Finite bounded Kraus instrument

Assume the Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a nonzero complex Hilbert space with inner product linear in its first argument (Hilbert space, Real and complex inner-product spaces and their induced length). Write $\mathcal{B}(H)$ for the bounded linear operators on $H$ (The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators); for $M\in \mathcal{B}(H)$, $M^{\ast }$ is its Hilbert adjoint, characterized by $⟨Mx,y⟩=⟨x,M^{\ast }y⟩$ (The Hilbert-space adjoint of a bounded operator).

Under this same ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis, Hilbert-adjoint identities gives $M^{\ast }\in \mathcal{B}(H)$ and $\mathrm{\parallel }{M}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }M\mathrm{\parallel }$.

A finite bounded Kraus instrument datum on $H$ consists of an integer $m\ge 1$, integers $q_a\ge 0$ for $a=1,\dots ,m$, and bounded operators $M_{a,r}\in \mathcal{B}(H)$ for $r=1,\dots ,q_a$. It is normalized when

${\sum }_{a=1}^m{\sum }_{r=1}^{q_a}{M}_{a,r}^{\ast }{M}_{a,r}=I_H\mathrm{.}$

Empty inner sums are zero. For each outcome $a$, define its outcome operator and operation by

$E_a:={\sum }_{r=1}^{q_a}{M}_{a,r}^{\ast }{M}_{a,r},{\mathcal{J}}_a(X):={\sum }_{r=1}^{q_a}{M}_{a,r}X{M}_{a,r}^{\ast }(X\in \mathcal{B}(H))\mathrm{.}$

These formulas take values in $\mathcal{B}(H)$: the adjoint theorem above makes each $M_{a,r}^{\ast }$ bounded, and $\mathrm{\parallel }{M}_{a,r}X{M}_{a,r}^{\ast }x\mathrm{\parallel }\le \mathrm{\parallel }{M}_{a,r}\mathrm{\parallel }\mathrm{\parallel }X\mathrm{\parallel }\mathrm{\parallel }{M}_{a,r}^{\ast }\mathrm{\parallel }\mathrm{\parallel }x\mathrm{\parallel }=\mathrm{\parallel }{M}_{a,r}{\mathrm{\parallel }}^2\mathrm{\parallel }X\mathrm{\parallel }\mathrm{\parallel }x\mathrm{\parallel }$; the sums are finite. This definition specifies only the displayed finite Kraus form; it is not a representation theorem for arbitrary instruments and does not assert that an apparatus realizes these data.