Information and disturbance in operational probabilistic theories

Abstract

Any measurement is intended to provide information on a system, namely knowledge about its state. However, we learn from quantum theory that it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. In this paper we address the issue of the interplay between information and disturbance for a general operational probabilistic theory. The traditional notion of disturbance considers the fate of the system state after the measurement. However, the fact that the system state is left untouched ensures that also correlations are preserved only in the presence of local discriminability. Here we provide the definition of disturbance that is appropriate for a general theory. Moreover, since in a theory without causality information can be gathered also on the effect, we generalise the notion of no-information test. We then prove an equivalent condition for no-information without disturbance-atomicity of the identity-namely the impossibility of achieving the trivial evolution-the identity-as the coarse-graining of a set of non trivial ones. We prove a general theorem showing that information that can be retrieved without disturbance corresponds to perfectly repeatable and discriminating tests. Based on this, we prove a structure theorem for operational probabilistic theories, showing that the set of states of any system decomposes as a direct sum of perfectly discriminable sets, and such decomposition is preserved under system composition. As a consequence, a theory is such that any information can be extracted without disturbance only if all its systems are classical. Finally, we show via concrete examples that no-information without disturbance is independent of both local discriminability and purification.

arXiv comment24 pages, 1 figure

References in corpus (25)

• Fermionic quantum computation
• Informational derivation of Quantum Theory
• Non-local correlations as an information theoretic resource
• Heisenberg's Uncertainty Principle
• Probabilistic theories with purification
• Quantum State Disturbance vs. Information Gain: Uncertainty Relations for Quantum Information
• A mathematical theory of resources
• Uncertainty Relations for Noise and Disturbance in Generalized Quantum Measurements
• Quantum root-mean-square error and measurement uncertainty relations
• General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks
• Limited Holism and Real-Vector-Space Quantum Theory
• All reversible dynamics in maximally non-local theories are trivial
• Distinguishability Measures and Entropies for General Probabilistic Theories
• No-hypersignaling principle
• Information-disturbance tradeoff in quantum measurements
• The Feynman problem and Fermionic entanglement: Fermionic theory versus qubit theory
• Strong nonlocality: A trade-off between states and measurements
• Fermionic computation is non-local tomographic and violates monogamy of entanglement
• If no information gain implies no disturbance, then any discrete physical theory is classical
• Classical theories with entanglement
• Determinism without causality
• Classicality without local discriminability: decoupling entanglement and complementarity
• Testing axioms for Quantum Mechanics on Probabilistic toy-theories
• No-free-information principle in general probabilistic theories
• Reversible Dynamics in Strongly Non-Local Boxworld Systems

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• Shannon theory beyond quantum: information content of a source
• Shannon theory for quantum systems and beyond: information compression for fermions
• Minimal operational theories: classical theories with quantum features