ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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226 13. Entanglement in Gaussian valence bond states s Figure 13.4. Pictorial representation of the entanglement between a probe (green) mode and its neighbor (magenta) modes on an harmonic ring with an underlying valence bond structure. As soon as the parameter s (encoding entanglement in the input port of the valence bond building block) is increased, pairwise entanglement between the probe mode and its farther and farther neighbors gradually appears in the corresponding output Gaussian valence bond states. By translational invariance, each mode exhibits the same entanglement structure with its respective neighbors. In the limit s → ∞, every single mode becomes equally entangled with every other single mode on the ring, independently of their relative distance: the Gaussian valence bond state is in this case fully symmetric. critical bosons, which in general do not fall in special subclasses of finite-bonded GVBS. 32 The valence bond picture however effectively captures the entanglement distribution in translationally invariant N-mode harmonic rings [GA13], as we are demonstrating in this Chapter. In this case the GVBS building blocks are equal at all sites, γ [i] ≡ γ ∀i, while the number of parameters Eq. (11.1) of the target state reduces, see Sec. 11.2.4, to the number of independent pairwise correlations (only functions of the distance between the two sites), which by our counting argument is ΘN ≡ (N − N mod 2)/2. The corresponding threshold for a GVBS representation becomes M ≥ IntPart[( √ 8ΘN + 1 − 1)/4]. As ΘN is bigger for even N, so it is the resulting threshold, which means that in general a higher number of EPR bonds is needed, and so more entanglement is inputed in the GVBS projectors and gets distributed in the target N-mode Gaussian state, as opposed to the case of an odd N. This finally clarifies why nearest-neighbor entanglement in ground states of pure translationally invariant N-mode harmonic rings (which belong to the class of states characterized by Proposition 1 of Sec. 11.2.2) is frustrated for odd N [272]. 13.2.2. Medium-range correlations Back to the main track, the connection between input entanglement in the building block and output correlation length in the destination GVBS, can be investigated in detail considering a general building block γ with s > smin. The GVBS CM in Eq. (13.3) can still be worked out analytically for a low number of modes, and 32 Recently, analytical progress on the area law issue (complementing the known results for the noncritical bosonic case [187]) has been obtained for the continuum limit of the real scalar Klein-Gordon massless field [60]. It is known [202] that the ground state of such critical model does not admit a GVBS representation with a finite number M of ancillary EPR bonds.
s 20 17.5 15 12.5 10 7.5 5 2.5 13.2. Entanglement distribution in Gaussian valence bond states 227 50 (a) (b) long-range entanglement next-n.n. nearest neighbors unphysical 2.5 5 7.5 10 12.5 15 17.5 20 x s 40 30 20 10 0 long-range entanglement next-n.n. nearest neighbors unphysical 2.5 5 7.5 10 12.5 15 17.5 20 x Figure 13.5. Entanglement distribution for a six-mode GVBS constructed from (a) infinitely entangled EPR bonds and (b) finitely entangled bonds given by two-mode squeezed states of the form Eq. (2.22) with r = 1.1. The entanglement thresholds sk with k = 1 (solid red line), k = 2 (dashed green line) and k = 3 (dotted blue line) are depicted as functions of the parameter x of the building block. For s > sk, all pairs of sites i and j with |i − j| ≤ k are entangled (see text for further details). numerically for higher N. Let us keep the parameter x fixed; we find that with increasing s the correlations extend smoothly to distant modes. A series of thresholds sk can be found such that for s > sk, two given modes i and j with |i − j| ≤ k are entangled. While trivially s1(x) = smin for any N (notice that nearest neighbors are entangled also for s = s1), the entanglement boundaries for k > 1 are in general different functions of x, depending on the number of modes. We observe however a certain regularity in the process: sk(x, N) always increases with the integer k. These considerations follow from analytic calculations on up to ten-modes GVBS, and we can infer them to hold true for higher N as well, given the overall scaling structure of the GVBS construction process. This entails the following remarkable result [GA13], pictorially summarized in Fig. 13.4. ➢ Entanglement distribution in Gaussian valence bond states. The maximum range of bipartite entanglement between two modes, i.e. the maximum distribution of multipartite entanglement, in a GVBS on a translationally invariant harmonic ring, is monotonically related to the amount of entanglement in the reduced two-mode input port of the building block. Moreover, no complete transfer of entanglement to more distant modes occurs: closer sites remain still entangled even when correlations between farther pairs arise. This feature will be precisely understood in the limit s → ∞. 13.2.2.1. Example: a six-mode harmonic ring. To clearly demonstrate the intriguing connection described above, let us consider the example of a GVBS with N = 6 modes. In a six-site translationally invariant ring, each mode can be correlated with another being at most 3 sites away (k = 1, 2, 3). From a generic building block Eq. (13.1), the 12 × 12 CM Eq. (13.3) can be analytically computed as a function of s and x. We can construct the reduced CMs γout i,i+k of two modes with distance k, and evaluate for each k the respective symplectic eigenvalue ˜νi,i+k of the corresponding partial transpose. The entanglement condition s > sk will correspond to the inequality ˜νi,i+k < 1. With this conditions one finds that s2(x) is the only acceptable solution to the equation: 72s8 −12(x2 +1)s6 +(−34x4 +28x2 −
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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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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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Part III Multipartite entanglement
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Part IV Quantum state engineering o
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Page 278 and 279: 264 Conclusion and Outlook compass
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Page 286 and 287: 272 List of Publications [GA11] G.
Page 288 and 289: 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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