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Finite channel, Kraus, Choi, and Stinespring equivalence

Statement

Let $H, K$ be finite-dimensional complex inner-product spaces, fix an ordered orthonormal basis $B$ of $H$ , and let $Φ : L (H) \to L (K)$ be complex-linear. The following four conditions are equivalent:

$Φ$ is a quantum channel.
$J_{B} (Φ) ⪰ 0$ and ${Tr}_{K} J_{B} (Φ) = I_{H}$ .
$Φ (X) = \sum_{a} K_{a} X K_{a}^{†}$ for a finite family of maps $K_{a} : H \to K$ with $\sum_{a} K_{a}^{†} K_{a} = I_{H}$ .
There are a finite-dimensional $E$ and an isometry $V : H \to K \otimes E$ with $Φ (X) = {Tr}_{E} (V X V^{†})$ for every $X$ .

The Choi order is input factor first, output factor second.

Facts & Assumptions

Given: Finite-dimensional complex inner-product spaces $H, K$ , an ordered orthonormal basis $B$ of $H$ , and a complex-linear map $Φ$ .

[F1]

Finite-dimensional quantum channel (same-page definition): condition 1 means complete positivity and trace preservation.

[F2]

Kraus and Choi characterizations of complete positivity (same-page theorem): complete positivity is equivalent to Choi non-negativity and to existence of a finite Kraus representation.

[F3]

Trace preservation is Kraus normalization (same-page theorem): trace preservation of a Kraus map is equivalent to $\sum_{a} K_{a}^{†} K_{a} = I_{H}$ .

[F4]

The Choi partial trace characterizes trace preservation (same-page theorem): for a completely positive $Φ$ , trace preservation is equivalent to ${Tr}_{K} J_{B} (Φ) = I_{H}$ .

[F5]

Every finite-dimensional channel has a finite isometric dilation (same-page theorem): every finite-dimensional channel has an isometric Stinespring representation.

[F6]

An isometry followed by partial trace is a channel (same-page theorem): every isometric Stinespring form is a channel.

[L1]

Choi matrix with input factor first (same-page definition): $J_{B} (Φ)$ uses the fixed input basis and input-first/output-second tensor convention.

[L2]

Isometric Stinespring representation (same-page definition): an isometric dilation has the displayed partial-trace form.

Proof

1.1givenF1F2F4L1

Condition 1 implies condition 2: its complete-positivity part and [F2] give $J_{B} (Φ) ⪰ 0$ , while its trace-preservation part and [F4] give ${Tr}_{K} J_{B} (Φ) = I_{H}$ . Conversely, condition 2 gives complete positivity by [F2] and trace preservation by [F4], hence condition 1 by [F1].

1.2givenF1F2F3L1

Conditions 1 and 3 are equivalent: by [F2], condition 1's complete positivity is equivalent to a finite Kraus representation, and by [F3] its trace-preservation clause is equivalent to the stated normalization. The same two results show condition 3 implies both clauses of condition 1.

1.3givenF5F6L2

By [F5], condition 1 implies condition 4. By [F6], condition 4 implies condition 1. Thus condition 4 is equivalent to condition 1.

2.1step 1.1step 1.2step 1.3L1L2∎

Steps 1.1, 1.2, and 1.3 connect each of conditions 2, 3, and 4 to condition 1 in both directions, so all four conditions are equivalent. Throughout, the Choi factor order remains the one in [L1], and the environment occupies the second factor of $K \otimes E$ in [L2].

Depends on

Used by

Qubit dephasing channel in Kraus and Choi forms Example

Dependency tree · two levels

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Sources

John Watrous, The Theory of Quantum Information (2018), §§2.2.2–2.2.3 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3, §§3.2.6–3.3.2 (standard reference, not scraped)