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Explicit construction for unitary extensions of CPTP maps?

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Given a completely positive and trace preserving map $\Phi : \textrm{L}(\mathcal{H})\to\textrm{L}(\mathcal{G})$, it is clear by the Kraus representation theorem that there exist $A_k \in \text{L}(\mathcal{H}, \mathcal{G})$ such that $\Phi(\rho) = \sum_k A_k \rho A_k^\dagger$ for all density operators $\rho$ on $\mathcal{H}$. (I'll consider the special case $\mathcal{H} = \mathcal{G}$ for simplicity.)

If we use then the system+environment model to express this action as $\Phi(\rho)=\text{Tr}_{\mathcal{H}_E} (Y\rho Y^\dagger)$ for an isometry $Y$ from $\mathcal{H}$ to $\mathcal{H}\otimes\mathcal{H}_E$, where $\mathcal{H}_E$ is an ancilla modelling the environment, what is an explicit construction for a unitary $U$ that has the same action on inputs of the form $\rho\otimes\left|0\right>\left<0\right|_E$? That is, how can I construct an explicit dilation of the map to a unitary acting on a larger space? I understand that this is possible by Steinspring's dilation theorem, but actually constructing an explicit form for the dialated unitary I have had much less success with.

This post has been migrated from (A51.SE)
asked Oct 29, 2011 in Theoretical Physics by Chris Granade (260 points) [ no revision ]
@Marcin: I deleted it because there was an error in the proof.

This post has been migrated from (A51.SE)
There used to be Joe Fitzsimmons' answer here, what happened?

This post has been migrated from (A51.SE)

1 Answer

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The isometry $Y:\mathcal H\rightarrow \mathcal H_E \otimes \mathcal H$ is $$ Y=\left(\begin{array}{c} A_1 \\ \vdots \\ A_K \end{array}\right) = \sum_k |k\rangle \otimes A_k\ . $$ Clearly, $$ \mathrm{tr}_E(Y\rho Y^\dagger) = \sum_{kl} \mathrm{tr}(|k\rangle\langle l|) A_k\rho A_l^\dagger = \sum_k A_k \rho A_k^\dagger \ , $$ as desired. Moreover, $Y$ is an isometry, $Y^\dagger Y=I$, i.e., its columns are orthonormal, which follows from the condition $\sum_k A_k^\dagger A_k=I$ (i.e., the map is trace preserving).

Now if you want to obtain a unitary which acts on $|0\rangle\langle 0|\otimes \rho$ the same way $Y$ acts on $\rho$, you have to extend the matrix $Y$ to a unitary by adding orthogonal column vectors. For instance, you can pick linearly independent vectors from your favorite basis and orthonormalize. (Clearly, $U$ is highly non-unique, as its action on environment states other than $|0\rangle$ is not well defined.)

This post has been migrated from (A51.SE)
answered Oct 30, 2011 by Norbert Schuch (290 points) [ no revision ]

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