scholarly journals Uniform probability distribution over all density matrices

2022 ◽  
Author(s):  
Eddy Keming Chen ◽  
Roderich Tumulka
Keyword(s):  
Hilbert Space ◽  
Probability Distribution ◽  
Probability Measure ◽  
Density Matrix ◽  
Uniform Distribution ◽  
Joint Distribution ◽  
Complex Hilbert Space ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Uniform Probability Distribution

AbstractLet $$\mathscr {H}$$ H be a finite-dimensional complex Hilbert space and $$\mathscr {D}$$ D the set of density matrices on $$\mathscr {H}$$ H , i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on $$\mathscr {D}$$ D that can be regarded as the uniform distribution over $$\mathscr {D}$$ D . We propose a measure on $$\mathscr {D}$$ D , argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.

scholarly journals Distinguishability of non-orthogonal density matrices does not imply violations of the second law

2019 ◽  
Author(s):  
PierGianLuca Porta Mana
Keyword(s):  
Hilbert Space ◽  
Density Matrix ◽  
Thermodynamic Analysis ◽  
Second Law Of Thermodynamics ◽  
The Other ◽  
Second Law ◽  
Density Matrices ◽  
Matrix Space ◽  
Von Neumann ◽  
Quantum Thermodynamic

The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.

scholarly journals Finite Dimensional Approximations to Some Flows on the Projective Limit Space of Spheres

1966 ◽  
Vol 28 ◽  
pp. 167-177
Author(s):  
Hisao Nomoto
Keyword(s):  
Probability Distribution ◽  
Probability Space ◽  
Essential Role ◽  
Projective Limit ◽  
Dimensional Sphere ◽  
Basic Space ◽  
Limit Space ◽  
Finite Dimensional ◽  
Uniform Probability ◽  
Uniform Probability Distribution

Let Ω be the projective limit space of a sequence of probability space Ωn which is a certain subset of (n − 1)-dimensional sphere with the usual uniform probability distribution on it. T. Hida [2], starting from a sequence of finite dimensional flows which are derived from some one-parameter subgroups of rotations of spheres, constructed a flow {Tt} on Ω as the limit of them. Observing his method, the concept of consistency of flows which approximate {Tt} seems to play an essential role in his work [2]. As will be made clear in the following sections, the concept of consistency is closely related to the projective limiting structure of our basic space Ω. The purpose of this paper is to determine all the flows on Ω which can be approximated in the sense of [2] by finite dimensional flows.

scholarly journals Nonlinear *-Lie Higher Derivations of Standard Operator Algebras

2018 ◽  
Vol 26 (1) ◽  
pp. 15-29
Author(s):  
Mohammad Ashraf ◽  
Shakir Ali ◽  
Bilal Ahmad Wani
Keyword(s):  
Hilbert Space ◽  
Operator Algebra ◽  
Operator Algebras ◽  
Complex Hilbert Space ◽  
Standard Operator ◽  
Standard Operator Algebra ◽  
Finite Dimensional ◽  
Standard Operator Algebras ◽  
Higher Derivation ◽  
Adjoint Operation

Abstract Let ℌ be an in finite-dimensional complex Hilbert space and A be a standard operator algebra on ℌ which is closed under the adjoint operation. It is shown that every nonlinear *-Lie higher derivation D = {δn}gn∈N of A is automatically an additive higher derivation on A. Moreover, D = {δn}gn∈N is an inner *-higher derivation.

scholarly journals Exponential Negation of a Probability Distribution

2021 ◽  
Author(s):  
Qinyuan Wu ◽  
Yong Deng ◽  
Neal Xiong
Keyword(s):  
Information Processing ◽  
Probability Distribution ◽  
Uniform Distribution ◽  
Probability Distributions ◽  
Numerical Examples ◽  
Entropy Increase ◽  
Intelligent Information Processing ◽  
Uniform Probability Distribution ◽  
Intelligent Information ◽  
Number Of Iterations

Abstract Negation operation is important in intelligent information processing. Different with existing arithmetic negation, an exponential negation is presented in this paper. The new negation can be seen as a kind of geometry negation. Some basic properties of the proposed negation are investigated, we find that the fix point is the uniform probability distribution. The proposed exponential negation is an entropy increase operation and all the probability distributions will converge to the uniform distribution after multiple negation iterations. The convergence speed of the proposed negation is also faster than the existed negation. The number of iterations of convergence is inversely proportional to the number of elements in the distribution. Some numerical examples are used to illustrate the efficiency of the proposed negation.

On triangle contractive operators in Hilbert spaces

1979 ◽  
Vol 85 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Dang Dinh Ang ◽  
Le Hoan Hoa
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Finite Dimensional ◽  
Fixed Line ◽  
Image Position

AbstractLet H be a finite dimensional real or complex Hilbert space. We denote by Λ(x, y, z) the area of the triangle with vertices x, y, z ∈ H. A map f: H → H is triangle contractive TC if 0 < α < 1 and for each x, y, z ∈ H eitherorandandWe prove that if f is TC either there is a fixed point w = f(w) or a fixed line L = ⊃ f(L) We characterize the f which are TC and continuous but have no fixed point.

scholarly journals On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

2019 ◽  
Vol 09 (01) ◽  
pp. 2040004
Author(s):  
Marco Chiani ◽  
Alberto Zanella
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Random Matrices ◽  
Probability Distribution Function ◽  
Joint Distribution ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Arbitrary Subset ◽  
Finite Dimensional ◽  
Joint Probability Distribution Function

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

scholarly journals QUANTUM INFORMATION, ENTANGLEMENT AND ENTROPY

2020 ◽  
Author(s):  
Siddharth Sharma
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Density Matrix ◽  
Pythagorean Theorem ◽  
Quantum Entropy ◽  
Density Matrices ◽  
Relative Quantum ◽  
Axiomatic Quantum Mechanics ◽  
Measure Entropy

In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix.

scholarly journals Conjugates, Filters and Quantum Mechanics

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 158 ◽  
Author(s):  
Alexander Wilce
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Probabilistic Models ◽  
Complex Hilbert Space ◽  
Bipartite State ◽  
Conjugate System ◽  
Jordan Structure ◽  
Finite Dimensional ◽  
Basic Measurement ◽  
The Cost

The Jordan structure of finite-dimensional quantum theory is derived, in a conspicuously easy way, from a few simple postulates concerning abstract probabilistic models (each defined by a set of basic measurements and a convex set of states). The key assumption is that each system A can be paired with an isomorphic conjugate system, A¯, by means of a non-signaling bipartite state ηA perfectly and uniformly correlating each basic measurement on A with its counterpart on A¯. In the case of a quantum-mechanical system associated with a complex Hilbert space H, the conjugate system is that associated with the conjugate Hilbert space H, and ηA corresponds to the standard maximally entangled EPR state on H⊗H¯. A second ingredient is the notion of a reversible filter, that is, a probabilistically reversible process that independently attenuates the sensitivity of detectors associated with a measurement. In addition to offering more flexibility than most existing reconstructions of finite-dimensional quantum theory, the approach taken here has the advantage of not relying on any form of the ``no restriction" hypothesis. That is, it is not assumed that arbitrary effects are physically measurable, nor that arbitrary families of physically measurable effects summing to the unit effect, represent physically accessible observables. (An appendix shows how a version of Hardy's ``subpace axiom" can replace several assumptions native to this paper, although at the cost of disallowing superselection rules.)

scholarly journals Finite Dimensional Approximations to Some Flows on the Projective Limit Space of Spheres II

1967 ◽  
Vol 29 ◽  
pp. 127-135 ◽  
Author(s):  
Hisao Nomoto
Keyword(s):  
Probability Distribution ◽  
Measure Space ◽  
Projective Limit ◽  
Limit Space ◽  
Finite Dimensional ◽  
Uniform Probability ◽  
Uniform Probability Distribution

In the previous paper [6], we have considered flows on the measure space (Ω, ℬ, P), which was the projective limit of a certain subspace (Ωn, ℬn, Pn) of the measure space (Sn, ℬ(Sn), Pn), where Sn is the (n − 1) · sphere with radius and Pn is the uniform probability distribution over Sn.

scholarly journals Cosine of angle and center of mass of an operator

2012 ◽  
Vol 62 (1) ◽  
Author(s):  
Kallol Paul ◽  
Gopal Das
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Center Of Mass ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Alternative Method

AbstractWe consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.

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