DURACIÓN DE CADA UNA LA ACTIVIDADES

In contrast with the classical evolution, the unitary evolution of quantum mechanics can be non-trivially interrupted by measurements that reduce the state. We follow here the theory of von Neumann, i.e., convential quantum mechanics, that adds a dynamic interpretation to the projections, see equation (1.27). If the measurement is performed at a time τ when the system was initialized in stateρ, then the reduction takes place with probability (1.26):

which can be generalized to a sequence of measurements. We extend here the results of the previous section to include interactions with a measurement apparatus.

3.3.1

Path-space measure

We consider a sequence of times 0, τ,2τ, . . . , nτ =t (evenly spaced for con- venience). A macroscopic trajectoryω assigns to each of these times a value for the macroscopic observables: ω = (ω0, . . . , ωn) with each ωi being equal

to some projectionPα on the linear subspaceHα ofH=⊕αHα. We consider

the operator

Gω≡ωnUτωn−1. . . ω1Uτω0

and, for a given density matrixρon_H, the matrix

Dρ(ω, ω′)≡Tr[Gωρ G†ω′]. (3.7) This resembles the correlation matrix that is used in the definition of quantum dynamical entropy, see [3, p.189], but again, here our partitioning corresponds to macrostates. It is easy to verify that_Dρis a density matrix:

• Positivity:

The positivity ofρis preserved by the addition of projectors and unitary operators under the trace.

• Normalisation: Tr[_Dρ] = X ω Dρ(ω, ω) = X ω0,...,ωn Tr[ωnUτ. . . ω1Uτω0ρ ω0Uτ†ω1. . . Uτ†ωn] = Tr[ρ] = 1.

We call it the path-space density matrix. It has appeared before in a similar form as the correlation matrix [3, p.189].

Dρ depends on the initial density matrix ρ and it also gives the probability

to find the system after time t = nτ of (measurement-)free evolution in the macrostateα: X ω,ω′ ωn=ω′ n=Pα Dρ(ω, ω′) = Tr[Pαρt], (3.8) whereρt≡UtρUt†.

More importantly, the probability to measure the trajectoryω, i.e., to see the system at the initial time in the macrostate represented by ω0, at time τ in

macrostateω1, and so on until timenτ =t, is given by the diagonal element

Its marginal at timet=nτ, ˆ µt(α)≡ X ω ωn=Pα Probρ[ω], (3.10)

is the probability of a macrostateαat timet after the unitary evolution was interrupted bynmeasurements. We recover equation (3.2) in the case n= 1.

3.3.2

Time-reversal

On trajectories, the time-reversal transformation Θ is (Θω)m≡πωn−mπ, m= 0,1, . . . , n.

Since the microscopic dynamics is time-reversal invariant, we immediately de- duce that

GΘω=πG†ωπ

and it is easy to verify that forρ=1/din equation (3.7)

D(Θω′,Θω) =D(ω, ω′) (3.11) In particular, for the diagonal elements (3.9) withρ=1/d, we find Prob[ω] =

Prob[Θω]. This identity is the generalization of (3.4) and it is the expression of (quantum) detailed balance. Furthermore, it shows that the measurements do not introduce a time-asymmetric element, see also [2]. We will go into more detail in section 4.4.

Yet, in general, the system could start in a nonequilibrium state and evolve towards equilibrium. This involves a change of entropy, that is its total entropy production for closed and thermally isolated systems. The time-reversal of the density matrixDρ, writtenDρΘ, is defined as

DρΘ(ω, ω′)≡ Dρ(Θω′,Θω), (3.12)

so that in the case (3.11), _DΘ =_D. This last equality is broken for a general

Dρ and it seems natural to estimate this breaking via the relative entropy

S(Dρ1|Dρ2Θ)≡Tr[Dρ1(lnDρ1−lnDρ2Θ)] (3.13)

In view of the classical results of Maes and Netoˇcn´y [77, 80], it is to be ex- pected that this relative entropy is related to the entropy production for the appropriate choices ofρ1 (as the initial state) and ofρ2 (as the time-reversal

of the final state).

3.3.3

Entropy production

We start again from ˆµ as the initial probability distribution, thus ρ(ˆµ) is the initial density matrix. The final probability distribution ˆµt is defined in

equation (3.10). The main result is now readily obtained starting with the analogue of equation (3.3): the logarithmic ratio of probabilities (3.9) of a macroscopic trajectory, one started from ˆµand the other started from ˆµtπ, is

abbreviated as

Rµˆ(ω)≡ln

Probµˆ[ω]

Probµˆtπ[Θω]

(3.14) This object will not always be well-defined for allω. We can suppose however that the ˆµ(α)₆= 0₆= ˆµt(α). A little calculation, very similar to section 3.2.3,

yields

Rµˆ(ω) = lndωn−lndω0−ln ˆµt(ωn) + ln ˆµ(ω0) (3.15)

In the notation of equation (2.20), the first difference in the right-hand side equals the change of the Boltzmann entropySB(αt)−SB(α0) for a trajectory

that starts in macrostateα0 and ends in macrostateαt. If the system is ini-

tially prepared in essentially one macrostate, ˆµ(α0)≃1, and if the evolution

on the level of macrostates is quasiautonomous until timet in the sense that ˆ

µt(αt)≃1, then only that change in Boltzmann entropies survives. The im-

portance of autonomy was discussed in section 2.4.3, we will give an explicit example of how this autonomy is realized in chapter 5.

Taking the expectation ofRˆµ(ω), we get, similar as in equation (3.6),

X

ω

Probµˆ[ω]Rµˆ(ω) =S(ˆµt)−S(ˆµ)>0, (3.16)

equal to the change of quantum entropy (2.27)–(2.28). The nonnegativity fol- lows from Jensen’s inequality applied to the left-hand side of equation (3.16), see the classical argument in equation (2.36) of section 2.3.2.

The change of quantum entropy, the right-hand side of equation (3.16), coin- cides with the relative entropy on the level of density matrices for trajectories. To show this, remark that:

S(Dµ|DµtπΘ) = Tr [DµlnDµ]−Tr [DµlnDµtπΘ] =X ω,ω′ Dµ(ω, ω′) lnDµ(ω′, ω)− Dµ(ω, ω′) lnDµtπΘ(ω ′_{, ω)} =X ω,ω′ Dµ(ω, ω′) ln Dµ (ω′_{, ω)} DµtπΘ(ω′, ω) We define a slight generalization of formula (3.14):

Rµ(ω, ω′) = ln Dµ(ω

′_{, ω)}

DµtπΘ(ω ′_{, ω)}

Since Dµ(ω, ω′) = 0 when ω0 6= ω0′ or ωn 6= ω′n, we obtain in a completely

analogous fashion to equation (3.15) that also for the generalization: Rµ(ω, ω′) = lnd(ωn)−lnd(ω0) + lnµ(ω0)−lnµt(ωn)

Inserting in the previous equation and grouping the terms with equal times, we get S(Dµ|DµtπΘ) = X ω,ω′ Dµ(ω, ω′) lnd(ωn)−lnµt(ωn) −X ω,ω′ Dµ(ω, ω′) lnd(ω0)−lnµ(ω0) = X ωn,ω′n X ω0...ωn−1 ω′ 0...ω′n−1 Dµ(ω, ω′) ! lnd(ωn)−lnµt(ωn) − X ω0,ω0′ X ω1...ωn ω′ 1...ω ′ n Dµ(ω, ω′) ! lnd(ω0)−lnµ(ω0)

In the above equation, we can replace the sum over allωn andωn′ by one sum

over allωn sinceDµ(ω, ω′) = 0 ifωn 6=ω′n. The same is true for the sum over

ω0 and ω0′. With equation (3.8) and an analogue expression, but with now

fixingω0instead ofωn in the sum, we get

S(Dµ|DµtπΘ) = X ωn Tr[ωnρt] lnd(ωn)−lnµt(ωn) −X ω0 Tr[ω0ρ0] lnd(ω0)−lnµ(ω0) =S(µt)−S(µ),

which, by equality (3.16) produces the desired result, as announced in equation (3.13).