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Physical Review A

Finite-time and infinite-time disentanglement of multipartite Greenberger-Horne-Zeilinger-type states under the collective action of different types of noise

Nguyen Ba An, Jaewan Kim

Published Feb 3, 2009• 19 citations
Publisher

Abstract

Disentanglement dynamics of multipartite Greenberger-Horne-Zeilinger-type states $\ensuremath{\mid}{\ensuremath{\Phi}}_{N}⟩=\ensuremath{\alpha}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}+\ensuremath{\beta}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}N}$ due to local environments is studied. It was shown [L. Aolita et al. , Phys. Rev. Lett. 100, 080501 (2008)] that for $\ensuremath{\mid}\ensuremath{\alpha}∕\ensuremath{\beta}\ensuremath{\mid}\ensuremath{\geqslant}1$ it takes an infinite time for states $\ensuremath{\mid}{\ensuremath{\Phi}}_{N}⟩$ to become completely separable if their qubits are only exposed either to a phase-damping dynamics or to an amplitude-damping one. We show, however, that if both phase damping and amplitude damping are simultaneously active then the transition to complete separability takes just a finite time even for $\ensuremath{\mid}\ensuremath{\alpha}∕\ensuremath{\beta}\ensuremath{\mid}\ensuremath{\geqslant}1$. We further show that application of appropriate local operators can circumvent such a finite-time disentanglement.

DOI
10.1103/physreva.79.022303

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Disentanglement dynamics of multipartite Greenberger-Horne-Zeilinger-type states \ensuremath∣\ensuremathΦN⟩=\ensuremathα\ensuremath∣0⟩\ensuremath⨂N+\ensuremathβ\ensuremath∣1⟩\ensuremath⨂N\ensuremath{\mid}{\ensuremath{\Phi}}_{N}⟩=\ensuremath{\alpha}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}+\ensuremath{\beta}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}N}\ensuremath∣\ensuremathΦN​⟩=\ensuremathα\ensuremath∣0⟩\ensuremath⨂N+\ensuremathβ\ensuremath∣1⟩\ensuremath⨂N due to local environments is studied. It was shown [L. Aolita et al. , Phys. Rev. Lett. 100, 080501 (2008)] that for \ensuremath∣\ensuremathα∕\ensuremathβ\ensuremath∣\ensuremath⩾1\ensuremath{\mid}\ensuremath{\alpha}∕\ensuremath{\beta}\ensuremath{\mid}\ensuremath{\geqslant}1\ensuremath∣\ensuremathα∕\ensuremathβ\ensuremath∣\ensuremath⩾1 it takes an infinite time for states \ensuremath∣\ensuremathΦN⟩\ensuremath{\mid}{\ensuremath{\Phi}}_{N}⟩\ensuremath∣\ensuremathΦN​⟩ to become completely separable if their qubits are only exposed either to a phase-damping dynamics or to an amplitude-damping one. We show, however, that if both phase damping and amplitude damping are simultaneously active then the transition to complete separability takes just a finite time even for \ensuremath∣\ensuremathα∕\ensuremathβ\ensuremath∣\ensuremath⩾1\ensuremath{\mid}\ensuremath{\alpha}∕\ensuremath{\beta}\ensuremath{\mid}\ensuremath{\geqslant}1\ensuremath∣\ensuremathα∕\ensuremathβ\ensuremath∣\ensuremath⩾1. We further show that application of appropriate local operators can circumvent such a finite-time disentanglement.
We investigate the disentanglement dynamics of generalized multiqubit Greenberger-Horne-Zeilinger-type states $\ensuremath{\mid}{\ensuremath{\Psi}}_{n,N}⟩=\ensuremath{\alpha}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}n}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}(N\ensuremath{-}n)}+\ensuremath{\beta}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}(N\ensuremath{-}n)}$ (with $1\ensuremath{\leqslant}n<N$) in a local decoherence environment. By adopting the time after which all the bipartite entanglements vanish or become negligibly small as the criterion of entanglement robustness, we show that states $\ensuremath{\mid}{\ensuremath{\Psi}}_{n,N}⟩$ are more robust than states $\ensuremath{\mid}{\ensuremath{\Psi}}_{N}⟩=\ensuremath{\alpha}{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}+\ensuremath{\beta}{\ensuremath{\mid}1⟩}^{\ensuremath{\bigotimes}N}$ in all respects. Since the two states $\ensuremath{\mid}{\ensuremath{\Psi}}_{N}⟩$ and $\ensuremath{\mid}{\ensuremath{\Psi}}_{n,N}⟩$ can be transformed into each other by local operations, we confirm that entanglement robustness can be enhanced by local operations though the amount of entanglement cannot.
We find an algebraic formula for the NNN-partite concurrence of NNN qubits in an XXX matrix. XXX matrices are density matrices whose only nonzero elements are diagonal or antidiagonal when written in an orthonormal basis. We use our formula to study the dynamics of the NNN-partite entanglement of NNN remote qubits in generalized NNN-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case in which each qubit interacts with a local amplitude damping channel. It is shown that only one type of GHZ state loses its entanglement in finite time; for the rest, NNN-partite entanglement dies out asymptotically. Algebraic formulas for the entanglement dynamics are given in both cases. We directly confirm that the half-life of the entanglement is proportional to the inverse of NNN. When entanglement vanishes in finite time, the time at which entanglement vanishes can decrease or increase with NNN depending on the initial state. In the macroscopic limit, this time is independent of the initial entanglement.
We study the robustness of the Greenberger-Horne-Zeilinger (``cat'') class of multipartite states under decoherence. The noise model is described by a general completely positive map for qubits independently coupled to the environment. In particular, the robustness of NNN-party entanglement is studied in the large NNN limit when (a) the number of spatially separated subsystems is fixed but the size of each subsystem becomes large and (b) the size of the subsystems is fixed while their number becomes arbitrarily large. We obtain conditions for entanglement in these two cases. Among our other results, we show that the parity of an entangled state (i.e., whether it contains an even or odd number of qubits) can lead to qualitatively different robustness of entanglement under certain conditions.
Entanglement of an open bipartite system is investigated. In particular, we calculate the entanglement of a subsystem AAA with kkk particles in an NNN atoms of two-level system $(k\ensuremath{\leqslant}N∕2)$ initially prepared in a mesoscopic superposition $\ensuremath{\mid}\ensuremath{\psi}⟩=[{\ensuremath{\mid}{\ensuremath{\phi}}_{1}({\ensuremath{\theta}}_{0})⟩}^{\ensuremath{\bigotimes}N}+{\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}N}]$, with $\ensuremath{\mid}{\ensuremath{\phi}}_{1}({\ensuremath{\theta}}_{0})⟩=\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{0}\ensuremath{\mid}0⟩+\mathrm{sin}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\theta}}_{0}\ensuremath{\mid}1⟩$, subject to the time evolution described by a dephasing channel. For an arbitrary number of particles NNN, numerical results are given for the full time evolution up to ten particles and $1\ensuremath{\leqslant}k\ensuremath{\leqslant}5$. Analytical results are obtained for short times and asymptotic time regimes. Entanglement is maximum when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}∕2$ [Greenberger-Horne-Zeilinger (GHZ) state], independently of the number of particles in each partition. As NNN gets larger, entanglement tends to 1∕21∕21∕2, and becomes independent of ${\ensuremath{\theta}}_{0}$. The ``thermodynamic'' regime ($N\ensuremath{\rightarrow}\ensuremath{\infty}$ with fNfNfN finite, where f=k∕Nf=k∕Nf=k∕N) is reached already for N=1000N=1000N=1000 particles. In the regime of k⪡Nk⪡Nk⪡N and $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the entanglement has its maximum value at ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}∕2$ and for ${\ensuremath{\theta}}_{0}=0,\ensuremath{\pi}$ its minimum value. The time evolution reduces the maximum value of entanglement, showing that the subsystem AAA loses coherence to the environment. Finally, we point out the differences between the entanglement rate obtained here from that one in Phys. Rev. Lett. 89, 210402 (2002), where only one NNN-particle subsystem is used.
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Complete sets of commuting observables (CSCOs) of the form ${\ensuremath{\Sigma}}_{N}={\ensuremath{\Pi}}_{i=1}^{N}{\ensuremath{\sigma}}_{i{\ensuremath{\alpha}}_{i}}\phantom{\rule{0.3em}{0ex}}({\ensuremath{\alpha}}_{i}=x,y,z)$ for an NNN-qubit system are extracted by a simple graphic approach. One can construct $2\ifmmode\times\else\texttimes\fi{}{3}^{N}$ sets of operators, each set consisting of KN{K}_{N}KN​ commuting ${\ensuremath{\Sigma}}_{N}$, ${K}_{N}={2}^{N\ensuremath{-}1}+1$ for even NNN, and ${2}^{N\ensuremath{-}1}$ for odd NNN. Any NNN functional-independent operators among the KN{K}_{N}KN​ operators may be adopted as a CSCO, whose simultaneous eigenstates (SEs) span an orthonormal basis of NNN-qubit space. These SEs have reduced density matrix of rank 2 and can be reduced to the Greenberger-Horne-Zeilinger (GHZ) state form of Eq. (2) in suitable representations. The all-versus-nothing demolition of the elements of reality holds for each basis of the form of Eq. (2) for NNN-qubit $(N\ensuremath{\geqslant}3)$ systems. ${\ensuremath{\Sigma}}_{N}$ may be considered as the infinitesimal operator of rotational operator $R({\ensuremath{\alpha}}_{1},{\ensuremath{\alpha}}_{2},\dots{}{\ensuremath{\alpha}}_{N})={\ensuremath{\Pi}}_{i=1}^{N}\mathrm{exp}[\ensuremath{-}i\ensuremath{\pi}{\ensuremath{\sigma}}_{i{\ensuremath{\alpha}}_{i}}∕2]$ , whose eigenvalue (signature) $r={e}^{\ensuremath{-}i\ensuremath{\pi}\ensuremath{\alpha}}$, or signature exponent $\ensuremath{\alpha}$, may be equivalently used for characterizing each basis.