© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Texts and readings in mathematics

Quantum Coding Theorems

Kalyanapuram Rangachari Parthasarathy

Published Jan 1, 2013• 0 citations

Abstract

We restrict ourselves to finite level systems. Elementary classical probability theory is woven around the notions of a finite sample space or an alphabet A, events which are subsets of A, random variables which are real or complex-valued functions on A and probability distributions which are, once again, functions on A with values in the unit interval adding to unity. An event E ⊂ A can be equivalently replaced by the two-valued indicator function 1 E which is one on the set E and zero on the complement E′ of E in A. Thus all the basic objects, namely, events, random variables and distributions are functions on A. Complex-valued functions on A constitute a commutative algebra A $$ \mathcal{A} $$ with complex conjugation as an involution. Thus an event can be identified as an element f ∈ A $$ \mathcal{A} $$ satisfying f = f2 = f̄ where the bar indicates complex conjugation. A real-valued random variable is an element f ∈ A $$ \mathcal{A} $$ satisfying f = f̄ whereas a probability distribution is an element p ∈ A $$ \mathcal{A} $$ satisfying p(x) ≥ 0 ∀ x ∈ A and ∑ x∈A p( x )=1 . $$\sum\limits_{x \in A} {p\left( x \right) = 1} .$$ . Note that every nonnegative function on A has the form f f̄ for some f ∈ A $$ \mathcal{A} $$ .

DOI: 10.1007/978-93-86279-59-0_4

Related papers

Events, Observables and States

Kalyanapuram Rangachari Parthasarathy

The source of inspiration for doing quantum probability lies in the ingenious methods adopted by physicists to compute probabilities of events concerning the subatomic world of elementary particles where the laws of classical mechanics break down and the distinction between a particle and a wave becomes vague. These methods lead to a generalisation of classical probability which may be described as a study of observable quantities concerning any system subject to chance and exhibiting the following basic features: (i) each observable quantity behaves like a classical real valued random variable on a probability space and hence has a probability distribution on the real line; (ii) it may be impossible to measure simultaneously two individually observable quantities in the sense that measurement of one may interfere with and destroy already existing information about the value of the other; (iii) bounded observable quantities constitute a real linear space in which the vectorial sum of two individually observable quantities interfering with each other is different from the arithmetic sum of their actual values. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Birkhäuser Basel eBooks•Jan 1, 1992•16 citations

Probability Theory on the Lattice of Projections in a Hilbert Space

Kalyanapuram Rangachari Parthasarathy

In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra \( \mathcal{B} \) X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on \( \mathcal{B} \) X constitute a convex set whose extreme points are degenerate probability measures.

Texts and readings in mathematics•Jan 1, 2005•0 citations

Probability Distributions and Hilbert Spaces: Quantum and Classical Systems

Vladimir I. Man’ko, Giuseppe Marmo

We use the fact that some linear Hamiltonian systems can be considered as "finite level" quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular class of linear Hamiltonian systems.

arXiv:quant-ph/9903021•Physica Scripta•Aug 1, 1999•15 citations

Upgrading Probability via Fractions of Events

Roman Frič, Martin Papčo

The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

Communications in Mathematics•Aug 20, 2016•8 citations

On a quantum martingale convergence theorem

Kyler Scott Bridgen Johnson, Michael J. Kozdron

It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.

arXiv:1504.03829•International Journal of Quantum Information•Oct 28, 2022•0 citations

Classical and Quantum Probability

Harish Parthasarathy

Classical probability spaces:

CRC Press eBooks•Sep 14, 2022•0 citations

We restrict ourselves to finite level systems. Elementary classical probability theory is woven around the notions of a finite sample space or an alphabet A, events which are subsets of A, random variables which are real or complex-valued functions on A and probability distributions which are, once again, functions on A with values in the unit interval adding to unity. An event E ⊂ A can be equivalently replaced by the two-valued indicator function 1 E which is one on the set E and zero on the complement E′ of E in A. Thus all the basic objects, namely, events, random variables and distributions are functions on A. Complex-valued functions on A constitute a commutative algebra A

\mathcal{A}

with complex conjugation as an involution. Thus an event can be identified as an element f ∈ A

\mathcal{A}

satisfying f = f2 = f̄ where the bar indicates complex conjugation. A real-valued random variable is an element f ∈ A

\mathcal{A}

satisfying f = f̄ whereas a probability distribution is an element p ∈ A

\mathcal{A}

satisfying p(x) ≥ 0 ∀ x ∈ A and ∑ x∈A p( x )=1 .

\sum\limits_{x \in A} {p\left( x \right) = 1} .

. Note that every nonnegative function on A has the form f f̄ for some f ∈ A

\mathcal{A}

.

The source of inspiration for doing quantum probability lies in the ingenious methods adopted by physicists to compute probabilities of events concerning the subatomic world of elementary particles where the laws of classical mechanics break down and the distinction between a particle and a wave becomes vague. These methods lead to a generalisation of classical probability which may be described as a study of observable quantities concerning any system subject to chance and exhibiting the following basic features: (i) each observable quantity behaves like a classical real valued random variable on a probability space and hence has a probability distribution on the real line; (ii) it may be impossible to measure simultaneously two individually observable quantities in the sense that measurement of one may interfere with and destroy already existing information about the value of the other; (iii) bounded observable quantities constitute a real linear space in which the vectorial sum of two individually observable quantities interfering with each other is different from the arithmetic sum of their actual values. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra

\mathcal{B}

X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on

\mathcal{B}

X constitute a convex set whose extreme points are degenerate probability measures.

The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.