Question 13

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Question 13.

In Notes 11, on encoding two classical bits as a Bell state it says: ‘We can encode two classical bits in the state of the two qubit system. This information can be recovered by performing a joint measurement (on the two qubits) which projects onto the Bell basis. This is called a Bell measurement and is a measurement in which the measurement operators are the Bell state projectors’.

Could you outline how this works? Surely if you make a measurement with one of the Bell projectors you can only tell whether the two qubits are in that Bell state or if they aren’t, after which you aren’t able to perform a second measurement and the state has collapsed to the post measurement state. So then you only have a 1 in 4 chance of recovering the information which is the equivalent of just guessing?

Ah, good to clarify this point. The measurement isn’t a two outcome measurement for each projector, asking “is it in this state or not”. Instead, it’s a four outcome measurement. The measurement is described by the set of four projectors \[ \{| \Phi^+ \rangle\langle \Phi^+ |_{AB}, | \Phi^- \rangle\langle \Phi^- |_{AB}, | \Psi^+ \rangle\langle \Psi^+ |_{AB}, | \Psi^- \rangle\langle \Psi^- |_{AB}\} \] and the four measurement outcomes correspond to the four Bell states. This measurement which will pick out the right state, without modifying it, iff the state being measured is indeed one of the four Bell states, as discussed in Question 10. This is a nonlocal measurement because the projectors involve both the \(A\) and \(B\) system (and in fact are entangled). So to perform such a measurement, one needs access to both parts of the state.

Maybe the following statements will help clarify things, in case this seems murky or maybe like cheating:

  1. Why? One way to see this is by equation (2.180) of Nielsen and Chaung: \(\operatorname{tr}[M \rho_A] = \operatorname{tr}[ (M\otimes I)\rho_{AB}]\) for any observable (Hermitian matrix) \(M\). An observable \(M\) on \(A\) alone has expectation value \(\operatorname{tr}[ (M\otimes I)\rho_{AB}]\), but we see these values are described exactly by the reduced density matrix. In fact, you can take this as the definition of the partial trace, since it is the unique function \(f\) from density matrices on \(AB\) to density matrices on \(A\) such that \(\operatorname{tr}( M f(\rho_{AB})) = \operatorname{tr}[ (M\otimes I) \rho_{AB}]\), as described in Box 2.6 of Nilsen and Chaung.