ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
254 14. Gaussian entanglement sharing in non-inertial frames We have that the residual Gaussian contangle [see Eq. (8.5)], G res τ (σLLN ¯ N) ¯ ≡ Gτ (σL|(LN ¯ N)) ¯ − Gτ (σL|L) ¯ (14.27) = arcsinh 2 [cosh 2 a + cosh(2s) sinh 2 a] 2 − 1 − 4a 2 , quantifies precisely the multipartite correlations that cannot be stored in bipartite form. Those quantum correlations, however, can be either tripartite involving three of the four modes, and/or genuinely four-partite among all of them. The tripartite portion (only present in equal content in the tripartitions ¯ L|L|N and L|N| ¯ N) can be estimated as in Fig. 8.2, and specifically it decays to zero in the limit of high acceleration. Therefore, in the regime of increasingly high a, eventually approaching infinity, any form of tripartite entanglement among any three modes in the state σLLN ¯ N¯ is negligible (exactly vanishing in the limit of infinite acceleration). It follows that, exactly like in Chapter 8, in the regime of high acceleration a, the residual entanglement Gres τ determined by Eq. (14.27) is stored entirely in the form of four-partite quantum correlations. Therefore, the residual entanglement in this case is a good measure of genuine four-partite entanglement among the four Rindler spacetime modes. It is now straightforward to see that Gres τ (σLLN ¯ N) ¯ is itself an increasing function of a for any value of s (see Fig. 8.3), and it diverges in the limit a → ∞. The four-mode state Eq. (14.27) obtained with an arbitrarily large acceleration a, consequently, exhibits a coexistence of unlimited genuine four-partite entanglement, and pairwise bipartite entanglement in the reduced two-mode states σL| L¯ and σN| N. ¯ This peculiar distribution of CV entanglement in the considered Gaussian state has been defined as infinitely promiscuous in Chapter 8, where its consequences are discussed in a practical optical setting [GA19]. It is interesting to note that in the relativistic analysis we present here [GA20], the genuine four-partite entanglement increases unboundedly with the observers’ acceleration. This is in fact in strong contrast with the case of an inertial observer and an accelerating one (Sec. 14.2), where we find that, in the infinite acceleration limit, the genuine tripartite entanglement saturates at 4s2 (i.e. the original entanglement encoded between the two inertial observers). In the scenario considered in this Section, the acceleration of Leo and Nadia creates ex novo entanglement (function of the acceleration) between the respective Rindler regions of both observers independently. The information loss at the double horizon is such that even an infinite entangled state in the inertial frame contains no quantum correlations when detected by two observers traveling at finite acceleration. If one considers even higher acceleration of the observers, it is basically the entanglement between the Rindler regions which is redistributed into genuine four-partite form. The tripartite correlations tend to vanish as a consequence of the thermalization which destroys the inertial bipartite entanglement. The multipartite entanglement, obviously, increases infinitely with acceleration because the entanglement between the Rindler regions increases without bound with acceleration. It is remarkable that such promiscuous distribution of entanglement can occur without violating the fundamental monogamy constraints on entanglement sharing (see Chapter 6). To give a simple example, suppose the bipartite entanglement in the inertial frame is given by 4s2 = 16 for s = 2. If both observers travel with an effective
14.3. Distributed Gaussian entanglement due to both accelerated observers 255 acceleration parameter a = 7, the four-partite entanglement [given by Eq. (14.27)] among all Rindler modes is 81.2 ebits, more than 5 times the inertial bipartite entanglement. At the same time, a bipartite entanglement of 4a 2 = 196 is generated between region I and region II of each observer. A final caveat needs to be stated. The above results show that unbounded entanglement is created by merely the observers’ motion. This requires of course an unlimited energy needed to fuel their spaceships. Unfortunately, in this setting such entanglement is mostly unaccessible, as both Leo and Nadia are confined in their respective Rindler region I. The only entangled resource that can be used is the degraded two-mode thermal squeezed state of modes λI and νI, whose entanglement soon vanishes for sufficiently high, finite acceleration. Let us remark, once more, that in the practical setting of quantum optics the same four-mode Gaussian states of light beams can be instead accessed and manipulated, as shown in Chapter 8. The role of the acceleration on the detection of the field is played in that case by the effects of a nonlinear crystal through the mechanism of parametric down-conversion. In such a non-relativistic context, the different types of entanglement can be readily used as a resource for bipartite and/or multipartite transmission and processing of CV quantum information [GA19]. 14.3.3. Mutual information It is very interesting to evaluate the mutual information I(σ L|N) between the states measured by Leo and Nadia, both moving with acceleration parameter a. In this case the symplectic spectrum of the reduced (mixed) two-mode CM σ L|N of Eq. (14.17) is degenerate as it belongs to the family of GMEMS (see Sec. 4.3.3.1), yielding ν−(σL|N) = ν+(σL|N ) = Det σL|N mutual information then reads I(σL|N) = f( Det σL) + f( Det Det σN) − 2f σL|N 1 4 . From Eq. (2.40), the 1 4 , (14.28) with f(x) defined by Eq. (2.39). Explicitly: I(σ L|N) = 2 cosh 2 (a) cosh 2 (s) log[cosh 2 (a) cosh 2 (s)]−[cosh(2s) cosh 2 (a)+sinh 2 (a)− 1] log{ 1 2 [cosh(2s) cosh2 (a) + sinh 2 (a) − 1]} + 1 3] 1 2 − 2} log{[2 cosh(2s) sinh 2 (2a) + cosh(4a) + 3] 1 2 − 2} − 1 2 {[2 cosh(2s) sinh2 (2a) + cosh(4a) + 2 {[2 cosh(2s) sinh2 (2a) + cosh(4a) + 3] 1 2 + 2} log{[2 cosh(2s) sinh 2 (2a) + cosh(4a) + 3] 1 2 + 2} + log(16). We plot the mutual information both directly, and normalized to the inertial entropy of entanglement, which is equal to Eq. (14.15), EV (σ p L|N ) = f(cosh 2s) , (14.29) with f(x) given by Eq. (2.39). We immediately notice another novel effect. Not only the entanglement is completely destroyed at finite acceleration, but also classical correlations are degraded, see Fig. 14.9(b). This is very different to the case of a single non-inertial observer where classical correlations remain invariant. The asymptotic state detected by Leo and Nadia, in the infinite acceleration limit (a → ∞), contains indeed some residual classical correlations (whose amount is an increasing function of the squeezing s). But these correlations are always smaller than the classical correlations in the inertial frame given by Eq. (14.29). Classical correlations are robust against the effects of the double acceleration only
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- Page 288 and 289: 274 Bibliography [23] C. H. Bennett
- Page 290 and 291: 276 Bibliography [79] J. Eisert, C.
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14.3. Distributed Gaussian entanglement due to both accelerated observers 255<br />
acceleration parameter a = 7, the four-partite entanglement [given by Eq. (14.27)]<br />
among all Rindler modes is 81.2 ebits, more than 5 times the inertial bipartite<br />
entanglement. At the same time, a bipartite entanglement of 4a 2 = 196 is generated<br />
between region I and region II of each observer.<br />
A final caveat needs to be stated. The above results show that unbounded<br />
entanglement is created by merely the observers’ motion. This requires of course an<br />
unlimited energy needed to fuel their spaceships. Unfortunately, in this setting such<br />
entanglement is mostly unaccessible, as both Leo and Nadia are confined in their<br />
respective Rindler region I. The only entangled resource that can be used is the<br />
degraded two-mode thermal squeezed state of modes λI and νI, whose entanglement<br />
soon vanishes for sufficiently high, finite acceleration.<br />
Let us remark, once more, that in the practical setting of quantum optics<br />
the same four-mode Gaussian states of light beams can be instead accessed and<br />
manipulated, as shown in Chapter 8. The role of the acceleration on the detection<br />
of the field is played in that case by the effects of a nonlinear crystal through the<br />
mechanism of parametric down-conversion. In such a non-relativistic context, the<br />
different types of entanglement can be readily used as a resource for bipartite and/or<br />
multipartite transmission and processing of CV quantum information [GA19].<br />
14.3.3. Mutual information<br />
It is very interesting to evaluate the mutual information I(σ L|N) between the states<br />
measured by Leo and Nadia, both moving with acceleration parameter a.<br />
In this case the symplectic spectrum of the reduced (mixed) two-mode CM<br />
σ L|N of Eq. (14.17) is degenerate as it belongs to the family of GMEMS (see<br />
Sec. 4.3.3.1), yielding ν−(σL|N) = ν+(σL|N ) = Det σL|N mutual information then reads<br />
I(σL|N) = f( Det σL) + f( Det Det σN) − 2f σL|N<br />
1<br />
4 . From Eq. (2.40), the<br />
1 <br />
4 , (14.28)<br />
with f(x) defined by Eq. (2.39).<br />
Explicitly:<br />
I(σ L|N) = 2 cosh 2 (a) cosh 2 (s) log[cosh 2 (a) cosh 2 (s)]−[cosh(2s) cosh 2 (a)+sinh 2 (a)−<br />
1] log{ 1<br />
2 [cosh(2s) cosh2 (a) + sinh 2 (a) − 1]} + 1<br />
3] 1<br />
2 − 2} log{[2 cosh(2s) sinh 2 (2a) + cosh(4a) + 3] 1<br />
2 − 2} − 1<br />
2 {[2 cosh(2s) sinh2 (2a) + cosh(4a) +<br />
2 {[2 cosh(2s) sinh2 (2a) +<br />
cosh(4a) + 3] 1<br />
2 + 2} log{[2 cosh(2s) sinh 2 (2a) + cosh(4a) + 3] 1<br />
2 + 2} + log(16).<br />
We plot the mutual information both directly, and normalized to the inertial<br />
entropy of entanglement, which is equal to Eq. (14.15),<br />
EV (σ p<br />
L|N ) = f(cosh 2s) , (14.29)<br />
with f(x) given by Eq. (2.39). We immediately notice another novel effect. Not only<br />
the entanglement is completely destroyed at finite acceleration, but also classical<br />
correlations are degraded, see Fig. 14.9(b). This is very different to the case of a<br />
single non-inertial observer where classical correlations remain invariant.<br />
The asymptotic state detected by Leo and Nadia, in the infinite acceleration<br />
limit (a → ∞), contains indeed some residual classical correlations (whose amount<br />
is an increasing function of the squeezing s). But these correlations are always<br />
smaller than the classical correlations in the inertial frame given by Eq. (14.29).<br />
Classical correlations are robust against the effects of the double acceleration only