REGLAMENTO DE QUEJAS Y DENUNCIAS DEL INSTITUTO FEDERAL ELECTORAL

In its broader meaning, the term decoherence2 indicates the effectively irre- versible disappearance of quantum coherence as a dynamical consequence of the interaction between a quantum system and the environment. The intuitive pic- ture of the process is that of an environment continuously “monitoring” the system, acquiring information about its state and consequently diluting this information content into a larger Hilbert space in the form of quantum correlations, i.e. entan- glement. Indeed, open quantum system dynamics in its general formulation can be thought of as a process in which the environment performs indirect measurements on the system. In such indirect measurements a non-destructive interaction between a probe-like environment E and the system S is followed by a projective measurement on E. Consider an initial joint state ρSE(0) = ρS(0) ⊗ |0ih0|E3 evolving unitarily

under the action of U (t). If {|ji} and {j} are the eigenvectors and associated eigen- values of the environment, the density matrix of S at time t conditioned on the outcome j of a projective measurement on E is:

ρ(j)_S (t) = TrE{[1 ⊗ |jihj|E]U (t)(ρS(0) ⊗ |0ih0|E)U †_{(t)[1 ⊗ |jihj|} E]} Prob(j|ρS(t)) = hjE|U (t)|0EiρS(0)h0E|U †_(t)|j Ei Prob(j|ρS(t)) . (2.17)

In almost all practical situations the degrees of freedom of the environment are out of the experimental control, namely, if the environment is big, the actual outcome of the measurement is out of reach and the density operator of the system must be described by the sum over all possible conditional states ρ(j)_S (t) weighted by their

2

The seminal papers on decoherence are [71, 72, 73]; For comprehensive reviews on the topic, see [74] and [75].

3

Let us remark that the assumption of an initial absence of correlations between system and environment is shared by both the quantum operations formalism and the master equation approach to open quantum systems dynamics. We also note that there is no loss of generality in considering an initial pure state for the environment in that by expanding the Hilbert space HEan initial mixed

probabilities Prob(j|ρS(t)). Upon defining the measurement operators Ej = hjE|U (t)|0Ei , with X i E_i†Ei=1 , (2.18) we thus have ρS(t) = X j Prob(j|ρS(t))ρ(j)_S (t) = X j EjρS(0)Ej†. (2.19)

The formal equivalence between Eq.(2.19) and the operator sum formalism (cf. Sec.(2.1)) implies that the effect of a general environmental interaction on the state of the system, when the environment is not read out, can be understood as an environmental monitoring on the system, resulting in an increase of entanglement between the system and the degrees of freedom of the environment. As discussed in Sec.(1.7), the more a bipartite quantum system ρSE gets entangled, the more

the information about ρS is delocalized into quantum correlation: Phase relations

become locally (i.e. with respect to S) inaccessible and the state of S becomes more mixed. In turn, purely quantum phenomena, like superpositions and interference, are suppressed and eventually become unobservable at the level of the system. This suggests that the decoherence mechanism provides an explanation to the problem of the quantum-to-classical transition, namely the emergence of classicality from an underlying world governed by the laws of quantum mechanics. Indeed, decoherence theory gives prescriptions to single out some preferred states, called pointer states, which persist in spite of environmental monitoring. These states, which lack coher- ence and therefore do not exhibit quantum behaviors, arise through a process called einselection [72, 73, 74] which effectively rules out nonclassical superposition states.

In light of the considerations above, an illustrative example of a stylized mech- anism accounting for the transition from the quantum to the classical world is rep- resented by the phase damping channel. One may interpret this channel as de- scribing the scattering interaction between a heavy “classical” particle and a bath of photons. The particle position is initially in a superposition of two eigenstates |ψi = _√1

2(|xi+|−xi). We can introduce a scattering rate τ such that (cf. Sec.(2.1.2))

factor is (1 − p)n = (1 − τ δt)t/δt which, in the limit δt → 0 takes an exponential form e−τ. Hence, after a time t  τ−1 coherence in the position basis is almost totally suppressed, and quantum effects like interference are no longer observable. We also remark that, whenever dealing with coherent superpositions of macroscopi- cally distinguishable states of a “heavy” object, the decoherence time scale is much quicker than that of dissipation [76]. Physically, following this interpretation of the phase damping channel, the position-eigenstate basis is selected because the particle- photon interactions are localized in space, and photons impinging on the particle get scattered into different (not necessarily mutually orthogonal) states according to the different distinguishable position of the particle (the “monitoring” process pre- viously described). In general, the spatial locality of the interaction of the system with its environment is the cause of the rise of a preferred basis for decoherence and the example just presented is considered to be representative of the transition between a quantum and a classical behavior in many physical situations.

Part II

Quantum cellular automata for

transport processes

Chapter 3

Excitation transfer through

noisy QCA

In this chapter, based on [77], we introduce a model of energy transfer via a noisy quantum cellular automaton construction on a one-dimensional qubit lattice. The model represents the first strictly local discrete-time dynamics approach to the problem, since prior to this all others were based on a master equation.

We begin in the first section by presenting a brief but self-contained introduction to quantum cellular automata, providing the fundamental notions underlying these systems and highlighting the interplay between their axiomatic definition and their privileged constructive representation. In the second section we will consider the problem of constructing a class of one-qubit CP maps that, in a certain limit, repro- duce all classical Markov transition matrices on dichotomic probability distributions. Then, in the third section, we will embed this construction into a quantum cellu- lar automaton structure and in the final section we will present a numerical study of the resulting dynamics applied to the problem of excitation transfer, compar- ing the performance of classical and quantum dynamics with equal local transition probabilities.