Bell Measurement on 3-qubit GHZ state?

[DodongoBongo]

How do you do a Bell measurement on a state that doesn't have a power of 2 number of qubits? I've got GHZ states like this:

$$|GHZ_{ijk}> = \frac{|0_{i}0_{j}0_{k}> + |1_{i}1_{j}1_{k}>}{\sqrt{2}}$$

And I'm trying to Bell measure the following state at qubits 2 and 3:

$$|GHZ_{012}>|GHZ_{345}>$$

I've permuted my Bell bases so I can measure those two qubits, but I'm not sure exactly what to do next. I've been doing a bit of research and I've been thinking of tracing away one of the qubits, but I'm not sure if that's really the right way to do it. Any insight or help is appreciated.

 

[azntoon]

To perform a Bell measurement on a state that doesn't have a power of 2 number of qubits, you can either use a generalized version of the Bell measurement or you can trace away one of the qubits. In order to do this for your GHZ state, you would first need to express the state in terms of the Bell basis. For a three qubit state, the Bell basis is given by: |B_{000}> = \frac{|00> + |11>}{\sqrt{2}} |B_{001}> = \frac{|00> - |11>}{\sqrt{2}}|B_{010}> = \frac{|01> + |10>}{\sqrt{2}} |B_{011}> = \frac{|01> - |10>}{\sqrt{2}}You can then decompose your GHZ state as follows: |GHZ_{012}>|GHZ_{345}> = \frac{1}{2}\Big(|B_{000}>|B_{000}> + |B_{001}>|B_{001}> + |B_{010}>|B_{010}> + |B_{011}>|B_{011}>\Big)Now, if you want to measure the state of qubits 2 and 3, you can trace away qubit 1 to obtain a two qubit state: |GHZ_{23}> = \frac{1}{2}\Big(|B_{000}> + |B_{001}> + |B_{010}> + |B_{011}>\Big)Finally, you can apply a two qubit Bell measurement to this state in order to measure the state of qubits 2 and 3.