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An isometry followed by partial trace is a channel

Statement

Let $H, K, E$ be finite-dimensional complex inner-product spaces and let $V : H \to K \otimes E$ be a linear isometry. Then $Φ (X) = {Tr}_{E} (V X V^{†}), X \in L (H),$ is a quantum channel from $H$ to $K$ .

Facts & Assumptions

Given: Finite-dimensional complex inner-product spaces $H, K, E$ and a linear isometry $V : H \to K \otimes E$ .

[F1]

Isometric Stinespring representation: the represented map is $X \mapsto {Tr}_{E} (V X V^{†})$ and the isometry satisfies $V^{†} V = I_{H}$ by Finite-dimensional linear isometries satisfy the adjoint identity.

[F2]

Ordered tensor registers, their inner product, and wire permutations and The product basis is orthonormal and tensor dimensions multiply: if $(f_{α})$ and $(u_{a})$ are orthonormal bases of $K$ and $E$ , then $(f_{α} \otimes u_{a})$ is an orthonormal basis of $K \otimes E$ with the displayed factor order.

[F3]

Partial trace over a finite tensor factor and The partial trace is characterized by the trace pairing and is basis-independent: for any such output and environment bases, $({Tr}_{E} R)_{α β} = \sum_{a} R_{(α, a), (β, a)}$ .

[F4]

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Every linear map between finite-dimensional inner product spaces has a unique adjoint, and Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ : finite-dimensional adjoints exist uniquely, obey their defining pairing identity, and reverse compatible compositions.

[F5]

Inner products separate vectors, and the induced norm is homogeneous: $∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$ : inner-product pairings separate vectors.

[F6]

A finite Kraus sum is completely positive: every finite Kraus sum is completely positive.

[F7]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : for compatible rectangular maps, the traces of the two products agree.

[F8]

Trace is a linear functional on $M_{n} (F)$ : trace is linear over finite sums.

[F9]

Finite-dimensional quantum channel: a channel is a complex-linear completely positive map that preserves trace on every input operator.

Proof

1.1givenF2F3F4F5chooseconstructalgebra

Choose ordered orthonormal bases $(f_{α})$ of $K$ and $(u_{a})$ of $E$ by finite-dimensional orthonormal-basis existence. For each $a$ , define the linear map $P_{a} : K \otimes E \to K$ on the product basis by $P_{a} (f_{α} \otimes u_{b}) = f_{α}$ if $b = a$ and $0$ otherwise. Its adjoint satisfies $P_{a}^{†} f_{α} = f_{α} \otimes u_{a}$ : both sides have the same pairing with every product basis vector by [F2] and [F4]. Linearity extends the equality of pairings to every first argument, and [F5] then makes the vectors equal. For any $R \in L (K \otimes E)$ , the $(α, β)$ entry of $\sum_{a} P_{a} R P_{a}^{†}$ is $\sum_{a} R_{(α, a), (β, a)}$ , which is the defining coordinate of ${Tr}_{E} R$ by [F3]. Thus ${Tr}_{E} R = \sum_{a} P_{a} R P_{a}^{†}$ .

2.1givenF1F4F6step 1.1algebra

Put $L_{a} = P_{a} V : H \to K$ . By [F4], $L_{a}^{†} = V^{†} P_{a}^{†}$ . Substituting $R = V X V^{†}$ into step 1.1 gives $Φ (X) = \sum_{a} P_{a} V X V^{†} P_{a}^{†} = \sum_{a} L_{a} X L_{a}^{†}$ for every $X$ . This is a finite Kraus representation, so [F6] makes $Φ$ completely positive.

3.1givenF1F4F7F8F9step 2.1algebra∎

On each product basis vector $f_{α} \otimes u_{b}$ , the sum $\sum_{a} P_{a}^{†} P_{a}$ acts as the identity; hence $\sum_{a} P_{a}^{†} P_{a} = I_{K \otimes E}$ . By [F4] and [F1], $\sum_{a} L_{a}^{†} L_{a} = V^{†} (\sum_{a} P_{a}^{†} P_{a}) V = V^{†} V = I_{H}$ . Rectangular trace cyclicity [F7] and trace linearity [F8] then give, for every $X \in L (H)$ , ${tr}_{K} Φ (X) = \sum_{a} {tr}_{K} (L_{a} X L_{a}^{†}) = {tr}_{H} (X \sum_{a} L_{a}^{†} L_{a}) = {tr}_{H} X$ . The complete positivity from step 2.1 and this trace identity are exactly the channel conditions [F9]. Empty bases and sums give the same identities when $E = 0$ ; the isometry equation then forces $H = 0$ .

Depends on

Used by

Complementary channels and coherent information Definition
Every finite extension is obtained from a purification by a channel Lemma
Flat-input coherent information is achievable by subspace codes Lemma
Finite channel, Kraus, Choi, and Stinespring equivalence Theorem
Fuchs Van De Graaf Theorem
Full Rank Pair No Broadcasting Theorem
Pure State No Cloning Theorem
The finite-dimensional erasure map is a quantum channel Theorem

Dependency tree · two levels

57 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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