Finite tensor density, observable and measurement language

Fix an integer $q\ge 1$, let $V={\mathbb{C}}^q$ with its standard ordered orthonormal basis $(\mathrm{\mid }j⟩{)}_{j=1}^q$, and use the inner-product convention that is linear in the first argument. For each integer $N\ge 0$, write $V_N=V^{\otimes N}$ with the displayed factor order; $V_0=\mathbb{C}$. The lexicographically ordered product vectors form the chosen basis of $V_N$, as in Ordered tensor registers, their inner product, and wire permutations.

A finite tensor density on $V_N$ is a density operator in the exact root-library sense of Density operator: a nonnegative endomorphism $\rho$ with $\mathrm{Tr}\rho =1$. In the standard column coordinates this means $\rho ={\rho }^{\ast }$ and $x^{\ast }\rho x\ge 0$ for every $x$.

A bounded observable matrix on $V_N$ is a Hermitian matrix $A=A^{\ast }$, with operator norm $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}={\sup }_{\mathrm{\parallel }x\mathrm{\parallel }=1}\mathrm{\parallel }Ax\mathrm{\parallel }$. Its mathematical expectation functional in $\rho$ is ${\omega }_{\rho }(A)=\mathrm{Tr}(\rho A)$. This is a finite matrix trace pairing; its physical interpretation is supplied separately by a measurement postulate.

A finite POVM with $r\ge 1$ outcomes is a family $M_1,\dots ,M_r$ of nonnegative matrices on $V_N$ with ${\sum }_{a=1}^r{M}_a=I$. Each $M_a$ is an effect, meaning $0\le M_a\le I$; the upper bound follows because $I-M_a={\sum }_{b\mathrm{\ne }a}{M}_b\ge 0$. The numbers $\mathrm{Tr}(\rho M_a)$ are its finite trace pairings. They are nonnegative and normalized when read as outcome probabilities under the adopted physical Born rule. Here $M\ge 0$ means $M=M^{\ast }$ and $x^{\ast }Mx\ge 0$ for every column vector $x$. This definition alone does not adopt the physical probability rule.