# Are the first and second qubits of the state $| 111 \rangle + | 010 \rangle + | 101 \rangle + | 000 \rangle$ entangled with each another?

State of qubits: $$\frac{1}{2} (| 111 \rangle + | 010 \rangle + | 101 \rangle + | 000 \rangle)$$

Are the first and second qubits of this register entangled with each another?

• What are your ideas? Nov 15, 2021 at 9:48
• I found info about 2 qubits here quantumcomputing.stackexchange.com/questions/2263/… but I don't understand it quite well Nov 15, 2021 at 9:54
• My hint would be to first try to reduce the problem to a two-qubit problem. And then you can try to check whether the two-qubit system is entangled, for instance you could use the PPT condition to check entanglement. Nov 15, 2021 at 9:58

So, what I suggest is to take each qubit in turn, and try factoring out that specific qubit. For instance, if I take the first qubit, $$|0\rangle(|00\rangle+|10\rangle)+|1\rangle(|11\rangle+|01\rangle).$$ Since the two terms in brackets are different, the first qubit is not separable from the other two. You could continue to do the same thing for each of the other two qubits. Or, perhaps you'll notice something about the above equation if you try simplifying it a little bit, which might push you in the right direction (I'm quite deliberately not giving you the answer).
You can see that by measuring the first qubit, the state of second qubit has equal probability of being either $$|0\rangle$$ or $$|1\rangle$$.