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Orthogonal error restrictions determine syndrome projections

Statement

Let $H_{L}, H_{O}$ be finite-dimensional complex inner-product spaces, and let $W_{i} : H_{L} \to H_{O}$ be finitely many isometries with pairwise orthogonal ranges. Then $P_{i} = W_{i} W_{i}^{*}$ are pairwise orthogonal projections, their sum $S = \sum_{i} P_{i}$ is an orthogonal projection satisfying $0 \leq S \leq I_{H_{O}}$ , and $Q = I_{H_{O}} - S$ is the orthogonal projection onto the residual complement. The empty family is allowed, with $S = 0$ and $Q = I_{H_{O}}$ .

Facts & Assumptions

Given: Finite-dimensional complex inner-product spaces $H_{L}, H_{O}$ and a finite family of isometries $W_{i} : H_{L} \to H_{O}$ with pairwise orthogonal ranges.

[F2]

An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint: a self-adjoint idempotent on a finite-dimensional inner-product space is an orthogonal projection.

[F3]

The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$ : an orthogonal projection selects the component in its range along the orthogonal complement.

[F4]

Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ : $(A B)^{*} = B^{*} A^{*}$ and $T^{* *} = T$ for compatible finite-dimensional maps.

[F5]

Non-negative and positive operators: $0 \leq T$ means $T$ is self-adjoint and $⟨ T x, x ⟩ \geq 0$ for every $x$ .

[F6]

Real and complex inner product spaces, with the inner product linear in the first argument: $⟨ x, x ⟩ = ∥ x ∥^{2}$ and the inner product is positive definite.

[F7]

An isometric quantum encoding gives an orthogonal code projector: every finite-dimensional linear isometry $W$ satisfies $W^{*} W = I$ .

[F8]

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ and Every linear map between finite-dimensional inner product spaces has a unique adjoint: adjoints exist and satisfy $⟨ W x, y ⟩ = ⟨ x, W^{*} y ⟩$ .

Proof

1.1givenF2F4F7F8algebra

By [F7, F8], each $W_{i}$ has an adjoint and $W_{i}^{*} W_{i} = I$ . For each $i$ , [F4] gives $P_{i}^{*} = (W_{i} W_{i}^{*})^{*} = W_{i} W_{i}^{*}$ , and $P_{i}^{2} = W_{i} (W_{i}^{*} W_{i}) W_{i}^{*} = W_{i} W_{i}^{*} = P_{i}$ by [F7]. The image of $P_{i}$ is contained in $ran W_{i}$ , while $P_{i} W_{i} x = W_{i} x$ for every $x$ ; hence its image equals $ran W_{i}$ . By [F2], $P_{i}$ is the orthogonal projection onto that range.

2.1givenstep 1.1F2F4F6F8algebra

If $i \neq j$ , orthogonality of the ranges gives $⟨ x, W_{i}^{*} W_{j} y ⟩ = ⟨ W_{i} x, W_{j} y ⟩ = 0$ for all $x, y \in H_{L}$ by [F8]. Taking $x = W_{i}^{*} W_{j} y$ shows $W_{i}^{*} W_{j} = 0$ by [F6]. Hence $P_{i} P_{j} = W_{i} (W_{i}^{*} W_{j}) W_{j}^{*} = 0$ . Thus $S^{*} = S$ and $S^{2} = \sum_{i} P_{i}^{2} + \sum_{i \neq j} P_{i} P_{j} = S$ ; [F2] makes $S$ an orthogonal projection. The same calculation gives $(I - S)^{*} = I - S = (I - S)^{2}$ , so [F2] makes $Q = I - S$ an orthogonal projection.

3.1step 1.1step 2.1F2F3F5F6algebra∎

Each $P_{i}$ fixes its own range and kills the other ranges, so $ran S$ is exactly the orthogonal sum of the $ran W_{i}$ . Since $Q = I - S$ is an orthogonal projection, its range is $\ker S$ , the orthogonal complement of the range of $S$ by [F3]. For every $x \in H_{O}$ , $⟨ S x, x ⟩ = ∥ S x ∥^{2} \geq 0$ and $⟨ Q x, x ⟩ = ∥ Q x ∥^{2} \geq 0$ , so [F5] gives $0 \leq S \leq I$ . For the empty family the sums give $S = 0, Q = I$ directly.

Depends on

Used by

An explicit syndrome recovery corrects a Knill–Laflamme error family Theorem

Dependency tree · two levels

22 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 Preskill, Quantum Information, Chapter 7: Quantum Error Correction (working draft, March 2026) (standard reference, not scraped)
David W. Kribs, Raymond Laflamme, David Poulin, and Maia Lesosky, Operator Quantum Error Correction (standard reference, not scraped)