Effective Kraus rank and branch weight

PDF p. 4: First paragraph, Fig. 2

Let $H$ and $K$ be finite-dimensional complex inner-product spaces with $d=\dim H>0$, and let $\mathcal{N}:\mathcal{L}(H)\to \mathcal{L}(K)$ be a quantum channel, meaning a completely positive, trace-preserving linear map. Here $\mathcal{L}(H)$ is the space of linear endomorphisms of $H$. Fix an ordered orthonormal basis $(e_i)$ of $H$ and form the input-first, unnormalized Choi matrix $J={\sum }_{i,j}\mathrm{\mid }{e}_i⟩⟨e_j\mathrm{\mid }\otimes \mathcal{N}(\mathrm{\mid }{e}_i⟩⟨e_j\mathrm{\mid })$.

Choose an orthonormal eigenvector $v_k$ for each positive eigenvalue ${\lambda }_k$ of $J$, repeated with multiplicity. The associated canonical Kraus operator $E_k:H\to K$ is defined by ${\sum }_i{e}_i\otimes E_k{e}_i=\sqrt{{\lambda }_k}{v}_k$. Its weight is $w_k=\mathrm{Tr}(E_k^{†}{E}_k)$. The article's effective Kraus rank is $r_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{\#}\{k:w_k>{10}^{-8}\}$, with a strict inequality. The exact Kraus rank is the minimum number of operators in any finite Kraus representation of the channel; equivalently, it counts all positive weights, including those at or below the threshold.

The branch associated with $E_k$ is the map $X\mapsto E_{k}X{E}_k^{†}$. For the maximally mixed input $I_H\mathrm{/}d$, its probability in this canonical instrument is $w_k\mathrm{/}d$. This is a property of the specified mathematical instrument; a canonical decomposition alone does not identify a unique physical measurement record.

Kraus rank and Choi representation

Canonical Choi eigenvectors, Kraus rank and branch weights

Exact versus thresholded Kraus rank protocol

By [F1], the canonical weights are precisely the positive eigenvalues of the unnormalized Choi matrix, counted with multiplicity. Consequently the count is finite and independent of the choice of orthonormal basis within a repeated-eigenvalue eigenspace, although the individual operators may change. The threshold count cannot exceed exact rank; it is smaller precisely when a positive weight is at most ${10}^{-8}$. By [F2], the stated branch probability follows for the specified instrument. By [F3], using $J\mathrm{/}d$ instead would divide the weights and the equivalent threshold by $d$; the numerical threshold in this definition applies to $J$ itself. Canonical Choi eigenvectors, Kraus rank and branch weights Exact versus thresholded Kraus rank protocol