On defining the Hamiltonian beyond quantum theory

Abstract

Energy is a crucial concept within classical and quantum physics. An essential tool to quantify energy is the Hamiltonian. Here, we consider how to define a Hamiltonian in general probabilistic theories, a framework in which quantum theory is a special case. We list desiderata which the definition should meet. For 3-dimensional systems, we provide a fully-defined recipe which satisfies these desiderata. We discuss the higher dimensional case where some freedom of choice is left remaining. We apply the definition to example toy theories, and discuss how the quantum notion of time evolution as a phase between energy eigenstates generalises to other theories.

arXiv commentAuthors' accepted manuscript for inclusion in the Foundations of Physics topical collection on Foundational Aspects of Quantum Information

References in corpus (39)

• Device-independent security of quantum cryptography against collective attacks
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• No Signalling and Quantum Key Distribution
• The role of quantum information in thermodynamics --- a topical review
• Fundamental limitations for quantum and nano thermodynamics
• From Bell's Theorem to Secure Quantum Key Distribution
• In defense of the epistemic view of quantum states: a toy theory
• Informational derivation of Quantum Theory
• Probabilistic theories with purification
• Work extraction and thermodynamics for individual quantum systems
• Quantum Mechanics as Quantum Measure Theory
• Bloch vectors for qudits
• A derivation of quantum theory from physical requirements
• Generalized Probability Theories: What determines the structure of quantum theory?
• Higher-order interference and single-system postulates characterizing quantum theory
• Entropy and Information Causality in General Probabilistic Theories
• Entropy in general physical theories
• Generalized Probabilistic Theories Without the No-Restriction Hypothesis
• Distinguishability Measures and Entropies for General Probabilistic Theories
• Three Slit Experiments and the Structure of Quantum Theory
• Generalised phase kick-back: the structure of computational algorithms from physical principles
• Entanglement and thermodynamics in general probabilistic theories
• Higher-order interference in extensions of quantum theory
• Deriving Grover's lower bound from simple physical principles
• Computation in generalised probabilistic theories
• Thermodynamics and the structure of quantum theory
• Ruling out higher-order interference from purity principles
• Microcanonical thermodynamics in general physical theories
• Operational axioms for diagonalizing states
• Entropy, majorization and thermodynamics in general probabilistic theories
• Unifying typical entanglement and coin tossing: on randomization in probabilistic theories
• A general framework for phase and interference
• Stabilizer notation for Spekkens' toy theory
• A toy model for quantum mechanics
• Bounds on the power of proofs and advice in general physical theories
• How uncertainty enables non-classical dynamics
• Interferometric computation beyond quantum theory
• The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer
• Entropies in General Probabilistic Theories and its Application to Holevo Bound

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• Some no-go results in Quantum Thermodynamics
• From reasonable postulates to generalised Hamiltonian systems
• Generalized dynamical theories in phase space and the hydrogen atom
• Simulating all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories
• Semi-device-independent information processing with spatiotemporal degrees of freedom
• Tsirelson's Inequality for the Precession Protocol is Maximally Violated by Quantum Theory
• Information-theoretic derivation of energy, speed bounds, and quantum theory