LemmaStatement: AI-generatedProof: AI-generated

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Equivalence of logical and code-supported physical decoding

Statement

Let $d \geq 1$ be an integer, let $H_{P}, H_{O}$ be nonzero finite-dimensional complex inner-product spaces, and let $V : C^{d} \to H_{P}$ be a linear isometry. Set $P = V V^{†}$ and ${Ad}_{V} (X) = V X V^{†}$ . CPTP logical decoders $R : L (H_{O}) \to L (C^{d})$ correspond bijectively to CPTP code-supported physical recoveries $Q : L (H_{O}) \to L (H_{P})$ by $Q = {Ad}_{V} \circ R$ and $R (Y) = V^{†} Q (Y) V$ . Code-supported means $P Q (Y) P = Q (Y)$ for every $Y \in L (H_{O})$ .

Fix a quantum channel $N : L (H_{P}) \to L (H_{O})$ , and let $D$ be any completed logical transpose decoder for $(V, N)$ . Here quantum channels are complex-linear, completely positive and trace-preserving, and $F_{e}$ denotes entanglement fidelity for the maximally mixed logical input $I_{d} / d$ . Corresponding $R$ and $Q$ have identical decoded logical channels after $N \circ {Ad}_{V}$ , hence equal entanglement fidelities. Their fidelity sets have the same real supremum $g$ . The logical transpose value $f = F_{e} (D \circ N \circ {Ad}_{V})$ is feasible, independent of the support completion, and satisfies $0 \leq f \leq g \leq 1$ .

Statement references

Code restriction and decoded channel types

Feasibility and range of transpose entanglement fidelity

Entanglement fidelity on a finite logical space

Logical Kraus operators of the completed transpose decoder

Facts & Assumptions

Given: The finite spaces, isometry, fixed quantum channel, decoder classes and completed logical transpose decoder in the Statement.

Goal: Prove the channel correspondence, equality of decoded fidelity sets, and the feasibility and bounds of the transpose value.

[F1]

The finite-dimensional channel, code-support and encoding conventions are those of code restriction: $V^{†} V = I$ , $P = V V^{†}$ , and the restricted noise and decoded compositions are channels. Encoding by an isometry and decoding a code-supported output by compression identify the logical and code spaces. Code restriction and decoded channel types

[F2]

The completed transpose decoder has a completion-independent feasible entanglement fidelity $f$ , and the real supremum $g$ of the logical-decoder fidelity set exists with $0 \leq f \leq g \leq 1$ . Feasibility and range of transpose entanglement fidelity

[F3]

For the finite-dimensional isometry $V$ and quantum channel $N$ , any completed transpose recovery compressed by $V^{†} (\cdot) V$ gives a CPTP logical decoder $D$ . Logical Kraus operators of the completed transpose decoder

[F4]

On a channel $E$ on $C^{d}$ , $F_{e} (E)$ is the overlap with the maximally entangled reference state after applying the identity on the reference and $E$ on the logical factor; its input convention is $I_{d} / d$ . Entanglement fidelity on a finite logical space

Proof

Given: The finite spaces, isometry, fixed quantum channel, decoder classes and completed logical transpose decoder in the Statement.

Goal: Prove the channel correspondence, equality of decoded fidelity sets, and the feasibility and bounds of the transpose value.

Proof technique: direct.

1.1givenF1

Conjugation by $V$ preserves positivity with every ancilla and preserves trace because $V^{†} V = I$ . Thus it sends a CPTP logical decoder to a CPTP code-supported physical recovery. Conversely compression is completely positive and preserves trace on code-supported outputs: in a basis adapted to the code its trace is the sum of the same code diagonal entries. It therefore gives a CPTP logical decoder from any such physical recovery.

2.1givenF1step 1.1

Composing the two constructions gives the identity: on the logical side use $V^{†} V = I$ ; on the physical side use $P Q (Y) P = Q (Y)$ . For every logical operator $X$ , substitution yields $V^{†} Q (N (V X V^{†})) V = R (N (V X V^{†}))$ . Thus the decoded maps after the fixed encoded noise coincide as linear maps.

3.1F1F2F3F4step 2.1∎

The coincident decoded maps in step 2.1 are channels by [F1]. Applying the overlap definition [F4] to equal channels gives equal entanglement fidelities for input $I_{d} / d$ . The bijection therefore identifies their entire fidelity sets. The completed logical transpose decoder $D$ is among these logical decoders by [F3]; [F2] supplies its completion-independent feasible value $f$ , the existence of the logical-decoder supremum $g$ , and $0 \leq f \leq g \leq 1$ . Equality of the fidelity sets gives the same supremum for code-supported physical recoveries.

Depends on

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