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Kraus rank equals Choi rank

PDF p. 3: Second paragraph, Ref. [43]

Statement

Let $H$ and $K$ be finite-dimensional complex inner-product spaces, with $d = \dim H > 0$ , and let $N : L (H) \to L (K)$ be a quantum channel, meaning a completely positive trace-preserving linear map. Fix an ordered orthonormal basis $(e_{i})_{i = 1}^{d}$ of $H$ . A Kraus representation is a finite family of linear maps $A_{a} : H \to K$ satisfying $N (X) = \sum_{a} A_{a} X A_{a}^{†}$ for every $X \in L (H)$ . Its Kraus rank is the minimum number of operators in such a representation. Define the input-first, unnormalized Choi matrix by $J = \sum_{i, j = 1}^{d} ∣ e_{i} ⟩ ⟨ e_{j} ∣ \otimes N (∣ e_{i} ⟩ ⟨ e_{j} ∣)$ . Then the Kraus rank of $N$ equals $rank J$ .

Statement references

Kraus rank and Choi representation

Facts & Assumptions

Given: The spaces, channel, basis and Choi matrix in the Statement.

Goal: Show that the minimum size of a Kraus representation equals $rank J$ .

[F1]

For these finite-dimensional spaces and this Choi convention, the canonical Choi decomposition gives a Kraus representation with $r = rank J$ operators, and every Kraus representation has at least $r$ operators. Canonical Choi eigenvectors, Kraus rank and branch weights

[F2]

The Kraus rank is the minimum size of a finite Kraus representation, and the displayed formula is the input-first, unnormalized Choi convention. Kraus rank and Choi representation

Proof

Given: The spaces, channel, basis and Choi matrix in the Statement.

Goal: Show that the minimum size of a Kraus representation equals $rank J$ .

1.1givenF1F2

The spaces, channel and basis satisfy the hypotheses of [F1], and the Choi matrix in [F1] is the matrix $J$ of the Statement by [F2]. Thus a Kraus representation with $r = rank J$ operators exists, while every such representation has at least $r$ operators.

2.1F2step 1.1∎

By the definition in [F2], the minimum representation size is therefore both at most and at least $r$ . It equals $r = rank J$ , as claimed.

Depends on

Used by · 2

Dependency tree · 2 levels

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 3, second paragraph