A spectral order for infinite-dimensional quantum spaces

2012 ◽  
Vol 23 (1) ◽  
pp. 95-130
Author(s):  
JOE MASHBURN
Keyword(s):  
Quantum Logic ◽  
Information Content ◽  
Shannon Entropy ◽  
Closed Subset ◽  
Quantum States ◽  
Spectral Order ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case

In this paper we extend the spectral order of Coecke and Martin to infinite-dimensional quantum states. Many properties present in the finite-dimensional case are preserved, but some of the most important are lost. The order is constructed and its properties analysed. Most of the useful measurements of information content are lost. Shannon entropy is defined on only a part of the model, and that part is not a closed subset of the model. The finite parts of the lattices used by Birkhoff and von Neumann as models for classical and quantum logic appear as subsets of the models for infinite classical and quantum states.

scholarly journals ENTANGLED MARKOV CHAINS GENERATED BY SYMMETRIC CHANNELS

2005 ◽  
Vol 08 (03) ◽  
pp. 497-504 ◽  
Author(s):  
TAKAYUKI MIYADERA
Keyword(s):  
Markov Chains ◽  
Entropy Density ◽  
Dimensional Case ◽  
Von Neumann Entropy ◽  
Quantum Random Walk ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Symmetric Channel ◽  
Finite Dimensional ◽  
Finite Dimensional Case

A notion of entangled Markov chain was introduced by Accardi and Fidaleo in the context of quantum random walk. They proved that, in the finite-dimensional case, the corresponding states have vanishing entropy density, but they did not prove that they are entangled. In this note this entropy result is extended to the infinite-dimensional case under the assumption of finite speed of hopping. Then the entanglement problem is discussed for spin-1/2, entangled Markov chains generated by a binary symmetric channel with hopping probability 1-q. The von Neumann entropy of these states, restricted on a sublattice is explicitly calculated and shown to be independent of the size of the sublattice. This is a new, purely quantum, phenomenon. Finally the entanglement property between the sublattices [Formula: see text] and [Formula: see text] is investigated using the PPT criterium. It turns out that, for q≠ 0, 1, ½ the states are non-separable, thus truly entangled, while for q = 0, 1, ½, they are separable.

scholarly journals Quantum logic as motivated by quantum computing

2005 ◽  
Vol 70 (2) ◽  
pp. 353-359 ◽  
Author(s):  
J. Michael Dunn ◽  
Tobias J. Hagge ◽  
Lawrence S. Moss ◽  
Zhenghan Wang
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Computer Science ◽  
Quantum Logic ◽  
Theoretical Computer Science ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Definition Of

§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection.The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.

Nonholonomic deformation of coupled and supersymmetric KdV equations and Euler–Poincaré–Suslov method

2015 ◽  
Vol 27 (04) ◽  
pp. 1550011 ◽  
Author(s):  
Partha Guha
Keyword(s):  
Kdv Equation ◽  
Infinite Dimensional ◽  
Kdv Equations ◽  
Finite Dimensional ◽  
Euler Top ◽  
Two Component ◽  
Finite Dimensional Case ◽  
Coupled Kdv Equations ◽  
The Right ◽  
Supersymmetric Kdv Equation

Recently, Kupershmidt [38] presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al. [31]. In this paper, we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler–Poincaré–Suslov (EPS) formulation. In a finite-dimensional case, we modify Kupershmidt's deformation of the Euler top equation to obtain the standard EPS construction on SO(3). We extend Kupershmidt's infinite-dimensional construction to construct a nonholonomic deformation of a wide class of coupled KdV equations, where all these equations follow from the Euler–Poincaré–Suslov flows of the right invariant L2 metric on the semidirect product group [Formula: see text], where Diff (S1) is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to the two-component Camassa–Holm equation. We also give a derivation of a nonholonomic deformation of the N = 1 supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler–Poincaré–Suslov (EPS) method.

scholarly journals A new proof of Donoghue's interpolation theorem

2004 ◽  
Vol 2 (3) ◽  
pp. 253-265 ◽  
Author(s):  
Yacin Ameur
Keyword(s):  
Hilbert Space ◽  
Radon Measure ◽  
Interpolation Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Positive Radon Measure ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
New Interpretation ◽  
Necessary And Sufficient

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

scholarly journals Further Insights Into the Damping-Induced Self-Recovery Phenomenon

2019 ◽  
Vol 141 (4) ◽  
Author(s):  
Tejas Kotwal ◽  
Roshail Gerard ◽  
Ravi Banavar
Keyword(s):  
Large Amplitude ◽  
Mechanical Systems ◽  
Dimensional Case ◽  
Theoretical Interpretation ◽  
Underactuated Mechanical Systems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Bicycle Wheel

In a series of papers, Chang et al. proved and experimentally demonstrated a phenomenon in underactuated mechanical systems, that they termed “damping-induced self-recovery.” This paper further investigates a few features observed in these demonstrated experiments and provides additional theoretical interpretation for the same. In particular, we present a model for the infinite-dimensional fluid–stool–wheel system, that approximates its dynamics to that of the better understood finite dimensional case, and comment on the effect of the intervening fluid on the large amplitude oscillations observed in the bicycle wheel–stool experiment.

scholarly journals Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces

2005 ◽  
Vol 77 (4) ◽  
pp. 589-594 ◽  
Author(s):  
Paolo Piccione ◽  
Daniel V. Tausk
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Countable Family ◽  
Spectral Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Lagrangian Subspaces ◽  
Finite Dimensional Case ◽  
Adjoint Operators

We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.

scholarly journals Irreducible locally nilpotent finitary skew linear groups

1995 ◽  
Vol 38 (1) ◽  
pp. 63-76 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
Linear Groups ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finitary Linear Group ◽  
Locally Nilpotent ◽  
Finitary Linear ◽  
Finite Dimensional Case

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

scholarly journals Correspondences, von Neumann Algebras and Holomorphic L2 Torsion

2000 ◽  
Vol 52 (4) ◽  
pp. 695-736 ◽  
Author(s):  
A. Carey ◽  
M. Farber ◽  
V. Mathai
Keyword(s):  
Complex Manifold ◽  
Von Neumann Algebras ◽  
Compact Complex Manifold ◽  
Dolbeault Cohomology ◽  
Hermitian Metrics ◽  
Variation Formula ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Finite Von Neumann Algebras

AbstractGiven a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic L2 torsion, which lies in the determinant line of the twisted L2 Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in [CFM]. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute ametric variation formula for the holomorphic L2 torsion, which shows that it is not in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic L2 torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

GENERALIZED ANALYTIC FUNCTIONS ON THE COUNTABLE PRODUCT AND DIRECT SUM OF THE COMPLEX NUMBERS

2001 ◽  
Vol 04 (04) ◽  
pp. 559-567
Author(s):  
HENRIK PETERSSON
Keyword(s):  
Exponential Type ◽  
Analytic Functions ◽  
Classical Result ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Borel Transform ◽  
Finite Dimensional ◽  
Countable Product ◽  
Finite Dimensional Case

A classical result states that, in n variables, the space of the entire functionals can be identified with the space of exponential type functions via the Fourier–Borel transform. Thus, in this way the spaces of the entire and exponential type functions can be put in duality, the Martineau duality. We give a proof that the entire functionals, on the countable direct product and direct sum of the field of complex numbers, can be identified with exponential type functions in the same way. In other words, we show that the infinite dimensional Fourier–Borel transform defines Martineau dualities analogous to the finite dimensional case.

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