The essential issue in the Hawking paradox boils down to quantum 'entanglement'. Here we explain the notion of entangled states.
We had seen that the vacuum is not really empty in quantum theory. Because of the uncertainty principle, every once in a while there can be a 'quantum fluctuation', which will cause a pair of particles to be created.
We can also imagine a quantum fluctuation in which the positron appeared on the left, and the electron on the right
In quantum mechanics, different configurations are called different states. Thus the above figures represent two different allowed states of the electron and the positron: one where the electron is on the left, and one where the positron is on the left.
Entangled states :   Now we recall a basic fact about quantum theory: if two different states are allowed, then so is their sum. In the present case this sum would give the state
What does such a state mean? In this state, is the electron on the left or on the right?
The answer is the following:
(a) There is a 50% probability that the electron is on the left, and a 50% probability that it is on the right. This is all we can say about the position of the electron. Similarly, there is a 50% chance that the positron on the left and a 50% chance that it is on the right.
(b) But we can actually say more when we talk about both the particles together: if the electron is on the left, then the positron is sure to be on the right, and if the electron is on the right, then the positron will be on the left. Thus the positions of the two particles are 'correlated'.
Such states are called 'entangled states'. In this example, the state of the electron is 'entangled' with the state of the positron. This entanglement is encoded in the if statement involved in the description of the state: if the electron is on the left, then the positron is on the right, and if the electron is on the right, then the positron is on the left. It makes no sense to describe the state of the electron by itself.
The state created by vacuum fluctuations is indeed a state of this entangled kind. Let us now see what this implies for the black hole problem.
Importance for the black hole problem :   We have seen that vacuum fluctuations produce particle pairs near the horizon. One member of the pair is inside, and one outside.
Suppose for concreteness that particles making up the pair are an electron and a positron. Is the electron inside the horizon, or is the positron inside the horizon?
From what we have seen about entangled states, we can guess that the state we will get is an entangled state of the form
Thus the electron is inside if the positron is outside, and the positron is inside if the electron is outside.
This observation is very important for the following reason. Suppose someone were to say: "I don't care about what happens inside the horizon. I live far from the black hole, so I care only about the particles that are far outside the horizon."
We can see however that such an attitude would be inconsistent. One member of the Hawking pair is far outside the hole, having escaped as 'Hawking radiation'. The person outside must be concerned with the state of this particle. But the state of this outer particle makes no sense by itself: the outer particle is an electron if the inner one is a positron, and it is a positron if the inner one is an electron. So we are forced to consider also the state of the particle which is inside the horizon.
We will see that this fact has crucial consequences for the physics of black holes.