ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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172 9. Two-mode Gaussian states in the lab Figure 9.9. Normalized noise variances at 3.5 MHz of the rotated modes while scanning the local oscillator phase for an angle of the plate of 0.3 ◦ , before and after the non-local operation. The homodyne detections are inquadrature. After this operation, squeezing is observed on orthogonal quadratures. In this instance one finds for the partially transposed symplectic eigenvalue ˜ν− 0.46 (corresponding to a logarithmic negativity between A1 and A2 much lower than the previous value: EN = 1.13), whereas the smallest eigenvalues read λ1 = λ2 0.40. The entanglement can thus be raised via passive operations to the optimal value EN = 1.32, corresponding to ˜ν− = √ λ1λ2. The passive transformation capable of optimizing the entanglement is easily found, according to the theoretical analysis of Sec. 9.2.2. If O is the rotation diagonalizing the 2 × 2 symmetric matrix α ′′ defined in Eq. (9.10), then the transformation K ≡ B T (1/2)( ⊕ O)B(1/2) turns the CM σ(ρ = 0.3 ◦ ) into ¯σ(ρ = 0.3 ◦ ), which is diagonal in the orthogonal quadratures A∓ and in a symmetric standard form in the quadratures A1,2. The entanglement of such a matrix is therefore optimal under passive operations. The optimal symplectic operation K consists in a ‘phase-shift’ of the rotated modes A1,2. In the experimental practice, such an operation can be readily performed on co-propagating, orthogonally polarized beams [219]. The minimal combination of waveplates needed can be shown to consist in three waveplates: two λ/4 waveplates denoted Q and one λ/2 waveplate denoted H. When using any combination of these three plates, the state can be put back into standard form which will maximize the entanglement. Fig. 9.9 gives the normalized noise variances before and after this operation. In agreement with the theory, the
9.2. Experimental production and manipulation of two-mode entanglement 173 CM is changed into: ¯σ(ρ = 0.3 ◦ ) = ⎛ ⎜ ⎝ 0.4 0 (0) (0) 0 12.59 (0) (0) (0) (0) 12.59 0 (0) (0) 0 0.4 ⎞ ⎟ ⎠ . (9.12) giving the expected optimal logarithmic negativity EN = 1.32 between A1 and A2, larger than the value before the operation. No more entanglement can be generated by passive operations on this Gaussian state, which has been experimentally transformed into the form which achieves the maximum possible bipartite quantum correlations. Let us remark again that this transformation is non-local in the sense of the EPR argument [73]: it has to be performed before spatially separating the entangled modes for a quantum communication protocol for instance. 9.2.6. Summary of the experiment We have given the flavor of the powerful tools underlying the description of CV systems in quantum optics. These tools allow for a nice pictorial view of twomode Gaussian entangled states. Specifically, we have experimentally achieved the following [GA8]. ➢ Experimental production and manipulation of two-mode entanglement. Continuous-variable entanglement in two-mode Gaussian states has been produced experimentally with an original device, a type-II optical parametric oscillator containing a birefringent plate; it has been quantitatively measured by homodyne reconstruction of the standard form covariance matrix, and optimized using purely passive operations. We have also studied quantitatively the influence of the noise, affecting the measurement of the elements of the CM, on the entanglement, showing that the most significant covariances (exhibiting the highest stability against noise) for an accurate entanglement quantification are the diagonal terms of the diagonal singlemode blocks, and the off-diagonal terms of the intermodal off-diagonal block, the latter being the most difficult to measure with high precision. Alternative methods have been devised to tackle this problem [87, 195] based on direct measurements of global and local invariants of the CM [GA2], as introduced in Sec. 4.4.1. Such techniques have been implemented in the case of pulsed beams [263] but no experiment to date has been performed for continuous-wave beams.
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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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Eq. (13.3) thus takes the form 13.1
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13.2. Entanglement distribution in
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s 20 17.5 15 12.5 10 7.5 5 2.5 13.2
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13.3. Optical implementation of Gau
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of the form Eq. (13.1), with 13.3.
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13.4. Telecloning with Gaussian val
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CHAPTER 14 Gaussian entanglement sh
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14.1. Entanglement in non-inertial
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14.2. Distributed Gaussian entangle
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6 s 4 14.2. Distributed Gaussian en
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2.5 0 0 5 10 IΣAR7.5 0 14.3. Distr
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0 entanglement condition 14.3. Dist
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4 3 a 2 1 14.4. Discussion and outl
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Part VI Closing remarks Entanglemen
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262 Conclusion and Outlook Gaussian
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264 Conclusion and Outlook compass
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266 A. Standard forms of pure Gauss
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272 List of Publications [GA11] G.
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274 Bibliography [23] C. H. Bennett
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276 Bibliography [79] J. Eisert, C.
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278 Bibliography [140] J. Laurat, T
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280 Bibliography [198] T. Roscilde,
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282 Bibliography [258] J. Von Neuma
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ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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