Applying a gate to entangle states

[adwodon]

Homework Statement

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

$\frac{1}{\sqrt{2}}$($\mid 0 \rangle$$_{A}$ + $\mid 1 \rangle$$_{A}$)$\mid 0 \rangle$$_{B}$$\mid 0 \rangle$$_{C}$

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

000 $\rightarrow$ 000
001 $\rightarrow$ 001
010 $\rightarrow$ 011
011 $\rightarrow$ 010
100 $\rightarrow$ 101
101 $\rightarrow$ 100
110 $\rightarrow$ 110
111 $\rightarrow$ 111

Which gives you

$\frac{1}{\sqrt{2}}$($\mid 0 \rangle$$_{A}$$\mid 0 \rangle$$_{C}$ + $\mid 1 \rangle$$_{A}$$\mid 1 \rangle$$_{C}$)$\mid 0 \rangle$$_{B}$

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.

Homework Equations

-

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.

 

[mark726]

Thank you for sharing your question with us. It seems like you have a good understanding of the first part of the question, where you apply the parity gate to the initial state and obtain an entangled state between A and C. Now, for the second part, you are correct in thinking that you need to put C in a superposition in order to entangle A and B. This can be achieved by applying a Hadamard gate to C, which will put it in a superposition of 0 and 1. This will change the state to:

\frac{1}{2}(\mid 0 \rangle_{A}\mid 0 \rangle_{C} + \mid 1 \rangle_{A}\mid 1 \rangle_{C} + \mid 0 \rangle_{A}\mid 1 \rangle_{C} + \mid 1 \rangle_{A}\mid 0 \rangle_{C})\mid 0 \rangle_{B}

Now, when you apply the parity gate, the resulting state will be:

\frac{1}{2}(\mid 0 \rangle_{A}\mid 0 \rangle_{C}\mid 0 \rangle_{B} + \mid 1 \rangle_{A}\mid 1 \rangle_{C}\mid 0 \rangle_{B} + \mid 0 \rangle_{A}\mid 1 \rangle_{C}\mid 1 \rangle_{B} + \mid 1 \rangle_{A}\mid 0 \rangle_{C}\mid 1 \rangle_{B})

As you can see, A and B are now entangled, and C remains in a superposition. I hope this helps you solve the problem. Good luck!