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If I have two independent (unentangled) qubits, let's say one in state |1> and the other one in some superposition state with equal amplitudes and arbitrary phases. If I measure the qubit that is in the superposition state (and let's say get 0), it seems that the remaining phase is kicked back to the first (non-measured) qubit. So the state of the first (non-measured) qubit changes as a result of the measurement of the second qubit. But the qubits are independent and are not supposed to know about each other and impact each other. So why is the phase transferred to the first qubit? enter image description here

# some arbitrary state:
theta = [np.pi/3,np.pi/4]
a = 1/np.sqrt(2)*np.array([np.exp(1j*theta[0]),np.exp(1j*theta[1])])

qc = QuantumCircuit(2,1)
qc.x(0)
qc.initialize(a,1)  # set q[1] to a
qc.measure(1,0)
execute(qc,svsim).result().get_statevector()  # get the post-measurement state
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  • $\begingroup$ You mentioned about measuring qubits that differ by a global phase. Global phase is not seen when qubits are measured. Also, Phase kickback refer to flip of the phase of control qubit in a circuit with CNOT sandwiched between two H gates. $\endgroup$
    – RSW
    Commented Nov 11, 2022 at 6:02
  • $\begingroup$ When I looked at the state $\begin{bmatrix} 0 \\ e^{i\theta} \end{bmatrix}$ I thought it has a relative not the global phase and that bothered me as relative phase can be detected (for example by applying Hadamard gate before measurement). And that looked very similar to the phase kickback which shall happen only when the qubits are connected by H CNOT H gates. But then I realised my mistake, probably more accurate notation would be $$\begin{bmatrix} 0*e^{i\theta} \\ 1*e^{i\theta} \end{bmatrix}$$. I has no practical difference, but it makes it clear that there is only a global phase present. $\endgroup$
    – EugeneB
    Commented Nov 11, 2022 at 9:00

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If two states differ by $e^{i \theta}$ for any real $\theta$ then the two states are indistinguishable.

The bug is in the mathematics we use to describe states. All states that differ by a multiple of $e ^ {i \theta}$ are the same state.

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