Let's say we have a system of 2 qubits, which are entangled in an unknown Bell basis configuration. Since the qubits are in a Bell configuration, each state is orthogonal to every other state, and thus must be distinguishable from each other.
My understanding is that there are 4 measurement operators (see below) that can be simultaneously applied to a single qubit (because they commute). And this is why we can unambiguously identify which Bell state the qubits are in, even if Bob and Alice measure their respective qubits with the 4 measurement operators, individually and concurrently with each other.
Question
How does one come up with a quantum circuit for something like this? How do we go about mapping an arbitrary measurement to one of the well known quantum gates?
On a related note, let's say I have an arbitrary unitary matrix. How does one map that to a quantum gate?
Bell configuration:
- $\vert{T_1}\rangle = \frac{1}{\sqrt{2}} (\vert{10}\rangle - \vert{01}\rangle)$
- $\vert{T_2}\rangle = \frac{1}{\sqrt{2}} (\vert{10}\rangle + \vert{01}\rangle)$
- $\vert{T_3}\rangle = \frac{1}{\sqrt{2}} (\vert{00}\rangle + \vert{11}\rangle)$
- $\vert{T_4}\rangle = \frac{1}{\sqrt{2}} (\vert{00}\rangle - \vert{11}\rangle)$
Distinguishability:
- $\langle T_i \vert T_j \rangle = \delta_{i,j}$
Measurement operators:
- $M_{T_i} = \vert{T_i}\rangle\langle{T_i}\vert$
- Commutation: $[M_{T_i}, M_{T_j}] = 0$