# What is the quantum map that describes a measurement where the measurement outcome is known?

My question is the other case of this question.

Suppose I do a measurement in the $$Z$$ basis with outcomes $$\lambda_0, \lambda_1$$ but I don't record the outcome. In this case, I can describe what happened with a quantum channel $$\mathcal{E}$$. It has Kraus operators $$\{\langle 0\vert \otimes \vert \lambda_0\rangle, \langle 1\vert \otimes \vert \lambda_1\rangle \}$$. Acting on a state $$\rho$$, I get

$$\mathcal{E}(\rho) = \langle 0\vert \rho \vert 0\rangle \otimes \vert \lambda_0\rangle\langle \lambda_0\vert + \langle 1\vert \rho \vert 1\rangle \otimes \vert \lambda_1\rangle\langle \lambda_1\vert$$

What if I do record the outcome? In this case, how do I describe the map? It has to take an input $$\rho$$ and give me an output $$\vert 0\rangle\langle 0\vert$$ if I see $$\lambda_0$$ and $$\vert 1\rangle\langle 1\vert$$ if I see $$\lambda_1$$

Note that the $$\mathcal E$$ is doing the opposite of what you want: it produces state $$|\lambda_i\rangle$$ if you observe result $$i$$.
• @Jorge if you replace your state with another state $\sigma$, then the map is $\mathcal{E}(\rho)=\mathrm{tr}(\rho)\sigma$ Commented Jul 24 at 15:37
• @TristanNemoz I understand, but in this case, isn't the replacement state $\sigma$ conditioned on the measurement outcome? Can one include that conditioning in the description of the channel or does it fail due to the non-determinism and non-linearity? Thanks! Commented Jul 24 at 15:38
• @Jorge Yes it is, this is just the general representation of the replacement channel. If you know that you measured $0$ for instance, you can describe the operation you performed as $\rho\mapsto\mathrm{tr}(\rho)|0\rangle\!\langle0|$. Fundamentally, this is what you did: you took a state $\rho$ and you transformed it into the other state. Another way of writing it is, if you get the measurement outcome $x$, then the quantum channel you applied is $\rho\mapsto\mathrm{tr}(\rho)|x\rangle\!\langle x|$ Commented Jul 24 at 15:59