The noise

Finally, let us conclude this chapter by discussing the role of noise in our system. In our dynamics the system-environment coupling is modeled through the action of three different general kinds of noise: amplitude damping, phase damping and “excitation pumping” -like noise. When transferring excitations only (SES), the for- mer two could represent an advantage over a pure unitary dynamics. In fact, the amplitude damping is needed in order to introduce an asymmetry between the two directions of propagation, acting as a driving force to draw the excitation towards the end of the chain, whereas the introduction of some dephasing may smooth the localization effect previously discussed (Fig.(4.2).a,b) and/or contrast negative co- herent effects on absorption (see Sec.(3.4.1)), allowing for a better transfer [77]. For quantum state transfer though, coherences play a fundamental role and, as expected, all the sources of noise we considered are always (apart from the very specific case of Fig.(4.3).d) detrimental. More specifically, amplitude damping and dephasing result in a damping of the coherent oscillations of hF (t)i (see Fig.(4.3).a and (4.4)), whereas the restrictions imposed by the causality condition on {wµ} and {zµ} (last

two lines of Eqs.(4.16)) make a quantum state transfer dynamics with {wµ} 6= 0

strongly unfavorable. Assuming either {zµ} ∈ R or {wµ} ∈ R (the case where both

belong to C only complicates the picture without changing the results), we have that the only solutions to Eq.(4.16) where {wµ} 6= 0 are such that zµ = z, ∀µ, and

we find that this condition, whatever the other parameters of the dynamics may be, always translates into very poor quantum state transfer performance.

For the sake of the argument, one can think of relaxing the causality requirement. In that case, as the third line in Eqs.(4.16) is just a matter of normalization, one can build the vectors by satisfying P

µK

(l) µ

†

Eqs.(4.5,4.6,4.7), reads:    q 1 −ξ₂√1 − ηw0₀+ q ξ 2 √ 1 − ηw0₁+√ηw₂1= 0 q 1 −ξ₂w1₀− q ξ 2w11 = 0 . (4.22)

where we introduced a notation such that w_µa stands for the component a, a = (0, 1) of each local vector pertaining to the Kraus operator µ. Of course, a whole set of solutions for Eqs.(4.22) is possible. In order not to give too big a bias among the different ˜Kµ driving the dynamics, it is reasonable to privilege the most “bal-

anced”solutions, the ones where |w_µa− wb

ν| ≤ d, ∀µ, ν, a, b, with the smallest possible

d. To this aim, it is thus necessary to avoid solutions proportional to (ξ/2)−1/2, (1 − η)−1/2 or η−1/2, where for some regimes of the dynamics some components w_µa could be arbitrarily big. One such solution reads:

         w₀0 = w₀1 = −√T 3N q ξ/2 1−ξ/2 w₁0 = √T 3N(1 − √ η); w₁1= −√T 3N w₂0 = w₂1 = √T 3Npξ/2 √ 1 − η (4.23)

Above, we assumed {wµ} ∈ R and we arbitrarily chose a normalization proportional

to T /√N , where the free parameter T tunes the strength of the noise. Notice that Eqs.(4.23) assure that {w_µa} ∈ h±T /√3Ni. As a result of relaxing causality, the components of {wµ} and {zµ} are now “decoupled” and a tuple {zµ} = (c, 0, 0) –

which allows for the best quantum state transfer performance – can be selected. However, as it can be appreciated in Fig.(4.4), the effect of this kind of noise is very similar to that of the dephasing.

0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 <F> time steps T>0;ξ=0 T=0;ξ=0.025 T>0;ξ=0.025

Figure 4.4: Noisy dynamics for a ring of N = 8 qubits and p = q = 1. In the depicted case T > 0 corresponds to the value T = 0.1 (see Eq.(4.23)). However, the magnitude chosen for the two parameters ξ and T does not acquire any particular meaning here, as the figure is intended to be only a representative example of the damping in hF (t)i resulting from introducing noise (cfr. Fig.(4.3).d).

Part III

Chapter 5

Thermalization of finite

quantum systems

In this brief chapter we introduce the concept of typicality in quantum thermo- dynamics. The attention will be focused on the issue of thermalization of quantum states. In Sec.(5.2) numerical results concerning thermalization of locally-interacting closed quantum systems are shown.

5.1

Thermalization and the typicality approach, in a

nutshell

It is a known, well-established empirical fact that a physical system weakly coupled to a large bath will eventually evolve to a state which is independent from all initial state, bath and bath-state interaction conditions, but only characterized by a few macroscopic parameters of the bath, like its temperature. This thermalization phenomenon holds for both classical and quantum systems. Furthermore, the final “local” (when system and environment are considered as a whole) state will no longer exhibit any macroscopically visible evolution, despite the underlying dynamics being unitary, hence not leading to any fixed points. This ubiquitous, generic behavior thus seems to be paradoxical, to some extent, when addressed within the framework of quantum theory: In a sense, asking whether closed quantum systems thermalize is equivalent to asking whether quantum mechanics alone – which is governed by the time-reversal symmetric Schr¨odinger equation – is capable of describing irreversable dynamics.

The typicality approach to the problem [46, 47, 45, 49] aims at giving an expla- nation relying on the concept that an overwhelmingly large region of the portion of the Hilbert space accessible1 to the evolving system is almost entirely filled with typical states, i.e. states that possess similar properties, like expectation values of a set of relevant observables, the probability to measure certain values for some functions, or reduced density matrices. Here we choose to privilege an approach fo- cused on estimating the form of the latter, mostly because reduced density matrices amount to the set of all observables that can be locally defined for the system. To demonstrate thermalization, one then wishes to show that for the vast majority of pure states within a certain energy interval, the reduced density matrices pertaining to a small subsystem are the same as the ones that would be obtained by tracing out from the microcanonical ensemble corresponding to the same energy interval [45, 46]. Let us stress that these arguments primarily address relative frequencies, while delegating the dynamical mechanism leading to thermalization to the reason- able expectation that, if what discussed above is verified, any dynamics will bring and leave the system within the region filled by such typical states.