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I got a review from a journal that

The problem with their proposal is that they wrongly assume that the inner product between a pair of quantum states can be determined exactly even when one is given a copy of such pair.

Can someone please help me in understanding this statement ? Can't we always determine the inner product of two quantum states?

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  • $\begingroup$ I think we need more information. Are they saying that, given $|\psi\rangle$ and $|\phi\rangle$, you can't calculate $\langle \psi|\phi \rangle?$. If you know what both states are, then calculating them is trivial. Is there more information regarding the states or how one obtains them? $\endgroup$ Nov 1, 2022 at 14:00
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    $\begingroup$ New tag for 'understanding referees'? $\endgroup$
    – R.W
    Nov 1, 2022 at 15:29

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I guess what they're trying to say is that there is no circuit that can compute inner products of arbitrary states. Indeed, if it existed, we could use it to calculate the inner product of the states $|\phi\rangle$ and $|\psi\rangle = e^{i\gamma}|\phi\rangle$ which is equal to $$\langle \phi|\psi\rangle = e^{i\gamma}\lVert|\phi\rangle\rVert^2=e^{i\gamma}$$ But this would give us a way to observe the global phase, which is unphysical.

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    $\begingroup$ Just to add, you can use swap test to calculate absolute value of inner product but not the product itself. $\endgroup$ Nov 2, 2022 at 7:20
  • $\begingroup$ I feel like the statement is referring to computing $|\langle\phi|\psi\rangle|$. Sure you can't compute its phase, but that's kinda obvious being the phase non-physical; it wouldn't even appear using a more careful formalism. Given that they're writing "one is given a copy of such pair", they're probably referring to the fact that you can't precisely estimate $|\langle\phi|\psi\rangle|$ without performing sufficient statistics. $\endgroup$
    – glS
    Nov 7, 2022 at 7:57
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I believe the key term here is "a copy of such pair". Does your proposal involve creating a copy of a quantum state? Aka, cloning?

Although the "no-cloning theorem" forbids perfect cloning, one can still perform partial cloning with a fidelity bounded by 0.5 + 1/(1+d) where d is the dimension of the Hilbert space for the system (for qubits, d=2).

Hence for such a copy or clone of a quantum state, one can't always find the inner product exactly. The fidelity of the states bounds the fidelity of the inner product.

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