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Mathematically what is the difference between pure separable state and entangled state ?

Can anyone explain with equations?

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Entangled States are the ones who doesn't have a classical analog. They are members of a tensor product of two Hilbert Spaces, say:

$$H_{A}\otimes H_{B}$$

So an entangled state of a composite system is the one state which can't be written as a simple product state. For instance:

Consider the entangled pure state:

$$\mid \psi^{+}\rangle = \frac{1}{\sqrt{2}}\Big(\mid 01\rangle_{AB} + \mid 10\rangle_{AB} \Big)$$

This is an example which you can't write as a product state.

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  • $\begingroup$ Nice. Could you also put the same lines with spaces and formulas for separable states? For the purpose of completness. $\endgroup$
    – jaromrax
    Commented Jan 30, 2019 at 12:15

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