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Vistas of Algebraic Probability: Quantum Computation and Information

Antonio Falcó, Hermann G. Matthies

TL;DR

This work develops an algebraic probability framework where random variables live in a unital *-algebra with a state, unifying classical and quantum-like randomness through noncommutativity. It builds representations via GNS, links states to densities, and generalizes measurements to POVMs and channels, using CP maps, Stinespring dilations, and the Jamiołkowski–Choi correspondence. The text then connects these mathematical structures to quantum computation by modeling Quantum Processor Units (QPUs) within finite-dimensional operator algebras, illustrating how quantum algorithms (e.g., Grover) can be described abstractly. The approach emphasizes structural features (projections, MASAs, centers) and dynamic information flow (instruments, conditional expectations), aiming to provide a physics-flavored yet purely probabilistic and algebraic toolkit for quantum information processing. Overall, it offers a cohesive algebraic lens on QC and QI that highlights density matrices, PVMs/POVMs, and CP maps as central computational primitives with practical implications for uncertainty, measurement, and information transmission.

Abstract

Kolmogorov's foundation of probability takes measure spaces, $σ$-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation.

Vistas of Algebraic Probability: Quantum Computation and Information

TL;DR

This work develops an algebraic probability framework where random variables live in a unital *-algebra with a state, unifying classical and quantum-like randomness through noncommutativity. It builds representations via GNS, links states to densities, and generalizes measurements to POVMs and channels, using CP maps, Stinespring dilations, and the Jamiołkowski–Choi correspondence. The text then connects these mathematical structures to quantum computation by modeling Quantum Processor Units (QPUs) within finite-dimensional operator algebras, illustrating how quantum algorithms (e.g., Grover) can be described abstractly. The approach emphasizes structural features (projections, MASAs, centers) and dynamic information flow (instruments, conditional expectations), aiming to provide a physics-flavored yet purely probabilistic and algebraic toolkit for quantum information processing. Overall, it offers a cohesive algebraic lens on QC and QI that highlights density matrices, PVMs/POVMs, and CP maps as central computational primitives with practical implications for uncertainty, measurement, and information transmission.

Abstract

Kolmogorov's foundation of probability takes measure spaces, -algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation.
Paper Structure (42 sections, 77 theorems, 218 equations, 4 figures, 4 tables)

This paper contains 42 sections, 77 theorems, 218 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Algebraic Formulation of Probability
  3. Basics
  4. Introduction to Probability Algebras:
  5. Non-Commutative Probability Algebra:
  6. Dirac's Bra-Ket Notation:
  7. Representations and Hilbert Space
  8. General Representations:
  9. States of the Probability Algebra:
  10. Associated Hilbert Space Representation:
  11. States and Densities:
  12. Convergence:
  13. Correlation and Independence:
  14. Sums and Products:
  15. Structure of Finite Dimensional Algebras
  16. ...and 27 more sections

Key Result

Proposition 2.8

If $\mathsans{\omega}$ is a non-zero *-character (i.e. an observation or sample, cf. Definition def:sample), then no element of the form $\mathsans{a} - \mathsans{\omega}(\mathsans{a})\mathsans{1}$ is invertible. Hence, for any $\mathsans{\omega} \in \mathsf{X}(\mathbb{C}[\mathsans{a}])$, one has $\

Figures (4)

  • Figure 1: Schrödinger picture / information flow due to interaction
  • Figure 2: Conditional expectation induced information flow due to interaction
  • Figure 3: Conditional expectation induced information flow due to observation
  • Figure 4: Geometric strategy of Grover's algorithm in the invariant two-dimensional subspace $\mathrm{span}\{\ket{a},\ket{a_\perp}\}$. The oracle $W_1$ and diffusion $W_2$ are reflections; their composition $U_2=W_2W_1$ yields a rotation by $2\theta$ that amplifies the probability of measuring the marked state.

Theorems & Definitions (176)

  • Definition 2.1: State, Expectation
  • Remark 2.2
  • Definition 2.3: Probability Algebra
  • Definition 2.4: *-Homomorphism
  • Definition 2.5: Random Variable (RV)
  • Definition 2.6: Sample, Observation, Character
  • Definition 2.7: Spectrum
  • Proposition 2.8: Spectrum and Sample
  • proof
  • Definition 2.9: Commutator, Lie Algebra
  • ...and 166 more