Entanglement properties of multipartite entangled states under the influence of decoherence

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 $N\to \mathrm{\infty }$. 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.

Figure 1

Upper bound on lifetime $\kappa \tau$ of $N$-party entanglement.

Figure 2

Same as Fig. 1 but with double-logarithmic axis. Note that the same figures also display an upper bound on the lifetime $\kappa \tau$ of $M$-party entanglement in systems with $N\to \mathrm{\infty }$ particles for different $M$, as discussed in Sec. 7. In this case the numbers on the $x$ axis have to be considered as the number $M$ of $M$-party entanglement in question.

Figure 3

(Color online) The graph states corresponding to the complete graph and the star graph are equivalent to the GHZ state in Eq. (22) up to some local unitaries [6].

Figure 4

(Color online) Cluster states in two and three dimensions form a universal resource for quantum computation in the framework of the one-way quantum computer [21].

Figure 5

(Color online) Ring with seven qubits.

Figure 6

(Color online) Under individual coupling due to the same depolarizing channel the lower bounds on $\kappa t$ to the lifetime of distillable $N$-party entanglement for the 1D, 2D, and 3D cluster states remain constant for arbitrary system sizes $N$. For the $N$-party GHZ state the lower bound as well as the exact value for $\kappa t$ according to Eq. (26), until which GHZ state remains distillable entangled, strictly decrease and go to zero as $N\to \mathrm{\infty }$.

Figure 7

(Color online) For the case of particles in rings of size $N⩽10$, which individually decohere according to the same depolarizing channel (14) with parameter $p$: the critical value $p_{\mathrm{crit}}$, after which the first (last) partition becomes PPT ▵ [◻], the lower bound according to Sec. 5B and the upper bounds according to Secs. 5C, 5E.

Figure 8

(Color online) Representatives of the equivalence classes under local unitaries and graph isomorphies of the connected graphs with $N=5$,6,7 vertices [6]. The first (second) column depicts a representative of the class, that is the last (first) class of a given size $N$ to become PPT with respect to some partition. Hence the first (second) column contains those graphs for which the PPT criterion indicates that the $N$-party distillable entanglement contained in these states might be most stable (unstable) (among all graphs with the same number of vertices). Similarly the third (fourth) column shows a graph of the equivalence class, that is, the last (first) of a given size $N$ to become PPT with respect to all partitions. Hence the third (fourth) column contains those graphs for which the PPT criterion indicates that these states might be the last (first) to become $N$-party separable.

Figure 9

(Color online) For the case of particles in rings of size $N⩽10$, which individually decohere according to the same depolarizing channel with parameter $p$: the critical value $p_{\mathrm{crit}}$, after which the first (last) partition becomes PPT ▵ (◻), the lower bound according to Sec. 5B, and the upper bounds according to Secs. 5C, 5E.

Figure 10

(Color online) For the case of individual depolarizing channels with parameter $p$ the threshold value $p_{\mathrm{crit}}=p_{>}∕(2+p_{>})$ [see Eq. (66)] as a function of the interaction phase ${\phi }_{kl}\in [0,\pi ]$ for edges $\{k,l\}$ between two vertices $k$ and $l$ with the same increasing degree $m=\mid N_k\mid =\mid N_l\mid =2,3,\mathrm{\dots },10$. The horizontal line depicts the upper bound according to Sec. 5C in this case.

Figure 11

For blockwise entanglement different partitionings of particles into groups are considered.