Badhani–Das noisy-unitary conditional min-entropy

H. Badhani and S. Das, Conditional channel entropy sets fundamental limits on thermodynamic quantum information processing, arXiv:2604.01217v2, Section VI.1, conjecture after Proposition 10:

We conjecture that Proposition 10 can be extended to all unitaries with added white noise. We leave this as an open problem.

Proposition 10 assumes $|B'|=|B|=2$. The statement here retains that hypothesis and takes $A'=A=\mathbb C^d$, for every positive natural $d$. For every unitary $U$ on $\mathbb C^d\otimes\mathbb C^2$ and every $p\in[0,1]$, put $N_p(Z)=p\operatorname{tr}(Z)I/(2d)+(1-p)UZU^\dagger$. Then

$$ S_\infty[A\mid B]{N_p}=S\infty(R_AA\mid R_BB)_{\Phi^{N_p}}-\log_2 d. $$

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | Arbitrary positive input/output dimension on A | The natural dimension and positivity binder in claim | No upper bound on $d$. | | Qubit input/output conditioning | Matrix.unitaryGroup (Fin d × Fin 2) ℂ | Both B systems have dimension two. | | Arbitrary unitary channel | The universally quantified unitary in claim | No selected matrix or spectrum. | | Added white noise | noisyUnitary | Literal replacement-plus-unitary expression. | | $p\in[0,1]$ | The two real inequalities in claim | Both endpoints included. | | Choi operator/state | choiN, choiState | Index order $(R_A,A),(R_B,B)$, normalization $1/(2d)$. | | State/channel conditional entropy | condMinEntropy, channelCondMinEntropy | Density states and quantum channels range over the existing repository carriers. The channel formula inlines its matrix-unit Choi expression. | | Subtract $\log|A'|$ | The base-two logarithm term in claim | $|A'|=d$. | | Universal equality | result | Full claim, without additional hypotheses. |

The public definitions retain the expressions prescribed in #19060. Dmax uses the real logarithm followed by coercion to EReal; at zero input it gives zero rather than an extended logarithm of zero. The settlement uses trace-one states and positive lower bounds on feasible costs, so that zero-input extension is outside its entropy-scaling argument.

This is a tier-one published conjecture. The settling GID is D5/S3/Quantum/Information/BadhaniDasNoisyUnitary.result, admitted as open-problem-resolution (#19060; Proved). No digestion atom is used. The source note is Library/Entropy/badhani2026conditionalchannel.md.

The literature screen recorded in #19060 reads the source v2, its two citing works arXiv:2510.23731v2 and arXiv:2506.24079v3, and precursor arXiv:2410.01740; neither citing work supplies this conjecture's proof. Searches cover the title and authors, arXiv conditional-channel-entropy results, MathDB, the pinned formal-conjectures register, and repository issues and problem dossiers. No settlement was found in that searched scope. Completeness is ASSUMED-UNVERIFIED.

The normalized reduced Choi matrix has two maximally mixed qubit marginals. In the magic basis it is real symmetric. A real spectral basis therefore becomes a maximally entangled eigenbasis. Each diagonal sign symmetry has a unitary partner on the other factor; their tensor products preserve both the pure Choi vector and white noise.

Averaging a feasible density-state optimizer over these signs pinches it in that eigenbasis and preserves its trace. Its maximally entangled eigenvectors make the pinched density matrix the normalized Choi matrix of a channel. The two entropy optimizations consequently have equal costs after the dimension scaling. Positive trace bounds justify the logarithmic scaling.

The private declarations maximally_entangled_eigenbasis, sign_symmetry_of_eigenbasis, replacement_of_sign_symmetry and entropy_identity_of_sign_symmetry supply these steps on result's proof path. They are consumed helpers under the declared bind-only classification; they are not independent escape witnesses.

A positive dimension, unitary and noise weight in the specified interval violating the displayed equality would refute the mathematical claim. A statement mismatch, an additional unaccounted hypothesis, or an axiom outside the standard closure would invalidate this formal settlement.

The Lean module is D5/S3/Quantum/Information/BadhaniDasNoisyUnitary.lean; its only public theorem is D5.S3.Quantum.Information.BadhaniDasNoisyUnitary.result. Its Scribe displays the complete expanded claim and records one OpenProblemResolutionClaim of kind Proved.

The escape audit is unfinished for that theorem: #19176. No successful four-slot registration is claimed. The missing evidence is a faithful dependent dimension/unitary/noise realization, the guarded source reconstruction, lawful enrolled-template readout and variation/source-selection proofs. The public owner theorems D5.S3.Quantum.Subentropy.PinchingGap.sign_unitary and D5.S3.Quantum.Subentropy.PinchingGap.signTwirl_eq_blockPinch are also unfinished audit targets under that issue: the finite coordinate/sign/matrix family requires a faithful realization, lawful enrolled-template readout, and compiled selection and variation evidence. Both statements have complete formulas in the owner Scribe. The noisy-unitary proof applies these owners after transporting four coordinates to four singleton blocks.

Experiment

Experiment at commit 5ceac7c1a27dc65493859cc5b5bad0056ad11a83. Portable command from the pinned directory: OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python3 check.py (NumPy, SciPy, CVXPY, CLARABEL and SCS). The pinned check.py has SHA-256 0842460a7bb8242a35c320ffb4873103ba748b03cfe1554bd08a5cb4d3ce6c03. Command: env OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 /tmp/op-venv-sdp/bin/python /tmp/op-bdnoisy/rework3/check.py, run on the byte-identical pinned script; exit 0, RESULT PASS. The solver emitted a solution-accuracy warning. There are 48 random noisy-unitary cases ($d=1,2,3$, four unitaries each, $p\in{0,0.25,0.6,1}$), nine named-unitary cases (CNOT, SWAP and identity at $p\in{0,0.5,0.9}$), and 20 magic-basis tests ($d=1,2,3,4$, five unitaries each). Random-channel and pure-replacement controls at $d=2$ give cost gaps between 0.004522 and 0.154180; the corresponding $d=1$ controls are numerically zero. These are approximate SDP readings.

What the settlement shows

| Item | Status and evidence | Boundary | | --- | --- | --- | | Noisy-unitary equality | Proved in Lean by result | Every $d\ge1$, every unitary and every $p\in[0,1]$; includes $d=1$ and both noise endpoints. | | Decisive mechanism | Kernel-checked: maximally_entangled_eigenbasis, sign_symmetry_of_eigenbasis, replacement_of_sign_symmetry, entropy_identity_of_sign_symmetry | Maximally entangled eigenbasis, joint sign symmetry, pinching to a channel Choi state, and dimension scaling. | | Escape witness classification | Open, no independent witness claimed | mixed_unitary_choi, eigenbasis_sign_partner and symmetry_pinching_feasible are consumed bind-only helpers in this delivery; admission uses the external-problem exception. | | Computational utility | None | Symbolic unbounded dimension and arbitrary complex unitary/real noise; finite polynomial identities are normalization steps, not certified finite instances. | | Numerical equality checks | Computed by the pinned experiment: 48 random cases and nine named-unitary cases pass | Numerical support only; not a uniform theorem. | | Magic-basis structural checks | Computed by the pinned experiment: 20 structural tests pass | Numerical support for the tested instances only. | | General CPTP maps and pure replacement noise | Computed by the pinned experiment: four random CPTP controls and one pure-replacement control at $d=2$ have positive gaps, ranging from 0.004522 to 0.154180 | These computed controls show failure outside the white-noise family; no exact counterexample is kernel-certified here. | | Mixtures $0.6U_1+0.4U_2$ | Open as an independently reproduced computation in this delivery | #19060 reports five failures with $d=2$; the pinned experiment does not contain that test, so those sample counts are not certified by the reproduced program. | | Depolarization on B alone | Open | #19060 reports six numerical agreements at $p=1/2$, $d=1,2$; the pinned experiment does not contain this test. No family theorem is inferred. | | Larger conditioning systems | Open | The maximally entangled eigenbasis argument uses qubits; conditioning dimension greater than two is not settled. | | Sharpness and source consequences | Kernel-checked equality: result; the comparison with Proposition 1 is a prose consequence | The source's general-channel upper bound is attained throughout the specified white-noise unitary family. Applications needing precisely this equality can use it. The source's other channel questions are not settled. | | Independent verification | Open beyond bounded independent formula, helper-consumption and reuse checks | The independent codex-cli mirror assessment approves the source and mirror; verification does not certify exhaustive literature or reuse completeness. | | Literature status | Open as an exhaustive completeness claim | The bounded screen found no settlement; completeness is ASSUMED-UNVERIFIED. | | Four-slot escape audit | Open, #19176 | No declared_validated registration. |

Literature-search completeness and novelty beyond the recorded search scope are unverified. Numerical solver results are approximate and do not establish exact counterexamples or uniform extensions. The zero-input extension of Dmax differs from an extended logarithm at zero; the settlement's trace-one bounds exclude that input. Bounded reuse searches do not prove exhaustive nonexistence of an upstream owner. Larger conditioning dimensions and the local-depolarization family remain open.