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Thinking about gate creating entangled state, it looks to me that the inputs to the gate look like marginal/independent distributions, and the output looks like their joint distribution. Does this process have any statistical meaning?

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  • $\begingroup$ what do you mean exactly with "the inputs look like the marginal/independent distributions"? How does an input/output "look like a distribution"? $\endgroup$
    – glS
    Dec 27, 2021 at 9:37
  • $\begingroup$ Squared qubit's amplitudes sum to one (as well as any product state), entanglement looks like dependent joint distribution, and no-entanglement mean that factoring to two separate states is possible (again, the same behavior as independent random variables). So the gate creates some statistical joint distribution (separable or not) - the question is, how can this process be described from statistical point of view. $\endgroup$
    – sitems
    Dec 27, 2021 at 11:56
  • $\begingroup$ that is still unclear. When you say "entanglement looks like dependent joint distribution", I assume you mean "entangled states produce correlated (marginal) probability distributions"? That is true, but misleading: separable states can also give correlated probability distributions: consider as a trivial such example a maximally mixed state. $\endgroup$
    – glS
    Dec 27, 2021 at 14:59

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The question is a bit ambiguous but yes you can think of unentangled states as independent and the process of entanglement as "adding" correlations. Take the following unentangled state for instance:

$$\vert 00 \rangle + \vert 10 \rangle$$

Then the probability distribution of each qubit is independent of the other:

$$P(q_1|q_0) = P(q_1)\>\> and \>\>P(q_0|q_1) = P(q_0)$$

But by performing a $C_{not}$ with $q_0$ as the target we put the states into complete entanglement or correlation (statistically):

$$\vert 00 \rangle + \vert 11 \rangle$$ $$P(q_1|q_0) = P(q_0)\>\> and \>\>P(q_0|q_1) = P(q_1)$$

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