Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Logical Difference in Quantum Mathematics Separating Pure States from Mixed States

Steve Faulkner

Published Mar 1, 2015• 0 citations
OpenAlex

Abstract

Abstract I give a short explanation of how mathematics representing pure states is logically distinct from mixed states. This is intended as understandable to the undergraduate.

Related papers

Quantum states and knowledge: Between pure states and density matrices

Sandu Popescu

In this note I will present a subtle interplay between density matrices and the knowledge about their preparation, and I will argue that there is a need to consider a new type of quantum state, in between pure states and density matrices.

arXiv:1811.05472•Nov 13, 2018•2 citations

A Basic Modal Logic for the Abstract Treatment of Quantum Mixed States

Tomoaki Kawano

Quantum logic has developed in various directions since its inception.However, many existing logical approaches either focus only on pure states, or use Hilbert-space models in which the abstract relationship between pure states, mixed states, and probabilities is not made explicit.In this work, we introduce a model designed to handle quantum mixed states abstractly.Furthermore, we construct a proof system for the proposed logic and establish its completeness with respect to the intended semantics.

Electronic Proceedings in Theoretical Computer Science•Oct 25, 2026•0 citations

The Mathematical Foundations of Quantum Indeterminacy

Steve Faulkner

Abstract: In 2008, Tomasz Paterek et al published ingenious research, proving that quantum randomness is the output of measurement experiments, whose input commands a logically independent response. Following up on that work, this paper develops a full mathematical theory of quantum indeterminacy. I explain how, the Paterek experiments imply, that the measurement of pure eigenstates, and the measurement of mixed states, cannot both be isomorphically and faithfully represented by the same single operator. Specifically, unitary representation of pure states is contradicted by the Paterek experiments. Profoundly, this denies the axiomatic status of Quantum Postulates, that state, symmetries are unitary, and observables Hermitian. Here, I show how indeterminacy is the information of transition, from pure states to mixed. I show that the machinery of that transition is unpreventable, logically circular, unitary-generating self-reference: all logically independent. Profoundly, this indeterminate system becomes apparent, as a visible feature of the mathematics, when unitarity --- imposed by Postulate --- is given up and abandoned. Keywords: foundations of quantum theory, quantum mechanics, quantum randomness, quantum indeterminacy, quantum information, prepared state, measured state, pure states, mixed states, unitary, redundant unitarity, orthogonal, scalar product, inner product, mathematical logic, logical independence, self-reference, logical circularity, mathematical undecidability.

•Apr 1, 2016•0 citations

Why the Disjunction in Quantum Logic is Not Classical

Diederik Aerts, Ellie D’Hondt, Liane M. Gabora

arXiv:quant-ph/0007041•Foundations of Physics•Sep 1, 2000•27 citations

An Introduction to Logical Operations on Classical and Quantum Bits

Xavier Sigaud

This article is a short review on the concept of information. We show the strong relation between Information Theory and Physics, and the differences between classical and quantum information, with emphasis in their manipulation through logical gates.This paper is intended to be read by non-specialists and undergraduate students of Computer Science, Mathematics and Physics, with knowledge in Linear Algebra and Quantum Mechanics.

•Jan 1, 2004•0 citations

The Generalized BEADS Representation: An Informationally Complete Visualization of Mixed Quantum States

Celina Bräutigam, Steffen J. Glaser, Dennis Huber

A deep understanding of quantum information theory is essential for quantum science and technology. However, quantum mechanics and particularly quantum information theory suffer from a lack of intuitive images. Visualization tools offer novel perspectives on many problems in quantum physics by revealing important properties of quantum states and processes that are otherwise obscured by abstract mathematical formalisms. Here, we extend the previously established BEADS representation of pure states to the so-called generalized BEADS representation of mixed quantum states. We introduce spherical functions, so-called beads to visualize correlation structures of mixed quantum states. A recursive optimization scheme is used to obtain minimum-entanglement pure state decompositions of all subsystems of a given N-qubit state. We show how correlation function components can be decomposed into individual contributions that directly correspond to different types of correlations of mixed states. This allows us to visually characterize the total correlation, genuine n-partite entanglement, nonclassical quantum-discord-type correlations as well as classical correlations. After outlining the mathematical details relevant for constructing the representation, we examine the generalized BEADS representations for various classes of mixed states including product, classically correlated, separable but nonclassically correlated, as well as entangled states. As exemplary applications of the generalized BEADS representation, we showcase entanglement distillation and the effects of noise channels.

quant-pharXiv:2610.19054•Oct 15, 2026•0 citations
Quantum logic has developed in various directions since its inception.However, many existing logical approaches either focus only on pure states, or use Hilbert-space models in which the abstract relationship between pure states, mixed states, and probabilities is not made explicit.In this work, we introduce a model designed to handle quantum mixed states abstractly.Furthermore, we construct a proof system for the proposed logic and establish its completeness with respect to the intended semantics.
Abstract: In 2008, Tomasz Paterek et al published ingenious research, proving that quantum randomness is the output of measurement experiments, whose input commands a logically independent response. Following up on that work, this paper develops a full mathematical theory of quantum indeterminacy. I explain how, the Paterek experiments imply, that the measurement of pure eigenstates, and the measurement of mixed states, cannot both be isomorphically and faithfully represented by the same single operator. Specifically, unitary representation of pure states is contradicted by the Paterek experiments. Profoundly, this denies the axiomatic status of Quantum Postulates, that state, symmetries are unitary, and observables Hermitian. Here, I show how indeterminacy is the information of transition, from pure states to mixed. I show that the machinery of that transition is unpreventable, logically circular, unitary-generating self-reference: all logically independent. Profoundly, this indeterminate system becomes apparent, as a visible feature of the mathematics, when unitarity --- imposed by Postulate --- is given up and abandoned. Keywords: foundations of quantum theory, quantum mechanics, quantum randomness, quantum indeterminacy, quantum information, prepared state, measured state, pure states, mixed states, unitary, redundant unitarity, orthogonal, scalar product, inner product, mathematical logic, logical independence, self-reference, logical circularity, mathematical undecidability.
A deep understanding of quantum information theory is essential for quantum science and technology. However, quantum mechanics and particularly quantum information theory suffer from a lack of intuitive images. Visualization tools offer novel perspectives on many problems in quantum physics by revealing important properties of quantum states and processes that are otherwise obscured by abstract mathematical formalisms. Here, we extend the previously established BEADS representation of pure states to the so-called generalized BEADS representation of mixed quantum states. We introduce spherical functions, so-called beads to visualize correlation structures of mixed quantum states. A recursive optimization scheme is used to obtain minimum-entanglement pure state decompositions of all subsystems of a given N-qubit state. We show how correlation function components can be decomposed into individual contributions that directly correspond to different types of correlations of mixed states. This allows us to visually characterize the total correlation, genuine n-partite entanglement, nonclassical quantum-discord-type correlations as well as classical correlations. After outlining the mathematical details relevant for constructing the representation, we examine the generalized BEADS representations for various classes of mixed states including product, classically correlated, separable but nonclassically correlated, as well as entangled states. As exemplary applications of the generalized BEADS representation, we showcase entanglement distillation and the effects of noise channels.