For every separable Hilbert space, every p∈[0,1], and all positive trace-class operators A≤ B satisfying operatornameTrA=p and operatornameTrB=1, prove the sharp inequality lVert[A,log B]rVert_1≤ h_2(p), where h_2(p)=-plog p-(1-p)log(1-p). Equivalently, the optimal dimension-independent constant in this bound is c=1.

Bravyi's small incremental mixing conjecture

Bravyi’s conjecture asks whether the mixing rate of a binary quantum ensemble is bounded by its binary entropy, with the sharp dimension-independent constant $1$ . It was introduced by Bravyi; the general case was open in the earlier literature.

Known results

Bravyi proved a bound with constant $6$ when the average density operator has at most two distinct eigenvalues (reported in 2013).
A dimension-independent bound with constant $4$ was proved for binary ensembles, weaker near $p=0$ than the conjectured entropy bound (2013).
Later work obtained constants $9$ , then $2$ , in finite dimensions, and $11$ for separable infinite-dimensional spaces; the sharp constant $1$ remained unproved.
Related entangling-rate conjectures were proved with nonsharp constants, but this did not settle the mixing conjecture.

September 2026 claimed proof

Alexander Stottmeister’s article Sharp Quantum Entropy Mixing Rates and the Operator Layer Cake Theorem claims the sharp bound and corresponding entropy-mixing rates via an operator layer-cake representation. No independent mathematical assessment or verification of this claim was retrieved.

Current status (as of October 2026): Nonsharp dimension-independent bounds are established, while the sharp constant- $1$ conjecture has only a recent unverified proof claim.

Bravyi's small incremental mixing conjecture

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Progress summary

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