Eigenvalues, Peres' separability condition and entanglement

The general expression with the physical significance and positive definite condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the $4\times 4$ density matrix is generally solved. The obvious expression of Peres' separability condition for an arbitrary state of two qubits is then given out and it is very easy to use. Furthermore, we discuss some applications to the calculation of the entanglement, the upper bound of the entanglement, and a model of the transfer of entanglement in a qubit chain through a noisy channel.Comment: 12 pages (Revtex); Revised Version; The example of transfer of entanglement is rewritte

Quantum mechanics in general quantum systems (II): Perturbation theory

We propose an improved scheme of perturbation theory based on our exact solution [An Min Wang, quant-ph/0611216] in general quantum systems independent of time. Our elementary start-point is to introduce the perturbing parameter as late as possible. Our main skills are Hamiltonian redivision so as to overcome a flaw of the usual perturbation theory, and the perturbing Hamiltonian matrix product decomposition in order to separate the contraction and anti-contraction terms. Our calculational technology is the limit process for eliminating apparent divergences. Our central idea is ``dynamical rearrangement and summation" for the sake of the partial contributions from the high order even all order approximations absorbed in our perturbed solution. Consequently, we obtain the improved forms of the zeroth, first, second and third order perturbed solutions absorbing the partial contributions from the high order even all order approximations of perturbation. Then we deduce the improved transition probability. In special, we propose the revised Fermi's golden rule. Moreover, we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example of two-state system to illustrate our scheme and show its advantages. All of this implies the physical reasons and evidences why our improved scheme of perturbation theory are actually calculable, operationally efficient, conclusively more accurate. Our improved scheme is the further development and interesting application of our exact solution, and it has been successfully used to study on open system dynamics [An Min Wang, quant-ph/0601051].Comment: 51 pages, no figure. The second paper in our serial studies. Its earlier version is quant-ph/060205

Quantum mechanics in general quantum systems (III): open system dynamics

We investigate the exact solution, perturbation theory and master equation of open system dynamics based on our serial studies on quantum mechanics in general quantum systems [An Min Wang, quant-ph/0611216 and quant-ph/0611217]. In a system-environment separated representation, a general and explicit solution of open system dynamics is obtained, and it is an exact solution since it includes all order approximations of perturbation. In terms of the cut-off approximation of perturbation and our improved scheme of perturbation theory, the improved form of the perturbed solution of open systems absorbing the partial contributions from the high order even all order approximations is deduced. Moreover, only under the factorizing initial condition, the exact master equation including all order approximations is proposed. Correspondingly, the perturbed master equation and its improved form different from the existed master equation are given. In special, the Redfield master equation is derived out without using Born-Markov approximation. The solution of open system dynamics in the Milburn model is also gained. As examples, Zurek model of two-state open system and its extension with two transverse fields are studied.Comment: 20 pages, no figure. Citations were revised. This is the third preprint in our serial studies. The previous two manuscripts are quant-ph/0611216 and quant-ph/061121

Bounds on the generalized entanglement of formation for multi-party systems

We present a general method to find the upper and lower bounds on the generalized entanglement of formation for multi-party systems. The upper and lower bounds can be expressed in terms of the bi-partite entanglements of formation and/or entropies of various subsystems. The examples for tri- and four-party systems in the both pure states and mixed states are given. We also suggest a little modified definition of generalized entanglement of formation for multi-party systems if EPR pairs are thought of belonging to the set of maximally entangled states.Comment: 4 pages, Revte

Entanglement versus observables

A general scheme to seek for the relations between entanglement and bservables is proposed in principle. In two-qubit systems with enough general Hamiltonian, we find the entanglement to be the functions of observables for six kinds of chosen state sets and verify how these functions be invariant with time evolution. Moreover, we demonstrate and illustrate the cases with entanglement versus a set of commutable observables under eight kinds of given initial states. Our conclusions show how entanglement become observable even measurable by experiment, and they are helpful for understanding of the nature of entanglement in physics.Comment: 4.3 pages, 8 figures, Revised figure caption

Construction of a Scalable, Uniform and Universal Quantum Network and Its Applications

We present a possible candidate of construction of a scalable, uniform and universal quantum network, which is built from quantum gates to elements of quantum circuit, again to quantum subnetworks and finally to an entire quantum network. Our scheme can overcome some difficulties of the existing schemes and makes improvements to different extent in the scale of quantum network, ability of computation, implementation of engineering, efficiency of quantum network, universality, compatibility, design principle, programmability, fault tolerance, error control, industrialization and commercialization {\it et. al} aspects. As the applications of this construction scheme, we obtain the entire quantum networks for Shor's algorithm, Grover's algorithm and solving Schr\"odinger equation in general. This implies that the scalable, uniform and universal quantum networks are able to generally describe the known main results and can be further applied to more interesting examples in quantum computations.Comment: 38 pages, Revised Version, Revte

Simplified and obvious expression of concurrence in Wootters' measure of entanglement of a pair of qubits

We obtain a simplified and obvious expression of "concurrence" in Wootters' measure of entanglement of a pair of qubits having no more than two non-zero eigenvalues in terms of concurrences of eigenstates and their simple combinations. It not only simplifies the calculation of Wootters' measure of entanglement, but also reveals some its general and important features. Our conclusions are helpful to understand and use quantum entanglement further.Comment: 5 page

Generalized GHZ-class and W-class concurrence and entanglement vectors of the multipartite systems consisting of qubits

We propose two classes of the generalized concurrence vectors of the multipartite systems consisting of qubits. Making use of them, we are able to, respectively, describe and quantify GHZ-class and W-class entanglement both in total and between arbitrary two partite in the multipartite system consisting of qubits. In the case of pure state of three qubits that one partite is separable, it is shown to exactly back to the usual Wootters' concurrence after introduce a whole concurrence vector. In principle, our method is applicable to any $N$-partite systems consisting of $N$ qubits.Comment: 4 pages, Revtex. Adding: to show how to exactly back to the usual Wootters' concurrenc

Quantum mechanics in general quantum systems (IV): Green operator and path integral

We first rewrite the perturbation expansion of the time evolution operator [An Min Wang, quant-ph/0611216] in a form as concise as possible. Then we derive out the perturbation expansion of the time-dependent complete Green operator and prove that it is just the Fourier transformation of the Dyson equation. Moreover, we obtain the perturbation expansion of the complete transition amplitude in the Feynman path integral formulism, and give an integral expression that relates the complete transition amplitude with the unperturbed transition amplitude. Further applications of these results can be expected and will be investigated in the near future.Comment: 6 pages, no figure. This is the fourth preprint in our serial studies. The previous three preprints are, respectively, quant-ph/0611216, quant-ph/0611217 and quant-ph/060105

Quantum CPU and Quantum Simulating

Making use of an universal quantum network or QCPU proposed by me [6], some special quantum networks for simulating some quantum systems are given out. Specially, it is obtained that the quantum network for the time evolution operator which can simulate, in general, Schr\"odinger equation.Comment: 8 pages, Revised Versio