Tightness of the 0-fidelity lower bound and the worst-case processes

K. Mayer, A short note on the 0-fidelity, arXiv:2109.09629v1, proves for every $n$-qubit process $\mathcal E$ that $1-\frac32(1-F_0)\le F\le F_0$, where $F$ is the process fidelity and $F_0$ the 0-fidelity on tensor products of single-qubit SIC-POVM states, and writes:

This leads us to conjecture that the lower bound in Th.~1 is tight for all $n$.

A description of the worst-case process is an open problem.

The verbatim statements are in the literature note. Issue #14119 reads the two statements for every $n\ge1$, every choice of single-qubit SIC-POVMs and every finite Kraus family: the lower bound holds with equality exactly when every Kraus operator lies in the span of the identity and the single-qubit Pauli operators, and every value $f\in[\frac13,1]$ of $F_0$ is attained with equality.

The 0-fidelity is estimated from product states only. If the lower bound is tight for every $n$, the worst-case process fidelity does not approach the 0-fidelity as $n$ grows, so an estimate of $F_0$ certifies only $1-\frac32(1-F_0)$; describing the worst cases identifies which errors the product-state estimate cannot see.

Issue #14119 records the literature check before any Lean: one arXiv version; arXiv full-text, web and Zenodo searches and the citing work arXiv:2312.08590 contain no proof of tightness and no description of the worst-case processes; the repository had no 0-fidelity result. not-found-in-searched-scope.

Each qubit SIC is a 2-design: $\sum_kP_k\otimes P_k=\frac23(I+\mathrm{SWAP})$. Expanding a Kraus operator in Pauli strings $\sigma_\beta$ (with $|\beta|$ non-identity factors) gives $F_0=\sum_\beta3^{-|\beta|}q_\beta$ and $F=q_0$, where $q_\beta=d^{-2}\sum_j|\mathrm{tr}(\sigma_\beta^\dagger K_j)|^2\ge0$ and, by trace preservation, $\sum_\beta q_\beta=1$. Hence $F-1+\frac32(1-F_0)=\frac12\sum_{\beta\ne0}(1-3^{1-|\beta|})q_\beta\ge0$, with equality iff $q_\beta=0$ for every $|\beta|\ge2$. Depolarizing noise on one qubit, ${\sqrt pI,\sqrt{(1-p)/3},\sigma_a^{(1)}}$, has $F=p$ and $F_0=p+\frac{1-p}3$.

The statement concerns the 0-fidelity of the source (product single-qubit SIC states, uniform weights) and finite Kraus families on $n$ qubits; other input ensembles or weightings are outside the claim.

The canonical source is D5/S3/Quantum/QuantumChannels/MayerZeroFidelityTightness.lean, with public IsQubitSIC, productState, zeroFidelity, processFidelity, HasWeightOneSupport, claim and result : claim. It contains the single-qubit SIC frame identity sic_frame (the 2-design property of SIC-POVMs, entered as a literature prerequisite; see the note). It reuses the frozen Pauli, pauliMatrix, tensorOp, wordOp and the tensor product law tensor_mul of D5/S3/Quantum/Information/StabilizerPairLocalUnitaryInequivalence, the frozen maximally entangled carrier maxEntangledVector, maxEntangled and its trace identity max_entangled_trace of D5/S3/QuantumBounds/PeritoTsirelson, the frozen two-qubit swap swapMatrix of D5/S3/Resource/CompositeConeProperness with its trace identity hswap_trace, Mathlib's hammingDist for the Pauli weight, the frozen Kraus carrier MatrixMap.of_kraus and the rank-one transport hsandwich of D5/S3/Quantum/Measurement/ExactConditionalPreparationCost, and the Pauli-word expansion word_expansion, the unit wordUnit and the identities tensor_one, word_one, tensor_trace, pauli_trace_pair, pauli_hermitian and word_hermitian of D5/S3/Quantum/Measurement/StabilizerPovmMaximalEntanglementRefutation. Each of these was a private declaration or a local step at its origin and is made public there, with its statement and proof unchanged. The axiom closure of result is exactly propext, Classical.choice and Quot.sound; there is no sorry, native_decide, or new axiom. The module statement is sha256:207677a8f33b9dd5698be1a72aaddf2cf8220c1ef9d80607297f8da5d317cfda, the result statement sha256:a5920d009739edf5671ef0802d3d3bc20d9adaeeed545c59b97a1c554535c065 and the claim statement sha256:0c567f7c43c7080bc672bd69401520dbe949740e5b88791c3428680b44d326cf. The Freeze event is sha256:e354c9094fed1af2bbf82833bc9c8804febc056779ed106613cb31df97dac7ac.

Tier 1 conjecture and open problem of a 2021 note, preregistered in issue #14119 before any Lean. theorem; resolution proved.

| declaration | proof_shape | escape_witness | admission_basis | | --- | --- | --- | --- | | result | content | channel_bound (the Pauli-weight form of $F-1+\frac32(1-F_0)$ and its equality case) and tight_channel | open-problem-resolution |

The other theorems of the module lie on the proof path of result (CLAUDE.md §3.2 「有消费的辅助声明」); each has free arguments. Utility is none. There is no digestion atom.

What the settlement shows

Proved by result: for every $n\ge1$, every choice of single-qubit SIC-POVMs and every finite Kraus family on $n$ qubits, $F\ge1-\frac32(1-F_0)$, with equality if and only if every Kraus operator has zero coefficient on every Pauli string with at least two non-identity factors; and every $f\in[\frac13,1]$ is the 0-fidelity of a process with $F=1-\frac32(1-f)$.

Established inside the proof, for general data. For every Kraus family, $F_0=\sum_\beta3^{-|\beta|}q_\beta$ and $F=q_0$ with the Pauli-twirl weights $q_\beta=d^{-2}\sum_j|\mathrm{tr}(\sigma_\beta^\dagger K_j)|^2$, which are nonnegative and, for a trace-preserving family, sum to one.

Argued, not formalized.

• Mechanism. Moving Pauli-twirl weight $\varepsilon$ from the identity to a Pauli string of weight $w$ lowers $F$ by $\varepsilon$ and $F_0$ by $(1-3^{-w})\varepsilon$; the decrease of $F_0$ is smallest, $\frac23\varepsilon$, for $w=1$. So for a given $F_0\in[\frac13,1]$ the process fidelity is lowest exactly when all error weight sits on strings acting on one qubit; errors of higher weight are more visible to the product-state average. Below $\frac13$ the affine bound is negative and is not the minimum (next item).
• The low-fidelity range. Since $q_\beta\ge0$ and $3^{-|\beta|}\ge3^{-n}$, every process has $F_0\ge3^{-n}$; for $f\in[3^{-n},\frac13]$ the minimum of $F$ is $0$, attained by Pauli channels supported on non-identity strings. So the tight lower boundary is $\max{0,\frac{3f-1}2}$ on $[3^{-n},1]$.
• Upper bound. The paper's upper bound $F\le F_0$ and its SDP improvement are not addressed here.

Open. A closed form of the best-case process fidelity for given $F_0$ (the paper's SDP upper bound) is not settled here.

Effect on the paper. The §III conjecture holds for every $n$, and the §IV open problem has an answer on $F_0\in[\frac13,1]$: the worst-case processes, that is, the equality cases of the bound $F\ge1-\frac32(1-F_0)$, are exactly those whose Kraus operators lie in the span of the identity and the single-qubit Pauli operators. For $F_0<\frac13$ the minimum of $F$ is $0$ (item above).

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.