# 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?

When you ask about an 'ideal' fidelity measure, it assumes that there is one measure which inherently is the most meaningful or truest measure. But this isn't really the case.

• 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.

So, which norm you want to use to describe distances on operators, depends on what those operators are and what you would like to say about them.

If one of them $$(U)$$ is a unitary while the other one $$(V)$$ is any matrix (with $$\| V \| = 1$$), then I think that

$$f = - tr((U - V)^{\dagger}(U - V))$$ would work well.

For example, if

1. $$V = U$$, then f would be 0.
2. $$V = 0$$, then f would be -1.
3. $$V = \lambda U$$, then f would be $$-|1 - \lambda|^{2}$$.

and so on.

I think that

$$f = \frac{tr(A^{\dagger}B)}{\sqrt{tr(A^{\dagger}A)}\sqrt{tr(B^{\dagger}B)}}$$ is a good useful measure.

Lets say $$B = e^{i \phi} A$$, then $$tr(A^{\dagger}B) = N e^{i \phi} tr(A^{\dagger}A)$$ and $$tr(B^{\dagger}B) = tr(A^{\dagger}A)$$, thus $$f = N e^{i \phi}$$

Also, this measure reduces to the Hilbert Schmidt norm when used for Unitary matrices.