. Michel, and Günter Mahler. "Quantum thermodynamics. Emergence of thermodynamic behavior within composite quantum systems. 2." (2009).
Petruccione, Francesco, and Heinz-Peter Breuer. The theory of open quantum systems. Oxford university press, 2002.