ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

More documents

Recommendations

Info

94 5. Multimode entanglement under symmetry The central observation of the present Chapter is embodied by following result [GA4, GA5], straightforwardly deducible from the discussions in Sec. 2.4.3 and Sec. 3.1.1. ➢ Unitarily localizable entanglement of bisymmetric Gaussian states. The bipartite entanglement of bisymmetric (M + N)-mode Gaussian states under M × N partitions is “unitarily localizable”, namely, through local unitary (reversible) operations, it can be completely concentrated onto a single pair of modes, each of them belonging respectively to the M-mode and to the N-mode blocks. Hence the multimode block entanglement (i.e. the entanglement between blocks of modes) of bisymmetric (generally mixed) Gaussian states can be determined as a two-mode entanglement. The entanglement will be quantified by the logarithmic negativity in the general instance because the PPT criterion holds, but we will also show some explicit nontrivial cases in which the entanglement of formation, Eq. (1.35), between M-mode and N-mode parties can be exactly computed. We remark that our notion of “localizable entanglement” is different from that introduced by Verstraete, Popp, and Cirac for spin systems [248]. There, it was defined as the maximal entanglement concentrable on two chosen spins through local measurements on all the other spins. 11 Here, the local operations that concentrate all the multimode entanglement on two modes are unitary and involve the two chosen modes as well, as parts of the respective blocks. 5.1. Bipartite block entanglement of bisymmetric Gaussian states In Sec. 2.4.3, the study of the multimode CM σ of Eq. (2.64) has been reduced to a two-mode problem by means of local unitary operations. This finding allows for an exhaustive analysis of the bipartite entanglement between the M- and N-mode blocks of a multimode bisymmetric Gaussian state, resorting to the powerful results available for two-mode Gaussian states (see Chapter 4). For any multimode Gaussian state with CM σ, let us define the associated equivalent two-mode Gaussian state ϱeq, with CM σeq given by σeq = + ν αM γ ′′ γ ′′T + ν βN , (5.1) where the 2×2 blocks have been implicitly defined in the CM, Eq. (2.65). As already mentioned, the entanglement of the bisymmetric state with CM σ, originally shared among all the M + N modes, can be completely concentrated by local unitary (symplectic) operations on a single pair of modes in the state with CM σeq. Such an entanglement is, in this sense, localizable. We now move on to describe some consequences of this result. A first qualificative remark has been explored in Sec. 3.1.1. Namely, PPT criterion turns out to be automatically necessary and sufficient for separability of (M + N)-mode Gaussian states under M × N bipartitions [GA5]. For a more quantitative investigation, the following symplectic analysis, which completes that of Sec. 2.4.3, will be precious. 11 This (non-unitarily) localizable entanglement will be also computed for (mixed) symmetric Gaussian states of an arbitrary number of modes, and it will be shown to be in direct quantitative connection with the optimal fidelity of multiparty teleportation networks [GA9] (see Sec. 12.2 and Fig. 12.4).
5.1. Bipartite block entanglement of bisymmetric Gaussian states 95 5.1.1. Symplectic properties of symmetric states As a preliminary analysis, it is useful to provide a symplectic parametrization for the standard form coefficients of any two-mode reduced state of a fully symmetric Nmode CM σβN , Eq. (2.60). Following the discussion in Sec. 2.4.1.1, the coefficients b, z1, z2 of the standard form are determined by the local, single-mode invariant Det β ≡ µ −2 β , and by the symplectic invariants Det σβ2 ≡ µ−2 β2 and ∆β2 ≡ ∆(σβ2). Here µβ (µ β2) is the marginal purity of the single-mode (two-mode) reduced states, while ∆β2 is the remaining seralian invariant, Eq. (2.34), of the two-mode reduced states. According to Sec. 4.1, this parametrization is provided, in the present instance, by the following equations b = 1 µβ , z1 = µβ 4 (ɛ− − ɛ+) , z2 = µβ 4 (ɛ− + ɛ+) , (5.2) with ɛ− = ∆2 β2 − 4 µ 2 β2 , and ɛ+ = ∆β2 − 4 µ 2 β 2 − 4 µ 2 β2 . This parametrization has a straightforward interpretation, because µβ and µ β 2 quantify the local mixednesses and ∆ β 2 regulates the entanglement of the twomode blocks at fixed global and local purities [GA2] (see Sec. 4.3.3). Moreover, we can connect the symplectic spectrum of σ β N , given by Eq. (2.61), to the known symplectic invariants. The (N − 1)-times degenerate eigenvalue ν − β is independent of N, while ν + β N can be simply expressed as a function of the single- mode purity µβ and the symplectic spectrum of the two-mode block with eigenvalues ν − β and ν+ β 2: (ν + βN ) 2 N(N − 2) = − µ 2 + β (N − 1) 2 N(ν + β2) 2 + (N − 2)(ν − β )2 . (5.3) In turn, the two-mode symplectic eigenvalues are determined by the two-mode invariants by the relation 2(ν ∓ β )2 = ∆ β 2 ∓ ∆ 2 β 2 − 4/µ 2 β 2 . (5.4) The global purity Eq. (2.37) of a fully symmetric multimode Gaussian state is µ βN ≡ −1/2 Det σβN = (ν − β )N−1ν + βN −1 , (5.5) and, through Eq. (5.3), can be fully determined in terms of the one- and two-mode parameters alone. Analogous reasonings and expressions hold of course for the fully symmetric M-mode block with CM σ α M given by Eq. (2.60). 5.1.2. Evaluation of block entanglement in terms of symplectic invariants We can now efficiently discuss the quantification of the multimode block entanglement of bisymmetric Gaussian states. Exploiting our results on the symplectic characterization of two-mode Gaussian states [GA2, GA3] (see Sec. 4.1) we can select the relevant quantities that, by determining the correlation properties of the two-mode Gaussian state with CM σeq, also determine the entanglement and
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 57 and 58: 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 and 102: Ν p Σopt 1 0.8 0.6 0.4 0.2 0 4.5
Page 103 and 104: GEF GEF 4 3 2 1 0 4.6. Summary and
Page 105: 4.6. Summary and further remarks 91
Page 110 and 111: 96 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 158 and 159:
144 7. Tripartite entanglement in t
Page 160 and 161:
146 7. Tripartite entanglement in t
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?