Determining the fundamental limits of nonlinear functionals of quantum measurement statistics is a crucial yet generally intractable nonconvex optimization problem. We introduce a generic support-function-based outer-approximation framework for solving concave-minimization (or convex-maximization) problems over the quantum state space. By mapping the problem onto a reduced $\mathcal{Z}$ space, we characterize the exact quantum boundary through supporting half space derived from the largest eigenvalues of effective observables. This yields an effective method that produces tight bounds for general measurements in finite-dimensional quantum systems with preassigned numerical precision. As an initial application, we recover the exact variance-based uncertainty relations of [PRL \textbf{119}, 170404 (2017)] and efficiently compute optimal entropic uncertainty relations (EURs). Our results reveal that standard analytical and majorization-based EUR bounds are fundamentally loose for generic measurements, and we show that the resulting exact bounds directly enhance quantum steering detection under asymmetric settings. We further apply the framework to determine the maximal athermality resource certifiable from a restricted measurement scenario. Our method thus provides a universal computational tool for exploring the boundaries of quantum state space and the limits of quantum resources.
Determining the fundamental limits of nonlinear functionals of quantum measurement statistics is a crucial yet generally intractable nonconvex optimization problem. We introduce a generic support-function-based outer-approximation framework for solving concave-minimization (or convex-maximization) problems over the quantum state space. By mapping the problem onto a reduced Z space, we characterize the exact quantum boundary through supporting half space derived from the largest eigenvalues of effective observables. This yields an effective method that produces tight bounds for general measurements in finite-dimensional quantum systems with preassigned numerical precision. As an initial application, we recover the exact variance-based uncertainty relations of [PRL \textbf{119}, 170404 (2017)] and efficiently compute optimal entropic uncertainty relations (EURs). Our results reveal that standard analytical and majorization-based EUR bounds are fundamentally loose for generic measurements, and we show that the resulting exact bounds directly enhance quantum steering detection under asymmetric settings. We further apply the framework to determine the maximal athermality resource certifiable from a restricted measurement scenario. Our method thus provides a universal computational tool for exploring the boundaries of quantum state space and the limits of quantum resources.
Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. Here we introduce methods for the quantification of resources in general probabilistic theories (GPTs), focusing in particular on the technical issues associated with infinite-dimensional state spaces. We define a universal resource quantifier based on the robustness measure, and show it to admit a direct operational meaning: in any GPT, it quantifies the advantage that a given resource state enables in channel discrimination tasks over all resourceless states. We show that the robustness acts as a faithful and strongly monotonic measure in any resource theory described by a convex and closed set of free states, and can be computed through a convex conic optimization problem. Specializing to continuous-variable quantum mechanics, we obtain additional bounds and relations, allowing an efficient computation of the measure and comparison with other monotones. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. In particular, we establish exact expressions for various classes of states, including Fock states and squeezed states in the resource theory of nonclassicality and general pure states in the resource theory of entanglement, as well as tight bounds applicable in general cases.
Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty Qρ(A), with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed.
We solve this fixed-state optimization problem by determining the largest coefficient copt(ρ) for which Vρ(A)≥copt(ρ)Qρ(A) holds for every observable A, and show that it depends only on the smallest and largest eigenvalues of ρ. The same optimization principle extends to a new power-commutator family 21∣[ρs,A]∣2(s≥1/2), providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework.
More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.
Uncertainty relations are usually stated as bounds on selected combinations of variances, but the full covariance matrix contains substantially richer information about the geometry of quantum state space and about the operational capabilities of quantum systems. Here we characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform, the natural analogue of position and momentum in a finite Hilbert space. We combine analytic arguments with convex-geometric and semidefinite-programming methods based on joint numerical ranges to describe the admissible region through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi parameter estimation protocols and separability criteria for finite-dimensional bipartite systems, including discrete analogues of continuous-variable EPR-type witnesses. Our results establish a systematic and versatile platform for connecting uncertainty relations, convex quantum geometry, metrology, and entanglement detection in finite-dimensional systems.
Many fundamental problems in quantum information theory can be formulated as optimization tasks over quantum states and measurements. While the convex structure of quantum theory often enables powerful semidefinite programming formulations, a major challenge persists: the exponential growth of Hilbert space dimension with system size. This dissertation discusses techniques to address these and related obstacles across several core problems in quantum information. We first study optimization over separable quantum states. Such problems are typically NP-hard and standard semidefinite programming hierarchies suffer from the curse of dimensionality. We introduce a hybrid algorithm that leverages a quantum co-processor to reduce the effective dimension of the problem. While several existing approaches provide only a numerical bound on the optimal value, our method is guaranteed to return an explicit feasible solution: a valid separable state, even in instances where it is not optimal. We apply this framework to approximate the minimum separable energy of many-body Hamiltonians, define an entanglement measure for their ground spaces, and demonstrate numerical scalability to systems of up to 28 qubits. We next consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their source code: the quantum circuit that prepares them. We show that the typical semidefinite program for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the variable dimensions from dL to NL, where d is the Hilbert space dimension, N is the number of candidate states, and L is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced problem from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures whose associated discrimination problems were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits. We also analyze structural measurement strategies. As a counterpart to the well-known pretty good measurement for minimum-error state discrimination, we introduce the pretty bad measurement for the exclusion variant, in which the goal is to identify a state that was not sent, rather than to identify the one that was, and prove that both are guaranteed to perform at least as well as blind guessing in their respective objectives. Finally, we consider evaluating Boolean functions under severe memory constraints and prove that a simple local "greedy" measurement strategy is globally optimal precisely for affine Boolean functions, while more generally achieving at least the square of the optimal global success probability. Together, these results demonstrate that high-dimensional quantum optimization problems often admit hidden structure enabling dimension reduction, hybrid computation, and provable approximation guarantees.
As a foundation of quantum physics, uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertainty relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position and momentum operators, Heisenberg’s uncertainty principle points exactly to the attainable area of the variances of position and momentum. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables and (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.