= Measurement backaction
{title2=$\rho\mapsto K_m\rho K_m^\dagger/\operatorname{Tr}(K_m\rho K_m^\dagger)$}
Measurement backaction is the effect of a <quantum measurement> on later predictions for the measured system. A <quantum instrument> with one <Kraus operator> $K_m$ per recorded outcome has
$$
p_m=\operatorname{Tr}(K_m\rho K_m^\dagger),\qquad
\rho_m=\frac{K_m\rho K_m^\dagger}{p_m},\qquad
\sum_m K_m^\dagger K_m=I.
$$
The conditional <density operator> $\rho_m$ exists only when $p_m>0$. Its nonlinear normalization expresses conditioning; the unnormalized map $\rho\mapsto K_m\rho K_m^\dagger$ is linear. More generally, an unresolved outcome can contain several <Kraus operators>: replace the numerator by $\sum_a K_{ma}\rho K_{ma}^\dagger$ and take its <trace> for $p_m$. https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf[Watrous, The Theory of Quantum Information, §2.3.2]{ref}.
Keeping the outcome gives a <selective quantum measurement>. Discarding it gives a <nonselective quantum measurement> with <quantum channel> $\rho\mapsto\sum_m p_m\rho_m$. The latter describes the whole measured population without its labels, whereas each $\rho_m$ describes a selected subpopulation. Neither description requires the measured quantity to have possessed its eventual value before the interaction.
For a concrete test, prepare the <Pauli X eigenstate> $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$. Its $X=+1$ <probability> is one. An ideal $Z$ <projective measurement> gives zero or one with equal <probability>, leaving $|0\rangle$ or $|1\rangle$ under the <Lüders rule>. Discarding the result leaves the <maximally mixed state> $I/2$. A subsequent $X$ measurement now gives either sign with <probability> one half, even if the $Z$ result was kept. This observable change in statistics is backaction. Measuring $X$ on the original $|+\rangle$ instead leaves it unchanged: disturbance depends on both the preparation and the instrument. <Measurement-based quantum feedback> uses the recorded outcome to choose a subsequent control operation, so both parts of the instrument matter.
Back to article page
