4 FIG. 2: Construction of a controlled phase gate for two time- bin qubits using uniform XPM. for ( δz (1) i , δz (2) j ) = (0 , 0) , (0 , δz ) , ( δz, 0) , ( δz, δz ) for ( i, j ) = (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1), respectively. The fac- tors v$_{i}$t represent an explicit time dependence in the def- inition of the logical states such that they are co-moving with the wavepackets at group velocity v$_{i}$ . We consider the case that the two photons are ini- tially prepared in a pure two-qubit state | ψ (0) 〉 = ∑ kl c$_{kl}$ | kl (0) 〉 . To characterize the performance of the CPG we need to obtain the logical components ρ$_{ij}$$_{;}$$_{kl}$ ( t ) = 〈 ij ( t ) | ρ ( t ) | kl ( t ) 〉 of the density matrix ρ ( t ) at time t . A somewhat tedious calculation that can be found in App. C 1 leads to We remark that φ$_{1}$ = φ$_{2}$ for unitary dynamics. Eq. (15) is precisely the result that one would obtain in a single- mode treatment. Hence, for unitary dynamics a mis- match in the group velocities of two pulses that allows one pulse to overtake the other in a Kerr medium will result in a uniform phase shift for QND measurements of the photon number. In practice, overtaking will impose a minimum requirement on the length of the Kerr medium and SPM will lead to a distortion of the signal. with The phase factor in Eq. (20) determines the XPM ef- fect. We now prove that it is only nontrivial for the logical state | 10 〉 because only in this case the associated wavepackets overtake each other. ρ$_{ij}$$_{;}$$_{kl}$ ( t ) = X ∗ $_{kl}$X$_{ij}$ , (19) X$_{ij}$ = c$_{ij}$ ∫ dz$_{1}$ dz$_{2}$ | ψ$_{1}$ ( z$_{1}$ ) ψ$_{2}$ ( z$_{2}$ + d ) | 2 × e − i η v 2 Δ $_{v}$θ$_{2}$ ( z$_{2}$ − z$_{1}$ + d + δz (2) j − δz (1) i , Δ vt $^{)}$. (20) U$_{CPG}$ | kl 〉 = e i k (1 − l ) $^{β}$| kl 〉 , k, l = 0 , 1 . (17) The phase factor is proportional to the function θ$_{2}$ ( z$_{2}$ − z$_{1}$ + d + δz (2) j − δz (1) i , Δ vt ) which, according to Appendix B, is nearly constant for a given range of its two vari- ables. Ignoring the variables z$_{i}$ for the moment we con- sider θ$_{2}$ ( d + δz (2) j − δz (1) i , Δ vt ) instead. For the four states 00 , 01 , 10 , 11 the first argument takes the values d, d + δz, d − δz , and d respectively. If we choose the time t for which the photons are interacting such that d − δz < Δ vt < d, d + δz then θ$_{2}$ = 0 for ij = 00 , 01 , 11 and θ$_{2}$ = V$_{2}$ for ij = 10. Hence only the latter combina- tion will experience an XPM phase shift. This conclusion also holds if we re-introduce the variables z$_{i}$ because they only vary over the support of the two wavepackets, which is smaller than d, δz ; the value of θ$_{2}$ therefore does not change. Thus we arrive at Our proposal to build a CPG extends previous designs to build quantum gates using XPM [16, 40] by ensuring a uniform XPM phase shift. In a controlled phase gate, two qubits with logical basis states | 0 〉 , | 1 〉 are manipulated in such a way that the state acquires a phase shift β only if both qubits are in the logical state | 1 〉 ⊗ | 0 〉 . In other words, a CPG is a unitary operator that maps the basis states | kl 〉 ≡ | k 〉 ⊗ | l 〉 to [42] B. Controlled-Phase Gate and the density matrix elements become With the uniform cross phase shifter discussed above, such a gate can be implemented for time-bin encoded photonic qubits as indicated in Fig. 2. In this case each optical field E$_{n}$ carries one photon. For each photon the logical basis states are encoded in two wavepackets, one of which is delayed compared to the other. The logical state | 0 〉 for qubit n ( n = 1 , 2) can be encoded in wavepackets ψ$_{n}$ ( z ) so that photons in this state are annihilated by operator a$_{n}$ = a$_{n}$ [0]. The logical state | 1 〉 can be encoded in wavepackets that are shifted to the right, ψ$_{n}$ ( z − δz ); photons in this state are annihilated by the operators a$_{n}$ [ δz ]. A CPG can then be constructed by arranging the initial distances between the wavepackets in such a way that the only states that do overlap inside the XPM medium are state | 1 〉 of the first qubit and state | 0 〉 of the second qubit. For all other logical states the wavepackets in the two field modes never overlap so that there is no XPM effect. The logical basis states for two time-bin encoded qubits at time t are given by the set Let us now work out the performance of the CPG by calculating the concurrence [43] of the density ma- trix (22) for a specific initial product state | ψ (0) 〉 = ( | 0 〉 + | 1 〉 ) ⊗ ( | 0 〉 + | 1 〉 ) / 2, which corresponds to c$_{ij}$ = 1 2 for all i, j . The concurrence C of the density matrix (22) then evaluates to C = | sin( φ$_{2}$/ 2) | . This indicates a perfect CPG because the initial product state is trans- formed into a maximally entangled state provided one can achieve a large phase shift φ$_{2}$ = π . X$_{ij}$ = c$_{ij}$e − i i (1 − j ) $^{φ$_{2}$}$, (21) ρ$_{ij}$$_{;}$$_{kl}$ ( t ) = c$_{ij}$c ∗ $_{kl}$e − i( i (1 − j ) − k (1 − l )) $^{φ$_{2}$}$. (22) | ij ( t ) 〉 = a † $_{1}$[ δz (1) i + v$_{1}$t ] a † $_{2}$[ δz (2) j + v$_{2}$t ] | 0 〉 , (18)