Optimising the relative entropy under semidefinite constraints

npj | quantum information Article Published in partnership with The University of New South Wales https://doi.org/10.1038/s41534-026-01184-4 Optimising the relative entropy under semidefinite constraints Check for updates 1234567890():,; 1234567890():,; Gereon Koßmann1 & René Schwonnek2 Finding the minimal relative entropy of two quantum states under semidefinite constraints is a pivotal problem located at the mathematical core of various applications in quantum information theory. An efficient method for providing provable upper and lower bounds is the central result of this work. Our primordial motivation stems from the essential task of estimating secret key rates for QKD from the measurement statistics of a real device. Further applications include the computation of channel capacities, the estimation of entanglement measures and many more. We build on a recently introduced integral representation of quantum relative entropy by [Frenkel, Quantum 7, 1102 (2023)] and provide reliable bounds as a sequence of semidefinite programs (SDPs). Our approach ensures provable sublinear convergence in the discretization, while also maintaining resource efficiency in terms of SDP matrix dimensions. Additionally, we can provide gap estimates to the optimum at each iteration stage. Within the last four decades, the field of quantum cryptography has undertaken a massive evolution. Originating from theoretical considerations by Bennet and Brassard in 19841 we are now in a world where technologies like QKD systems and Quantum random number generators are on the edge of being a marked ready reality. Moreover, there is an ongoing flow (see e.g.2,3 and references therein) of demonstrator setups and proof-ofprinciple experiments within the academic realm that bears a cornucopia of cryptographic quantum technologies that may reach a next stage in a not too far future. Despite these gigantic leaps on the technological side, we have to constitute that the theoretical security analysis of quantum cryptographic systems is still in a process of catching up with these developments. To the best of our knowledge, there are yet no commercial devices with a fully comprehensive, openly accessible, and by the community verified security proof. Nevertheless, theory research has taken the essential steps in providing the building blocks for a framework that allows to do this4. Most notably, the development of the entropy accumulation theorem5,6 and comparable techniques7, allow us to deduce reliable guarantees on an εsecure extractable finite key in the context of general quantum attacks requiring only bounds on an asymptotic quantity such as the conditional von Neumann entropy as input. The pivotal problem, and the input to this framework, is to find a good lower bound on the securely extractable randomness that a cryptographic device offers in the presence of a fully quantum attacker8. Mathematically, this quantity is expressed by the conditional von Neumann entropy H(X∣E). Using Claude Shannon’s intuitive description, it can be understood as the uncertainty an attacker E has about the outcome of a measurement X, which is performed by the user of a device. There are several existing numerical techniques for estimating this quantity given a set of measurement data provided by a device9–14. We will add to this collection, by providing a practical and resource efficient method for this problem, which interpolates between an executable tool and theoretical bounds on the relative entropy by convex interpolation. At the core of our work stands a recently described15,16, and pleasingly elegant, integral representation of the quantum (Umegaki) relative entropy17 (see also18) that we employ in order to formulate the problem of reliably bounding H(X∣E) as an instance of semidefinite programs (SDP) by discretizing integrals. Our method comes with a provable sublinear convergence guarantee in the discretization, whilst staying resource efficient with the matrix dimension of the underlying SDPs. We furthermore can provide an estimate for the gap to the optimum for any discretization stage. To this end, let H ffi Cd be a finite-dimensional Hilbert space. Write BðHÞ for the (bounded) linear operators on H and SðHÞ :¼ fω 2 BðHÞ : ω ≥ 0; tr½ω ¼ 1g for the set of quantum states (density operators). Let hi : BðHÞ × BðHÞ ! R be affine maps, for i = 1, …, n. The central mathematical problem considered here—more general than estimating a conditional entropy H(X∣E) and not limited to QKD—is: inf Dðρ k σÞ s: t: hi ðρ; σÞ ≥ 0; i ¼ 1; . . . ; n; μ σ ≤ ρ ≤ λ σ; ρ; σ 2 SðHÞ; ð1Þ where quantum relative entropy is Dðρ k σÞ :¼ 
the (Umegaki)  tr ρ log ρ  log σ . The constraint ρ ≤ λ σ (with finite λ) enforces supp(ρ) 1 Institute for Quantum Information, RWTH Aachen University, Aachen, Germany. 2Institute for theoretical Physics, Leibniz University Hannover, e-mail: Hannover, Germany. npj Quantum Information | (2026)12:23 1 Article https://doi.org/10.1038/s41534-026-01184-4 ⊆ supp(σ), ensuring the relative entropy to be finite; if, in addition, μ > 0, then supp(ρ) = supp(σ). Despite being convex, this optimisation problem is highly non-linear and contains the analytically benign, but numerically problematic matrix logarithm. Thus, for general instances, (1) can not be solved directly by existing standard methods. The construction of a converging sequence of reliable lower bounds on the value c in (1) is the central technical contribution of this work. Our focus task of estimating key-rates can be cast as an instance of this (see the last section of IV and Supplementary Note 7). Here, lower bounds on (1) directly translate into lower bounds on the key-rate, which is exactly the direction of an estimate needed for a reliable security proof. There is however a long list of further problems that can be formulated as an instance of (1). It includes for example the optimisation over all types of entropies which are expressible as relative entropies. For example we provide the calculation of the entanglement-assisted classical capacity of a quantum channel in the Supplementary Note 8 where one has to optimise in fact the mutual information of a bipartite system. The optimization problem (1) naturally generalizes from relative entropies to general f-divergences. With minimal adjustments, our method can also tackle this class. Despite not being the focus of this work, as a detailed numerical analysis is left for future work, we already formulated the relevant technical parts of the Methods section IV from this more general perspective. Results In the following, we denote by BðHÞ the set of (bounded) linear operators on a finite-dimensional Hilbert space H and SðHÞ the set of quantum states on H, i.e. all positive operators with unit trace. The trace on BðHÞ is denoted as tr½. Moreover, any self adjoint operator A 2 BðHÞ, can be uniquely decomposed as a difference A = A+ − A− of Hilbert-Schmidt orthogonal positive operators A+ an (...truncated)