# What would be an ideal fidelity measure to determine the closeness between two non unitary matrices?

The Hibert Schmidt norm $$\mathrm {tr}(A^{\dagger}B)$$ works well for unitaries. It has a value of one when the matrices are equal and less than one otherwise. But this norm is absolutely unsuitable for non-unitary matrices. I thought maybe $$\frac{\mathrm{tr}(A^{\dagger}B)}{\sqrt{\mathrm{tr}(A^{\dagger}A)} \sqrt{\mathrm{tr}(B^{\dagger}B)}}$$ would be a good idea?

• For unitary operators, our analysis of the error used in approximating one unitary by another involves the distance induced by the operator norm: $$\bigl\lVert U - V \bigr\rVert_\infty := \max_{\substack{\lvert \psi\rangle \ne \mathbf 0}} \frac{\bigl \lVert (U - V) \lvert \psi \rangle \bigr\rVert_2}{\bigl \lVert \lvert \psi \rangle \bigr\rVert_2}$$ That is, it is the greatest factor by which the Euclidean norm (or 2-norm) of a vector will be increased by the action of $$(U - V)$$: if the two operators are very nearly equal, this factor will be very small.
I know you asked for norms on non-unitary matrices, but if a norm is useful for non-unitary matrices, you might hope that it would also be useful for unitary matrices, and the point here is that the 'operator norm' is. It is also useful for (non-unitary) observables: for two Hermitian operators $$E$$ and $$F$$ — representing evolution Hamiltonians, for instance, or measurement projectors — the operator norm $$\lVert E - F \rVert$$ conveys how similar $$E$$ and $$F$$ are in a way which directly relates to how easily you can operationally distinguish one from the other.
• On the other hand, for density operators $$\rho$$ and $$\sigma$$, the best distance measure to describe how easily you can distinguish them is the trace norm: $$\bigl\lVert \rho - \sigma \bigr\rVert_{\mathrm{tr}} := \mathrm{tr} \Bigl( \sqrt{(\rho - \sigma) ^2} \Bigr)$$ which is the same as (in fact, it's just a fancy way of writing) the sum of the absolute values of the eigenvalues of $$(\rho - \sigma)$$: if the two operators are very nearly equal, this sum will be very small.