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On this page IMBQ docs, until the sentence '..and since the global phase of a quantum state is not detectable..' I follow everything. However 'quantum phase' is introduced without any explaination? What is this?

glS
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Tim
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    be sure not to confuse the term "global quantum phase" with "quantum phase" in the context of "phase transitions" these are not the same "phase". The "quantum phase" you are referring to is the former. – Condo Jul 19 '21 at 14:37
  • possible dupicates: https://quantumcomputing.stackexchange.com/q/13689/55, https://quantumcomputing.stackexchange.com/q/5125/55 – glS Jul 19 '21 at 20:49

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It is a term of the form $e^{i\phi}$. So, the point is that two states $|\psi\rangle$ and $|\phi\rangle=e^{i\phi}|\psi\rangle$ differ only by this phase. We call it a global phase because it's affecting the whole of the state rather than just part of it.

You can easily see that this has no observable consequences. For example, if we measure in the standard basis, what's the probability of getting the answer $|0\rangle$? $$ p_0=|\langle0|\phi\rangle|^2=|e^{i\phi}\langle0|\psi\rangle|^2=|\langle0|\psi\rangle|^2 $$ The probability is the same for both cases. As it will be for any measurement basis and outcome.

DaftWullie
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    "It is a term of the form eiϕ" that does not enlight me. What is it? Where does it come from? What's the added context? – Tim Jul 19 '21 at 14:20
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    Also, is the phase represented on the bloch sphere? – Tim Jul 19 '21 at 14:23
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    The global phase does not appear on the Bloch sphere. – DaftWullie Jul 19 '21 at 14:47
  • Where does it come from? It 's just a common factor. In the same way that you can write $77+55=11(7+5)$. It's just that if you impose that state vectors have to have length 1, the only factor you could possibly pull out is of the form $e^{i\phi}$. For example, any single-qubit state is $\alpha e^{i\phi}|0\rangle+\beta e^{i\psi}|1\rangle$ where $\alpha$ and $\beta$ are real. It looks like we need 4 parameters to describe the state. But I could just write it as $e^{i\phi}(\alpha|0\rangle+e^{i(\psi-\phi)}\beta|1\rangle)$ at which point the claim is that you can neglect the $e^{i\phi}$ and only – DaftWullie Jul 19 '21 at 14:52
  • use 3 parameters (which you reduce further by the constraint $\alpha^2+\beta^2=1$) – DaftWullie Jul 19 '21 at 14:52