Find operation from GHZ state to one qubit state

Question

Suppose I had some phase shifted GHZ state $\frac1{\sqrt2}(|00\cdots0\rangle+e^{i\phi}|11\cdots1\rangle)$, is there a way apply some operation such that I can get the same phase shift for some one qubit state ${\psi}$.

Answer 1

Note that up to global phase, you can construct the $n$-qubit state you've provided as$^1$

\begin{align} |\phi\rangle &= \frac{1}{\sqrt{2}}\left( |0^n\rangle + e^{i\phi}|1^n\rangle\right)\tag{1} \\&\sim \text{CNOT}^{(n-1, n)} \text{CNOT}^{(n-2, n-1)} \cdots \text{CNOT}^{(1, 2)} R_z^{(1)}(\phi)|+\rangle|0^{n-1}\rangle\tag{2} \end{align} where $\text{CNOT}^{(a, b)}$ denotes a Controlled-Not controlled by qubit $a$ with qubit $b$ as the target and $a,b \in \{1, \dots, n\}$ and $R_z^{(1)}(\phi)$ is a Z-rotation by $\phi$ acting on qubit $1$, and $$ |+\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle + |1\rangle\right) \tag{3} $$ So assuming that you're trying to construct the state \begin{equation} |\psi\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle + e^{i\phi}|1\rangle\right) \sim R_z(\phi)|+\rangle\tag{4} \end{equation} then you can just reverse the effect of the $\text{CNOT}$ gates in Eq. (2):

\begin{align} |\psi\rangle|0^{n-1}\rangle = \text{CNOT}^{(1, 2)} \cdots\text{CNOT}^{(n-2, n-1)} \text{CNOT}^{(n-1, n)} |\phi\rangle \tag{5} \end{align}

where we've used the fact that $\text{CNOT}^2 = I$, and the remaining $n-1$ qubits can be safely thrown away since they're not entangled with $|\psi\rangle$ anymore.

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$^1$ I use "$\sim$" to denote equivalence up to global phase.