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Information-theoretic derivation of energy, speed bounds, and quantum theory

Lorenzo Giannelli, Giulio Chiribella

TL;DR

The work builds an information-theoretic reconstruction of quantum theory from collision-model dynamics, showing that reversible evolution arises from informational nonequilibrium and that energy observables are in one-to-one correspondence with generators of dynamics. By embedding OPTs in a finite-dimensional, convex framework and imposing causality, classical decomposability, purity preservation, and strong symmetry, the authors derive a duality between states, observables, and generators and recover quantum theory on complex Hilbert spaces. A central achievement is an information-theoretic derivation of the Mandelstam–Tamm speed bound, linking the evolution speed to the variance of the energy observable, and clarifying how energy and dynamics emerge from information processing. The results further distinguish complex quantum theory (which satisfies DIN) from real quantum theory (which does not), and provide a coherent pathway from information principles to the dynamical structure and speed limits that underpin quantum mechanics and its potential generalizations.

Abstract

We provide a derivation of quantum theory in which the existence of an energy observable that generates the reversible dynamics follows directly from information-theoretic principles. Our first principle is that every reversible dynamics is implementable through a collision model, i.e. a sequence of fast collisions with an array of identically prepared systems. Combined with four additional principles, known as causality, classical decomposability, purity preservation, and strong symmetry, the collision model pins down the quantum framework, sets up a one-to-one correspondence between energy observables and generators of the dynamics, and provides an information-theoretic derivation of the Mandelstam-Tamm bound on the speed of quantum evolutions.

Information-theoretic derivation of energy, speed bounds, and quantum theory

TL;DR

The work builds an information-theoretic reconstruction of quantum theory from collision-model dynamics, showing that reversible evolution arises from informational nonequilibrium and that energy observables are in one-to-one correspondence with generators of dynamics. By embedding OPTs in a finite-dimensional, convex framework and imposing causality, classical decomposability, purity preservation, and strong symmetry, the authors derive a duality between states, observables, and generators and recover quantum theory on complex Hilbert spaces. A central achievement is an information-theoretic derivation of the Mandelstam–Tamm speed bound, linking the evolution speed to the variance of the energy observable, and clarifying how energy and dynamics emerge from information processing. The results further distinguish complex quantum theory (which satisfies DIN) from real quantum theory (which does not), and provide a coherent pathway from information principles to the dynamical structure and speed limits that underpin quantum mechanics and its potential generalizations.

Abstract

We provide a derivation of quantum theory in which the existence of an energy observable that generates the reversible dynamics follows directly from information-theoretic principles. Our first principle is that every reversible dynamics is implementable through a collision model, i.e. a sequence of fast collisions with an array of identically prepared systems. Combined with four additional principles, known as causality, classical decomposability, purity preservation, and strong symmetry, the collision model pins down the quantum framework, sets up a one-to-one correspondence between energy observables and generators of the dynamics, and provides an information-theoretic derivation of the Mandelstam-Tamm bound on the speed of quantum evolutions.
Paper Structure (50 sections, 51 theorems, 174 equations, 1 figure)

This paper contains 50 sections, 51 theorems, 174 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Collisional models in general probabilistic theories.
  3. Informational equilibrium.
  4. Duality between observables and generators.
  5. Derivation of quantum theory.
  6. Speed bound.
  7. Conclusions.
  8. Overview of the framework of Operational Probabilistic Theories (OPTs)
  9. Systems
  10. Transformations
  11. Tests
  12. States, effects, and scalars
  13. Probabilistic structure
  14. Embedding into real vector spaces
  15. Coarse-graining and pure transformations
  16. ...and 35 more sections

Key Result

Theorem 1

For every reference state $\sigma$, the collisional dynamics $(\mathcal{U}_{t,\sigma})_{t\in \mathbb{R}}$ is a one-parameter Lie group, of the form $\mathcal{U}_{t,\sigma} = e^{ G_\sigma t}$, with $G_\sigma := ( \mathcal{I}_{\textnormal{A}} \otimes u_{\textnormal{A}'} ) \frac{ {\rm d} \mathcal

Figures (1)

  • Figure 1: Collision model for reversible dynamics. A target system in an initial state $\rho$ undergoes a sequence of interactions with a set of systems, independently and identically prepared in the reference state $\sigma$. Here we require that every reversible dynamics of the target system can be obtained by varying the state $\sigma$, in the limit of instantaneous and infinitely frequent interactions.

Theorems & Definitions (72)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3: Observable-generator duality
  • Definition 1: Speed of state change
  • Theorem 4: Speed limit
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Definition 3: Pure transformation
  • ...and 62 more