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arXiv:2512.04345

The operator layer cake theorem is equivalent to Frenkel's integral formula

Hao-Chung Cheng, Gilad Gour, Ludovico Lami, Po-Chieh Liu

quant-phcs.ITmath-phmath.FASubmitted Dec 4, 2025• Announced Dec 5, 2025• 0 citations
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Abstract

The operator layer cake theorem provides an integral representation for the directional derivative of the operator logarithm in terms of a family of projections [arXiv:2507.06232]. Recently, the related work [arXiv:2507.07065] showed that the theorem gives an alternative proof to Frenkel's integral formula for Umegaki's relative entropy [Quantum, 7:1102 (2023)]. In this short note, we find a converse implication, demonstrating that the operator layer cake theorem is equivalent to Frenkel's integral formula.

Comments
This note is related to arXiv:2208.12194, arXiv:2507.06232, and arXiv:2507.07065

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The operator layer cake theorem provides an integral representation for the directional derivative of the operator logarithm in terms of a family of projections [arXiv:2507.06232]. Recently, the related work [arXiv:2507.07065] showed that the theorem gives an alternative proof to Frenkel's integral formula for Umegaki's relative entropy [Quantum, 7:1102 (2023)]. In this short note, we find a converse implication, demonstrating that the operator layer cake theorem is equivalent to Frenkel's integral formula.
Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum Rényi and fff-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the Rényi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.
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.
In this work, we prove a one-shot random coding bound for classical-quantum channel coding, a problem conjectured by Burnashev and Holevo in 1998. By choosing the optimal input distribution, the bound implies the optimal error exponent (i.e., the reliability function) of classical-quantum channels for rates above the critical rate, even in infinite-dimensional Hilbert spaces. Our result extends to various quantum packing-type problems, including classical communication over any fully quantum channel with or without entanglement-assistance, constant composition codes, and classical data compression with quantum side information via fixed-length or variable-length coding. Our technical ingredient is to establish an operator layer cake theorem - the directional derivative of an operator logarithm admits an integral representation of certain projections. This shows that a kind of pretty-good measurement is equivalent to a randomized Holevo-Helstrom measurement, which provides an operational explanation of why the pretty-good measurement is pretty good.
In this paper, we prove a novel trace inequality involving two operators. As applications, we sharpen the one-shot achievability bound on the relative entropy error in a wealth of quantum covering-type problems, such as soft covering, privacy amplification, convex splitting, quantum information decoupling, and quantum channel simulation by removing some dimension-dependent factors. Moreover, the established one-shot bounds extend to infinite-dimensional separable Hilbert spaces as well. The proof techniques are based on the recently developed operator layer cake theorem and an operator change-of-variable argument, which are of independent interest.
We generalize Frenkel's integral formula for traces of operators to operators. The resulting formula holds for bounded self-adjoint positive operators and ppp-Schatten class of compact positive operators.
We show that Frenkel’s integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes–Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of ’filtered’ relative entropy distances.