On the Brukner–Zeilinger approach to information in quantum measurements

We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.

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Quantum gravity, dynamical phase-space and string theory

International Journal of Modern Physics D ◽

10.1142/s0218271814420061 ◽

2014 ◽

Vol 23 (12) ◽

pp. 1442006 ◽

Cited By ~ 33

Author(s):

Laurent Freidel ◽

Robert G. Leigh ◽

Djordje Minic

Keyword(s):

Quantum Theory ◽

Phase Space ◽

String Theory ◽

Quantum Gravity ◽

Momentum Space ◽

Symplectic Geometry ◽

Complex Geometry ◽

Hamiltonian Dynamics ◽

Natural Extension ◽

Dynamical Phase

In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

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On the transferability of electron density in binary vanadium borides VB, V3B4 and VB2

Acta Crystallographica Section B Structural Science Crystal Engineering and Materials ◽

10.1107/s2052520615018363 ◽

2015 ◽

Vol 71 (6) ◽

pp. 777-787 ◽

Cited By ~ 1

Author(s):

Bürgehan Terlan ◽

Lev Akselrud ◽

Alexey I. Baranov ◽

Horst Borrmann ◽

Yuri Grin

Keyword(s):

Quantum Theory ◽

Crystal Structures ◽

Electron Density ◽

Critical Points ◽

Atoms In Molecules ◽

Model Systems ◽

Suitable Model ◽

Interatomic Distances ◽

Systematic Analysis ◽

A Charge

Binary vanadium borides are suitable model systems for a systematic analysis of the transferability concept in intermetallic compounds due to chemical intergrowth in their crystal structures. In order to underline this structural relationship, topological properties of the electron density in VB, V3B4 and VB2 reconstructed from high-resolution single-crystal X-ray diffraction data as well as derived from quantum chemical calculations, are analysed in terms of Bader's Quantum Theory of Atoms in Molecules [Bader (1990). Atoms in Molecules: A Quantum Theory, 1st ed. Oxford: Clarendon Press]. The compounds VB, V3B4 and VB2 are characterized by a charge transfer from the metal to boron together with two predominant atomic interactions, the shared covalent B—B interactions and the polar covalent B—M interactions. The resembling features of the crystal structures are well reflected by the respective B—B interatomic distances as well as by ρ(r) values at the B—B bond critical points. The latter decrease with an increase in the corresponding interatomic distances. The B—B bonds show transferable electron density properties at bond critical points depending on the respective bond distances.

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Harmonic maps with torsion

Science China Mathematics ◽

10.1007/s11425-020-1744-9 ◽

2020 ◽

Author(s):

Volker Branding

Keyword(s):

Riemannian Manifolds ◽

Critical Points ◽

Mathematical Analysis ◽

Harmonic Maps ◽

Natural Extension ◽

Energy Functional ◽

Target Manifold

AbstractIn this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan (1924). The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.

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A contribution to modern ideas on the quantum theory

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1927.0086 ◽

1927 ◽

Vol 115 (770) ◽

pp. 208-214

Keyword(s):

Quantum Theory ◽

Kinetic Energy ◽

Line Element ◽

Natural Extension ◽

Space Time ◽

Relativity Theory ◽

Unit Mass ◽

The World ◽

Free Particles ◽

Theory Of Gravitation

The relativity theory of gravitation indicates that space-time is a four dimensional continuum in which the line element is measured by the equation ( ds ) 2 = g mn dx m dx n , (1) the notation being that generally adopted. The world-lines or natural tracks of free particles in this space are geodesics. From (1) we have g mn dx m /ds . dx n /ds = 1, (2) the quantity on the left being an expression corresponding to the kinetic energy of ordinary dynamics for a particle of unit mass. This correspondence is readily appreciated if it be noted that dx m /ds is the natural extension of the velocity, dx m /dt .

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Mixing properties of (α,β)-expansions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708000709 ◽

2009 ◽

Vol 29 (4) ◽

pp. 1119-1140 ◽

Cited By ~ 4

Author(s):

KARMA DAJANI ◽

YUSUF HARTONO ◽

COR KRAAIKAMP

Keyword(s):

Dynamical System ◽

Invariant Measure ◽

Lebesgue Measure ◽

Natural Extension ◽

Mixing Properties ◽

Special Values ◽

Image Position ◽

Relationship Of ◽

The Relationship

AbstractLet 0<α<1 andβ>1. We show that everyx∈[0,1] has an expansion of the formwherehi=hi(x)∈{0,α/β}, andpi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values ofα, we give the relationship of this expansion with the greedyβ-expansion.

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The complete set of uniform polyhedra

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1975.0022 ◽

1975 ◽

Vol 278 (1278) ◽

pp. 111-135 ◽

Cited By ~ 19

Keyword(s):

Natural Extension ◽

Regular Polygons ◽

Complete Set ◽

Definition Of ◽

Uniform Polyhedron

A definitive enumeration of all uniform polyhedra is presented (a uniform polyhedron has all vertices equivalent and all its faces regular polygons). It is shown that the set of uniform polyhedra presented by Coxeter, Longuet-Higgins & Miller (1953) is essentially complete. However, after a natural extension of their definition of a polyhedron to allow the coincidence of two or more edges, one extra polyhedron is found, namely the great disnub dirhombidodecahedron.

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Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

Journal of Applied Mathematics ◽

10.1155/2013/147921 ◽

2013 ◽

Vol 2013 ◽

pp. 1-15 ◽

Cited By ~ 5

Author(s):

Muhammad Ayub ◽

Masood Khan ◽

F. M. Mahomed

Keyword(s):

Lie Algebras ◽

Kepler Problem ◽

Natural Extension ◽

Three Dimensional ◽

Second Order ◽

Differential Invariants ◽

Canonical Forms ◽

Solvable Lie Algebra ◽

Complete Set ◽

Invariant Representations

We present a systematic procedure for the determination of a complete set ofkth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of twokth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case ofk= 2 and 31 classes for the case ofk≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of twokth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

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$\beta$-transformation, natural extension and invariant measure

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700000699 ◽

2000 ◽

Vol 20 (5) ◽

pp. 1271-1285 ◽

Cited By ~ 7

Author(s):

GAVIN BROWN ◽

QINGHE YIN

Keyword(s):

Invariant Measure ◽

Lebesgue Measure ◽

Natural Extension ◽

Connected Region ◽

Two Dimensional ◽

Constant Multiple ◽

Unit Square ◽

Simply Connected Region ◽

Simply Connected ◽

Dimensional Lebesgue Measure

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

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Quantum Measurements with, and Yet without an Observer

Entropy ◽

10.3390/e22101185 ◽

2020 ◽

Vol 22 (10) ◽

pp. 1185

Author(s):

Dmitri Sokolovski

Keyword(s):

Wave Function ◽

Quantum Theory ◽

Quantum Measurements ◽

Material Objects ◽

Wave Function Collapse

It is argued that Feynman’s rules for evaluating probabilities, combined with von Neumann’s principle of psycho-physical parallelism, help avoid inconsistencies, often associated with quantum theory. The former allows one to assign probabilities to entire sequences of hypothetical Observers’ experiences, without mentioning the problem of wave function collapse. The latter limits the Observer’s (e.g., Wigner’s friend’s) participation in a measurement to the changes produced in material objects, thus leaving his/her consciousness outside the picture.

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