ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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256 14. Gaussian entanglement sharing in non-inertial frames 2.5 0 0 5 10 IΣLN 7.5 0 1 a (a) 2 3 0 1 2 s 3 0.5 0 0 1 1.5 2 IΣLN p EVΣLN 0 1 a (b) 2 3 0 Figure 14.9. Total correlations between the two non-inertial observers Leo and Nadia, traveling with equal, uniform acceleration given by the effective squeezing parameter a. In the inertial frame, the modes are in a two-mode squeezed state with squeezing degree s. Plot (a) shows the mutual information I(σL|N ), given by Eq. (14.28), as a function of a and s. In plot (b) the same quantity is normalized to the entropy of entanglement perceived by inertial observers, EV (σ p ), Eq. (14.29). Notice in (a) how the mutual information is L|N an increasing function of the squeezing parameter s and saturates to a nonzero value in the limit of infinite acceleration; in contrast, the entanglement vanishes at finite acceleration (see Fig. 14.8). Plot (b), shows that this asymptotic value is smaller than the entropy of entanglement in the Minkowski frame (which is equal to the classical correlations detected by the inertial observers). Therefore, classical correlations are also degraded when both observers are accelerated, in contrast to the case where only one observer is in uniform acceleration (see Fig. 14.8). when the classical correlations in the inertial frame are infinite (corresponding to infinite shared entanglement in the inertial frame, s → ∞). The entanglement, however, is always fragile, since we have seen that it is completely destroyed at a finite, relatively small acceleration parameter a. Another intriguing fact is that, comparing Figs. 14.5(a) and 14.9(a), one sees that in both cases (either one or two non-inertial observers) the mutual information between the two “real” observers is a function of the acceleration parameter and of the initial squeezing. In the case of both accelerated observers, however, the mutual information is always smaller, as we have just discussed. We can study the difference between them, once we set for ease of comparison equal acceleration parameters, r = a, where r regulates Rob’s acceleration when Alice is inertial, and a is related to the acceleration of both Leo and Nadia in the present situation: D(a, s) = I(σ A|R) r=a − I(σ L|N) . (14.30) The quantity D(a, s) is plotted in Fig. 14.10: surprisingly, it is strictly bounded. It increases both with s and a, but in the asymptotic limit of infinite inertial shared entanglement, D(a, s → ∞) saturates exactly to 1 (as it can be checked analytically) for any a > 0. We remark that both mutual informations I(σ A|R) and I(σ L|N) diverge in this limit: yet their difference is finite and equal to one. Clearly, the small deficit of the mutual information seen when both observers are accelerated, is detected as loss of classical correlations, as plotted in Fig. 14.9(b). Mysteriously, the Unruh thermalization affects classical correlations when both observers are 1 2 s 3
4 3 a 2 1 14.4. Discussion and outlook 257 0 0 1 2 3 s 1 0.75 0.5 IΣABraIΣLN 0.25 0 4 Figure 14.10. Plot, as a function of the acceleration parameter a and the squeezing parameter s, of the difference between the mutual information shared by the inertial Alice and the non-inertial Rob, and the mutual information shared by the uniformly accelerating Leo and Nadia, as given by Eq. (14.30). accelerated: however, it degrades at most one absolute unit of classical correlations. This means that in the case when both Leo and Nadia escape the fall into a black hole, not only their entanglement is degraded but there is also a loss of classical information. 14.4. Discussion and outlook In this Chapter, based on Ref. [GA20], we presented a thorough study of classical and quantum correlations between modes of a scalar field measured by observers in relative acceleration. By considering the state of the field in the inertial frame in the simplest multi-mode squeezed state possible (the two-mode case) we were able to investigate in detail the entanglement in all partitions of the system. We considered two observers carrying single mode detectors and discussed the correlations on their measurements when both observers are in uniform acceleration and when only one of them is non-inertial. We find that in both settings entanglement is degraded with acceleration and we explain this degradation as an effect of re-distribution of the entanglement measured in an inertial frame. Our main results can be summarized as follows. When one of the observers is non-inertial the entanglement lost between the modes measured by him and the inertial observer is re-distributed in tripartite correlations. No entanglement is generated between the modes measured by the inertial observer and the modes in the causally disconnected region II. This shows that indeed the behavior for bosonic fields is very different to the Dirac case where the entanglement lost in the noninertial frame is re-distributed not into tripartite correlations but into bipartite correlations between the mode measured by the inertial observer and the mode in region II. The analysis of the mutual information shows that in this case classical correlations are conserved independently of the acceleration. The situation changes drastically by considering that both observers are non-inertial. In this case the entanglement lost between two non-inertial observers is re-distributed into mainly four-partite correlations although some tripartite correlations exist for finite acceleration. The surprising result here (though expected in the framework of distributed entanglement, as the additional fourth mode comes into play) is that entanglement vanishes completely at a finite acceleration. This is also strikingly different to the
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ENTANGLEMENT OF GAUSSIAN STATES Ger
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Life’s Entanglement. CR Studio In
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Abstract This Dissertation collects
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x Contents 2.2.2.2. Symplectic repr
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xii Contents 7.2.3. Tripartite enta
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xiv Contents Chapter 13. Entangleme
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Introduction About eighty years aft
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Introduction 5 Gaussian states. Imp
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Introduction 7 The companion Part V
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10 1. Characterizing entanglement a
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12 1. Characterizing entanglement w
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14 1. Characterizing entanglement W
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16 1. Characterizing entanglement F
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18 1. Characterizing entanglement a
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20 1. Characterizing entanglement i
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22 1. Characterizing entanglement e
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24 1. Characterizing entanglement n
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26 1. Characterizing entanglement p
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CHAPTER 2 Gaussian states: structur
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2.1. Introduction to continuous var
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2.2. Mathematical description of Ga
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2.2. Mathematical description of Ga
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2.3. Degree of information encoded
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2.3. Degree of information encoded
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2.4. Standard forms of Gaussian cov
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Part II Bipartite entanglement of G
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52 3. Characterizing entanglement o
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54 3. Characterizing entanglement o
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CHAPTER 4 Two-mode entanglement Thi
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4.2. Entanglement and symplectic ei
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4.3. Entanglement versus Entropic m
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1 0.75 0.5 SL1 0.25 0 (a) 4.3. Enta
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Μ Μ1Μ2 3 2 1 1 4.3. Entanglemen
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global generalized entropy S3 globa
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4.4. Quantifying entanglement via p
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4.4. Quantifying entanglement via p
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4.5. Gaussian entanglement measures
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Ν p Σopt 1 0.8 0.6 0.4 0.2 0 4.5
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GEF GEF 4 3 2 1 0 4.6. Summary and
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4.6. Summary and further remarks 91
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94 5. Multimode entanglement under
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Part III Multipartite entanglement
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110 6. Gaussian entanglement sharin
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122 7. Tripartite entanglement in t
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148 8. Unlimited promiscuity of mul
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Part IV Quantum state engineering o
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160 9. Two-mode Gaussian states in
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CHAPTER 10 Tripartite and four-part
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10.1. Optical production of three-m
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(out) (in) TRITTER 10.1. Optical pr
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10.2. How to produce and exploit un
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CHAPTER 11 Efficient production of
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11.2. Generic entanglement and stat
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11.3. Economical state engineering
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Part V Operational interpretation a
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Page 276 and 277: 262 Conclusion and Outlook Gaussian
Page 278 and 279: 264 Conclusion and Outlook compass
Page 280 and 281: 266 A. Standard forms of pure Gauss
Page 286 and 287: 272 List of Publications [GA11] G.
Page 288 and 289: 274 Bibliography [23] C. H. Bennett
Page 290 and 291: 276 Bibliography [79] J. Eisert, C.
Page 292 and 293: 278 Bibliography [140] J. Laurat, T
Page 294 and 295: 280 Bibliography [198] T. Roscilde,
Page 296: 282 Bibliography [258] J. Von Neuma
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ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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