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A degenerate code with a non-diagonal Knill–Laflamme matrix

Example

Take $H_{L} = C^{2}$ , $H_{P} = C^{2} \otimes C^{2}$ , with standard basis columns $e_{0} = (1, 0)^{T}$ and $e_{1} = (0, 1)^{T}$ , and define $V ψ = e_{0} \otimes ψ$ and $W ψ = e_{1} \otimes ψ$ . Put $P_{0} = e_{0} e_{0}^{*} = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ and $P_{1} = e_{1} e_{1}^{*} = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array})$ . On the first factor use $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ and $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ . Define a channel with Kraus operators

$E_{1} = \frac{I_{4}}{\sqrt{3}}, E_{2} = \frac{X \otimes I_{2}}{\sqrt{3}}, E_{3} = \frac{Z \otimes I_{2}}{\sqrt{3}} .$

The code stores the logical qubit in the second factor. The errors $E_{1}$ and $E_{3}$ have the same restriction to the code, while $E_{2}$ moves the code to the orthogonal first-factor subspace.

Facts & Assumptions

Given: The finite-dimensional encoding and the three explicitly displayed error operators in the Example.

[F1]

Quantum code as an isometric encoding and its code projector supplies the isometric encoding and code-projector convention.

[F2]

Finite-dimensional quantum channel, Finite Kraus representation, and Trace preservation is Kraus normalization specify finite Kraus channels and their trace-preserving normalization.

[F3]

An explicit syndrome recovery corrects a Knill–Laflamme error family: the displayed recovery construction corrects every finite family satisfying the scalar-compression condition.

[F4]

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

[F5]

Trace preservation is Kraus normalization: the adjoint-square sum of a trace-preserving Kraus family is the identity.

Verification

1.1givenF2F4F5algebra

Each of $I_{4}$ , $X \otimes I_{2}$ , and $Z \otimes I_{2}$ is unitary. Hence $\sum_{a = 1}^{3} E_{a}^{*} E_{a} = \frac{1}{3} (I_{4} + I_{4} + I_{4}) = I_{4}$ , so the displayed Kraus map is trace-preserving; it is completely positive because it is a finite Kraus sum.

2.1givenF1step 1.1algebra

With $P = V V^{*}$ , direct compression gives $V^{*} E_{a}^{*} E_{b} V = C_{a b} I_{L}, C = \frac{1}{3} (\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}) .$ Since $P = V V^{*}$ , this is equivalently $P E_{a}^{*} E_{b} P = C_{a b} P$ for every $a, b$ . In particular, $C$ is non-diagonal and has rank $2$ .

3.1step 2.1algebra

The orthonormal eigenvectors $u_{+} = (1, 0, 1) / \sqrt{2}$ , $u_{2} = (0, 1, 0)$ , and $u_{0} = (1, 0, - 1) / \sqrt{2}$ have eigenvalues $2 / 3, 1 / 3, 0$ , respectively. The corresponding mixed Kraus operators are $F_{+} = (I_{4} + Z \otimes I_{2}) / \sqrt{6} = \sqrt{2 / 3} (P_{0} \otimes I_{2})$ , $F_{2} = E_{2}$ , and $F_{0} = (I_{4} - Z \otimes I_{2}) / \sqrt{6} = \sqrt{2 / 3} (P_{1} \otimes I_{2})$ . The columns $u_{+}, u_{2}, u_{0}$ form a unitary matrix, so expanding the Kraus sum gives $\sum_{i} F_{i} Y F_{i}^{*} = \sum_{a} E_{a} Y E_{a}^{*}$ for every operator $Y$ .

4.1step 3.1algebra

By the definition of $W$ , one has $F_{+} V = \sqrt{2 / 3} V$ , $F_{2} V = \sqrt{1 / 3} W$ , and $F_{0} V = 0$ . The restrictions $V, W$ are isometries with orthogonal ranges, so the positive-eigenvalue error basis has exactly two syndromes; the zero eigenvalue error annihilates the code.

5.1step 4.1algebra

Define recovery Kraus operators $R_{+} = V^{*}$ and $R_{2} = W^{*}$ . Their adjoint products are $V V^{*} = P_{0} \otimes I_{2}$ and $W W^{*} = P_{1} \otimes I_{2}$ , so $R_{+}^{*} R_{+} + R_{2}^{*} R_{2} = I_{4}$ and the recovery is trace-preserving.

6.1step 2.1step 3.1step 5.1F1F3F5algebra∎

For the mixed errors, $R_{+} F_{+} V = \sqrt{2 / 3} I_{L}$ and $R_{2} F_{2} V = \sqrt{1 / 3} I_{L}$ ; all other products $R_{i} F_{j} V$ vanish. Thus for every $X \in B (H_{L})$ the corrected channel returns $(2 / 3 + 1 / 3) X = X$ . The unitary Kraus change preserves the original channel, so this recovery corrects the displayed three-error channel, including its degeneracy; this is the recovery construction in [F3].

Depends on

Used by

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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)