[Yajirobe]

So why does the Hadamard+CNOT acting on two qubits give an entangled state, but a Hadamard acting on a single qubit does not give an entangled state?

[nonsapreiche]

I think you need 2 qbit to have them entangled

[onorton]

Exactly. The point of quantum entanglement is that the state of two (or more) qubits cannot be separated. To entangle a single qubit is meaningless.

For two qubits, the simplest entangled states are the Bell States[0] (generated from a CNOT and Hadamard gate). The article gives an example of one of them.

[0] https://en.wikipedia.org/wiki/Bell_state

[Yajirobe]

What's the difference between an entangled state and a mixed state?

[onorton]

Knowledge is a bit rusty (took a module in Uni) but I'll try to answer.

A mixed state is that which is a linear combination of pure states e.g. a|0> + b|1>

What determines an entangled state is that qubit values will correlate exactly with each other. |00> + |11> would be an example of an entangled state as measuring one qubit determines the value of the other with certainty. If you measure |0> for the first qubit, the second will definitely be |0> and vice versa.

They are also not mutually exclusive as mixed entangled states exist.

[Yajirobe]

> A mixed state is that which is a linear combination of pure states e.g. a|0> + b|1>

Here https://en.wikipedia.org/wiki/Qubit#Mixed_state it says that 'Mixed states can be represented by points inside the Bloch sphere'. However, points ON the sphere correspond to linear combinations of the pure states. How to reconcile this?

[onorton]

Like I said, my knowledge is rusty.

I've gotten pure states mixed up with |0>, |1>

I believe what it's saying is that with a mixed state |a|^2 + |b|^2 does not need to equal 1.

So pure states are a|0> + b|1> where |a|^2 + |b|^2 = 1