ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

More documents

Recommendations

Info

88 4. Two-mode entanglement the negativity. For these states, mGMEMMS 2s opt (s, ˜ν−) = 1 − ˜ν 2 . (4.79) − + 2˜ν−s The further optimization over s is straightforward because m GMEMMS opt is an increasing function of s, so its global maximum is attained for s → ∞. In this limit, one has simply mGMEMMS max (˜ν−) = 1 . (4.80) ˜ν− From Eq. (4.65), one thus finds that for all GLEMS the following bound holds ˜ν−(σ p opt) ≥ 1 1 − 1 − ˜ν ˜ν−(σ) 2 −(σ) . (4.81) One can of course perform a similar analysis for GMEMS. But, after analogous reasonings and computations, what one finds is exactly the same result. This is not so surprising, keeping in mind that GMEMS, GLEMS and all two-mode Gaussian states with generic s and d but with global mixedness g saturating Ineq. (4.69), collapse into the same family of two-mode Gaussian states, the GMEMMS, completely determined by the local single-mode properties (they can be viewed as a generalization of the pure two-mode states: the symmetric GMEMMS are in fact pure). Hence, the bound of Ineq. (4.81), limiting the Gaussian EMs from above at fixed negativities, must hold for all GMEMS as well. At this point, it is tempting to conjecture that Ineq. (4.81) holds for all twomode Gaussian states. Unfortunately, the lack of a closed, simple expression for the Gaussian EM of a generic state makes the proof of this conjecture impossible, at the present time. However, one can show, by analytical power-series expansions of Eq. (4.64), truncated to the leading order in the infinitesimal increments, that, for any infinitesimal variation of the parameters of a generic CM around the limiting values characterizing GMEMMS, the Gaussian EMs of the resulting states lie always below the boundary imposed by the corresponding GMEMMS with the same ˜ν−. In this sense, the GMEMMS are, at least, a local maximum for the Gaussian EM versus negativity problem. Furthermore, extensive numerical investigations of up to a million CMs of randomly generated two-mode Gaussian states, provide confirmatory evidence that GMEMMS attain indeed the global maximum (see Fig. 4.9). We can thus quite confidently conjecture, however, at the moment, without a complete formal proof of the statement, that GMEMMS, in the limit of infinite average local mixedness (s → ∞), are the states of maximal Gaussian EMs at fixed negativities, among all two-mode Gaussian states. A direct comparison between the two prototypical representatives of the two families of entanglement measures, respectively the Gaussian entanglement of formation GEF and the logarithmic negativity EN , is plotted in Fig. 4.10. For any fixed value of EN , Ineq. (4.78) provides in fact a rigorous lower bound on GEF , namely GEF ≥ h[exp(−EN )] , (4.82) while Ineq. (4.81) provides the conjectured lower bound GEF ≤ h exp(EN ) 1 − 1 − exp(−2EN ) where we exploited Eqs. (4.15,4.66), and h[x] is given by Eq. (4.18). , (4.83)
GEF GEF 4 3 2 1 0 4.6. Summary and further remarks 89 0 1 2 3 4 E E Figure 4.10. Comparison between the Gaussian entanglement of formation GE F and the logarithmic negativity EN for two-mode Gaussian states. Symmetric states accomodate on the lower boundary (solid line), determined by the saturation of Ineq. (4.82). GMEMMS with infinite, average local mixedness, lie on the dashed line, whose defining equation is obtained from the saturation of Ineq. (4.83). All GMEMS and GLEMS lie below the dashed line. The latter is conjectured, with strong numerical support, to be the upper boundary for the Gaussian entanglement of formation of all two-mode Gaussian states, at fixed negativity. The existence of lower and upper bounds on the Gaussian EMs at fixed negativities (the latter strictly proven only for extremal states), limits to some extent the inequivalence arising between the two families of entanglement measures, for nonsymmetric two-mode Gaussian states. We have thus demonstrated the following. ➢ Ordering two-mode Gaussian states with entanglement measures. The Gaussian entanglement measures and the negativities induce inequivalent orderings on the set of entangled, nonsymmetric, two-mode Gaussian states. This inequivalence is however constrained: at fixed negativities, the Gaussian measures of entanglement are bounded from below (the states which saturate this bound are simply symmetric two-mode states); moreover, we provided some strong evidence suggesting that they are as well bounded from above. 4.6. Summary and further remarks Summarizing, in this Chapter we focused on the simplest conceivable states of a bipartite CV system: two-mode Gaussian states. We have shown that, even in this simple instance, the theory of quantum entanglement hides several subtleties and reveals some surprising aspects. Following Refs. [GA2, GA3, GA6], we have pointed out the existence of both maximally and minimally entangled two-mode Gaussian states at fixed local and global generalized p−entropies. The analytical properties of such states have been
Page 1 and 2:
ENTANGLEMENT OF GAUSSIAN STATES Ger
Page 3:
Life’s Entanglement. CR Studio In
Page 7:
Abstract This Dissertation collects
Page 10 and 11:
x Contents 2.2.2.2. Symplectic repr
Page 12 and 13:
xii Contents 7.2.3. Tripartite enta
Page 14 and 15:
xiv Contents Chapter 13. Entangleme
Page 17 and 18:
Introduction About eighty years aft
Page 19 and 20:
Introduction 5 Gaussian states. Imp
Page 21:
Introduction 7 The companion Part V
Page 24 and 25:
10 1. Characterizing entanglement a
Page 26 and 27:
12 1. Characterizing entanglement w
Page 28 and 29:
14 1. Characterizing entanglement W
Page 30 and 31:
16 1. Characterizing entanglement F
Page 32 and 33:
18 1. Characterizing entanglement a
Page 34 and 35:
20 1. Characterizing entanglement i
Page 36 and 37:
22 1. Characterizing entanglement e
Page 38 and 39:
24 1. Characterizing entanglement n
Page 40 and 41:
26 1. Characterizing entanglement p
Page 43 and 44:
CHAPTER 2 Gaussian states: structur
Page 45 and 46:
2.1. Introduction to continuous var
Page 47 and 48:
2.2. Mathematical description of Ga
Page 49 and 50:
2.2. Mathematical description of Ga
Page 51 and 52: 2.3. Degree of information encoded
Page 53 and 54: 2.3. Degree of information encoded
Page 55 and 56: 2.4. Standard forms of Gaussian cov
Page 63: Part II Bipartite entanglement of G
Page 66 and 67: 52 3. Characterizing entanglement o
Page 68 and 69: 54 3. Characterizing entanglement o
Page 71 and 72: CHAPTER 4 Two-mode entanglement Thi
Page 73 and 74: 4.2. Entanglement and symplectic ei
Page 75 and 76: 4.3. Entanglement versus Entropic m
Page 77 and 78: 1 0.75 0.5 SL1 0.25 0 (a) 4.3. Enta
Page 81 and 82: Μ Μ1Μ2 3 2 1 1 4.3. Entanglemen
Page 87 and 88: global generalized entropy S3 globa
Page 89 and 90: 4.4. Quantifying entanglement via p
Page 91 and 92: 4.4. Quantifying entanglement via p
Page 93 and 94: 4.5. Gaussian entanglement measures
Page 101: Ν p Σopt 1 0.8 0.6 0.4 0.2 0 4.5
Page 105: 4.6. Summary and further remarks 91
Page 108 and 109: 94 5. Multimode entanglement under
Page 121: Part III Multipartite entanglement
Page 124 and 125: 110 6. Gaussian entanglement sharin
Page 136 and 137: 122 7. Tripartite entanglement in t
Page 152 and 153:
138 7. Tripartite entanglement in t
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
148 8. Unlimited promiscuity of mul
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 171:
Part IV Quantum state engineering o
Page 174 and 175:
160 9. Two-mode Gaussian states in
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 189 and 190:
CHAPTER 10 Tripartite and four-part
Page 191 and 192:
10.1. Optical production of three-m
Page 193 and 194:
(out) (in) TRITTER 10.1. Optical pr
Page 195 and 196:
10.2. How to produce and exploit un
Page 197 and 198:
CHAPTER 11 Efficient production of
Page 199 and 200:
11.2. Generic entanglement and stat
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
11.3. Economical state engineering
Page 207:
Part V Operational interpretation a
Page 210 and 211:
196 12. Multiparty quantum communic
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 233 and 234:
CHAPTER 13 Entanglement in Gaussian
Page 235 and 236:
13.1. Gaussian valence bond states
Page 237 and 238:
Eq. (13.3) thus takes the form 13.1
Page 239 and 240:
13.2. Entanglement distribution in
Page 241 and 242:
s 20 17.5 15 12.5 10 7.5 5 2.5 13.2
Page 243 and 244:
13.3. Optical implementation of Gau
Page 245 and 246:
of the form Eq. (13.1), with 13.3.
Page 247 and 248:
13.4. Telecloning with Gaussian val
Page 249 and 250:
CHAPTER 14 Gaussian entanglement sh
Page 251 and 252:
14.1. Entanglement in non-inertial
Page 253 and 254:
14.2. Distributed Gaussian entangle
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
6 s 4 14.2. Distributed Gaussian en
Page 261 and 262:
2.5 0 0 5 10 IΣAR7.5 0 14.3. Distr
Page 263 and 264:
Page 265 and 266:
0 entanglement condition 14.3. Dist
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
4 3 a 2 1 14.4. Discussion and outl
Page 273:
Part VI Closing remarks Entanglemen
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 282 and 283:
Page 284 and 285:
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
show all

ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?