Notes - Quantum Information HT24, Misc
Flashcards
Suppose we undergo some process and end up with a quantum state that is either $ \vert \alpha\rangle$ with probability $p(\alpha)$ or $ \vert \beta\rangle$ with probability $p(\beta) = 1 - p(\alpha)$. Given that $p(\alpha) \to 1$ as the process is extended to infinity, how could we show mathematically that we converge to the state $ \vert \alpha\rangle$?
Consider the state represented as a density matrix:
\[\rho = p(\alpha)|\alpha\rangle\langle\alpha| + (1 - p(\alpha))|\beta\rangle\langle \beta|\]Then if $p(\alpha) \to 1$, we see
\[\rho \to |\alpha\rangle \langle \alpha|\]