ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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190 11. Efficient production of pure N-mode Gaussian states 11.3. Economical state engineering of arbitrary pure Gaussian states? Borrowing the ideas leading to the state engineering of block-diagonal pure Gaussian states [GA14], see Fig. 11.1, we propose here a scheme [GA18], involving (N 2 − 2N) independent optical elements, to produce more general N-mode pure Gaussian states, encoding correlations between positions and momentum operators as well. To this aim, we introduce ‘counter-beam-splitter’ transformations, named “seraphiques”, which, recovering the phase space ordering of Sec. 2.1, act on two modes j and k as ⎛ √ √ τ 0 0 1 − τ ⎜ √ √ Cj,k(τ) = ⎜ 0 √ τ − 1 − τ 0 ⎝ √ 0 1 − τ τ 0 − √ ⎞ ⎟ ⎠ , (11.2) √ 1 − τ 0 0 τ where the amplitude τ is related to an angle θ in phase space by τ = cos 2 θ. Such operations can be obtained from usual beam-splitters Bj,k(τ), Eq. (2.26), by applying a π/2 phase shifter Pk on only one of the two considered modes. Pk is a local rotation mapping, in Heisenberg picture, ˆqk ↦→ −ˆpk and ˆpk ↦→ ˆqk. In phase space, one has Cj,k(τ) = P T k Bj,k(τ)Pk. Notice that, even though Cj,k(τ) is equal to the product of single-mode operations and beam-splitters, this does not mean that such a transformation is “equivalent” to a beam-splitter in terms of state generation. In fact, the local operations do not commute with the beam-splitters, so that a product of the kind Bj,k(τ ′ )Cj,k(τ ′′ ) cannot be written as Bj,k(τ)Sl for some local operation Sl and τ. The state engineering scheme runs along exactly the same lines as the one for the block-diagonal states, Sec. 11.2.3, the only modification being that for each pair of modes except the last one (N − 1, N), a beam-splitter transformation is followed by a seraphique. In more detail (see Fig. 11.2): first of all (step 1), one squeezes mode 1 of an amount s, and mode 2 of an amount 1/s (i.e. one squeezes the first mode in one quadrature and the second, of the same amount, in the orthogonal quadrature); then one lets the two modes interfere at a 50 : 50 beamsplitter. One has so created a two-mode squeezed state between modes 1 and 2, which corresponds to the Schmidt form of the pure Gaussian state with respect to the 1 × (N − 1) bipartition. The second step basically corresponds to a redistribution of the initial two-mode entanglement among all modes; this task can be obtained by letting each additional k mode (k = 3 . . . N) interact step-by-step with all the previous l ones (l = 2 . . . k − 1), via beam-splitters and seraphiques (which are in turn combinations of beam-splitters and phase shifters). It is easy to see that this scheme is implemented with minimal resources. Namely, the state engineering process is characterized by one squeezing degree (step 1), plus N − 2 individual squeezings, together with N−2 i=1 i = (N − 1)(N − 2)/2 beam-splitter transmittivities, and [ N−2 i=1 i] − 1 = N(N − 3)/2 seraphique amplitudes, which amount to a total of (N 2 −2N) quantities, exactly the ones parametrizing a generic pure Gaussian state of N ≥ 3 modes up to local symplectic operations, Eq. (2.56). While this scheme (Fig. 11.2) is surely more general than the one for blockdiagonal states (Fig. 11.1), as it enables to efficiently create a broader class of pure Gaussian states for N > 3, we will leave it as an open question to check if it is general enough to produce all pure N-mode Gaussian states up to local unitaries. Verifying this analytically leads to pretty intractable expressions already
11.3. Economical state engineering of arbitrary pure Gaussian states? 191 1 2 3 4 N-1 N s s -1 1/2 … r3 b2,3 c2,3 b 2,N c 2,N b2,4 c2,4 b3,4 c3,4 b 3,N c 3,N r 4 … … … … … … r N … b N-1,N 1 2 3 4 N-1 N Figure 11.2. Possible scheme to create a general N-mode pure Gaussian state. White balls are vacua, while each color depicts a different single-mode determinant (i.e. different degrees of local mixedness). Vertical arrows denote single-mode squeezing operations with squeezing parameters rj, horizontal circle-ended lines denote beam-splitting operations, Eq. (2.26), with transmittivity bi,j between modes i and j, and horizontal diamond-ended lines denote two-mode seraphiques, Eq. (11.2), with amplitudes ci,j. See text for details. for N = 4. Instead, it would be very interesting to investigate if the average entanglement of the output Gaussian states numerically obtained by a statistically significant sample of applications of our scheme with random parameters, matches
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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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Page 193 and 194: (out) (in) TRITTER 10.1. Optical pr
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Page 197 and 198: CHAPTER 11 Efficient production of
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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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