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Since we do not treat finite size effects, the fraction is arbitrary, although realistically it should be large enough to suppress Poissonian errors on the estimation of error rates.

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The assumption of a top-hat profile can be relaxed without changing our analysis, but the calculation in Appendix A would require numerics.

Other phase modulation schemes such as four-wave mixing in waveguides [54] or sum-frequency generation in nonlinear crystals [51] could of course be used. Here we consider electro-optic phase modulation, which can be implemented with off-the-shelf telecoms components efficiently and with low loss.

Eq. (27) in [71] corresponds to setting I= 0 with c→ 0, M→ Nand p→eBN.

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In general the key distillation will have non-unit efficiency βsuch that I= βIBA− IBE. However high efficiencies can be achieved, with β> 90%. Although the formulas are simplified by assuming β= 1, our security analysis can be trivially extended to cover the general case.

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