Infinite dimensionality of the post-processing order of measurements on a general state space

Abstract

For a partially ordered set $(S,⪯)$, the order (monotone) dimension is the minimum cardinality of total orders (respectively, real-valued order monotone functions) on $S$ that characterize the order $⪯$. In this paper we consider an arbitrary generalized probabilistic theory and the set of finite-outcome measurements on it, which can be described by effect-valued measures, equipped with the classical post-processing orders. We prove that the order and order monotone dimensions of the post-processing order are (countably) infinite if the state space is not a singleton (and is separable in the norm topology). This result gives a negative answer to the open question for quantum measurements posed in [Guff T \textit{et al.\/} 2021 \textit{J.\ Phys.\ A: Math.\ Theor.} \textbf{54} 225301]. We also consider the quantum post-processing relation of channels with a fixed input quantum system described by a separable Hilbert space $\mathcal{H}$ and show that the order (monotone) dimension is countably infinite when $\dim \mathcal{H}\ge 2$.

arXiv comment32 pages, 2 figures. New appendix (Appendix A) which proves the BSS theorem is added. New references are added

References in corpus (21)

• Quantum entanglement
• Quantum Resource Theories
• The Physics and Mathematics of the Second Law of Thermodynamics
• A mathematical theory of resources
• Generalized Probability Theories: What determines the structure of quantum theory?
• Clean Positive Operator Valued Measures
• Robustness of Measurement, discrimination games and accessible information
• Generalized Probabilistic Theories Without the No-Restriction Hypothesis
• Local asymptotic normality in quantum statistics
• Incompatibility of quantum channels
• Degradable channels, less noisy channels, and quantum statistical morphisms: an equivalence relation
• Sharp and fuzzy observables on effect algebras
• Comparison of Information Structures and Completely Positive Maps
• A Resource Theory of Quantum Measurements
• Non-classical correlations in quantum mechanics and beyond
• The Quantum Blackwell Theorem and Minimum Error State Discrimination
• Quantum incompatibility of channels with general outcome operator algebras
• Minimal sufficient positive-operator valued measure on a separable Hilbert space
• Order preserving maps on quantum measurements
• Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements
• Directed-completeness of quantum statistical experiments in the randomization order