Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements

Abstract

We introduce the post-processing preorder and equivalence relations for general measurements on a possibly infinite-dimensional general probabilistic theory described by an order unit Banach space $E$ with a Banach predual. We define the measurement space $\mathfrak{M}(E)$ as the set of post-processing equivalence classes of continuous measurements on $E.$ We define the weak topology on $\mathfrak{M}(E)$ as the weakest topology in which the state discrimination probabilities for any finite-label ensembles are continuous and show that $\mathfrak{M}(E)$ equipped with the convex operation corresponding to the probabilistic mixture of measurements can be regarded as a compact convex set regularly embedded in a locally convex Hausdorff space. We also prove that the measurement space $\mathfrak{M}(E)$ is infinite-dimensional except when the system is $1$-dimensional and give a characterization of the post-processing monotone affine functional. We apply these general results to the problems of simulability and incompatibility of measurements. We show that the robustness measures of unsimulability and incompatibility coincide with the optimal ratio of the state discrimination probability of measurement(s) relative to that of simulable or compatible measurements, respectively. The latter result for incompatible measurements generalizes the recent result for finite-dimensional quantum measurements. Throughout the paper, the fact that any weakly$\ast$ continuous measurement can be arbitrarily approximated in the weak topology by a post-processing increasing net of finite-outcome measurements is systematically used to reduce the discussions to finite-outcome cases.

arXiv comment83 pages. A new result on the maximal success probability of simulation have been added

References in corpus (27)

• The Physics and Mathematics of the Second Law of Thermodynamics
• Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation
• An Invitation to Quantum Incompatibility
• Generalized Probability Theories: What determines the structure of quantum theory?
• Quantifying quantum resources with conic programming
• All sets of incompatible measurements give an advantage in quantum state discrimination
• Quantum Incompatibility Witnesses
• Clean Positive Operator Valued Measures
• Robustness of Measurement, discrimination games and accessible information
• Incompatibility robustness of quantum measurements: a unified framework
• All quantum resources provide an advantage in exclusion tasks
• Operational framework for quantum measurement simulability
• Cloning and Broadcasting in Generic Probabilistic Theories
• Comparison of quantum statistical models: equivalent conditions for sufficiency
• Local asymptotic normality in quantum statistics
• Operational interpretation of weight-based resource quantifiers in convex quantum resource theories of states
• Quantum measurements on finite dimensional systems: relabeling and mixing
• All measurements in a probabilistic theory are compatible if and only if the state space is a simplex
• Simulating all quantum measurements using only projective measurements and postselection
• Simulability of observables in general probabilistic theories
• Incompatible measurements in a class of general probabilistic theories
• Optimal measurements in quantum mechanics
• Completely positive maps on modules, instruments, extremality problems, and applications to physics
• Non-classical correlations in quantum mechanics and beyond
• Quantum incompatibility of channels with general outcome operator algebras
• Weight of informativeness, state exclusion games and excludible information
• Directed-completeness of quantum statistical experiments in the randomization order

Cited by in corpus (5)

• Operational quantification of continuous-variable quantum resources
• Quantum guessing games with posterior information
• Random access test as an identifier of nonclassicality
• A tensor norm approach to quantum compatibility
• Infinite dimensionality of the post-processing order of measurements on a general state space