Applying a gate to entangle states

adwodon

1. The problem statement, all variables and given/known data

Part of a past paper, I can do all of the question a-d without problems but the second part of e) is giving me trouble.

So you have the following state

[itex]\frac{1}{\sqrt{2}}[/itex]([itex]\mid 0 \rangle[/itex][itex]_{A}[/itex] + [itex]\mid 1 \rangle[/itex][itex]_{A}[/itex])[itex]\mid 0 \rangle[/itex][itex]_{B}[/itex][itex]\mid 0 \rangle[/itex][itex]_{C}[/itex]

You apply a three bit parity gate, which you do in previous parts, the table looks as follows

000 [itex]\rightarrow[/itex] 000
001 [itex]\rightarrow[/itex] 001
010 [itex]\rightarrow[/itex] 011
011 [itex]\rightarrow[/itex] 010
100 [itex]\rightarrow[/itex] 101
101 [itex]\rightarrow[/itex] 100
110 [itex]\rightarrow[/itex] 110
111 [itex]\rightarrow[/itex] 111

Which gives you

[itex]\frac{1}{\sqrt{2}}[/itex]([itex]\mid 0 \rangle[/itex][itex]_{A}[/itex][itex]\mid 0 \rangle[/itex][itex]_{C}[/itex] + [itex]\mid 1 \rangle[/itex][itex]_{A}[/itex][itex]\mid 1 \rangle[/itex][itex]_{C}[/itex])[itex]\mid 0 \rangle[/itex][itex]_{B}[/itex]

So A and C are entangled.

The question then asks you to chose a new initial state so that after applying G and taking a measurement of C, A & B are entangled.

2. Relevant equations

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3. The attempt at a solution

I'm not entirely sure what to do, the gate only affects C, taking a measurement of C suggests that C should be in a superposition? Pretty confused on this one, but it is the last part of a question so it's supposed to be more a head-scratcher rather than bookwork.