= Unitary dilation of a measurement instrument
For <Kraus operators> satisfying $\sum_iA_i^\dagger A_i=I$, the map $V|\phi\rangle=\sum_iA_i|\phi\rangle\otimes|i\rangle$ is an <isometry>. Implement it by a <unitary operator> on the system and a <quantum ancilla>, initially in an extra unused state $|0\rangle$. Projecting the output <quantum ancilla> onto $|i\rangle$ gives joint unnormalized state $A_i\rho A_i^\dagger\otimes|i\rangle\langle i|$. The outcome <probability> is $\operatorname{Tr}(A_i\rho A_i^\dagger)$, and the conditional system state is $A_i\rho A_i^\dagger$ divided by this <probability>. The unused level can be merged into the first <orthogonal projection> without changing any outcome on the prepared state, giving a complete <projective measurement> with exactly the original outcome labels.
// Target: quantum-theory.bigb
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