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

Acta Physica Sinica

Quantum mechanics mixed state representation

Fan Hong-Yi, Lou Sen‐Yue, Xiao‐Yin Pan, Da Cheng

Published Jan 1, 2014• 0 citations
Publisher

Abstract

In the previous studies and literatures, quantum mechanics representations are a pure state representation, while in this paper beginning with reasonable ordering of operators in quantum mechanics and the normal distribution thought in probability statistics, we propose the concept of quantum mechanics mixed state representation firstly,and prove its completeness and orthogonality. The advantages of mixed state representation is that it can reflect a variety of expression of quantum mechanics operators and the corresponding operators ordering rules.

DOI
10.7498/aps.63.190302

Related papers

Quantum Theory

Jochen Rau

From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.

Oxford University Press eBooks•Nov 22, 2017•1 citation

Interference of Quantum States and Superposition Principle in Probability Representation of Quantum Mechanics

Margarita A. Man’ko, Vladimir I. Man’ko

The superposition of pure quantum states explicitly expressed in terms of a nonlinear addition rule of state density operators is reviewed. The probability representation of density matrices of qudit states is used to formulate the interference of the states as a combination of the probability distributions describing pure states. The formalism of quantizer–dequantizer operators is developed. Examples of spin-1/2 states and f-oscillator systems are considered.

Open Systems & Information Dynamics•Sep 1, 2019•2 citations

Research on quantum mechanical representations via the two-dimensional normal distribution function in statistics

Fan Hong-Yi

Based on the two-dimensional normal distributions in statistics and the technique of integration within an ordered product of operators(IWOP),the explicit forms in the Fock space of some quantum representations are simply obtained such as coordinate representation,momentum representation,coordinate-momentum intermediate representation,and coherent representation.In addition,their corresponding completeness is also easily proved.

College Physics•Jan 1, 2011•0 citations

Representation theory of quantum states

Tao Xiang

A number of quantum states form an orthonormal and complete basis set if they are normalized and mutually orthogonal and any quantum state can be expressed as a linear combination. In classical mechanics, any two physical variables are commutable. However, this property is no longer valid for physical operators. Commutation of operators is an important property of quantum mechanics. Although there are many representations of wave functions and observables connected by unitary transformations, it is useful to distinguish certain classes of representations, called pictures, which differ in the way the time evolution of the system is treated. The Schrodinger picture is one in which the operators are time-independent but the wave function is time dependent. The virial theorem relates the kinetic energy to its potential in an energy eigenstate.

CRC Press eBooks•Mar 31, 2022•0 citations

Observables, interference phenomenon and Born’s rule in the probability representation of quantum mechanics

Margarita A. Man’ko, Vladimir I. Man’ko

The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.

International Journal of Quantum Information•Jan 31, 2020•11 citations

Discussion on the Application of Representation in Quantum Mechanics

Xiuping Zhang

Representation is an abstract concept in quantum mechanics.Rationally,choosing a representation,which is similar to the coordinate system in geometry,can make the problem simply.To solve the same question,different representation in quantum system of micro-world is chosen,but its conclusion will not be changed.

Journal of Heihe University•Jan 1, 2011•0 citations
In the previous studies and literatures, quantum mechanics representations are a pure state representation, while in this paper beginning with reasonable ordering of operators in quantum mechanics and the normal distribution thought in probability statistics, we propose the concept of quantum mechanics mixed state representation firstly,and prove its completeness and orthogonality. The advantages of mixed state representation is that it can reflect a variety of expression of quantum mechanics operators and the corresponding operators ordering rules.
From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.
A number of quantum states form an orthonormal and complete basis set if they are normalized and mutually orthogonal and any quantum state can be expressed as a linear combination. In classical mechanics, any two physical variables are commutable. However, this property is no longer valid for physical operators. Commutation of operators is an important property of quantum mechanics. Although there are many representations of wave functions and observables connected by unitary transformations, it is useful to distinguish certain classes of representations, called pictures, which differ in the way the time evolution of the system is treated. The Schrodinger picture is one in which the operators are time-independent but the wave function is time dependent. The virial theorem relates the kinetic energy to its potential in an energy eigenstate.
The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.