= Quantum state ensemble
{title2=$\{p_i,|\psi_i\rangle\},\quad\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$}
= Quantum ensemble
{synonym}
A quantum state ensemble specifies a random preparation: normalized <pure states> $|\psi_i\rangle$ are selected with <probabilities> $p_i\geq0$, $\sum_i p_i=1$. Without an accessible preparation label, its measurement statistics are those of the <density operator> $\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$. Indeed every <POVM> effect $E$ has probability $\sum_i p_i\langle\psi_i|E|\psi_i\rangle=\operatorname{Tr}(E\rho)$. Different ensembles can therefore have identical operational statistics: equal mixtures of computational-basis qubits or Hadamard-basis qubits both have density operator $I/2$. The ensemble decomposition contains information about the preparation, not additional information accessible from the system alone.
Back to article page
