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Register a new userGenuinely Multipartite Concurrence of N-qubit X-matrices
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Added by
Seyed Mohammad Hashemi Rafsanjani
Publication date
2012
fields
Physics
and research's language is
English
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We find an algebraic formula for the N-partite concurrence of N qubits in an X-matrix. X- matricies are density matrices whose only non-zero elements are diagonal or anti-diagonal when written in an orthonormal basis. We use our formula to study the dynamics of the N-partite entanglement of N remote qubits in generalized N-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case when each qubit interacts with a partner harmonic oscillator. It is shown that only one type of GHZ state is prone to entanglement sudden death; for the rest, N-partite entanglement dies out momentarily. Algebraic formulas for the entanglement dynamics are given in both cases.
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We present a new kind of monogamous relations based on concurrence and concurrence of assistance. For $N$-qubit systems $ABC_1...C_{N-2}$, the monogamy relations satisfied by the concurrence of $N$-qubit pure states under the partition $AB$ and $C_1...C_{N-2}$, as well as under the partition $ABC_1$ and $C_2...C_{N-2}$ are established, which give rise to a kind of restrictions on the entanglement distribution and trade off among the subsystems.
By focusing on the X-matrix part of a density matrix of two qubits we provide an algebraic lower bound for the concurrence. The lower bound is generalized for cases beyond two qubits and can serve as a sufficient condition for non-separability for bipartite density matrices of arbitrary dimension. Experimentally, our lower bound can be used to confirm non-separability without performing a complete state tomography.
The problems of genuine multipartite entanglement detection and classification are challenging. We show that a multipartite quantum state is genuine multipartite entangled if the multipartite concurrence is larger than certain quantities given by the number and the dimension of the subsystems. This result also provides a classification of various genuine multipartite entanglement. Then, we present a lower bound of the multipartite concurrence in terms of bipartite concurrences. While various operational approaches are available for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and classify the genuine multipartite entanglement. As applications, the genuine multipartite entanglement of tripartite systems is analyzed in detail.
I generalize the concept of balancedness to qudits with arbitrary dimension $d$. It is an extension of the concept of balancedness in New J. Phys. {bf 12}, 075025 (2010) [1]. At first, I define maximally entangled states as being the stochastic states (with local reduced density matrices $id/d$ for a $d$-dimensional local Hilbert space) that are not product states and show that every so-defined maximal genuinely multi-qudit entangled state is balanced. Furthermore, all irreducibly balanced states are genuinely multi-qudit entangled and are locally equivalent with respect to $SL(d)$ transformations (i.e. the local filtering transformations (LFO)) to a maximally entangled state. In particular the concept given here gives the maximal genuinely multi-qudit entangled states for general local Hilbert space dimension $d$. All genuinely multi-qudit entangled states are an element of the partly balanced $SU(d)$-orbits.
We examine entanglement dynamics via concurrence among four two-state systems labeled $A, ~a, ~B, ~b$. The four systems are arranged on an addressable lattice in such a way that $A$ and $a$ at one location labeled $Aa$ can interact with each other via excitation exchange, and the same for $B$ and $b$ at location $Bb$. The $Aa$ location is prepared entangled with the $Bb$ location, but their mutual complete isolation prevents interaction in the interval between actions of an external addressing agent. There are six pairwise concurrences on the lattice, and we follow their evolution in the interval between external actions. We show how entanglement evolves and may exhibit the non-analytic effect termed entanglement sudden death (ESD), with periodic recovery. These loss and gain processes may be interpreted as entanglement transfer between the subsystems.
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