In this work, we expose the results from the literature that extract the thermodynamics from the collision model when the particles composing it are in a thermal state. This is done in connection with the second law, which is the change in the relative entropy of .
Entropy as an information measure
Relative entropy
A useful tool for deriving relationships and building intuition in information theory is relative entropy, or Kullback-Leibler divergence [39]. As you can imagine, the state that returns the greatest amount of information after a measurement is the maximally mixed state:
Subadditivity and mutual information
Considering only the action of K1 on ρ, it is observed that it does nothing: it is simply a rescaling of the identity:. the condition is the same until normalization. This interpretation of the operator sum decomposition is called the dynamics of quantum jumps, where each Kraus operator is associated with an action on the system, or jump [45].
Environment perspective
As mentioned before, the original formulation for a mapΦ is the average over the possible density matricesρn:. More will be covered on the operator-sum representation in the following sections, nevertheless we conclude this presentation with an important property of this type of map: different Kraus operators can refer to the same dynamics. Although the map is unique, the operator sum representation is not. with the postulates of quantum dynamics.
Thus, the effective dynamics of a closed system partition, which is described by the usual postulates of quantum mechanics, is a quantum operation. Both evidences taken together show that Eq. 3.18) represents a quantum operation that is equally valid as Eq. 3.2), and it is called the Stinespring dilatation.
Axiomatic definition of a quantum channel
Quantum coherence is a property that resembles entanglement in many ways, but nevertheless does not depend on any partition of the Hilbert space of the system. While such properties are common in the quantum realm, the inhabitants of the classical world do not experience these characteristics. The mechanism behind this is related to the existence of reservoirs: Hilbert spaces much larger than the system's own Hilbert space, which trigger the relaxation of the local system to a target state on a basis dictated by the interaction between the system and the reservoir, as will happen. discussed in Chapter 5.
In this chapter, Section 4.1 will take advantage of the physical appeal of entanglement resource theory to develop the basic elements of a general resource theory. A quantifier associated with this task will establish an order of the set of entangled states, where its maximum value will correspond to the maximally entangled state.
Resource theory of coherence
Maximally coherent state
Here we prove the existence of a state from which one can honestly reach any other state through incoherent operations [52]. Note that the set formed by the Kraus operators (4.5) forms a family of maps parameterized by the coefficientsci. This completes the proof as it was shown that any mixed state can be prepared through and incoherent operations.
Coherence measures
This is complemented by several measurements, especially for relative entropy (2.7) and all metrics. It is shown that, under certain conditions, the reservoir degrees of freedom are compressed into an additional structure in the reduced von Neumann equation for the system. The GKSL equation decomposes some subspaces of the original Hilbert space of the system, and its decomposition rate is related to aγk, justifying their name [46].
From these examples, one can draw some general conclusions about how the choice of jump operators affects the dynamics of the system: they represent the transitions caused by the system's environment. But from a qualitative perspective, the relevant message is that jump operators represent the transitions caused by the bath on the system.
Microscopic derivation of a master equation
The overall Hamiltonian of the system is H = HS +HE+V, where HS and HE are the corresponding free Hamiltonians for the system and the environment, and V is the interaction between them. This is not a limitation of the method, in fact it is chosen so that thermodynamics can be derived from it later. This is called the Nakajima-Zwanzig equation and is exact for the local state of the system.
The continuous calculations are beyond the scope of this discussion and can be found in Ref. However, most of the results obtained in the following sections still apply to the general map Eq.
Reservoir engineering
Energy preserving interactions and its parametrization
When the interaction is energy-conserving, we can always write it as whereLk and Ak are eigenoperators of the system and respectively Hamiltonian ancilla, with frequency of the opposite Bohr sign: ωk for Lk and −ωk for Ak. This guarantees that every quantum leaving/entering the system has the same energy as entering/leaving the cell, keeping it. Thus it is shown that in the continuum limit the collisional model thermalizes the system without any need for approximations.
Consider a string mancillaeρm, which interacts with the system during timeτ/mand Vrescaled asm/√. The first and second collisions yield, from Eq. 6.26). With two different thermal ancillae, for example, we can lead the system to a non-equilibrium steady state [32].
Thermodynamics
In the collision model setup, where ancillae are discarded after the interaction with the system, we can see two mechanisms for information loss. This form for the entropy production is valid for any collision model as general as Eq. In the last equality, we used the fact that the global evolution is unitary and thus conserves entropy.
We have successfully written the first and second laws of thermodynamics in terms of the system information alone in the collision model. The results correspond to the desired effects for a system in contact with a thermal reservoir in the weak coupling limit: they are the same not only from a dynamic point of view, as shown in the last section, but also from a thermodynamic one .
Weakly coherent collisional models
Hamiltonian engineering
In the continuum limit they will decay, so no change is expected in the dissipative part of the dynamics. For concreteness it is interesting to cast G in terms of eigenoperators, as done with the dissipator in Eq. where hBjiχ = Tr{Bj χ} is the mean of the eigenoperators of the ancilla Hamiltonian with. thatG is hollow in the energy base of the system. The choice of the elements can be made through the interaction, and their relative intensities can be chosen independently from χ: with weak coherences, the collision model can be used to perform Hamiltonian engineering.
The dissipator is the same, however there will be an extra termG. Thus the entire master equation will read. 6.58). In the derivation it was clear that the additive property arises from the fact that each junction is of higher order inτ and thus vanishes in the continuum limit.
Thermodynamics
For detailed calculation, refer to Appendix A. The flux of energy entering the system now has two different contributions: one through a dissipator and another through a Hamiltonian. We argue here that since the nature of the former is dissipative, it is an incoherent source of energy like heat. But with the help of G a transformation process takes place and the bath coherently injects a fraction of its lost energy into the system.
Summing the two contributions and again taking the continuity limitτ→ 0, we finally write note here that both incoherent heat and coherent work play the usual roles within the entropy production functional form. Thus, the amount of work done on the system is limited by the free energy difference experienced by the system. 6.59) signals a straightforward extension of the first and second laws to multiple reservoirs.
Resource interconversion
In the first law, a work-like term appears, even though the evolution of the global system is time-independent. To address such bathing, one can set the arrival time of the ancillae as a Poisson process [21]. In this appendix we provide the updated status of the ancillae after they have interacted with the system.
Using the general map (A.1), we can calculate the energy changes of the system and auxiliary elements defined as ∆HS =tr. Note that the structure of these results is completely independent of any particular choice for the ancillae states.
Von Neumann entropy
To calculate the entropy production defined in equation 6.35) of the main text, several entropy quantities must be calculated depending on these states. Since we are interested in the limit τ → 0, we can calculate these quantities using perturbation theory, which becomes exact in the limit τ → 0. We begin by stating some general results on the perturbative expansions of the von Neumann entropy, the relative coherence entropy, and the quantum Kullback-Leibler divergence (relative entropy). thepi are not degenerate, we can then write, up to order 2, Pi = pi+σii+2X. B.2), by expanding PilogPi to second order and using the fact that trσ=0, we find that.
Relative entropy of coherence
Quantum relative entropy
The populationsσii contribute both at order og2, whereas the coherences (off-diagonals) only start to contribute at order 2. whereσandµ are arbitrary, but both depend onρ0 to order0. B.8). To calculate the last term, we need not only the perturbation theory for the eigenvalues of ρ [Eq. B.3)], but also for its eigenvectors. Pi|˜iih˜i|let us write. B.10) In addition to writing logPi as a power series, we will also have to expand h˜i|ρ0|˜ii. Inserting this result into Eq. B.10) and expanding all terms in then finally leads to tr(ρ0logρ)= X.
Moreover, the result depends on both the populations and the coherences, and both in the same order2. Because system and environment always start uncorrelated and because global dynamics are unitary, the mutual information developed in the map (A.1) can be written as.
The term proportional to logZA disappears, for it is proportional to tr(ρ0A−ρA), and the second is therefore nothing but the total energy change of the ancilla in Eq.
But this, in turn, is minus the change in energy in the system. difference between the relative entropies of coherence of ρ0A and ρA respectively [Eq. Translation of Ludwig Boltzmann's article on the relationship between the second fundamental theorem of mechanical heat theory and probability calculations regarding the conditions of thermal equilibrium sitzungberichte der kaiserlichen akademie der wistschaften. A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains.
The Theory of Relaxation Processes* *this work was started while the author was at Harvard University, and was then supported in part by joint service contract n5ori-76, project order i.Collision Model-Based Approach to Non-Markovian Quantum Dynamics. Physics review A - Atomic, molecular and optical physics.
