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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...

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    $\begingroup$ 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. $\endgroup$ – glS May 8 at 1:05
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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.

This gets at part of what makes entanglement "special," i.e., different from classical correlation. If you have a machine that spits out EPR pairs, what distinguishes them from classically correlated variables is the fact that you get correlation no matter what basis you measure in. If you have a machine that, say, spits out pairs of electrons whose spins are either both up or both down (along some particular direction), each with probability 0.5, then if you measure along that direction you'll indeed find correlation. If you were to measure along a perpendicular direction you'd get no correlation at all, and this would allow you to determine that the machine is not producing entanglement, but if you are constrained to just measure in the original basis, there is no way to tell.

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