Multiparticle entanglement under the influence of decoherence
O. Gühne,1,2F. Bodoky,3and M. Blaauboer31
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2
Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria
3
Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 共Received 9 June 2008; published 10 December 2008兲
We present a method to determine the decay of multiparticle quantum correlations as quantified by the geometric measure of entanglement under the influence of decoherence. With this, we compare the robustness of entanglement in Greenberger-Horne-Zeilinger共GHZ兲, cluster, W, and Dicke states of four qubits and show that the Dicke state is the most robust. Finally, we determine the geometric measure analytically for decaying GHZ and cluster states of an arbitrary number of qubits.
DOI:10.1103/PhysRevA.78.060301 PACS number共s兲: 03.67.Mn, 03.65.Yz, 03.67.Pp
The decoherence of quantum states is a process in quan-tum dynamics that is relevant for the discussion of funda-mental issues like the transition from quantum to classical physics关1兴. Also from a practical point of view decoherence phenomena have to be studied as they occur in experiments involving entanglement and their suppression is of vital im-portance for any implementation of quantum information processing.
Due to this importance, the influence of decoherence on the entanglement of multiparticle systems has been studied from several perspectives 关2,3兴. These investigations con-cerned either the lifetime of entanglement or the entangle-ment properties of the bipartite system which arise if the multiparticle system is split into two parts. The lifetime of entanglement, however, gives no quantitative information about the decay of entanglement关3兴. Moreover, as a highly entangled multiparticle state may be separable with respect to each bipartition关4兴, considering bipartite aspects only may not lead to a full understanding of the decoherence process. It is therefore highly desirable to investigate a full multipar-tite entanglement measure under the influence of decoher-ence. Unfortunately, all known entanglement measures for multiparticle entanglement are defined via complicated opti-mization procedures 关5兴, which makes it practically impos-sible to compute them for a given mixed quantum state.
In this paper we present a method to investigate the decay of quantum correlations which can be used to overcome these difficulties. We study different four-qubit states and use our method to compare their robustness against decoherence, using a phenomenological model described below. Our ap-proach allows us to compute the entanglement for Greenberger-Horne-Zeilinger 共GHZ兲 and cluster states of an arbitrary number of qubits and thereby to investigate the scaling behavior for these states under decoherence. As we will further see, our results can be directly tested in nowa-days experiments with photons or trapped ions. Finally, from the viewpoint of pure quantum information theory, our re-sults represent one of the few cases where the computation of a relevant entanglement measure for mixed states can be per-formed关6兴.
We consider the following situation: a pure quantum state 兩典 is prepared at time t=0 and in the presence of noise evolves to a mixed state 共t兲. Our task is to quantitatively investigate the time evolution of the entanglement E共t兲 = E关共t兲兴 and its dependence on the initial state and the num-ber of qubits.
As entanglement quantifier, we use the geometric measure of entanglement 关7兴. This is a popular entanglement mono-tone for multiparticle systems, which is related to the dis-crimination of multiparticle states with local means 关8兴 and has been investigated from several perspectives 关9–11兴. For pure states, it is defined as
EG共兩典兲 = 1 − max
兩典=兩a典兩b典兩c典¯兩具兩典兩
2, 共1兲
i.e., as one minus the maximal overlap of 兩典 with fully separable states 兩典. It is extended to mixed states by the convex roof construction
EG共兲 = min pk,兩k典
兺
kpkEG共兩k典兲, 共2兲 where the minimization is taken over all convex decomposi-tions of —i.e., over all probabilities pk and states 兩k典 which fulfill 兺kpk兩k典具k兩=. Clearly, the optimization in Eq. 共1兲 and especially in Eq. 共2兲 is difficult to perform.
Our method can be summarized as follows: since any set of probabilities pkand states兩k典 in Eq. 共2兲 results in a valid upper bound, we obtain a good upper bound by choosing them appropriately. Then we use the results of Refs.关12,13兴 to obtain a lower bound on EG共t兲. There it has been shown how the geometric measure can be estimated if the mean value of a single or a few observables is given关14兴. We show that the lower and upper bounds coincide for the multipar-ticle states we investigate below, allowing for a precise de-termination of EG共t兲.
The noise we consider is described by a master equation for the matrix elements klas they are used in phenomeno-logical models of decoherence for, e.g., electron spin qubits 关15兴: tkl=
再
兺
i⫽k 共Wikii− Wkikk兲 for k = l, − Vklkl for k⫽ l.冎
共3兲 We consider a global dephasing process, where the relax-ation of the diagonal elements plays no role 共Wij= 0∀ i, j兲 and the off-diagonal terms are affected by a global dephasing rate Vkl⬅␥. This process leads to exponentially decaying off-diagonal components kl共t兲=xkl共0兲, where here and in the following x⬅e−␥t. At the end of the paper we will discussPHYSICAL REVIEW A 78, 060301共R兲 共2008兲
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extensions of our method to other 共e.g., local兲 decoherence models.
Before presenting our results for different four-qubit states in detail, note that by construction the geometric mea-sure is a convex quantity—i.e., EG关1+共1−兲2兴 艋EG共1兲+共1−兲EG共2兲. From this, it follows directly that EG共t兲 is monotonically decreasing. Moreover, EG共x兲 is con-vex in the parameter x 关since 共x兲 depends linearly on x兴, and consequently EG共t兲 is convex in the time t.
Let us start our discussion with the four-qubit GHZ state 关16兴, given by
兩GHZ4典 = 共兩0000典 + 兩1111典兲/
冑
2. 共4兲 The geometric measure of the GHZ states equals 1/2 as the maximal overlap with product states is 1/2 关7兴.In order to estimate the entanglement from below, note that the time-dependent fidelity is given by F共t兲 = Tr关共t兲兩GHZ4典具GHZ4兩兴=共1+e−␥t兲/2. Using the results of Ref. 关12兴 we can estimate the geometric measure from F共t兲. Explicitly, it has been shown that if for an arbitrary the fidelity F of a state 兩0典 is given, then EGis bounded by
EG共兲 艌 sup r艌0
再
1 + rF共t兲 −1 2关1 + r +冑
共1 + r兲 2− 4rE G共0兲兴冎
. 共5兲 Applying this to 共t兲 and the fidelity of the GHZ state leads to共using x=e−␥t兲EG共兲 艌 1
2共1 −
冑
1 − x2兲. 共6兲
In order to obtain an upper bound, we consider the two states 兩1典=c兩0000典+s兩1111典 and 兩2典=s兩0000典+c兩1111典, with c = cos共␣兲 and s = sin共␣兲, and write 共t兲 =共1/2兲兺k=1,2兩k典具k兩. Then, x/2=共2cs兲/2 has to hold, and using the fact that the geometric measure for the states兩k典 is given by EG= min兵s2, c2其 关17兴 one arrives at an upper bound for EG共t兲 which is given by the right-hand side of Eq. 共6兲. Hence, EG共兲=共1−
冑
1 − x2兲/2 for the four-qubit GHZ state 共4兲 under the influence of noise. This function is shown in Fig.1.Second, let us discuss the four-qubit cluster state 关18兴 兩CL4典 = 共兩0000典 + 兩0011典 + 兩1100典 − 兩1111典兲/2, 共7兲 which has a geometric measure of EG共兩Cl4典兲=3/4 关10兴. For this state, the decay of the fidelity is given by F共t兲=共1+3x兲/4, which 关combined with Eq. 共5兲兴 leads to the lower bound
E共兲 艌3
8关1 + x −
冑
1 +共2 − 3x兲x兴. 共8兲 For the upper bound, we consider four trial vectors for the decomposition. The first is given by 兩1典=c兩0000典 + s共兩1100典+兩0011典−兩1111典兲/冑
3, and the other three are ob-tained from this by permuting the four terms. We choose c 艌s; then, any of the four states has a geometric measure of EG共兩i典兲=s2. With the ansatz 共t兲=共1/4兲兺k=14 兩
k典具k兩 we ob-tain as a condition on s and c that x/4=关共2cs/
冑
3兲 +共2s2/3兲兴/4. From this, c can be determined. This leads after a short calculation to the insight that the right-hand side of Eq.共8兲 also constitutes an upper bound on the entanglement and thus describes exactly the time evolution of the entangle-ment.Third, a four-qubit W state is given by 关19兴
兩W4典 = 共兩0001典 + 兩0010典 + 兩0100典 + 兩1000典兲/2, 共9兲 for which the geometric measure equals 37/64 关7兴. Let us first derive the upper bound. We take as test states the state 兩1典=c兩1000典+s共兩0100典+兩0010典+兩0001典兲/
冑
3 and permuta-tions thereof. Using a symmetry argument关9兴, their geomet-ric measure is determined to be EG共兩i典兲=关5+3c2共c4+ c2 − 3兲兴/关共3−4c2兲2兴 for c艋1/冑
2 and EG共兩i典兲=1−c2 for c 艌1/
冑
2关17兴. Then, one can derive an upper bound as for the cluster state.It turns out, however, that this upper bound is not convex in x. Since we know that EGhas to be convex, we can take the convex hull共in x兲 of this upper bound:
EG艋 37共81x − 37兲 2816 for x艌 x0, EG艋 3 8关1 + x −
冑
1 +共2 − 3x兲x兴 for x 艋 x0, 共10兲 with x0= 2183/2667. Physically, taking the convex hull just means that for short times共when x艌x0兲 the optimum in the convex roof in Eq. 共2兲 is of the form 共x兲=p兩W4典具W4兩+ 共1−p兲共x0兲, with 共x0兲=共1/4兲兺k兩k典具k兩. Note that for longer times共x艋x0兲 the upper bound 共10兲 is the same as for the cluster state.In order to see that this upper bound is optimal, let us derive a lower bound. Here, we not only take the fidelity of the W state into account, but we use as a second observable the projector onto the space with one excitation, P1 = 兩0001典具0001兩 + 兩0010典具0010兩+兩0100典具0100兩+兩1000典具1000兩. Using the fact that the fidelity of 兩W4典 is given by F共t兲 =共1+3x兲/4 and that it is always in the space spanned by P1
( i.e., Tr关共t兲P1兴=1), we can use the methods of Ref. 关12兴 to obtain a lower bound from these two expectation values关20兴. It turns out that, within numerical accuracy, the lower bound
FIG. 1. 共Color online兲 The geometric measure EG共t兲 for four-qubit GHZ, cluster, W, and Dicke states for the case␥=1. For the W and Dicke states, the upper bounds are shown, as the curves of the lower bounds coincide with that. For tⲏ2 the values for the W and cluster states coincide关see Eqs. 共8兲 and 共10兲兴.
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coincides with the upper bound, giving strong numerical evi-dence that Eq. 共10兲 is the exact expression for the decay of quantum correlations in the W state共9兲.
As a last example of a four-qubit state, let us discuss the symmetric Dicke state 关21兴 given by
兩D4典 = 共兩0011典 + 兩0101典 + 兩1001典 + 兩1100典 + 兩0110典
+兩1010典兲/
冑
6, 共11兲which has a geometric measure EG= 5/8 关7兴. To obtain the upper bound, we proceed similarly as for the W state: We take six states, the first one being 兩1典=c兩0011典+s共兩0101典 +兩1001典+兩1100典+兩0110典+兩1010典兲/
冑
5 and other ones ob-tained by permuting the terms. Then we make the ansatz 共t兲=共1/6兲兺k=16 兩
k典具k兩. This leads to an upper bound which is not convex in x, and subsequently taking the convex hull leads to EG艋 5共3x − 1兲 16 for x艌 5 7, EG艋 5 18关1 + 2x −
冑
1 +共4 − 5x兲x兴 for x 艋 5 7. 共12兲 The lower bound is found analogously to the W state as well, with the projectorP2 onto the space with two excitations as second observable. The resulting bound coincides again with the upper bound, giving strong evidence that Eq. 共12兲 de-scribes the time evolution of the entanglement.For a comparison between the different states, we con-sider the logarithmic derivative =t(ln关EG共t兲兴) =tEG共t兲/EG共t兲, which describes the relative decay of en-tanglement 关22兴. The values of this quantity are plotted in Fig.2. One can clearly see that the Dicke state is the most robust state, while the GHZ state is the most fragile state 关23兴. It is an interesting open question as to which properties of the Dicke state are responsible for the high robustness.
For N qubits, we restrict our attention to GHZ and linear cluster states, as they are highly relevant for applications like quantum metrology or measurement-based quantum compu-tation 关24兴. In the decoherence model, one might keep the dephasing rate ␥=␥0 constant for any number of qubits 共where ␥0 is the single-qubit dephasing rate兲 or scale it as
␥= N␥0共as would occur for the GHZ state in a local dephas-ing model兲. For the GHZ state, nothdephas-ing changes and all the formulas obtained in the above for the four-qubit case apply. Concerning the cluster state, we consider linear cluster states with N = 2n qubits. The linear cluster state is given by
兩CLN典 = 丢 k=1
n
关兩00典 + 兩11典共x丢1兲兴/
冑
2, 共13兲 where this formula should be understood as an iteration, with the operator共x丢1兲 acting on the Bell state of the next two qubits. Explicitly, we have 兩CL2典=共兩00典+兩11典兲/冑
2 and 兩CL4典⬃兩00典共兩00典+兩11典兲+兩11典关x丢1共兩00典+兩11典兲兴 = 兩0000典 +兩0011典+兩1110典+兩1101典 关25兴. Note that the maximal overlap of the cluster state with fully separable states is 1/2n关10兴.We calculate the geometric measure for this state under the effect of decoherence in the same way as for the four-qubit case. The lower bound is obtained from the fidelity F共t兲=关1+共2n− 1兲x兴/2n and yields E
G艌 1
2N(共2−3⫻2
n+ 2N兲x + 2共2n− 1兲兵1−
冑
共1−x兲关1+共2n− 1兲x兴其). For the upper bound, we consider 2n test states similar to the ones before and arrive at an upper bound which coincides again with the lower bound.To investigate the scaling behavior, we consider the time t1/2when the entanglement has decreased to half of the initial value. These times can be directly computed for the N-particle GHZ and cluster states. Figure 3 shows t1/2 as a function of the number of qubits N for the constant and linear models of␥. In both cases the time t1/2of the cluster state is, in the limit N→⬁, larger than that of the GHZ state by a factor of ln共4兲/ln共4/3兲⬇4.82, giving quantitative evidence for the higher robustness against dephasing of the cluster state.
In the calculations presented in this paper we concentrated on a global decoherence model. However, our results can also be applied to other models. First, for the W and GHZ states, our model is equivalent to a local dephasing noise, as occurs in multiphoton experiments. Given the experimental availability of W and GHZ states with photons关26兴, our re-sults can be directly tested with present-day technology. Sec-ond, for local decoherence models where relaxation is the dominant process, a small modification of our scheme allows the calculation of the geometric measure for certain states,
FIG. 2. 共Color online兲 Logarithmic derivative of EG共t兲 共for ␥
= 1兲 for different four-qubit states. The nonanalytic behavior for the Dicke and W states stems from the convex hull in Eqs. 共10兲 and 共12兲.
FIG. 3. 共Color online兲 Comparison between the GHZ state and the cluster state as a function of the number of qubits N. The half lifetime t1/2is shown for the case that ␥=4 and for the case that
␥=N increases linearly with the number of qubits.
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such as the W state关27兴. The fact that this type of decoher-ence is dominant in ion traps关28兴 combined with the possi-bility to generate such states关29兴 opens another way for an experimental test.
To conclude, our results provide a versatile method to determine the decay of multiparticle entanglement for quan-tum states under the influence of decoherence. Our results
can be tested in multiqubit experiments and may therefore lead to a better understanding of decoherence as a fundamen-tal obstacle in quantum information processing.
We thank A. Miyake for discussions. This work has been supported by the FWF 共START prize兲, the EU 共SCALA, QICS, OLAQUI兲, and the Netherlands Organisation for Sci-entific Research 共NWO兲.
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关4兴 For instance, thermal states of a Heisenberg model of few spins have this property for certain temperatures. Therefore, if decoherence drives a system into such a state, multiparticle entanglement measures must be considered. See G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Phys. Rev. Lett. 99, 250405共2007兲.
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关13兴 O. Gühne, M. Reimpell, and R. F. Werner, Phys. Rev. A 77, 052317共2008兲.
关14兴 In a nutshell, E can be estimated from the mean values of two observables A and B as follows 关12,13兴: for a given pair 共1,2兲 one defines the observable X=1A +2B and considers
its Legendre transform Eˆ 共X兲=sup兵Tr共X兲−E共兲其. Then, for any state E共兲艌Tr共X兲−Eˆ共X兲, and more generally E共兲 艌sup1,2Tr共X兲−Eˆ共X兲. The calculation of Eˆ共X兲 can often be
simplified or analytically performed关12,13兴, and Eq. 共5兲 is an example thereof.
关15兴 H.-A. Engel and D. Loss, Phys. Rev. B 65, 195321 共2002兲; M. Blaauboer and D. P. DiVincenzo, Phys. Rev. Lett. 95, 160402
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suffices to derive an upper bound of E关共t兲兴. Such an upper bound is easy to obtain by finding a product vector with a high overlap in Eq.共1兲.
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关23兴 One may ask at this point whether this conclusion is justified, as our decoherence model is not invariant under local changes of the basis 共as any dephasing model兲, leading to different decoherence rates in different bases. In our case, however, the chosen basis for the GHZ and cluster states maximizes the long-time fidelity F共t=⬁兲=Tr关兩典具兩共t=⬁兲兴=兺kkk2, as the
maximal overlap with product states bounds the diagonal ele-ments of , and hence kk艋1/2 共GHZ; respectively, 艋1/4 for
the cluster state兲. From this it follows that the fidelity in any other basis will be smaller at any time, and hence lower bounds based on the fidelity will only be worse关12,13兴. This is also a justification of the cluster-state representation in Eq. 共13兲, which is in this sense superior to the one in Ref. 关18兴. 关24兴 V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330
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关25兴 One can iteratively check that the resulting state is an eigen-state of the operators S1=zz11. . .; S2=xxx11. . .; S3
=1zzz1 . . . ; . . .; SN=1 . . . 1xx and thereby fulfills共up to a
local rotation兲 the definition of a linear cluster state 关18兴. 关26兴 M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H.
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