Pauli measurement branch semantics

Let $H_n=({\mathbb{C}}^2{)}^{\otimes n}$ be an ordered $n$-qubit register, let $\rho \in M_{2^n}(\mathbb{C})$ be a density operator, and let $M\in M_{2^n}(\mathbb{C})$ be a Hermitian Pauli observable with $M^2=I$. For $b\in \{0,1\}$ define

$P_b:=\frac{I+(-1{)}^{b}M}{2},p_b:=\mathrm{Tr}(\rho P_b),{\mathcal{I}}_b(\rho ):=P_b\rho P_b\mathrm{.}$

The outcome $b$ has probability $p_b$. Its conditional density operator is ${\rho }_b:={\mathcal{I}}_b(\rho )\mathrm{/}{p}_b$ when $p_b>0$; it is $\mathrm{\perp }$ (no assigned conditional density operator) when $p_b=0$. The branch map ${\mathcal{I}}_b$ is the projective measurement instrument for the $(-1{)}^b$ eigenspace of $M$. A non-Hermitian phase-aware Pauli is outside this definition. For $n\ge 1$, Hermiticity of $P(k,x,z)=i^k{X}^x{Z}^z$ means $k\equiv x\cdot z(\mathrm{m}\mathrm{o}\mathrm{d}2)$; in particular $Y=iXZ$ is admitted. For $n=0$, the admitted observables are $I_{H_0}$ and $-I_{H_0}$.

The hypotheses and meanings of density operator, Pauli observable, projective measurement, POVM outcome law, and instrument are the cited suppliers. The local projector calculation is Pauli measurement projectors and total probability ↗.

Ronald de Wolf, Quantum Computing: Lecture Notes, §20.2, PDF pp. 186–188, defines the two Pauli eigenspace projectors and their outcome probabilities: https://arxiv.org/pdf/1907.09415.

John Watrous, Understanding Quantum Information and Computation, Lesson 14, §14.1, printed pp. 405–407 (PDF pp. 411–413), treats a Hermitian Pauli tensor as a two-outcome observable: https://jhwatrous.github.io/UQIC.pdf.