## The Other Superoperator Isomorphism

A few months ago, I spent two posts describing the Choi-Jamiolkowski isomorphism between linear operators from M_{n} to M_{m} (often referred to as “superoperators“) and linear operators living in the space M_{n} ⊗ M_{m}. However, there is another isomorphism between superoperators and regular operators — one that I’m not sure of any name for but which has just as many interesting properties.

Recall from Section 1 of this post that any superoperator Φ can be written as

for some operators {A_{i}} and {B_{i}}. The isomorphism that I am going to focus on in this post is the one given by associating Φ with the operator

The main reason that M_{Φ} can be so useful is that it retains the operator structure of Φ. In particular, if you define vec(X) to be the vectorization of the operator X, then

In other words, if you treat X as a vector, then M_{Φ} is the operator describing the action of Φ on X. From this it becomes simple to compute some basic quantities describing Φ. For example, the induced Frobenius norm,

is equal to the standard operator norm of M_{Φ}. If n = m then we can define the eigenvalues {λ} and the eigenmatrices {V} of Φ in the obvious way via

Then the eigenvalues of Φ are exactly the eigenvalues of M_{Φ}, and the corresponding eigenvectors of M_{Φ} are the vectorizations of the eigenmatrices of Φ. It is similarly easy to check whether Φ is invertible (by checking whether or not det(M_{Φ}) = 0), find the inverse if it exists, or find the nullspace (and a pseudoinverse) if it doesn’t.

Finally, here’s a question for the interested reader to think about: why is the transpose required on the B_{i} operators for this isomorphism to make sense? That is, why can we not define an isomorphism between Φ and the operator

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