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In quantum computing, quantum circuits are a common model for visualizing unitary evolution. If I want to represent a unitary $U$ acting on a $2$-qubit state $|00\rangle$ followed by a measurement in the computational basis then I can draw the following picture:

enter image description here

This representation is concise and unambiguous: $U$ is a unitary process, $|00\rangle$ and $U$ are both defined over $2$ qubits, both measurements are applied to single qubits in the computational basis, etc.

Now suppose I want to represent an arbitrary quantum channel $\mathcal{E}: \mathbb{C}^{4 \times 4} \rightarrow \mathbb{C}^{4\times 4}$ (completely positive, trace preserving). What are ways to visualize this (non-unitary) process $\mathcal{E}$ acting on an input $U|\psi\rangle$? I would appreciate suggestions that are used in publications or other main-stream references.


Possible approaches I have considered and their drawbacks:

  • I could just add another "gate" $\mathcal{E}$ to the diagram above, and comment that $\mathcal{E}$ is a CPTP map. But this seems dangerous because $\mathcal{E}$ is represented in the same way as other unitary operators even though it represents a more general quantum process. This violates readers' expectations and could cause confusion. For example:

enter image description here

  • I could define $V$ to be an isometric extension of $\mathcal{E}$ and add an ambiguous number of ancillary qubits so that my operators are still unitary. However this seems like overkill, since I might choose to define $\mathcal{E}$ in terms of Kraus operators and don't to provide the formalism for an isometry for $\mathcal{E}$ for the sole purpose of providing a picture of my process:

enter image description here

  • I could introduce a different style of box to represent $\mathcal{E}$, but this seems non-standard so it might defeat the purpose of providing a clear description of the process $\mathcal{E}$. And its not clear what style of box one should use if they do take this approach. For example:

enter image description here

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    $\begingroup$ I think usually people would just operate with density matrices from the start if the overall dynamics is nonunitary. Just replace $|0\rangle\to|0\rangle\!\langle 0|$ etc and you can use essentially the same notation, maybe replacing $U$ with something like $\mathcal E_U$ to indicate a unitary channel rather than a unitary gate $\endgroup$
    – glS
    Commented May 19, 2021 at 19:56
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    $\begingroup$ If you are willing to step away from the standard quantum circuit model you can do a lot with categorical quantum mechanics (eg arxiv.org/abs/quant-ph/0510032 and lots of other publicly available material from Bob Coecke) $\endgroup$ Commented May 19, 2021 at 20:58
  • $\begingroup$ The mainstream way is to represent $\mathcal E$ via its purification $V$. $\endgroup$ Commented May 19, 2021 at 21:37

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