5 IV. XPM WITH NON-HAMILTONIAN COUPLING despite [ E$_{1}$ ( z, 0) , E$_{2}$ ( z $^{′}$, 0)] = 0. The conventional way to deal with non-Hamiltonian dynamics in the Heisenberg picture is to introduce Langevin noise operators. In the context of nonlinear optics this has first been done by Boivin et al. [46] in the case of time-dependent self-phase modulation. We now show that a similar approach can also be made for instantaneous XPM between two light pulses with dif- ferent group velocities. Following Ref. [46] we introduce Hermitian decoherence operators m$_{i}$ ( z, t ) such that the dynamical equations are modified to In the previous section we have devised a scheme to create uniform XPM for quantized light pulses for the special case of unitary dynamics for which v$_{1}$V$_{1}$ ( z ) = v$_{2}$V$_{2}$ ( z ). The theory behind this scheme corresponds to a regularized quantization of the classical theory (1) for the case that the XPM coefficients are related by v$_{1}$γ$_{1}$ = v$_{2}$γ$_{2}$ . However, in classical optics this relation between the two XPM coefficients is not generally adopted. In this section we therefore study the quantization of the classical model with v$_{1}$γ$_{1}$ = v$_{2}$γ$_{2}$ and its implications for quantum information processing. Because the model needs to be regularized, we consider the general case with two non-local interaction potentials that fulfill v$_{1}$V$_{1}$ ( z ) = v$_{2}$V$_{2}$ ( z ). The decoherence operators commute with all field op- erators. We adopt the frequently used assumption of δ -correlated decoherence, The discussion in Sec. II has revealed that the quantum dynamics in this case must be non-unitary because there is no Hamiltonian that can generate the equations of mo- tion. As Eq. (2) is based on a phenomenological model that does not explicitly include absorption or dissipation, the origin for this non-unitarity cannot be identified in an unambiguous way. There might be implicit absorp- tion processes hidden in the model, although we strongly doubt that this is the case because the dynamics does not have a structure that is comparable to the Lindblad form [44]. Another possibility is that the external fields that usually are needed for EIT-based XPM media may induce energy fluctuations that lead to non-Hamiltonian dynamics. Yet another possibility is that the direct quan- tization of the classical theory (1) ignores the averaging processes that are associated with a macroscopic decrip- tion of electrodynamics [45]. Such averaging procedures lead to loss of information, which may induce decoher- ence in a quantum system. ( 1 v$_{i}$ ∂$_{t}$ + ∂$_{z}$$^{)}$ E$_{i}$ ( z, t ) = − i ( ∫ ∞ −∞ dz $^{′}$I$_{3}$$_{−}$$_{i}$ ( z $^{′}$, t ) (25) × V$_{i}$ ( z − z $^{′}$) + m$_{i}$ ( z, t ) $^{)}$E$_{i}$ ( z, t ) . [ m$_{i}$ ( z, t ) , m$_{j}$ ( z $^{′}$, t$^{′}$)] = i η ( V$_{i}$ ( z − z $^{′}$) v$_{j}$ − V$_{j}$ ( z − z $^{′}$) v$_{i}$ $^{)}$δ ( t − t $^{′}$) . (26) E$_{i}$ ( z, t ) = R$_{i}$ ( z, t ) U$_{3}$$_{−}$$_{i}$ ( z, t ) E$_{i}$ ( z − v$_{i}$t, 0) , (27) R$_{i}$ ( z, t ) = exp$^{(}$ − i v$_{i}$ ∫ t 0 dt ′ m$_{i}$ ( z − v$_{i}$ ( t − t $^{′}$) , t $^{′}$)$^{)}$ . (28) The solution of Eq. (25) can be written as Using the commutation relations (26) it is not hard to show that As a consequence, solution (27) fulfills the equal-time commutation relations. Introducing the decoherence term in Eq. (25) has thus led to a consistent quantum field theory of XPM for light pulses with different group velocities. R † $_{1}$( z, t ) R$_{2}$ ( z $^{′}$, t ) R$_{1}$ ( z, t ) = R$_{2}$ ( z $^{′}$, t ) e iΦ( z $^{′}$− z,t $^{)}$, R † $_{2}$( z, t ) R$_{1}$ ( z $^{′}$, t ) R$_{2}$ ( z, t ) = R$_{1}$ ( z $^{′}$, t ) e − iΦ( z − z $^{′}$,t ) . (29) A. Controlled-Phase Gate Despite this ambiguity with respect to the cause of non-unitary dynamics it is worthwhile studying this sit- uation because one can gain a better understanding of the quantization of a new class of classical models. Further- more, even if the theory that we develop below cannot include the microscopic details behind the XPM interac- tion it nevertheless should help to estimate the effects of a mismatch between the XPM coefficients. To characterize the performance of a CPG in the case of non-unitary dynamics we can repeat the steps dis- cussed in Sec. III B with solution (5) replaced by Eq. (27). Finding the logical density matrix elements must then be done within an open system approach that is described in App. C 2. As a result, the factors X$_{ij}$ become opera- tors on the Hilbert space of the environment (on which the decoherence operators act) and Eq. (21) is replaced by In the case v$_{1}$V$_{1}$ ( z ) = v$_{2}$V$_{2}$ ( z ), Eq. (5) still provides an exact solution to the dynamical equation of motion. However, this solution is inconsistent with basic require- ments of quantum field theory. To illustrate this point we consider the equal-time commutation relations be- tween the Heisenberg field operators, which should agree with the commutation relations between the respective Schr¨ odinger operators. In our case this principle is vio- lated because X$_{ij}$ = c$_{ij}$e − i i (1 − j ) φ$_{2}$ ∫ dz$_{1}$ dz$_{2}$ | ψ$_{1}$ ( z$_{1}$ ) ψ$_{2}$ ( z$_{2}$ + d ) | 2 × R$_{1}$ ( z$_{1}$ + v$_{1}$t + δz (1) i , t ) R$_{2}$ ( z$_{2}$ + d + v$_{2}$t + δz (2) j , t ) . (30) E$_{2}$ ( z $^{′}$, t ) E$_{1}$ ( z, t ) = E$_{1}$ ( z, t ) E$_{2}$ ( z $^{′}$, t ) e − iΦ( z $^{′}$− z,t $^{)}$, (23) Φ( z ′ − z, t ) ≡ η Δ v$^{(}$ v$_{1}$θ$_{1}$ ( z ′ − z + Δ vt, Δ vt ) − v$_{2}$θ$_{2}$ ( z ′ − z + Δ vt, Δ vt ) $^{)}$, (24)