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In the period finding algorithm (see the picture), through the standard procedure in quantum computing algorithms, we have $$U_f|\Psi\rangle|0^n\rangle=\frac{1}{\sqrt{2^n}}\left(\sum_{x=0}^{2^n-1}|x\rangle\right)\otimes|f(x)\rangle.$$ Then the next key step, we apply the measurement. My question is, if we apply the measurement with computational basis $P_m$ only in the second component (the space where $|f(x)\rangle$ lives), then this measurement with respect to the whole composite system is $I\otimes P_m$.

Then follow this viewpoint, the "first" component should not be changed after the measurement right? (Since $\left(\sum_{x=0}^{2^n-1}|x\rangle\right)$ and $|f(x)\rangle$ are clearly not entangled, i.e. it can be written as the form $|\phi\rangle\otimes|\psi\rangle$). Therefore the resulting state should be something like $\underbrace{\left(\sum_{x=0}^{2^n-1}|x\rangle\right)}_{\text{unchanged}}\otimes|f_0\rangle$.

Where did I have the mistake?

enter image description here

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    $\begingroup$ they are entangled. The parenthesis should also include the $|f(x)\rangle$ term. Otherwise what would the $x$ in $|f(x)\rangle$ mean? $\endgroup$
    – glS
    May 15 at 19:19
  • $\begingroup$ @gIS I think your comment should be an answer :-) $\endgroup$ May 15 at 19:37
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The state is indeed entangled. The parenthesis should include the $|f(x)\rangle$ factor in the equation (otherwise, what would the $x$ in $f(x)$ mean?).

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