The minimum change principle provides an information-theoretic characterization of the Bayes reversal channel in classical probability theory and has recently been proposed as a framework for extending Bayes'rule to quantum information theory. Using quantum relative entropy, we investigate a minimum change principle for the setting of quantum statistical inference. Specifically, we consider a forward process based on a classical-to-quantum preparation channel and a reverse process based on a quantum-to-classical measurement channel. We establish a closed-form characterization of measurements that are optimal for this principle, and this optimal measurement can be found via a dual formulation involving a single unconstrained Hermitian variable. This perspective allows us to recover some notable measurements within the same framework, including pretty good measurements and Fermi-Dirac thermal measurements, and we use it to discover a novel family that we call softmin thermal measurements. We further show that softmin thermal measurements arise as optimal solutions to entropy-regularized semidefinite optimization problems, demonstrating that they play a role for measurements analogous to that of thermal states in statistical mechanics. Finally, we prove an additivity property for the relative-entropy minimum change principle and investigate the performance of Fermi-Dirac thermal measurements for quantum hypothesis testing.
A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence \&purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures.
Quantum relative entropy is a core concept in physics, governing the limits of communication, thermodynamic irreversibility and quantum resource conversion. However, the requirement that physical processes cannot increase state distinguishability, the data-processing inequality, permits an infinite family of alternative divergence measures. Here we show that quantum relative entropy is uniquely selected by a sharper operational principle. We evaluate distinguishability through binary guessing games, in which an observer discriminates between pairs of quantum states using the optimal measurement. We prove that any additive measure that respects the odds revealed by these optimal measurements must coincide with the Umegaki relative entropy. This rigidity is a purely quantum phenomenon. Whereas classical theory permits a continuous family of valid divergence measures, including R\'enyi divergences, quantum noncommutativity. collapses this mathematical freedom. The result is exact, requiring neither a thermodynamic limit of infinitely many copies nor super-additivity assumptions for correlated states. It establishes quantum relative entropy not merely as an asymptotic quantity, but as the unique additive distinguishability measure compatible with single-shot quantum discrimination.
This work presents a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis, and formalizes the DPI for the sandwiched R\'enyi relative entropy for positive semidefinite operators on finite-dimensional quantum systems.
Kazumi Kasaura, Kei Tsukamoto, Kento Mori et al.· 2 citations
It is proved that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders.
Quantum theory challenges the view that individual measurement outcomes are predefined and independent of the measurement context. Yet the quantum state itself -- the catalogue of probabilities for all possible measurements -- is usually assumed to be well defined. We argue that this assumption tacitly relies on measurements being performed relative to ideal, infinitely-resourceful reference frames. We show that, when measurements are made relative to non-ideal quantum reference frames, the probabilities themselves become indefinite: even in the limit of arbitrarily large number of runs, the relative frequencies may remain uncertain. The uncertainty is irreducible in a quantum-mechanical sense, as we show by proving a Bell-type theorem for relative frequencies. We further propose a quantum-optical implementation of these relational measurements based on pulsed homodyne detection. Our findings motivate an extension of the notion of the quantum state to regimes constrained by finite resources. We expect them to be especially relevant at the interface between quantum theory and general relativity, where the resources and information available in a bounded region of spacetime are fundamentally limited.
Esteban Castro-Ruiz, Nathan Cohen, L. Barbado et al.· 0 citations
Coherent states bridge the gap between quantum and classical physics, but their overcomplete and nonorthogonal nature makes it difficult to identify the minimal discrete set needed to reconstruct quantum information. Finite spin-coherent tomography and discrete coherent-state operator bases are known, but here we address the more specific rank-resolved problem of preserving the canonical contravariant-symbol representation. We show that the canonical finite coherent-state formula reconstructs every operator in the rank-$S$ sector exactly if and only if the sampling points form a spherical $(2J+S)$-design. We call the associated configurations spin-coherent quantum designs. We further give a fully explicit positive-weight Gauss-Legendre construction that avoids the need for an equal-weight spherical design. Together, these results establish a unified framework for reading out physical observables from a handful of measurement samples, playing for spin systems the role that the so-called von Neumann lattice plays for canonical coherent states. Finally, we derive practical protocols for estimating moments of spin operators from these constructions, with direct applications to polarimetry, magnetometry, and quantum state tomography.
Marcin Rudzinski, A. Z. Goldberg, A. B. Klimov et al.· 0 citations