Modelación de incertidumbre con diagramas de influencia

We now study the entanglement in the state (4.2.1c).

The density matrix for the subsystem Alice-Rob is obtained from ∣Ψ¹⟩⟨Ψ¹∣ by tracing over region II: As in subsection 4.2.1, it is not possible to find an analytic expression for the eigenvalues of (4.2.11). We calculate N numerically. We show our numerical results in Fig. 4.3.

Chapter 4: Entanglement redistribution between charged bosonic field modes in relativistic settings

0.2 0.4 0.6 0.8 1.0 1.2 1.4

r

0.1 0.2 0.3 0.4 0.5

N

q_R= 0.1 q_R= 0.3 q_R= 0.5 qR= 0.6 q_R= 0.7 q_R= 0.9 q_R= 1.0

Figure 4.3: NegativityN as a function of r for the state ρ^{P T}A−R. Curves are for qR= 1, 0.9, 0.7, 0.6, 0.5, 0.3, 0.1from top to bottom.

We find once more that entanglement is degraded in all cases and vanishes in the limit of infinite acceleration.

The standard result addressing how entanglement behaves as a function of accelera-tion of one of the two parties have, in the majority of cases, shown that entanglement monotonically decreases. A rare counterexample was found in [51]. There the authors found that, in analogous settings as described in this Chapter and for fermionic systems, there are values of q_R that allow for non-monotonically decrease of entanglement as a function of the acceleration. In the present we find again a similar behavior as in [51].

The physical motivations for such feature to arise are not yet understood, although we might conjecture the following: when q_R= 1, there is initial entanglement between Alice and Right Unruh modes as analyzed by Bob, while when q_R≠ 1, Alice is entangled with both Right Unruh modes (Right Unruh degrees of freedom) and Left Unruh modes (Left Unruh degrees of freedom). On one hand, when qR = 1, we can perform a change of basis from Unruh to Rindler bases and by tracing over degrees of freedom in region II we will always lose entanglement (see (3.1.1)). On the other hand, when qR≠ 1 we can

Chapter 4: Entanglement redistribution between charged bosonic field modes in relativistic settings

still perform the same change of basis and trace over degrees of freedom in region II, but in this case, while increasing r decreases the contribution from the Right modes in region I (see (3.1.1)), it also increases the contribution of the Left modes in region I, therefore creating two competing effects. For this reason, one can expect that in principle entanglement is not monotonically degraded¹.

We focus on a region 0< r < 0.25 of 4.3. We find numerically that for some values of

∣q^R∣ entanglement is not a monotonically decreasing function of r. We choose to show a sample for∣q^R∣ = 0.9.

0.05 0.10 0.15 0.20

r

0.398 0.400 0.402 0.404 0.406

N

q_R= 0.9

Figure 4.4: NegativityN as a function of r in the range 0 < r < 0.25 from figure 4.3.

We choose∣q^R∣ = 0.9 as a sample.

1We thank Gerardo Adesso at the University of Nottingham for indicating that this might be the cause and that Local Operations might lie at the basis of such effect

Chapter 4: Entanglement redistribution between charged bosonic field modes in relativistic settings

Assuming now that Rob only analyzes particles, we trace over antiparticles in region I and obtain

−ρ^{P T}_A−R=1

2 ∑_n T²ⁿ{

∣+⟩ ⟨+∣ ⊗ [∣q_R∣²

C⁴ (n + 1) ∣n + 1⟩ ⟨n + 1∣ + ∣q_L∣²

C² ∣n⟩ ⟨n∣]

+ ∣−⟩ ⟨−∣ ⊗ [∣q_L∣²

C⁴ (n + 1) + ∣q_R∣²

C² ] ∣n⟩ ⟨n∣

+ ∣+⟩ ⟨−∣ ⊗ [ T C⁴

√(n + 1)(n + 2)qLq_R^∗∣n⟩ ⟨n + 2∣ + h.c.]} (4.2.12)

We are able to analytically find the eigenvalues of the state (4.2.12). Unlike the case for the state (4.2.6), where all the eigenvalues could be negative when (4.2.7) is satisfied, here we find that only a finite subset of the eigenvalues can be negative and this subset depends on r. The entanglement for this scenario is plotted in Fig. 4.5.

0.2 0.4 0.6 0.8 1.0 1.2 1.4

r

0.01 0.02 0.03

N

qR= 0.1 q_R= 0.3 q_R= 0.5 q_R= 0.6 q_R= 0.7 q_R= 0.9 q_R= 1.0

Figure 4.5: NegativityN as a function of r for the state ρ^{P TA−R}. Curves are for qR= 1, 0.9, 0.7, 0.6, 0.5from top to bottom.

Assuming that Rob looks only at antiparticles yields analogous results.

We find that entanglement behaves very differently to the corresponding fermionic case

Chapter 4: Entanglement redistribution between charged bosonic field modes in relativistic settings

where entanglement between Alice’s and Rob’s particle (or antiparticle) sector is iden-tically zero [60]. However, here the entanglement grows with acceleration and reaches a maximum value after which it degrades. We trace the difference between the two cases down to extra terms of the form ∣n⟩ ⟨n + 2∣ which appear in (5.2.6). Clearly, no such fermionic Fock state ∣(n + 2)^±Ω⟩ can exist due to Pauli exclusion principle. This behavior is again a rare case of non monotonically decrease of entanglement with increasing r.

4.3 Conclusions

In this chapter we have analyzed the entanglement tradeoff between the particles and antiparticles when the initial maximally entangled states are composed by charged bosonic fields. Including antiparticles in the study of field mode entanglement in non-inertial frames has deepened our understanding of key features which explain the differ-ence in behavior of entanglement in the fermionic and bosonic case. It was shown in [60]

that in the fermionic case an entanglement redistribution between particle and antipar-ticle modes is responsible for the finite value of entanglement in the infinite acceleration limit. In particular, the relative redistribution for different particle and antiparticle bi-partitions could be used to explain the behavior of the entanglement when particles and antiparticles were considered as a whole system. Here we have analyzed the charged bosonic case and computed the entanglement in the partitions that correspond to those considered for fermions in [60]. We showed that, due to the bosonic statistics, there are substantial differences in the entanglement behavior when particles or antiparticles are not taken into account. We also find rare cases of non-monotonically decrease of entanglement as a function of acceleration. However, we confirmed that entanglement is always degraded in the infinite acceleration limit independently of the redistribution of entanglement between the particle and antiparticle bipartitions.

The main difference with the results found in [60] is the following: while in the fermionic case the redistribution of entanglement between particle and antiparticle sec-tors can explain the survival of entanglement for infinite accelerations, in the bosonic case such redistribution allows the entanglement to degrade at infinite accelerations. Such behavior is better explained by analyzing Figures 4.3 and 4.5, where the entanglement vanishes in the limit of infinite acceleration for both cases.

Part II

Chapter 5

Entanglement degradation of cavity

modes due to motion

Chapter 5: Entanglement degradation of cavity modes due to motion

A common way to implement quantum information tasks involves storing information in cavity field modes. How the motion of the cavities affects the stored information is a question that could be of practical relevance in space-based experiments [52, 53]. At present, there has been no success in concretely addressing this question in the framework of RQI, although a first step was attempted in [11]. From a completely different per-spective and for almost four decades, the community working on the dynamical Casimir effect has been seeking an experimentally implementable model which would allow for the demonstration of the creation of particles as predicted by QFT in Casimir-like settings, where the wall of a cavity confining the quantum electromagnetic field in its vacuum state oscillates rapidly and excites the vacuum of the field therefore producing pairs of correlated particles. Any input in this direction could increase corroboration of QFT and provide indication that motion dopes affect quantum resources.

In this chapter we introduce a scheme that allows us to confine a relativistic quantum field in a cavity and determine how modes before any motion occurs are related to the modes in the cavity after it has undergone non inertial motion. The cavity walls are modeled by Dirichlet boundary conditions on the field. In particular, we find a suitable parameter which enables us to employ perturbation techniques and obtain analytical re-sults to the lowest contributing order. This perturbation parameter is directly related to the physical variables of the problem and controls the size of the cavity vs the magnitude of the proper acceleration.

We use this setup to analyze the degradation of initially maximal entangled state of couples of uncharged scalar field modes each within a cavities in Minkowski space. One cavity will remain inertial while the other will undergo motion that need not be sta-tionary. Our analysis therefore combines the explicit confinement of a quantum field to a finite size cavity and a freely adjustable time-dependence of the cavity’s accel-eration. This allows observers within the cavities to implement quantum information protocols in a way that is localized both in space and in time [11]. In particular, our system-environment split is manifestly causal and invokes no horizons or other notions that would assume acceleration to persist into the asymptotic, post-measurement future (cf. [54, 55]). By the equivalence principle, the analysis can be regarded as a model of gravity effects on entanglement.

Chapter 5: Entanglement degradation of cavity modes due to motion

5.1 Cavity prototype configuration