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 State-vector and Probabilities

Henry Krips

Published Feb 20, 1992• 0 citations
Publisher

Abstract

Abstract One of the features of modern physics is the extensive use of quantitative concepts to enrich qualitative concepts. For example a comparative qualitative judgement that A is hotter than B will be redescribed quantitatively by saying that the temperature value of A is greater than that of B. This quantitative language has much greater expressive power than the qualitative language it replaces. For example it enables sense to be made of such claims as ‘A is as much hotter than B as B is hotter than C’. For our purposes, however, the important feature of quantitative language is that it enables the development of the idea of a possible state.

DOI
10.1093/oso/9780198242802.003.0003

Related papers

Mathematical Background

F. S. Levin

The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.

Cambridge University Press eBooks•Dec 3, 2001•0 citations

The concepts of truth, possibility and probability in the language of quantum physics

Peter Mittelstaedt

•Jan 1, 1981•2 citations

Quantum information theory

Stephen Mark Barnett

The astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.

Oxford University Press eBooks•May 28, 2009•24 citations

The Method of Relativized Conceptualization, A Theory of Factual Probabilities and A Factual Wave Quantum Mechanics

Mioara Mugur-Schächter

•Jan 1, 2025•0 citations

Qrisp: A Framework for Compilable High-Level Programming of Gate-Based Quantum Computers

Raphael Seidel, Bock, Sebastian, René Zander +4 more

While significant progress has been made on the hardware side of quantum computing, support for high-level quantum programming abstractions remains underdeveloped compared to classical programming languages. In this article, we introduce Qrisp, a framework designed to bridge several gaps between high-level programming paradigms in state-of-the-art software engineering and the physical reality of today's quantum hardware. The framework aims to provide a systematic approach to quantum algorithm development such that they can be effortlessly implemented, maintained and improved. We propose a number of programming abstractions that are inspired by classical paradigms, yet consistently focus on the particular needs of a quantum developer. Unlike many other high-level language approaches, Qrisp's standout feature is its ability to compile programs to the circuit level, making them executable on most existing physical backends. The introduced abstractions enable the Qrisp compiler to leverage algorithm structure for increased compilation efficiency. Finally, we present a set of code examples, including an implementation of Shor's factoring algorithm. For the latter, the resulting circuit shows significantly reduced quantum resource requirements, strongly supporting the claim that systematic quantum algorithm development can give quantitative benefits.

arXiv:2406.14792•Jun 20, 2024•15 citations

The Method of Relativized Conceptualization, A Theory of Factual Probabilities and A Factual Wave Quantum Mechanics

Mioara Mugur-Schächter

•Jan 1, 2026•0 citations
Abstract One of the features of modern physics is the extensive use of quantitative concepts to enrich qualitative concepts. For example a comparative qualitative judgement that A is hotter than B will be redescribed quantitatively by saying that the temperature value of A is greater than that of B. This quantitative language has much greater expressive power than the qualitative language it replaces. For example it enables sense to be made of such claims as ‘A is as much hotter than B as B is hotter than C’. For our purposes, however, the important feature of quantitative language is that it enables the development of the idea of a possible state.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
The astute reader might have formed the impression that quantum in formation science is a rather qualitative discipline because we have not, as yet, explained how to quantify quantum information. There are three good reasons for leaving this important question until the final chapter. Firstly, quantum information theory is technically demanding and to treat it at an earlier stage might have suggested that our subject was more complicated than it is. Secondly, there is the fact that many of the ideas in the field, such as teleportation and quantum circuits, are unfamiliar and it was important to present these as simply as possible. Finally, and most importantly, the theory of quantum information is not yet fully developed. It has not yet reached, in particular, the level of completeness of its classical counterpart. For this reason we can answer only some of the many questions we would like a quantum theory of information to address. Having said this, we can say that however, there are beautiful and useful mathematical results and it seems certain that these will continue to form an important part of the theory as it develops. We noted in the introduction to Chapter 1 that ‘quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed’. A presentation of at least the beginnings of a quantitative theory is the objective of this final chapter. The entropy or information derived from a given probability distribution is, as we have seen, a convenient measure of the uncertainty associated with the distribution. If many of the probabilities are large, so that many of the possible events are comparably likely, then the entropy will be large. If one probability is close to unity, however, then the entropy will be small. It is convenient to introduce entropy in quantum mechanics as a measure of the uncertainty, or lack of knowledge, of the form of the state vector. If we know that our system is in a particular pure state then the associated uncertainty or entropy should be zero. For mixed states, however, it will take a non-zero value.
While significant progress has been made on the hardware side of quantum computing, support for high-level quantum programming abstractions remains underdeveloped compared to classical programming languages. In this article, we introduce Qrisp, a framework designed to bridge several gaps between high-level programming paradigms in state-of-the-art software engineering and the physical reality of today's quantum hardware. The framework aims to provide a systematic approach to quantum algorithm development such that they can be effortlessly implemented, maintained and improved. We propose a number of programming abstractions that are inspired by classical paradigms, yet consistently focus on the particular needs of a quantum developer. Unlike many other high-level language approaches, Qrisp's standout feature is its ability to compile programs to the circuit level, making them executable on most existing physical backends. The introduced abstractions enable the Qrisp compiler to leverage algorithm structure for increased compilation efficiency. Finally, we present a set of code examples, including an implementation of Shor's factoring algorithm. For the latter, the resulting circuit shows significantly reduced quantum resource requirements, strongly supporting the claim that systematic quantum algorithm development can give quantitative benefits.