Cayley transform and the Kronecker product of Hermitian matrices

- Hardy, Yorick, Fošner, Ajda, Steeb, Willi-Hans

**Authors:**Hardy, Yorick , Fošner, Ajda , Steeb, Willi-Hans**Date:**2013-05-05**Subjects:**Hermitian matrices , Cayley transform , Kronecker product**Type:**Article**Identifier:**uj:5976 , http://hdl.handle.net/10210/8441**Description:**We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices.**Full Text:**

**Authors:**Hardy, Yorick , Fošner, Ajda , Steeb, Willi-Hans**Date:**2013-05-05**Subjects:**Hermitian matrices , Cayley transform , Kronecker product**Type:**Article**Identifier:**uj:5976 , http://hdl.handle.net/10210/8441**Description:**We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices.**Full Text:**

An eigenvalue problem for a fermi system and lie algebras

- Steeb, Willi-Hans, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2013-04-15**Subjects:**Schrodinger equation , Eigenvalues , Lie algebras , Hamilton operators , Fermi Hamilton operators**Type:**Article**Identifier:**uj:5977 , http://hdl.handle.net/10210/8442**Description:**Please refer to full text to view abstract**Full Text:**

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2013-04-15**Subjects:**Schrodinger equation , Eigenvalues , Lie algebras , Hamilton operators , Fermi Hamilton operators**Type:**Article**Identifier:**uj:5977 , http://hdl.handle.net/10210/8442**Description:**Please refer to full text to view abstract**Full Text:**

Exceptional points, nonnormal matrices, hierarchy of spin matrices and an eigenvalue problem

- Steeb, Willi-Hans, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2013-02-08**Subjects:**Hamilton operators , Spin matrices , Eigenvalues**Type:**Article**Identifier:**uj:5979 , http://hdl.handle.net/10210/8444**Description:**Exceptional points are studied for non-hermitian Hamilton operators given by a hierarchy of spin-operators.**Full Text:**

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2013-02-08**Subjects:**Hamilton operators , Spin matrices , Eigenvalues**Type:**Article**Identifier:**uj:5979 , http://hdl.handle.net/10210/8444**Description:**Exceptional points are studied for non-hermitian Hamilton operators given by a hierarchy of spin-operators.**Full Text:**

Spin-Hamilton operator, graviton-photon coupling and an eigenvalue problem

- Hardy, Yorick, Steeb, Willi-Hans

**Authors:**Hardy, Yorick , Steeb, Willi-Hans**Date:**2012-09-28**Subjects:**Spin-Hamilton operator , Graviton-photon coupling , Eigenvalues**Type:**Article**Identifier:**uj:5973 , http://hdl.handle.net/10210/8418**Description:**We solve exactly the eigenvalue problem for a spin Hamilton operator describing graviton-photon coupling. Entanglement of the eigenstates are also studied.**Full Text:**

**Authors:**Hardy, Yorick , Steeb, Willi-Hans**Date:**2012-09-28**Subjects:**Spin-Hamilton operator , Graviton-photon coupling , Eigenvalues**Type:**Article**Identifier:**uj:5973 , http://hdl.handle.net/10210/8418**Description:**We solve exactly the eigenvalue problem for a spin Hamilton operator describing graviton-photon coupling. Entanglement of the eigenstates are also studied.**Full Text:**

Hamilton operators, discrete symmetries, brute force and symbolicC++

- Steeb, Willi-Hans, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2012-08-23**Subjects:**Hamilton operators , Spin-Hamilton operators , Quantum theory**Type:**Article**Identifier:**uj:5974 , http://hdl.handle.net/10210/8419**Description:**To find the discrete symmetries of a Hamilton operator ˆH is of central importance in quantum theory. Here we describe and implement a brute force method to determine the discrete symmetries given by permutation matrices for Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi systems are considered as examples. A computer algebra implementation in SymbolicC++ is provided.**Full Text:**

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2012-08-23**Subjects:**Hamilton operators , Spin-Hamilton operators , Quantum theory**Type:**Article**Identifier:**uj:5974 , http://hdl.handle.net/10210/8419**Description:**To find the discrete symmetries of a Hamilton operator ˆH is of central importance in quantum theory. Here we describe and implement a brute force method to determine the discrete symmetries given by permutation matrices for Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi systems are considered as examples. A computer algebra implementation in SymbolicC++ is provided.**Full Text:**

A sequence of quantum gates

- Steeb, Willi-Hans, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2012-02-10**Subjects:**Hilbert spaces , Hamilton operators , Cayley transform , Quantum gates**Type:**Article**Identifier:**uj:5978 , http://hdl.handle.net/10210/8443**Description:**We study a sequence of quantum gates in finite-dimensional Hilbert spaces given by the normalized eigenvectors of the unitary operators. The corresponding sequence of the Hamilton operators is also given. From the Hamilton operators we construct another hierarchy of quantum gates via the Cayley transform.**Full Text:**

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2012-02-10**Subjects:**Hilbert spaces , Hamilton operators , Cayley transform , Quantum gates**Type:**Article**Identifier:**uj:5978 , http://hdl.handle.net/10210/8443**Description:**We study a sequence of quantum gates in finite-dimensional Hilbert spaces given by the normalized eigenvectors of the unitary operators. The corresponding sequence of the Hamilton operators is also given. From the Hamilton operators we construct another hierarchy of quantum gates via the Cayley transform.**Full Text:**

Spin Hamilton operators, symmetry breaking, energy level crossing and entanglement

- Steeb, Willi-Hans, Hardy, Yorick, De Greef, Jacqueline

**Authors:**Steeb, Willi-Hans , Hardy, Yorick , De Greef, Jacqueline**Date:**2012-01-27**Subjects:**Hilbert spaces , Hamilton operators , Spin matrices , Eigenvalues**Type:**Article**Identifier:**uj:5975 , http://hdl.handle.net/10210/8440**Description:**© Willi-Hans Steeb, Yorick Hardy and Jacqueline de Greef, 2012 Available online: http://arxiv.org/abs/1110.4204v2 , 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.**Full Text:**

**Authors:**Steeb, Willi-Hans , Hardy, Yorick , De Greef, Jacqueline**Date:**2012-01-27**Subjects:**Hilbert spaces , Hamilton operators , Spin matrices , Eigenvalues**Type:**Article**Identifier:**uj:5975 , http://hdl.handle.net/10210/8440**Description:**© Willi-Hans Steeb, Yorick Hardy and Jacqueline de Greef, 2012 Available online: http://arxiv.org/abs/1110.4204v2 , 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.**Full Text:**

Vector product and an integrable dynamical system

- Steeb, Willi-Hans, Tanski, Igor, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Tanski, Igor , Hardy, Yorick**Date:**2011**Subjects:**Vector product , Nambu mechanics , Differential equations , Integrals**Type:**Article**Identifier:**uj:5808 , ISSN 0253-6102 , http://hdl.handle.net/10210/7816**Description:**We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.**Full Text:**

**Authors:**Steeb, Willi-Hans , Tanski, Igor , Hardy, Yorick**Date:**2011**Subjects:**Vector product , Nambu mechanics , Differential equations , Integrals**Type:**Article**Identifier:**uj:5808 , ISSN 0253-6102 , http://hdl.handle.net/10210/7816**Description:**We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.**Full Text:**

Classical and quantum computing.

**Authors:**Hardy, Yorick**Date:**2008-05-29T08:32:10Z**Subjects:**recursion theory , coding theory , turing machines , neural networks , quantum computers , quantum theory , algorithms , boolean algebra , data encryption (computer science)**Type:**Thesis**Identifier:**uj:2441 , http://hdl.handle.net/10210/489**Description:**Prof. W.H. Steeb**Full Text:**

**Authors:**Hardy, Yorick**Date:**2008-05-29T08:32:10Z**Subjects:**recursion theory , coding theory , turing machines , neural networks , quantum computers , quantum theory , algorithms , boolean algebra , data encryption (computer science)**Type:**Thesis**Identifier:**uj:2441 , http://hdl.handle.net/10210/489**Description:**Prof. W.H. Steeb**Full Text:**

Entanglement and quantum communication complexity.

**Authors:**Hardy, Yorick**Date:**2007-12-07T07:50:15Z**Subjects:**computational complexity , quantum communication , entropy (information theory)**Type:**Thesis**Identifier:**uj:14740 , http://hdl.handle.net/10210/175**Description:**Keywords: entanglement, complexity, entropy, measurement In chapter 1 the basic principles of communication complexity are introduced. Two-party communication is described explicitly, and multi-party communication complexity is described in terms of the two-party communication complexity model. The relation to entropy is described for the classical communication model. Important concepts from quantum mechanics are introduced. More advanced concepts, for example the generalized measurement, are then presented in detail. In chapter 2 the di erent measures of entanglement are described in detail, and concrete examples are provided. Measures for both pure states and mixed states are described in detail. Some results for the Schmidt decomposition are derived for applications in communication complexity. The Schmidt decomposition is fundamental in quantum communication and computation, and thus is presented in considerable detail. Important concepts such as positive maps and entanglement witnesses are discussed with examples. Finally, in chapter 3, the communication complexity model for quantum communication is described. A number of examples are presented to illustrate the advantages of quantum communication in the communication complexity scenario. This includes communication by teleportation, and dense coding using entanglement. A few problems, such as the Deutsch-Jozsa problem, are worked out in detail to illustrate the advantages of quantum communication. The communication complexity of sampling establishes some relationships between communication complexity, the Schmidt rank and entropy. The last topic is coherent communication complexity, which places communication complexity completely in the domain of quantum computation. An important lower bound for the coherent communication complexity in terms of the Schmidt rank is dervived. This result is the quantum analogue to the log rank lower bound in classical communication complexity. , Prof. W.H. Steeb**Full Text:**

**Authors:**Hardy, Yorick**Date:**2007-12-07T07:50:15Z**Subjects:**computational complexity , quantum communication , entropy (information theory)**Type:**Thesis**Identifier:**uj:14740 , http://hdl.handle.net/10210/175**Description:**Keywords: entanglement, complexity, entropy, measurement In chapter 1 the basic principles of communication complexity are introduced. Two-party communication is described explicitly, and multi-party communication complexity is described in terms of the two-party communication complexity model. The relation to entropy is described for the classical communication model. Important concepts from quantum mechanics are introduced. More advanced concepts, for example the generalized measurement, are then presented in detail. In chapter 2 the di erent measures of entanglement are described in detail, and concrete examples are provided. Measures for both pure states and mixed states are described in detail. Some results for the Schmidt decomposition are derived for applications in communication complexity. The Schmidt decomposition is fundamental in quantum communication and computation, and thus is presented in considerable detail. Important concepts such as positive maps and entanglement witnesses are discussed with examples. Finally, in chapter 3, the communication complexity model for quantum communication is described. A number of examples are presented to illustrate the advantages of quantum communication in the communication complexity scenario. This includes communication by teleportation, and dense coding using entanglement. A few problems, such as the Deutsch-Jozsa problem, are worked out in detail to illustrate the advantages of quantum communication. The communication complexity of sampling establishes some relationships between communication complexity, the Schmidt rank and entropy. The last topic is coherent communication complexity, which places communication complexity completely in the domain of quantum computation. An important lower bound for the coherent communication complexity in terms of the Schmidt rank is dervived. This result is the quantum analogue to the log rank lower bound in classical communication complexity. , Prof. W.H. Steeb**Full Text:**

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