# What are some standard ways to visualize non-unitary processes

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:

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:

• 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:

• 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:

• 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
– glS
May 19 '21 at 19:56
• 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) May 19 '21 at 20:58
• The mainstream way is to represent $\mathcal E$ via its purification $V$. May 19 '21 at 21:37