ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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270 A. Standard forms of pure Gaussian states under local operations the antisymmetric condition (A.6): this number turns out to match the number of free parameters in σqp for N = 3, but it is no longer enough to make σqp null for pure states with N ≥ 4. This is further discussed in Chapter 11. Summing up, we have rigorously determined the number of “locally irreducible” free parameters of pure Gaussian states [GA18], unambiguously showing that the quantification and qualification of the entanglement (which, by definition, is preserved under local unitary operations) in such states of N modes is completely determined by 1 parameter for N = 2 and (N 2 − 2N) parameters for N > 2, as reported in Eq. (2.56).
List of Publications [GA1] G. Adesso, F. Illuminati, and S. De Siena, Characterizing entanglement with global and marginal entropic measures, Phys. Rev. A 68, 062318 (2003), e–print quant-ph/0307192. [GA2] G. Adesso, A. Serafini, and F. Illuminati, Determination of Continuous Variable Entanglement by Purity Measurements, Phys. Rev. Lett. 92, 087901 (2004), e–print quant-ph/0310150. [GA3] G. Adesso, A. Serafini, and F. Illuminati, Extremal entanglement and mixedness in continuous variable systems, Phys. Rev. A 70, 022318 (2004), e–print quant-ph/0402124. [GA4] G. Adesso, A. Serafini, and F. Illuminati, Quantification and Scaling of Multipartite Entanglement in Continuous Variable Systems, Phys. Rev. Lett. 93, 220504 (2004), e–print quant-ph/0406053. [GA5] A. Serafini, G. Adesso, and F. Illuminati, Unitarily localizable entanglement of Gaussian states, Phys. Rev. A 71, 023249 (2005), e–print quant-ph/0411109. [GA6] G. Adesso, A. Serafini, and F. Illuminati, Entanglement, Purity, and Information Entropies in Continuous Variable Systems, Open Sys. & Information Dyn. 12, 189 (2005), e–print quant-ph/0506049. [GA7] G. Adesso and F. Illuminati, Gaussian measures of entanglement versus negativities: Ordering of two-mode Gaussian states, Phys. Rev. A 72, 032334 (2005), e–print quant-ph/0506124. [GA8] J. Laurat, G. Keller, J. A. Oliveira-Huguenin, C. Fabre, T. Coudreau, A. Serafini, G. Adesso, and F. Illuminati, Entanglement of two-mode Gaussian states: characterization and experimental production and manipulation, J. Opt. B: Quantum Semiclass. Opt. 7, S577 (2005), e–print quant-ph/0507067. [GA9] G. Adesso and F. Illuminati, Equivalence between Entanglement and the Optimal Fidelity of Continuous Variable Teleportation, Phys. Rev. Lett. 95, 150503 (2005), e–print quant-ph/0412125. [GA10] G. Adesso and F. Illuminati, Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems, New J. Phys. 8, 15 (2006), e–print quant-ph/0410050. 271
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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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196 12. Multiparty quantum communic
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Page 233 and 234: CHAPTER 13 Entanglement in Gaussian
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Page 249 and 250: CHAPTER 14 Gaussian entanglement sh
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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: 268 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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