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I'm now studying quantum ML and now studying about fidelity ($\mathcal{F}$).

To my knowledge, fidelity means the distance between two quantum states, $\textit{i.e.,}$ if $\mathcal{F} ==1$, then the two quantum states are identical.

From this, suppose initial quantum states $|0\rangle$ goes on arbitrary two unitary gates $U_1$ and $U_2$ ; and $\mathcal{F}$ of two outputs equals 1, $\mathcal{F}(U_1|0\rangle, U_2|0\rangle)=1$.

Then, can I tell $U_1 =U_2$? IF not, can I make it true with finite requiements?

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I think you mean that if $\mathcal{F}=1$, the two states are identical.

Now to your question: is it true that $$ \mathcal{F}(U_1|0\rangle,U_2|0\rangle)=0\implies U_1=U_2. $$ This is not true. You need more conditions! As a simply counter example, consider $$ U_1=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right),\qquad U_2=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right). $$ These two unitaries have the same effect on the basis state $|0\rangle$. Thus, you also need to add to your list of requirements that you can make a similar statement for a complete basis of states. Even then it might not be enough: in my previous example, if you also tried the $|1\rangle$ state, you would also conclude that the outputs have fidelity 1, because fidelity doesn't see a global phase. In this case, you'd have to also test something like a $|+\rangle$ state.

In the case of a single qubit, it is sufficient to test two states which are neither parallel nor orthogonal. For a larger Hilbert space dimension ($d$), off the top of my head, you should require $d$ vectors, but you have to be careful with your choice of states to capture enough relative phase information. It is sufficient to select $d-1$ orthonormal vectors (this checks that, up to a relative phase, all but one of the columns of the unitaries are the same with respect to a particular orthonormal basis), and then a final one which has some support on all the previous ones and yet also completes the basis (checking the final column is correct, and the relative phases of the previous columns).

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  • $\begingroup$ Thank you! I understand the concept of fidelity by your counter example. Can I ask one more minor question? Then, is there any other metric that can show the relativeness between the arbitrary unitaries in quantum mechanics? not the state! $\endgroup$
    – JERMY
    Sep 1, 2022 at 10:25
  • $\begingroup$ What is the setting? Do you have a representation of these untiaries stored somewhere (e.g. on a computer) that you need to compare, or is it that there are two unknown unitaries that you are given an implementation of, and you have to do experiments to determine if they are the same? $\endgroup$
    – DaftWullie
    Sep 1, 2022 at 12:22
  • $\begingroup$ Oh! In my experiment, I define identical unitaries $U_1, U_2$ and want to check the relativeness/closeness of these unitaries. I want to make $U_1$ and $U_2$ go far away although they are the same at the first time. $\endgroup$
    – JERMY
    Sep 2, 2022 at 1:28

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