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From the Stinespring dilation, we have that the dual or complementary channel can be observed in/expressed with the environment.

Can we reconstruct any channel for environments with $\text{dim}>1$ or does the dimension need to be higher? Also, how would one go forward with doing so?

glS
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Pink Elephants
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    what do you mean with "the complementary channel can be expressed with the environment"? Stinespring's dilation just tells you that any channel can be described as a unitary/isometric evolution in a larger space, ie $\Phi(\rho)=\operatorname{tr}_2[V\rho V^\dagger]$, and that the complementary channel can be described via the same isometric evolution by looking at the "environment output", ie $\Phi^c(\rho)=\operatorname{tr}_1[V\rho V^\dagger]$. – glS Mar 27 '24 at 15:35
  • To reformulate: what is the minimal dimention of the environment so that we can represent every quantum channel for a fixed Euclidean space? – Pink Elephants Mar 28 '24 at 07:35

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If a completely positive map $\Phi$ sends states in $\mathbb{C}^n$ to states in $\mathbb{C}^m$, then its Choi is a linear operator acting in an $nm$-dimensional space, and thus its maximum possible rank is $nm$. The rank of the Choi corresponds to the minimum number of Kraus operators needed to represent $\Phi$, which in turn corresponds to the minimal dimension of the environment to represent $\Phi$ via Stinespring dilation. See for example the first part of this answer, and the links there, for more details on why this is so.

It follows that all completely positive maps of this form can be represented using an environment of dimension $nm$.

glS
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  • Moreover, $mn$ is the smallest number which works for all channels because one can easily construct channels such that the Choi matrix has rank $mn$. The simplest example is, arguably, the unital reset channel $X\mapsto {\rm tr}(X)\frac{{\bf1}m}m$ as its Choi matrix equals $\frac{{\bf1}{mn}}{m}$ – Frederik vom Ende Mar 29 '24 at 07:22