Kraus rank and Choi representation

PDF p. 3: Second paragraph

Let $H$ and $K$ be finite-dimensional complex inner-product spaces, with $d=\dim H>0$, and fix an ordered orthonormal basis $(e_i{)}_{i=1}^d$ of $H$. Write $\mathcal{L}(H)$ for the linear endomorphisms of $H$. A quantum channel $\mathcal{N}:\mathcal{L}(H)\to \mathcal{L}(K)$ is a complex-linear, completely positive, trace-preserving map: complete positivity means that ${\mathrm{id}}_{\mathcal{L}(E)}\otimes \mathcal{N}$ takes positive operators to positive operators for every finite-dimensional complex inner-product space $E$, and trace preservation means $\mathrm{Tr}\mathcal{N}(X)=\mathrm{Tr}X$ for every $X$.

A finite Kraus representation of $\mathcal{N}$ is a finite family of linear maps $A_a:H\to K$ such that $\mathcal{N}(X)={\sum }_a{A}_{a}X{A}_a^{†}$ for every $X\in \mathcal{L}(H)$, where $A_a^{†}$ is the adjoint. The Kraus rank is the minimum number of operators in such a representation; zero operators are allowed but cannot reduce this minimum.

The input-first, unnormalized Choi matrix relative to the chosen basis is $J={\sum }_{i,j=1}^d\mathrm{\mid }{e}_i⟩⟨e_j\mathrm{\mid }\otimes \mathcal{N}(\mathrm{\mid }{e}_i⟩⟨e_j\mathrm{\mid })$. Given an orthonormal family of eigenvectors $v_k$ for all positive eigenvalues ${\lambda }_k$ of $J$, repeated according to multiplicity, define the associated canonical Kraus operators $K_k:H\to K$ by ${\sum }_i{e}_i\otimes K_k{e}_i=\sqrt{{\lambda }_k}{v}_k$. For a linear map $T:H\to K$, its finite-dimensional Hilbert–Schmidt norm is $\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}=({\sum }_{i=1}^d\mathrm{\parallel }T{e}_i{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$; this value is independent of the orthonormal input basis. Their weights are $w_k=\mathrm{Tr}(K_k^{†}{K}_k)$, the squares of their Hilbert–Schmidt norms.

Finite-dimensional quantum channel

Finite Kraus representation

Choi matrix with input factor first

Canonical Choi eigenvectors, Kraus rank and branch weights

Hilbert–Schmidt operator and Hilbert–Schmidt norm

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Given: The finite-dimensional spaces, ordered orthonormal input basis, channel and canonical operators in the Definition.

1.1 For every linear map $T:H\to K$, the finite-basis specialization of the Hilbert–Schmidt definition is $\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2={\sum }_i\mathrm{\parallel }T{e}_i{\mathrm{\parallel }}^2$. The trace of $T^{†}T$ is the sum of its diagonal coefficients in this orthonormal basis, and the adjoint identity gives each coefficient as $\mathrm{\parallel }T{e}_i{\mathrm{\parallel }}^2$. Hence $\mathrm{Tr}(T^{†}T)={\sum }_i\mathrm{\parallel }T{e}_i{\mathrm{\parallel }}^2=\mathrm{\parallel }T{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2$. Computing in any other orthonormal input basis gives the same basis-independent trace, so the sum and its nonnegative square root are independent of that basis. Hilbert–Schmidt operator and Hilbert–Schmidt norm The basis-independent trace of an endomorphism of a finite-dimensional vector space Finite Kraus representation [given, algebra]

2.1 The canonical Choi lemma applies to precisely the spaces, channel and Choi convention in the Definition. It supplies the constructed finite Kraus representation and states directly that its size $r=\mathrm{rank}J$ is the minimum among all finite Kraus representation sizes. Thus the minimum used to define Kraus rank exists. The lemma also gives $\mathrm{Tr}(K_k^{†}{K}_l)={\lambda }_k{\delta }_{kl}$, so step 1.1 yields $w_k={\lambda }_k=\mathrm{\parallel }{K}_k{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2$. The construction and weights are well defined after choosing the eigenvectors. As the lemma explains, a repeated eigenvalue permits an orthonormal eigenbasis change, so individual canonical operators need not be unique; the positive eigenvalues with multiplicities, and hence the weights, remain unchanged. Canonical Choi eigenvectors, Kraus rank and branch weights [given, step 1.1] ∎