A hierarchy of Hamilton operators and entanglement

Open Physics ◽  
2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Willi-Hans Steeb ◽  
Yorick Hardy
Keyword(s):  
Energy Level ◽  
Time Evolution ◽  
Hilbert Spaces ◽  
Unitary Operator ◽  
Entangled States ◽  
Level Crossing ◽  
Finite Dimensional

AbstractWe consider a hierarchy of Hamilton operators Ĥ N in finite dimensional Hilbert spaces $$ \mathbb{C}^{2^N } $$. We show that the eigenstates of Ĥ N are fully entangled for N even. We also calculate the unitary operator U N(t) = exp(—Ĥ N t/ħ) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.

scholarly journals Spin Hamilton Operators, Symmetry Breaking, Energy Level Crossing, and Entanglement

2012 ◽  
Vol 67 (10-11) ◽  
pp. 608-612
Author(s):  
Willi-Hans Steeb ◽  
Yorick Hardy ◽  
Jacqueline de Greef
Keyword(s):  
Energy Spectrum ◽  
Energy Level ◽  
Symmetry Breaking ◽  
Hilbert Spaces ◽  
Spin Systems ◽  
Level Crossing ◽  
Finite Dimensional ◽  
Breaking Energy ◽  
Coupled Spin Systems

We study finite-dimensional product Hilbert spaces, coupled spin systems, entanglement, and energy level crossing. The Hamilton operators are based on the Pauli group.We show that swapping the interacting term can lead from unentangled eigenstates to entangled eigenstates and from an energy spectrum with energy level crossing to avoided energy level crossing.

UNITARY OPERATORS, ENTANGLEMENT, AND GRAM–SCHMIDT ORTHOGONALIZATION

2009 ◽  
Vol 20 (06) ◽  
pp. 891-899
Author(s):  
YORICK HARDY ◽  
WILLI-HANS STEEB
Keyword(s):  
Quantum Computing ◽  
Computer Algebra ◽  
Hilbert Spaces ◽  
Unitary Transformation ◽  
Unitary Operator ◽  
Unitary Operators ◽  
Unitary Matrices ◽  
Finite Dimensional ◽  
Schmidt Orthogonalization

We consider finite-dimensional Hilbert spaces [Formula: see text] with [Formula: see text] with n ≥ 2 and unitary operators. In particular, we consider the case n = 2m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces [Formula: see text] are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.

Energy Level Crossing and Entanglement

2009 ◽  
Vol 64 (7-8) ◽  
pp. 445-447 ◽  
Author(s):  
Willi-Hans Steeb
Keyword(s):  
Hilbert Space ◽  
Energy Level ◽  
Level Crossing ◽  
Hamilton Operator ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Finite Dimensional Hilbert Space

AbstractWe consider a Hamilton operator in a finite dimensional Hilbert space with energy level crossing. We discuss the question how energy level crossing and entanglement of states in this Hilbert space are intertwined. Since energy level crossing is related to symmetries of the Hamilton operator we also derive these symmetries and give the reduction to the invariant Hilbert subspaces.

scholarly journals A Gutzwiller trace formula for large hermitian matrices

2017 ◽  
Vol 29 (08) ◽  
pp. 1750027
Author(s):  
Jens Bolte ◽  
Sebastian Egger ◽  
Stefan Keppeler
Keyword(s):  
Time Evolution ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Semiclassical Approximation ◽  
Integral Operators ◽  
Fourier Integral ◽  
Quantum Systems ◽  
Quantization Condition ◽  
Fourier Integral Operators ◽  
Finite Dimensional

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schrödinger equation. In the framework of Weyl quantization, we construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantization condition for eigenvalues as well as some simple examples.

Density Matrices and Their Time Evolution

2008 ◽  
Vol 15 (02) ◽  
pp. 109-121 ◽  
Author(s):  
E. Brüning ◽  
F. Petruccione
Keyword(s):  
Time Dependence ◽  
Time Evolution ◽  
Explicit Form ◽  
Hilbert Spaces ◽  
Density Matrices ◽  
Nonlinear Constraint ◽  
Finite Dimensional ◽  
Independent Parameters ◽  
Special Case ◽  
Lack Of Knowledge

Already in the case of finite dimensional Hilbert spaces [Formula: see text] the general form of density matrices ρ is not known. The main reason for this lack of knowledge is the nonlinear constraint for these matrices. We propose a representation of density matrices on finite dimensional Hilbert spaces in terms of finitely many independent parameters. For dimensions 2, 3, and 4 we write down this representation explicitly. As a further application of this representation we study the time dependence of density matrices ρ(t) which in our case is implemented through time dependence of the independent parameters. Under obvious differentiability assumptions the explicit form of [Formula: see text] is determined. As a special case we recover, for instance, the Lindblad form.

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

scholarly journals Preparation of Approximate Eigenvector by Unitary Operations on Eigenstate in Abrams-Lloyd Quantum Algorithm

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Latha S. Warrier
Keyword(s):  
Time Evolution ◽  
Programming Language ◽  
Spin Chain ◽  
Evolution Operator ◽  
Unitary Operator ◽  
Language Model ◽  
Quantum Algorithm ◽  
Basis Vector ◽  
Unitary Matrix ◽  
Time Evolution Operator

The Abrams-Lloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector Va. The eigenstate is a basis vector in the orthonormal eigenspace. Finding another eigenvalue, using a random approximate eigenvector, may require many trials as the trial may repeatedly result in the eigenvalue measured earlier. We present a method involving orthogonalization of the eigenstate obtained in a trial. It is used as the Va for the next trial. Because of the orthogonal construction, Abrams-Lloyd algorithm will not repeat the eigenvalue measured earlier. Thus, all the eigenvalues are obtained in sequence without repetitions. An operator that anticommutes with a unitary operator orthogonalizes the eigenvectors of the unitary. We implemented the method on the programming language model of quantum computation and tested it on a unitary matrix representing the time evolution operator of a small spin chain. All the eigenvalues of the operator were obtained sequentially. Another use of the first eigenvector from Abrams-Lloyd algorithm is preparing a state that is the uniform superposition of all the eigenvectors. This is possible by nonorthogonalizing the first eigenvector in all dimensions and then applying the Abrams-Lloyd algorithm steps stopping short of the last measurement.

Indecomposability of entanglement witnesses

2015 ◽  
pp. 478-488
Author(s):  
Xiao-Fei Qi ◽  
Jin-Chuan Hou
Keyword(s):  
Entangled States ◽  
Finite Dimensional ◽  
Entanglement Witnesses ◽  
Ppt Criterion

We present a way to construct indecomposable entanglement witnesses from any permutations pi with pi^2 not equal to id for any finite dimensional bipartite systems. Some new bounded entangled states are also found, which can be detected by such witnesses while cannot be distinguished by PPT criterion, realignment criterion, etc.

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