scholarly journals A Classical Interpretation of the Scrooge Distribution

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 619 ◽  
Author(s):  
William Wootters
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Distribution ◽  
Density Matrix ◽  
Pure State ◽  
Quantum Mechanical ◽  
Classical Probability ◽  
Pure States ◽  
Probability Simplex ◽  
Shed Light

The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state compared with all other distributions that have the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept with no natural classical interpretation. In this paper, we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Classical Probability ◽  
Quantum Process ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years Man’ko and co-authors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely that mathematically the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

Probability in quantum mechanics

1978 ◽  
Vol 10 (4) ◽  
pp. 725-729 ◽  
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

Probability in quantum mechanics

1978 ◽  
Vol 10 (04) ◽  
pp. 725-729
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

scholarly journals THE SCHWINGER REPRESENTATION OF A GROUP: CONCEPT AND APPLICATIONS

2006 ◽  
Vol 18 (08) ◽  
pp. 887-912 ◽  
Author(s):  
S. CHATURVEDI ◽  
G. MARMO ◽  
N. MUKUNDA ◽  
R. SIMON ◽  
A. ZAMPINI
Keyword(s):  
Quantum Mechanics ◽  
Lie Group ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Direct Sum ◽  
Pure States ◽  
Group Concept ◽  
Set Up ◽  
Multiplicity Free

The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the Schwinger oscillator construction for SU(2), and its relevance in several quantum mechanical contexts is highlighted. The Schwinger representations for SU(2), SO(3) and SU(n) for all n are constructed via specific carrier spaces and group actions. In the SU(2) case, connections to the oscillator construction and to Majorana's theorem on pure states for any spin are worked out. The role of the Schwinger Representation in setting up the Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group is brought out.

Probability relations between separated systems

1936 ◽  
Vol 32 (3) ◽  
pp. 446-452 ◽  
Author(s):  
E. Schrödinger
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Pure State ◽  
Relativistic Quantum ◽  
Present Theory ◽  
Relativistic Quantum Mechanics ◽  
Quantum Mechanical ◽  
Alternative Possibility ◽  
Linearly Independent ◽  
Statistical Operator

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.

scholarly journals Probabilistic frames for non-Boolean phenomena

2016 ◽  
Vol 374 (2058) ◽  
pp. 20150102 ◽  
Author(s):  
Louis Narens
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Quantum Probability ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Foundations Of Probability ◽  
Quantifying Uncertainty ◽  
Behavioural Sciences ◽  
Generalized Probability ◽  
Provided Examples

Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Classical Probability ◽  
Quantum Process ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

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