Two more intimidating words, one simple idea. Then you are done with this track.
Multiply most arrows by a grid and they swing round to point somewhere new. But a few special arrows do not turn at all — they only get longer or shorter.
Those are the grid's eigenvectors, and how much they stretch is the eigenvalue.
- before the operator
- after the operator
Why anyone cares
Three facts, and together they explain most of the notation in this course:
- The results you can get are the eigenvalues of whatever question you asked.
- After you look, the qubit becomes the matching eigenvector. That is "collapse", stated exactly.
- The chance of each result is the overlap from two lessons ago, squared.
So "measuring the normal way" means asking the question Z, whose special directions happen to be |0⟩ and |1⟩.
Worked out in one line
Z is the grid [[1, 0], [0, −1]].
Z|0⟩ = +1 × |0⟩— so |0⟩ is special, with value +1Z|1⟩ = −1 × |1⟩— so |1⟩ is special, with value −1
Ask Z and you get +1 or −1. Relabel those as 0 and 1 and you have the ordinary measurement you already knew.
Ask X instead and the special directions are |+⟩ and |−⟩. That is the "sideways" measurement — and it is exactly the second basis used in the BB84 protocol in the Cryptography track.
Where this shows up next
Pattern finding
The engine inside Shor's codebreaking algorithm is an eigenvalue hunt
Chemistry
Finding a molecule's lowest energy means finding a smallest eigenvalue
Error fixing
The 'do these agree?' checks are eigenvalue questions in disguise
Three of the most important techniques in the whole field are, underneath, this one idea.
Worth remembering
- Eigenvectors are the arrows a grid does not turn; eigenvalues are how much they stretch.
- Measurement results are eigenvalues, and afterwards the qubit sits on the matching eigenvector.
- Measuring the usual way is asking Z; measuring sideways is asking X.
- Codebreaking, chemistry and error-fixing are all this idea wearing different hats.