Calculating effective resistances on underlying networks of association schemes

Recently, in Refs. \cite{jsj} and \cite{res2}, calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity. In this paper, evaluation of effective resistances on more general networks which are underlying networks of association schemes is considered, where by using the algebraic combinatoric structures of association schemes such as stratification and Bose-Mesner algebras, an explicit formula for effective resistances on these networks is given in terms of the parameters of corresponding association schemes. Moreover, we show that for particular underlying networks of association schemes with diameter $d$ such that the adjacency matrix $A$ possesses $d+1$ distinct eigenvalues, all of the other adjacency matrices $A_i$, $i\neq 0,1$ can be written as polynomials of $A$, i.e., $A_i=P_i(A)$, where $P_i$ is not necessarily of degree $i$. Then, we use this property for these particular networks and assume that all of the conductances except for one of them, say $c\equiv c_1=1$, are zero to give a procedure for evaluating effective resistances on these networks. The preference of this procedure is that one can evaluate effective resistances by using the structure of their Bose-Mesner algebra without any need to know the spectrum of the adjacency matrices.Comment: 41 page

Perfect transfer of m-qubit GHZ states

By using some techniques such as spectral distribution and stratification associated with the graphs, employed in [1,2] for the purpose of Perfect state transfer (PST) of a single qubit over antipodes of distance-regular spin networks and PST of a $d$-level quantum state over antipodes of pseudo-distance regular networks, PST of an m-qubit GHZ state is investigated. To do so, we employ the particular distance-regular networks (called Johnson networks) J(2m,m) to transfer an m-qubit GHZ state initially prepared in an arbitrary node of the network (called the reference node) to the corresponding antipode, perfectly. Keywords: Perfect state transferenc, GHZ states, Johnson network, Stratification, Spectral distribution PACs Index: 01.55.+b, 02.10.YnComment: 17 page

Perfect transference of a d-level quantum state over pseudo-distance-regular networks

Following the prescription of Ref. \cite{PST} in which perfect state transference (PST) of a qubit over distance regular spin networks was discussed, in this paper PST of an arbitrary $d$-level quantum state (qudit) over antipodes of more general networks called pseudo distance-regular networks, is investigated. In fact, the spectral analysis techniques used in the previous work \cite{PST}, and algebraic structures of pseudo distance-regular graphs are employed to give an explicit formula for suitable coupling constants in the Hamiltonians so that the state of a particular qudit initially encoded on one site will evolve freely to the opposite site without any dynamical control, i.e., we show that how to derive the parameters of the system so that PST can be achieved. Keywords:Perfect state transfer, $d$-level quantum state, Stratification, Pseudo-distance-regular network PACs Index: 01.55.+b, 02.10.YnComment: 28 pages, 5 figure

A general algorithm for manipulating non-linear and linear entanglement witnesses by using exact convex optimization

A generic algorithm is developed to reduce the problem of obtaining linear and nonlinear entanglement witnesses of a given quantum system, to convex optimization problem. This approach is completely general and can be applied for the entanglement detection of any N-partite quantum system. For this purpose, a map from convex space of separable density matrices to a convex region called feasible region is defined, where by using exact convex optimization method, the linear entanglement witnesses can be obtained from polygonal shape feasible regions, while for curved shape feasible regions, envelope of the family of linear entanglement witnesses can be considered as nonlinear entanglement witnesses. This method proposes a new methodological framework within which most of previous EWs can be studied. To conclude and in order to demonstrate the capability of the proposed approach, besides providing some nonlinear witnesses for entanglement detection of density matrices in unextendible product bases, W-states, and GHZ with W-states, some further examples of three qubits systems and their classification and entanglement detection are included. Also it is explained how one can manipulate most of the non-decomposable linear and nonlinear three qubits entanglement witnesses appearing in some of the papers published by us and other authors, by the method proposed in this paper. Keywords: non-linear and linear entanglement witnesses, convex optimization. PACS number(s): 03.67.Mn, 03.65.UdComment: 37 page

Generating GHZ state in 2m-qubit spin network

We consider a pure 2m-qubit initial state to evolve under a particular quantum me- chanical spin Hamiltonian, which can be written in terms of the adjacency matrix of the Johnson network J(2m;m). Then, by using some techniques such as spectral dis- tribution and stratification associated with the graphs, employed in [1, 2], a maximally entangled GHZ state is generated between the antipodes of the network. In fact, an explicit formula is given for the suitable coupling strengths of the hamiltonian, so that a maximally entangled state can be generated between antipodes of the network. By using some known multipartite entanglement measures, the amount of the entanglement of the final evolved state is calculated, and finally two examples of four qubit and six qubit states are considered in details.Comment: 22 page

Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm

In papers\cite{js,jsa}, the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm. Moreover, it is shown that probability amplitude of observing walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. Finally the capability of Lanczos-based algorithm for evaluation of walk on arbitrary graphs (GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at infinite limit of number of vertices, are in agreement with those of central limit theorem of Ref.\cite{nko}.Comment: 29 pages, 4 figure

Evaluation of effective resistances in pseudo-distance-regular resistor networks

In Refs.[1] and [2], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on the stratification of the network and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity. In this paper, evaluation of effective resistances on more general networks called pseudo-distance-regular networks [21] or QD type networks \cite{obata} is investigated, where we use the stratification of these networks and show that the effective resistances between a given node such as $\alpha$ and all of the nodes $\beta$ belonging to the same stratum with respect to $\alpha$ ($R_{\alpha\beta^{(m)}}$, $\beta$ belonging to the $m$-th stratum with respect to the $\alpha$) are the same. Then, based on the spectral techniques, an analytical formula for effective resistances $R_{\alpha\beta^{(m)}}$ such that $L^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta}$ (those nodes $\alpha$, $\beta$ of the network such that the network is symmetric with respect to them) is given in terms of the first and second orthogonal polynomials associated with the network, where $L^{-1}$ is the pseudo-inverse of the Laplacian of the network. From the fact that in distance-regular networks, $L^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta}$ is satisfied for all nodes $\alpha,\beta$ of the network, the effective resistances $R_{\alpha\beta^{(m)}}$ for $m=1,2,...,d$ ($d$ is diameter of the network which is the same as the number of strata) are calculated directly, by using the given formula.Comment: 30 pages, 7 figure

Bell-states diagonal entanglement witnesses for relativistic and non-relativistic multispinor systems in arbitrary dimensions

Two kinds of Bell-states diagonal (BSD) entanglement witnesses (EW) are constructed by using the algebra of Dirac $\gamma$ matrices in the space-time of arbitrary dimension $d$, where the first kind can detect some BSD relativistic and non-relativistic $m$-partite multispinor bound entangled states in Hilbert space of dimension $2^{m\lfloor d/2\rfloor}$, including the bipartite Bell-type and iso-concurrence type states in the four-dimensional space-time ($d=4$). By using the connection between Hilbert-Schmidt measure and the optimal EWs associated with states, it is shown that as far as the spin quantum correlations is concerned, the amount of entanglement is not a relativistic scalar and has no invariant meaning. The introduced EWs are manipulated via the linear programming (LP) which can be solved exactly by using simplex method. The decomposability or non-decomposability of these EWs is investigated, where the region of non-decomposable EWs of the first kind is partially determined and it is shown that, all of the EWs of the second kind are decomposable. These EWs have the preference that in the bipartite systems, they can determine the region of separable states, i.e., bipartite non-detectable density matrices of the same type as the EWs of the first kind are necessarily separable. Also, multispinor EWs with non-polygon feasible regions are provided, where the problem is solved by approximate LP, and in contrary to the exactly manipulatable EWs, both the first and second kind of the optimal approximate EWs can detect some bound entangled states. Keywords: Relativistic entanglement, Entanglement Witness, Multispinor, Linear Programming, Feasible Region. PACs Index: 03.65.UdComment: 62 page