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:
\[\rho = p(\alpha) \vert \alpha\rangle\langle\alpha \vert + (1 - p(\alpha)) \vert \beta\rangle\langle \beta \vert\]Then if $p(\alpha) \to 1$, we see
\[\rho \to \vert \alpha\rangle \langle \alpha \vert\]