There is one more piece of notation, and like the others it hides something simple.
⟨φ|ψ⟩ is one number. It says how much these two states overlap: how alike they are.
What the number means
Zero — no overlap at all. The two states are opposites, and one measurement can tell them apart perfectly, every time.
One — same state, give or take an invisible phase.
Anything in between — partly alike, and you will sometimes mistake one for the other. No cleverness fixes this. It is a fact about the universe, not about your equipment.
Two famous results fall straight out of this
You cannot copy a qubit. A copier would have to work on any state you hand it. But copying would change how much states overlap, and gates are not allowed to do that. So no such machine can exist. Three physicists proved it in 1982, and it is called the no-cloning theorem.
Quantum key sharing works. In the Cryptography track, Alice deliberately sends states that partly overlap. An eavesdropper cannot tell them apart, cannot copy them, and cannot avoid leaving fingerprints. All of it traces back to this one number.
Orthogonal
The posh word for "no overlap at all" — the zero case. Orthogonal states are the only ones you can reliably tell apart.
And the Born rule, one more time
The amount of 0 in a state is exactly ⟨0|ψ⟩. So the rule from the last lesson can be written P(0) = |⟨0|ψ⟩|².
That looks like showing off, but it is genuinely useful: written this way, the rule works for any way of measuring, not just the usual one.
Worth remembering
- ⟨φ|ψ⟩ is a single number measuring how alike two states are.
- Zero overlap means perfectly distinguishable. Anything else means guesswork.
- Copying a qubit is impossible, and this is why.
- The Born rule written with overlaps works for any kind of measurement.