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arXiv:2404.17016 · npj Quantum Information

Optimising the relative entropy under semidefinite constraints

Gereon Koßmann, René Schwonnek

Published Jan 23, 2026• 6 citations
arXivHTMLPDF

Abstract

Abstract 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.

DOI
10.1038/s41534-026-01184-4

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Abstract 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.
This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Koßmann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.
Abstract We present an iterative algorithm based on semidefinite programming (SDP) for computing the quantum smooth max-mutual information Imax⁡ε(ρAB)I^\varepsilon _{\max }(\rho _{AB})Imaxε​(ρAB​) I max ε ( ρ AB ) of bipartite quantum states in any dimension. The algorithm is accurate if a rank condition for marginal states within the smoothing environment is satisfied and provides an upper bound otherwise. Central to our method is a novel SDP, for which we establish primal and dual formulations and prove strong duality. With the direct application of bounding the one-shot distillable key of a quantum state, this contribution extends SDP-based techniques in quantum information theory. Thereby it improves the capabilities to compute or estimate information measures with application to various quantum information processing tasks.
Abstract The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum–classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semidefinite programming (SDP), by making use of variational formulas for the measured relative entropies of states and semidefinite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semidefinite programs provide the optimal values of the measured relative entropies of states and channels, but also the numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of$\epsilon ininin\mathrm {poly}(1/\epsilon, \text {input dimension})time,comparedtothetime, compared to thetime,comparedtothe\exp (1/\epsilon, \text {input dimension})$running time required for direct computation.
Semidefinite programs (SDPs) are a class of optimisation problems that find application in numerous areas of physics, engineering and mathematics. Semidefinite programming is particularly suited to problems in quantum physics and quantum information science. Following a review of the theory of semidefinite programming, the book proceeds to describe how it can be used to address a wide range of important problems from across quantum information science. Specific applications include quantum state, measurement, and channel estimation and discrimination, entanglement detection and quantification, quantum distance measures, and measurement incompatibility. Though SDPs have become an increasingly important tool in quantum information science it’s not yet the kind of mathematics students learn routinely. Assuming only a basic knowledge of linear algebra and quantum physics and quantum information, this graduate-level book provides a unified and accessible presentation of one of the key numerical methods used in quantum information science. Whilst the focus is on the theoretical machinery of SDPs, the authors have provided an accompanying GitHub repository containing example code, covering some of the SDPs studied in this book. Part of IOP Series in Quantum Technology. Key features • Accessible for graduate students in science and mathematics. • A unified and accessible presentation of one of the key numerical methods used in quantum information science. • Written by leading researchers on the topic. • Accompanying GitHub repository with sample code.
Sometimes a mathematical tool emerges as uniquely useful and even a defining feature within some branch of physics such as Feynman diagrams. In quantum information theory, the semidefinite program (SDP) has emerged as such a tool. SDP is an optimization task in which a linear objective function is maximized over a set of Hermitian matrices with positive eigenvalues. This review discusses the highly efficient algorithms available for the SDP, and shows how comprehensively the SDP has been deployed in problems of entanglement characterization, quantum nonlocality, quantum channel capacities, and the bounding of ground-state energies.