Efficient Linear Optics Quantum Computation: Supplementary Information

In the paper we claimed that teleportation with |tn〉 satisfies that if 0 < k < n + 1, then the teleported state appears in mode n + k and only needs to be corrected by applying a phase shift. The modes 2n − l are in state 1 for 0 ≤ l < (n − k) and can be reused in future preparations requiring single photons. The modes 2n − l are in state 0 for n − k < l < n. If k = 0 we learn that the input state was |0〉0 and if k = n + 1, that it was |1〉0 . The probability of these two events is 1/(n + 1), regardless of the input. Both the necessary correction and which mode we teleported to are unknown until after the measurement. To prove the claim, consider a superposition α0|0〉0 + α1|1〉0 in the mode to be teleported. Suppose that k (k 6= 0, k 6= n + 1) photons are detected, more specifically, that the measurement detected rj ( ∑ j rj = k) photons in mode j. The effect of the measurement can be seen as a projection onto α0|x0〉|y0〉+α1|x1〉|y1〉 followed by a measurement of the first n+1 modes, where |x0〉 = F̂n+1|0〉|1〉k|0〉n−k, |y0〉 = |0〉k|1〉n−k, |x1〉 = F̂n+1|1〉k|0〉n−k+1 and |y1〉 = |0〉k−1|1〉n−k+1. Observe that applying P2πl/(n+1) to mode l for 0 ≤ l ≤ n after applying F̂n+1 is equivalent to shifting modes 0 . . . n circularly right before applying F̂n+1. This means that the states |x0〉 and |x1〉 differ only by phases in the number basis. Thus the measurement cannot distinguish between the two states. The relative phase of the detected number state in |x1〉 with respect to |x0〉 is given by ∏ j ω rjj . Thus the output state is α0|y0〉 + α1 ∏ j ω j|y1〉 which has the desired properties after correcting this relative phase by applying a phase shift to mode n + k. The cases k = 0 and k = n + 1 can be analyzed similarly.