Quantum probability distributions in the early Universe. I. Equilibrium properties of the Wigner equation

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

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On a Conjecture regarding Fisher Information

Advances in Mathematical Physics ◽

10.1155/2015/120698 ◽

2015 ◽

Vol 2015 ◽

pp. 1-4 ◽

Cited By ~ 4

Author(s):

Angelo Plastino ◽

Guido Bellomo ◽

Angel Ricardo Plastino

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Probability Distributions ◽

Free Particle ◽

Quantum Probability ◽

Time Dependent ◽

Theoretical Physics ◽

One Dimension ◽

Pure States ◽

Time Dependent Solution

Fisher’s information measureIplays a very important role in diverse areas of theoretical physics. The associated measuresIxandIp, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The productIxIphas been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimensionIxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifiesIxIp→0fort→∞.

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Probabilistic aspects of the SU(2)-harmonic oscillator

Journal of Applied Probability ◽

10.1017/s0021900200015461 ◽

2000 ◽

Vol 37 (01) ◽

pp. 306-314

Author(s):

Shunlong Luo

Keyword(s):

Harmonic Oscillator ◽

Coherent States ◽

Probability Distributions ◽

Quantum Probability ◽

Dimensional Sphere ◽

Large Radius ◽

Two Dimensional ◽

Bargmann Transform ◽

Gaussian Distributions ◽

Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

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Roots of quantum computing supremacy: superposition, entanglement, or complementarity?

The European Physical Journal Special Topics ◽

10.1140/epjs/s11734-021-00061-9 ◽

2021 ◽

Author(s):

Andrei Khrennikov

Keyword(s):

Quantum Computing ◽

Quantum Algorithms ◽

Probability Distributions ◽

Statistical Tests ◽

Joint Probability ◽

Quantum Probability ◽

Quantum Nonlocality ◽

Total Probability ◽

Probabilistic Algorithms ◽

Quantum Phenomena

AbstractThe recent claim of Google to have brought forth a breakthrough in quantum computing represents a major impetus to further analyze the foundations for any claims of superiority regarding quantum algorithms. This note attempts to present a conceptual step in this direction. I start with a critical analysis of what is commonly referred to as entanglement and quantum nonlocality and whether or not these concepts may be the basis of quantum superiority. Bell-type experiments are then interpreted as statistical tests of Bohr’s principle of complementarity (PCOM), which is, thus, given a foothold within the area of quantum informatics and computation. PCOM implies (by its connection to probability) that probabilistic algorithms may proceed without the knowledge of joint probability distributions (jpds). The computation of jpds is exponentially time consuming. Consequently, classical probabilistic algorithms, involving the computation of jpds for n random variables, can be outperformed by quantum algorithms (for large values of n). Quantum probability theory (QPT) modifies the classical formula for the total probability (FTP). Inference based on the quantum version of FTP leads to a constructive interference that increases the probability of some events and reduces that of others. The physical realization of this probabilistic advantage is based on the discreteness of quantum phenomena (as opposed to the continuity of classical phenomena).

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Geometrical structures for classical and quantum probability spaces

International Journal of Quantum Information ◽

10.1142/s021974991740007x ◽

2017 ◽

Vol 15 (08) ◽

pp. 1740007 ◽

Cited By ~ 3

Author(s):

Florio Maria Ciaglia ◽

Alberto Ibort ◽

Giuseppe Marmo

Keyword(s):

Affine Space ◽

Vector Fields ◽

Probability Distributions ◽

Quantum Probability ◽

Finite Sample ◽

Gradient Vector ◽

Tensor Fields ◽

Geometrical Structures ◽

Geometrical Properties ◽

Probability Spaces

On the affine space containing the space [Formula: see text] of quantum states of finite-dimensional systems, there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding the relevant geometrical properties of [Formula: see text]. Guided by Dirac’s analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyze their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.

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