ERRATUM: The D-Wave machine is a qubit-based quantum computer. I wanted to contrast it with gate-based, universal quantum computers.
Addition to “The core idea” slide: Here, the phrase “multiple classical states” means classical states that are mutually exclusive, that is, according to our everyday intuition, shouldn’t co-exist simultaneously. Also, isolation is very important: formally, quantum mechanics assigns a state (vector) only to isolated physical systems. So, when I later say that a system “has a quantum state”, it’s implicitly understood that the system is isolated (with the caveat that when the quantum state happens to be a classical one, we are more lenient and allow that the system might not be isolated, as no “extreme effort” is needed to create a classical state).
Addition to the “Entanglement – The EPR pair” slides: Quantum mechanics guarantees that any isolated system has a quantum state of its own. To make the example clearer, note that the EPR pair as a whole is isolated (as it has a quantum state). Then, the fact that the two individual qubits cannot have their own states implies that they are NOT isolated from each other (within the EPR pair system).
Addition to the “Quantum gates & information” slides: We assume that before applying the gate U, the state of the environment is independent of |ψ>. So if our “detective” is only allowed to examine the environment to figure out |ψ>, then applying a quantum gate won’t help her, as she still won’t be able to correlate |ψ> with the new state of the environment.