## What is the unitary operator realizing a given CPTP operator

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Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary operator on the system after adding few ancilla bits. I would like to know how to realize a given CPTP operator.

Choi's theorem states that any CPTP operator $$\Phi(\cdot) : C^*_{n\times n} \rightarrow C^*_{m \times m}$$ can be expressed as $$\Phi(\rho) = \sum_{j=1}^r F_j^\dagger \rho F_j$$, for some $$n \times m$$ matrices $$F_j$$ such that $$\sum_j F_j F_j^\dagger = I_n$$.

Using this fact, can we come up with unitary operation corresponding to the given CPTP operator $$\Phi$$?

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An isometry $$U:S\rightarrow S\otimes E$$, where $$S$$ is your system and $$E$$ is an environment, such that

$$\mathrm{Tr}_E(U\rho U^\dagger)=\Phi(\rho)$$ is called a Stinespring dilation of $$\Phi$$. An easy way to construct a Stinespring dilation from the Kraus operators is to consider $$\mathcal{H}_E=\mathrm{span}\{|j\rangle\}_{j=1}^r$$ and

$$U=\sum_j F_j^S\otimes|j\rangle^E$$

it is easy to see that

$$\mathrm{Tr}_E(U\rho U^\dagger)= \mathrm{Tr}_E\left(\sum_jF_j\rho F_k^\dagger \otimes |j\rangle\langle k|\right)= \sum_jF_j\rho F_j^\dagger=\Phi(\rho)$$

but notice that since the set of Kraus operators for a channel is not unique, neither is the Stinespring dilation.