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A source constantly produces a stream of photons in one of the following states

$$|\varphi_1\rangle=\dfrac{1}{\sqrt2}(a|00\rangle+ b|01\rangle+c|10\rangle+d|11\rangle)$$

$$|\varphi_2\rangle=\dfrac{1}{\sqrt2}(b|00\rangle+ a|01\rangle+d|10\rangle+ c|11\rangle)$$

where $a,b,c,d$ are some (valid) scalar values you know beforehand and $a\ne b\ne c\ne d$.

You are only sent the first qubit and have no access to the second qubit.

How to design a measurement (on the first qubit) that allow yout to tell whether the source is producing $|\varphi_1\rangle$ or $|\varphi_2\rangle$?

Assume you can measure as many times as you want.

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Write down the two reduced density matrices of the single qubits that you have access to. Apply the Helstrom measurement (there are several descriptions of this on the site already).

The problem is that, in this case, the two reduced density matrices are the same. That means that you cannot tell them apart. More explicitly, $$ |\varphi_2\rangle=(I\otimes X)|\varphi_1\rangle $$ The only different between the two states is on the second qubit.

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