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Does the following non-separable wavefunction represent an entangled state?

$\psi(x_1,x_2)$ = $\exp[i b x_{1}x_{2}]\phi(x_{1})\phi(x_{2})$

This state can not be factorized into functions of $x_{1}$ and $x_{2}$, but one might argue that an overall phase factor is physically irrelevant and only consider $\phi(x_{1})\phi(x_{2})$, is it valid to ignore the phase factor in this case?

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  • $\begingroup$ It depends a bit what you mean by phi and x, but generally: yes, this can be entangled. (Note that if x are positions, the product of the phase factors is rather unnatural, as it does not transform nicely.) $\endgroup$
    – Norbert Schuch
    Commented Jun 13, 2022 at 17:07
  • $\begingroup$ How can the phase factor here be "overall" when it varies between the parts of the state? $\endgroup$
    – HTNW
    Commented Jun 13, 2022 at 18:03
  • $\begingroup$ By overall I meant, unlike the case where if you had a superposition of terms in Psi(x1, x2) with different phases, leading to interference and other effects, this phase here will simply disappear when computing observables. But I think such an argument might not be valid while determining whether the wavefunction represents an entangled state, so I am guessing this state is entangled then. $\endgroup$
    – Paranoid
    Commented Jun 13, 2022 at 18:36
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    $\begingroup$ Only the global phase is unphysical. This $e^{ibx_1x_2}$ has physical effect : if you try to compute the expectation value of the momentum $P_1 = -i\hbar \partial_{x_1}$ for example $\endgroup$
    – SolubleFish
    Commented Jun 13, 2022 at 20:07

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