Appendix C — Quantum Reset is a CPTP map

We prove that the set of Kraus operators defined in Chapter 5 form a CPTP map.

We are required to show that

\[\sum_{i}\mathcal{E}_i^\dagger \mathcal{E}_i = \mathbf{1}\]

We start by evaluating \[\mathcal{E}_i^\dagger \mathcal{E}_i = |0\rangle\langle 0|\otimes I_2 \otimes \mathcal{R_i}^\dagger \mathcal{R_i} + \frac{1}{N}|1\rangle\langle 1| \otimes I_n = |0\rangle\langle 0|\otimes I_2 \otimes |i\rangle\langle i| + \frac{1}{N}|1\rangle\langle 1| \otimes I_n \]

Summing over \(i\)

\[\sum_{i} \mathcal{E}_i^\dagger \mathcal{E}_i = |0\rangle\langle 0 | \otimes I_2 \otimes I_n + |1\rangle\langle 1| \otimes I_2 \otimes I_n = I_{4n}\]

We are done.