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arXiv:2609.39814

Sharp Quantum Entropy Mixing Rates and the Operator Layer Cake Theorem

Alexander Stottmeister

quant-phmath-phmath.OASubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble $\{(1\!-\!p,ρ_1),(p,ρ_2)\}$ by $c\,\|{H}\|h_{2}(p)$ (with the binary entropy $h_{2}$). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality $\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)$ for positive trace-class operators $A\!\leq\! B$ with $\text{Tr} A\!=\!p$ and $\text{Tr} B\!=\!1$ on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be $c\!=\!1$. Here, I prove the latter conjecture using an exact integral representation of the commutator $[A,\log B]$ obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by $(2\log d\!+\!1)\|H\|$. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.

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At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble {(1 ⁣− ⁣p,ρ1),(p,ρ2)}\{(1\!-\!p,ρ_1),(p,ρ_2)\}{(1−p,ρ1​),(p,ρ2​)} by c ∥H∥h2(p)c\,\|{H}\|h_{2}(p)c∥H∥h2​(p) (with the binary entropy h2h_{2}h2​). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality ∥[A,log⁡B]∥1 ⁣≤ ⁣c h2(p)\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)∥[A,logB]∥1​≤ch2​(p) for positive trace-class operators A ⁣≤ ⁣BA\!\leq\! BA≤B with TrA ⁣= ⁣p\text{Tr} A\!=\!pTrA=p and TrB ⁣= ⁣1\text{Tr} B\!=\!1TrB=1 on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be c ⁣= ⁣1c\!=\!1c=1. Here, I prove the latter conjecture using an exact integral representation of the commutator [A,log⁡B][A,\log B][A,logB] obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by (2log⁡d ⁣+ ⁣1)∥H∥(2\log d\!+\!1)\|H\|(2logd+1)∥H∥. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.
We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Rényi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commutator, which can be expected to find other implementations. We apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with Rényi and Tsallis entanglement entropies taken as measures of entanglement in the system.
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We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most δδδ and d=dim⁡Ad=\dim Ad=dimA, the optimal dimension-only modulus of continuity is h2(δ)+δlog⁡(d2−1)h_2(δ)+δ\log(d^2-1)h2​(δ)+δlog(d2−1) up to δ=1−d−2δ=1-d^{-2}δ=1−d−2 and 2log⁡d2\log d2logd thereafter, where h2h_2h2​ denotes the binary entropy. When dim⁡B≥d\dim B\ge ddimB≥d, this bound is tight for every δ∈[0,1]δ\in[0,1]δ∈[0,1]. The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji \& Smith [IEEE ISIT (2020)], which follows a conceptually different approach.
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