Qiskit - Statevector measurement of single qubit in GHZ state collapses the entire state

Question

I have a GHZ state. I want to measure the third qubit in Hadamard basis, after which the state left behind should be a maximally entangled state as mentioned here.

But when I measure the third qubit, the entire state collapses into either $|000\rangle$ or $|{111}\rangle$

I am representing the quantum state as a statevector then measuring that.

#Create a circuit to generate GHZ state
circ = QuantumCircuit(3)

circ.h(0)
circ.cx(0, 1)
circ.cx(0, 2)

#Get the statevector from circuit
ghz_statevec = Statevector(circ)

H_matrix = 1/np.sqrt(2)*np.array([[1, 1],
                                [1,-1]])

#To measure in X basis, apply the Hadamard transform
evolved_state = ghz_statevec.evolve(H_matrix, [0])

evolved_state.draw('latex')

The output after applying Hadamard to the third qubit is $ \frac{1}{2} |{000}\rangle + \frac{1}{2} \ |001\rangle + \frac{1}{2} |110\rangle - \frac{1}{2} |111\rangle$

outcome, state = ghz_statevec.measure([0])

state.draw('latex')

But the state after measuring the third qubit is $|000\rangle$

When I use the circuit representation and do measurement, I get the expected outcome but not in the statevector representation.

Answer 1

The problem is that you measure ghz_statevec instead of evolved_state.

The following line gives the correct result:

outcome, state = evolved_state.measure([0])

Be aware though that this will return the complete system, including the qubit which has been measured. For instance, if outcome is equal to '1', then your resulting state would be $\frac{|001\rangle-|111\rangle}{\sqrt{2}}=\frac{|00\rangle-|11\rangle}{\sqrt{2}}\otimes|1\rangle$.

Answer 2

What outcome are you seeing? What did you expect?

I'm also not sure whether you meant ghz_statevec.measure([0]) or evolved_state.measure([0]) in the next to last line.

In either case, I'm seeing exactly what I expect. If I measure on ghz_statevec, then I either get an outcome of 0 or 1 and a corresponding state of $|000\rangle$ or $|111\rangle$

If I measure on evolved_state, then bit 0 has been decoupled from the entanglement, I still get an outcome of 0 or 1, but the other two bits can still both be 00 or both be 11.