8 We now concentrate on the factor 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, with θ ( z ) the step function. 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 The expectation value in the last line can be considerably simplified. X$_{ij}$ ≡ 〈 0 | a$_{1}$ [ δz (1) i + v$_{1}$t, t ] a$_{2}$ [ δz (2) j + v$_{2}$t, t ] | ψ (0) 〉 = 1 η ∫ dz$_{1}$ dz$_{2}$ ψ ∗ $_{1}$( z$_{1}$ − v$_{1}$t − δz (1) i ) × ψ ∗ $_{2}$( z$_{2}$ − v$_{2}$t − δz (2) j ) 〈 0 | U$_{2}$ ( z$_{1}$, t ) ˆ E$_{1}$ ( z$_{1}$ − v$_{1}$t, 0) × U$_{1}$ ( z$_{2}$, t ) ˆ E$_{2}$ ( z$_{2}$ − v$_{2}$t, 0) | ψ (0) 〉 = 1 η ∫ dz$_{1}$ dz$_{2}$ ψ ∗ $_{1}$( z$_{1}$ − v$_{1}$t − δz (1) i ) × ψ ∗ $_{2}$( z$_{2}$ − v$_{2}$t − δz (2) j ) 〈 0 | U † $_{1}$( z$_{2}$, t ) ˆ E$_{1}$ ( z$_{1}$ − v$_{1}$t, 0) × U$_{1}$ ( z$_{2}$, t ) ˆ E$_{2}$ ( z$_{2}$ − v$_{2}$t, 0) | ψ (0) 〉 = 1 η ∫ dz ′ 1 dz ′ 2 ψ ∗ $_{1}$( z ′ $_{1}$) ψ ∗ $_{2}$( z ′ 2 + d ) × e − i η v 2 Δ $_{v}$θ$_{2}$ ( z ′ $_{2}$− z ′ $_{1}$+ d + δz (2) j − δz (1) i , Δ vt ) × 〈 0 | ˆ E$_{1}$ ( z ′ 1 + δz (1) i , 0) ˆ E$_{2}$ ( z ′ 2 + d + δz (2) j , 0) | ψ (0) 〉 . (C6) θ$_{i}$ ( z$_{1}$, z$_{2}$ ) ≈ ( θ ( z$_{1}$ ) θ ( z$_{2}$ − z$_{1}$ ) − θ ( − z$_{1}$ ) θ ( z$_{1}$ − z$_{2}$ )) V$_{i}$ , (B1) Appendix C: Derivation of the action of a CPG on the density matrix 1. Unitary dynamics 〈 0 | ˆ E$_{1}$ ( z ′ 1 + δz (1) i , 0) ˆ E$_{2}$ ( z ′ 2 + d + δz (2) j , 0) | ψ (0) 〉 = ∑ k,l c$_{kl}$ 〈 0 | ˆ E$_{1}$ ( z ′ 1 + δz (1) i , 0) ˆ E$_{2}$ ( z ′ 2 + d + δz (2) j , 0) × a † $_{1}$[ δz (1) k ] a † $_{2}$[ δz (2) l ] | 0 〉 = η ∑ k,l c$_{kl}$ψ$_{1}$ ( z ′ 1 + δz (1) i − δz (1) k ) × ψ$_{2}$ ( z ′ 2 + d + δz (2) j − δz (2) l ) . (C7) ρ (0) = ∑ i $^{′}$,j $^{′}$,k,$^{′}$l′ c ∗ k $^{′}$l′ c$_{i}$ $^{′}$j ′ | i $^{′}$j $^{′}$(0) 〉〈 k $^{′}$l$^{′}$(0) | . (C1) 〈 ij ( t ) | U ( t ) = 〈 0 | U $^{†}$( t ) a$_{1}$ [ δz (1) i + v$_{1}$t ] a$_{2}$ [ δz (2) j + v$_{2}$t ] U ( t ) . (C2) 〈 0 | ˆ E$_{1}$ ( z ′ 1 + δz (1) i , 0) ˆ E$_{2}$ ( z ′ 2 + d + δz (2) j , 0) | ψ (0) 〉 = ηc$_{ij}$ψ$_{1}$ ( z ′ $_{1}$) ψ$_{2}$ ( z ′ 2 + d ) , (C8) 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, 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 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 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 We thus find for the logical matrix elements 2. Non-Hamiltonian Dynamics 〈 ij ( t ) | U ( t ) = 〈 0 | a$_{1}$ [ δz (1) i + v$_{1}$t, t ] a$_{2}$ [ δz (2) j + v$_{2}$t, t ] , (C3) a$_{i}$ [ z, t ] ≡ 1 √ η ∫ ∞ −∞ dz ψ ∗ $_{i}$( z − u ) ˆ E$_{i}$ ( z, t ) . (C4) ρ$_{ij}$$_{;}$$_{kl}$ ( t ) = 〈 ψ (0) | a † $_{2}$[ δz (2) l + v$_{2}$t, t ] a † $_{1}$[ δz (1) k + v$_{1}$t, t ] | 0 〉 × 〈 0 | a$_{1}$ [ δz (1) i + v$_{1}$t, t ] a$_{2}$ [ δz (2) j + v$_{2}$t, t ] | ψ (0) 〉 . (C5)