I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase?
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2 Answers
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As clearly evident from the Euler's form $z=re^{i\theta}$, a phase has something to do with rotation in the Argand plane but not affect the amplitude of a complex number. You can make a set of set of infinite complex numbers with the same magnitude. It can just be regarded as the extra degree of freedom for a given complex number.
In the perspective of Quantum Information/Computing, the observable quantities are the probabilities which are proportional to the complex number amplitudes $|z|^2=|re^{i\theta}|^2=(re^{i\theta})(r^*e^{-i\theta})=r^2$ which clearly doesn't care about the the phase $\theta$.
Let's consider the most simple non-trivial example. For any quantum state with two degrees of freedom (qubit): \begin{equation} |\psi\rangle=r_1e^{i\theta_1}|0\rangle+r_2e^{i\theta_2}|1\rangle \end{equation} This is described by two complex numbers with phases $\theta_1$ and $\theta_2$ respectively. It can be rewritten as: \begin{equation} |\psi\rangle=e^{i\theta_1}(r_1|0\rangle+r_2e^{i(\theta_2-\theta_1)}|1\rangle) \end{equation}
Now, if you calculate the amplitude $|\psi|^2$, the factor $e^{i\theta_1}$ in front will vanish by the argument above. This is called a global phase which is an overall phase in front. The relative phase is the quantity $\theta_2-\theta_1$ or $\theta_1-\theta_2$, however defined.
The relative phase is an observable quantity in Quantum Theory and it can be changed when a state evolved in accordance with the Schrodinger's equation $i\hbar\frac{d}{dt}|\psi\rangle=\hat{H}|\psi\rangle$.
The relative phase has also great importance when we consider the density matrix for a state defined as $\rho=|\psi\rangle \langle \psi|$ which for the example above is: \begin{equation} \rho=r_1^2|0\rangle\langle0|+r_1r_2e^{i(\theta_1-\theta_2)}|0\rangle\langle1|+r_2r_1e^{i(\theta_2-\theta_1)}|1\rangle\langle0|+r_2^2|1\rangle\langle1| \end{equation}
where it is only the relative phase that appears and not the global phase. In Quantum Information point of view, this relative phase appearing in the off-diagonal terms of the above matrix carries the information of coherence of the system which is one of the most unique properties of quantum systems.
These are some general concerns of relative and global phases. It does not make any sense to talk about a relative phase for a single complex number $z$.
Also, please see the wiki articles of such concepts, they clear enough content on these as a good start. Here you can refer to https://en.wikipedia.org/wiki/Qubit, mainly the Bloch sphere section.
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$\begingroup$ Can you please explain how $\vert \psi \vert^2$ is calculated to make $e^{i\theta_1}$ vanish? $\endgroup$ Commented Jul 15, 2020 at 21:46
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$\begingroup$ If $|\psi\rangle=e^{i\theta_1}C$, where $C$ is any complex number. Then $|\psi|^2=\langle\psi|\psi\rangle=e^{-i\theta_1}C^* \times e^{i\theta_1}C=C^*C=|C|^2$. $\endgroup$ Commented Jul 15, 2020 at 21:50
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$\begingroup$ Thank you so much for the explanation and the fast reply! $\endgroup$ Commented Jul 15, 2020 at 21:56
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$\begingroup$ Sorry for digging up this old answer, but it is a reference question ! You say that the relative phase arises from states evolving in accordance to the Schrödinger equation. Do you have any work to cite for this ? As I wrote it in a piece of work and would like to know where to find this information. $\endgroup$ Commented Dec 4, 2020 at 21:26
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1$\begingroup$ @Eenoku He mixed the complex number and quantum states. Better wirte as this for quamtum states: $|\psi\rangle=e^{i\theta}|\phi\rangle, \langle \psi |\psi\rangle =e^{-i\theta}\langle\phi| e^{i\theta}|\phi\rangle=\langle \phi |\phi\rangle$ .For complex number $C'= e^{i\theta}C$, you can check $|C'|^2=|C|^2$. $\endgroup$ Commented Oct 6, 2021 at 6:29
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From a physical point of view, there couldn't be a bigger difference.
Global phases are artefacts of the mathematical framework you are using, and have no physical meaning. Two states differing only by a global phase represent the same physical system. Indeed, a more careful treatment of quantum mechanics would involve defining quantum states as elements of a projective Hilbert space, in which all elements differing only by a phase are identified as equal.
On the other hand, relative phases are in some sense the core of quantum mechanics. States differing by a relative phase are different systems that evolve in different ways, although they will appear identical if only measured in the measurement basis in which they only differ by such relative phase.
