Estructura de las Representaciones Sociales y sus dimensiones

Contrary to the case of entangled qubit pairs there are, so far, no applicable entanglement mea- sures for systems with three or more qubits. Although there exist various proposals [139, 8] how to quantify many-body entanglement, none of these measures can satisfy the following criteria: 1.) general validity for pure and in particular mixed states. 2.) simple accessibility and 3.) scala- bility. In particular for mixed states, which are the central objects of investigation, entanglement measures exist only as convex-roof constructions of pure state measures. These constructions re- quires an optimization over infinitely many decompositions of the mixed state [see the definition of the entanglement of formation Eq.(6.3)].

Fortunately it turns out that the lack of an adequate entanglement measure does not restrict one to investigate the question whether a many-body state is still entangled or not. Entangle- ment is a phenomenon which depends on the quantum coherences in a system. As long as the system contains such coherences it can in priciple contain few-body entanglement. On the other hand the total lost of coherences due to system-enviroment interactions denotes the point of a total disentanglement. The results from the previous section, where I could monitor the time- evolution of entanglement and decoherence, are in complete agreement with this upper bound on the entanglement [see Fig.(6.9) and Fig.(6.10)].

In the following, I concentrate on two famous examples of maximally entangled many-body states , the generalW-state [199],

|W_ni∝|1000...i+|0100...i+|0010...i+.... , (6.23)

and the general Greenberger-Horne-Zeilinger, orGHZ-state[116],

|GHZ_ni∝|0000...i+|1111...i , (6.24)

where ndenotes the number of qubits in S, i.e. heren=N_S. The two previously studied Bell states|Ψ+iSand|Φ+iSare obviously incorporated in the above definitions of|Wniand|GHZni. This allows to combine the results for the time evolution of decoherence with respect to an entanglement breakdown for 2 qubits with the case of 3 and 4 qubits interacting with a global bathB. The simulations has been performed takingN_B=44 bath spins into account.

Going back to Figs. (6.9) and (6.10) in the previous chapter one can see that entanglement breaking occurs at the same time when the von Neumann entropyE(ρS) saturates for the first

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2

Concurrence and Entropy

t E[$(t)(|W4>)] E[$(t)(|W3>)] E[$(t)(|^+>)] C[lS(t)]/2, |^ + > γ=0 !" #$%#&'(&)%)*) +_, -./"0'"* +_,'1 -./"0'"*2345,'1 -./"0'"*234₆,'1 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 3

Concurrence and Entropy

t E[$(t)(|W >)] 3 4 E[$(t)(|W >)] ф>W_і!@ C(lS)/2, _і! γ=2 + + !" #$%#&'(&)%)*) +, -./"0'"* +_,'1 -./"0'"*2345,'1 -./"0'"*2346,'1

Figure 6.12: Evolution of Concurrence and the normalized von Neumann entropy for|Wi-states in a global bath. The bath length isNB=40, the number of entangled qubits isNS=2,3,4 and∆i=−1. The left and the right figure represent the two cases of non-decaying,γ=0, and decaying interactions,γ=2. In both pictures one can see the universal behavior of the normalized entropies, thus a linear growth followed by a convergence to the maximal possible value of entropy. This maximum marks the point of total entanglement breaking. To stress this fact I have also plotted the Concurrence of|Ψ+i. The biggerNSthe sooner the entropy deviates from the linear part. Thus, the fragility of many-body entanglement increases with the number of entangled parts. Due to the finite system size one observes recurrences of entanglement.

time: the two qubit stateSis completely entangled with the bathB. The total information, which was initially stored in the non-local correlations between the two qubits, must be now located in the quantum correlations between the systemSand the bathB. Thus, the entanglement between S and B serves as an indirect measure of the entanglement within S only. This observation is independent of whetherγ =0 orγ =2 and can, of course , be expanded to bigger systemsS.

Figure (6.12) compares the time evolution ofE(ρS)forW-like initial states for system sizes N_S= 2,3,4. In order to compare the entropy for different numbers of qubits I will plot the entropy per qubit, i.e. E(ρS)/NS, in the following. The evolution of entropy can be divided in three steps. First, at short time scales, it growth slowly followed then by a linear increase at intermediate times and finally by a saturation to some maximal value. The deviation from the linear increase of the normalized entropy thereby depends on the number of qubitsNS. As long as the entropyE(ρS) growths linearly the systemS still contains entanglement. With the beginning of the saturation ofE(ρS)the entanglement of the system is definitely lost and the state

ρS cannot be used anymore as a quantum resource. The total coherence is distributed amongn particles which all interact independenly with a bath. Roughly speaking the influence of the bath on the coherence ofSis threefold. Therefore, it is no surprise that with the increasing number of qubits (and assuming that all qubits interact) the coherences disappear faster.

The above observations are also valid in the case of the GHZ-like states. Figure (6.13) compares E(ρS)/NS the von Neumann entropies normalized by the number of central qubits N_S. Also in this case the saturation of the entropy marks the time when the central system S is completely disentangled and thus not usable anymore as quantum memory. This behavior is completely independent on whetherγ =0 orγ=2.

In general, the linear increase of the entropy agrees with the bounds given by the so-called Lieb-Robinson theorem [197], a theorem about the entanglement growth in dynamical systems

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 2.5 3

Concurrence and Entropy

t E [$ (t)(|GHZ₄>)] E [ $(t)(|GHZ3>)] E [ $(t)(|\+>)] C [lS1(t)]/2, |\ + > γ=0 !" #$%#&'(&)%)*) +_, -./"0'"* +,'1 -./"0'"*234₅,'1 -./"0'"*2346,'1 !" !"#$ !"#% !"#& !"#' !"#( !"#) !" !"#( !$ !$#( !% *+,-.//0,-0!1,2!3,4/+56 4 37894:9;<=>'?:@ !!39894:9;<=> ?:@ 39894:9;\A?:@ *9lB$CB%:D%C!;\A? & γ=2 !" #$%#&'(&)%)*) +_, -./"0'"* +_,'1 -./"0'"*2345,'1 -./"0'"*234₆,'1

Figure 6.13: Evolution of Concurrence and the normalized von Neumann entropy for|GHZi-states in a global bath. The bath’ length isNB=40, the number of entangled qubits isNS=2,3,4 and∆i=−1. The left and the right figure represent the two cases of non-decaying,γ=0, and decaying interactions,γ=2. The universal linear behavior of the normalized entropy, already discussed in Fig.(6.12), can been seen also for theGHZ-like state, independently of γ. Also here recurrences arise due to the finite system size.

[see Chapter??]. The saturation of the entropy corresponds to the time scales at whichρS(t)has minimal fidelity with the initial state. The state itself is definitely a mixed state from the very first contact with the environmentB. However, for short times it is still possible to extract an entangled pure state out this mixture ρS(t) using appropriate purification techniques[85]. Nevertheless, the question, how much mixedness is tolerable inρS, will depend on the algorithms and error correction schemes.

The attempt to extract a general law forT(N_S)the time of the complete entanglement break- ing was not successful. however, the simulations suggest thatT(N_S)does not grow exponentially fast. Whether the correspondence is polynomial or even logarithmic could not be extracted from the small number of data points. On the other hand, a fixed time T gives a non-exponential bound to the maximal storable number of qubits N_S (if one tolerates the total loss of system’s entanglement).

However, it is more realistic to believe that one can tolerate a certain entanglement loss only. The exact amount of entanglement does certainly not depend in a general way on the used algorithms [257] and error correction schemes [85]. Let assume that a certain loss per qubit is tolerable. Then the total storage time remains a constant up to maximal number of qubitsN_Smax. Otherwise the storage time will definitely reduce by a further factor NS. As long as the von Neumann entropy growth linearly this factor would look like 1/N_S. However, at very small time scales one observes a slow transient effect which suggests that f(N_S)>1/N_Sand thus a weaker suppression of the time scale of a successful storage.

At least for the here considered central systems and baths one alway finds an increasing fragility of the saved many-body state, which, however, is not exponentially fast. Due to the limited methods of investigation it remains an open question whether this non-exponential result is still valid for N_S →∞. Nevertheless, this thermodynamical limit has no relevance for the