content_image/414422.jpg content_image/414426.jpg content_image/414427.jpg content_image/414436.jpg content_image/414437.jpg content_image/414438.jpg content_image/414421.jpg content_image/414425.jpg content_image/414428.jpg
The only region where θ$_{i}$ ( z$_{1}$, z$_{2}$ ) varies significantly are bands of width 2 r around the lines z$_{1}$ = 0 and z$_{1}$ = z$_{2}$ , and only this transition region depends on the details of the potential. If the range r is much smaller than the optical wavelength we can ignore the transition region so that θ$_{i}$ ( z$_{1}$, z$_{2}$ ) can be replaced by the plateau value between the lines, 

<img src='content_image/414422.jpg'>

with θ ( z ) the step function. 

## Appendix C: Derivation of the action of a CPG on the density matrix 

The evolution of the density matrix ρ can be expressed with the aid of a unitary operator U ( t ) such that ρ ( t ) = U ( t ) ρ (0) U $^{†}$( t ), with 

<img src='content_image/414426.jpg'>

The logical matrix elements of the density matrix can then be expressed as ρ$_{ij}$$_{;}$$_{kl}$ ( t ) = 〈 ij ( t ) | U ( t ) ρ (0) U $^{†}$( t ) | kl ( t ) 〉 . Using the definition (18) of the logical basis states we are then led to consider the expression 

<img src='content_image/414427.jpg'>

We here inserted an additional factor U $^{†}$( t ) which does not change the result because U ( t ) | 0 〉 = | 0 〉 . The op- erators a$_{i}$ [ z ] are defined using the Schr¨ odinger operator E$_{i}$ ( z, 0), see Eq. (4). The effect of U † · · · U ( t ) is now to re- place the Schr¨ odinger operator by the respective Heisen- berg operator E$_{i}$ ( z, t ) so that 

<img src='content_image/414436.jpg'>

<img src='content_image/414437.jpg'>

We thus find for the logical matrix elements 

<img src='content_image/414438.jpg'>

## 1. Unitary dynamics 

We now concentrate on the factor 

<img src='content_image/414421.jpg'>

The expectation value in the last line can be considerably simplified. 

<img src='content_image/414425.jpg'>

The wavefunctions in Eq. (C6) imply that z ′ i ≪ δz be- cause z ′ i can only vary over the support of the wavefunc- tions. This in turn implies in Eq. (C7) that δz (1) i = δz (1) k and δz (2) j = δz (2) l because otherwise the argument of the wavefunctions would be outside of their support. Hence, 

<img src='content_image/414428.jpg'>

from which Eq. (20) follows. 

The derivation of the logical density matrix elements for non-unitary evolution follows essentially the same steps as in the unitary case described in App. C 1. How- ever, because the decoherence operators act on a differ- ent Hilbert space H$_{E}$ associated with the environment, the calculations have to be made on an enlarged Hilbert space H$_{S+E}$ = H$_{S}$ ⊗ H$_{E}$ that includes our system of pho- ton pulses as well as the environment. The dynamics on this enlarged Hilbert space is described by a unitary 

## 2. Non-Hamiltonian Dynamics 

