# How to identify whether a quantum state is entangled or separable if you can measure it only in one basis?

Suppose you have two qubits and don't know whether they're in an entangled state $$\frac{1}{\sqrt{2}}\left(|0,0\rangle + |1,1\rangle\right)$$ or in an equal mixture of states $$|0,0\rangle$$ and $$|1,1\rangle$$.

It'd be trivial to find out which alternative is true if you could measure both qubits in another basis, for example, the diagonal basis: measurement results would be correlated for the entangled state, but not for the separable mixed state.

Is there any way to discern the entangled state from the mixed state in the example above if you can measure the two qubits only in the computational basis?

Measurement in the computational basis would result in the same correlation regardless of whether the state is entangled or not: if the first qubit is measured as 0, the other one will also be 0. Assuming you can prepare and measure (in the computational basis) the same state many times, I'm wondering if you could use any unitary transformations before the measurement that would allow you to identify the entangled state. Or any other trick short of measuring in another basis...

• more generally, you cannot tell apart a superposition $\lvert0\rangle+\lvert1\rangle$ from a mixture $\lvert0\rangle\!\langle0\rvert+\lvert1\rangle\!\langle1\rvert$, if you only measure in the computational basis. For the same exact reason, the answer to the question is no, you cannot, if you are only measuring in the omputational basis the states you mention. – glS May 8 '19 at 1:05

If you are allowing unitary transformations prior to measurement, then we can effectively change our measurement basis to any basis we choose (a change of basis is a unitary transformation, and vice versa). Without doing any unitary transformation/change of basis, no, you cannot distinguish between the entangled case and the classical correlation (equal mixture of $$|00\rangle$$ and $$|11\rangle$$) just by measuring in the computational basis.