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arXiv:quant-ph/0408165 · Physical Review A

Entanglement properties of multipartite entangled states under the influence of decoherence

M. Hein, Wolfgang Dür, Hans J. Briegel

Published Mar 30, 2005• 193 citations
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Abstract

We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for Greenberger-Horne-Zeilinger (GHZ) -type superposition states decreases with the size of the system, while for a class of other states---namely, all graph states with constant degree---the lifetime is independent of the system size. We show that these results are largely independent of the specific decoherence model and are in particular valid for all models which deal with individual couplings of particles to independent environments, described by some quantum optical master equation of Lindblad form. For GHZ states, we derive analytic expressions for the lifetime of distillable entanglement and determine when the state becomes fully separable. For all graph states, we derive lower and upper bounds on the lifetime of entanglement. The lower bound is based on a specific distillation protocol, while upper bounds are obtained by showing that states resulting from decoherence in general become nondistillable or even separable after a finite time. This is done using different methods, namely, (i) the map describing the decoherence process (e.g., the action of a thermal bath on the system) becomes entanglement breaking, (ii) the resulting state becomes separable, and (iii) the partial transposition with respect to certain partitions becomes positive. To this aim, we establish a method to calculate the spectrum of the partial transposition for all mixed states which are diagonal in a graph-state basis. We also consider entanglement between different groups of particles and determine the corresponding lifetimes as well as the change of the kind of entanglement with time. This enables us to investigate the behavior of entanglement under rescaling and in the limit of large number of particles $N\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}$. Finally we investigate the lifetime of encoded quantum superposition states and show that one can define an effective time in the encoded system which can be orders of magnitude smaller than the physical time. This provides an alternative view on quantum error correction and examples of states whose lifetime of entanglement (between groups of particles) in fact increases with the size of the system.

DOI
10.1103/physreva.71.032350

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We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for Greenberger-Horne-Zeilinger (GHZ) -type superposition states decreases with the size of the system, while for a class of other states---namely, all graph states with constant degree---the lifetime is independent of the system size. We show that these results are largely independent of the specific decoherence model and are in particular valid for all models which deal with individual couplings of particles to independent environments, described by some quantum optical master equation of Lindblad form. For GHZ states, we derive analytic expressions for the lifetime of distillable entanglement and determine when the state becomes fully separable. For all graph states, we derive lower and upper bounds on the lifetime of entanglement. The lower bound is based on a specific distillation protocol, while upper bounds are obtained by showing that states resulting from decoherence in general become nondistillable or even separable after a finite time. This is done using different methods, namely, (i) the map describing the decoherence process (e.g., the action of a thermal bath on the system) becomes entanglement breaking, (ii) the resulting state becomes separable, and (iii) the partial transposition with respect to certain partitions becomes positive. To this aim, we establish a method to calculate the spectrum of the partial transposition for all mixed states which are diagonal in a graph-state basis. We also consider entanglement between different groups of particles and determine the corresponding lifetimes as well as the change of the kind of entanglement with time. This enables us to investigate the behavior of entanglement under rescaling and in the limit of large number of particles N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}N\ensuremath→\ensuremath∞. Finally we investigate the lifetime of encoded quantum superposition states and show that one can define an effective time in the encoded system which can be orders of magnitude smaller than the physical time. This provides an alternative view on quantum error correction and examples of states whose lifetime of entanglement (between groups of particles) in fact increases with the size of the system.
The multipartite Greenberger-Horne-Zeilinger (GHZ) state is a paradigmatic example of a highly entangled multipartite state with distinct quantum features. However, the GHZ state is very sensitive to generic decoherence processes, where its quantum features and, in particular, its entanglement diminish rapidly, thereby hindering possible practical applications. In this paper, we discuss GHZ-like quantum states with a block-local structure and show that they exhibit a drastically increased stability against noise for certain choices of block-encoding, thereby extending results of [Phys. Rev. Lett. 106, 110402 (2011)]. We analyze in detail the decay of the interference terms, the entanglement in terms of distillable entanglement, and negativity as well as the notion of macroscopicity as measured by the so-called index qqq and provide general bounds on these quantities. We focus on an encoding where logical qubits are themselves encoded as GHZ states, which leads to so-called concatenated GHZ (C-GHZ) states. We compare the stability of C-GHZ states with other types of encodings, thereby showing the superior stability of the C-GHZ states. Analytic results are complemented by numerical studies, where tensor network techniques are used to investigate the quantum properties of multipartite entangled states under the influence of decoherence.
We consider graph states of an arbitrary number of particles undergoing generic decoherence. We present methods to obtain lower and upper bounds for the system's entanglement in terms of that of considerably smaller subsystems. For an important class of noisy channels, namely, the Pauli maps, these bounds coincide and thus provide the exact analytical expression for the entanglement evolution. All of the results apply also to (mixed) graph-diagonal states and hold true for any convex entanglement monotone. Since any state can be locally depolarized to some graph-diagonal state, our method provides a lower bound for the entanglement decay of any arbitrary state. Finally, this formalism also allows for the direct identification of the robustness under size scaling of graph states in the presence of decoherence, merely by inspection of their connectivities.
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.
Multipartite quantum states may exhibit different types of quantum entanglement in that they cannot be converted into each other by local quantum operations only, and fully understanding mathematical structures of different types of multipartite entanglement is a very challenging task. In this paper, from the viewpoint of Hardy's nonlocality, we compare W and GHZ (Greenberger-Horne-Zeilinger) states and show a couple of crucial different behaviors between them. Particularly, by developing a geometric model for Hardy's nonlocality problem of W states, we derive an upper bound for its maximal violation probability, which turns out to be strictly smaller than the corresponding probability of GHZ states. This gives us a new comparison between these two quantum states, and the result is also consistent with our intuition that GHZ states are more entangled. Furthermore, we generalize our approach to obtain an asymptotic characterization for general NNN-qubit W states, revealing that when NNN goes up, the speed that the maximum violation probabilities decay is exponentially slower than that of general NNN-qubit GHZ states. We provide some numerical simulations to verify our theoretical results.