ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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182 10. Tripartite and four-partite state engineering 1 2 3 4 s a a Figure 10.4. Preparation of the four-mode Gaussian states σ of Eq. (8.1). Starting with four initially uncorrelated modes of light, all residing in the respective vacuum states (yellow beams), one first applies a two-mode squeezing transformation (light blue box), with squeezing s, between the central modes 2 and 3, and then two additional two-mode squeezing transformations (light pink boxes), each one with equal squeezing a, acting on the pair of modes 1,2 and 3,4 respectively. The resulting state is endowed with a peculiar, yet insightful bipartite entanglement structure, pictorially depicted in Fig. 8.1. It is interesting to observe that the amount of producible squeezing in optical experiments is constantly improving [224]. Only technological, no a priori limitations need to be overcome to increase a and/or s to the point of engineering excellent approximations to the demonstrated promiscuous entanglement structure, elucidated in Chapter 8, in multimode states of light and atoms (see also [222]). To make an explicit example, already with realistic squeezing degrees like s = 1 and a = 1.5 (corresponding to ∼ 3 dB and 10 dB, respectively, where decibels are defined in footnote 20 on page 144), one has a bipartite entanglement of Gτ (σ 1|2) = Gτ (σ 3|4) = 9 ebits (corresponding to a Gaussian entanglement of formation [270], see Sec. 3.2.2, of ∼ 3.3 ebits), coexisting with a residual multipartite entanglement of Gτ res (σ) 5.5 ebits, of which the tripartite portion is at most Gτ bound (σ 1|2|3) 0.45 ebits. This means that one can simultaneously extract at least 3 qubit singlets from each pair of modes {1, 2} and {3, 4}, and more than a single copy of genuinely four-qubit entangled states (like cluster states [192]). Albeit with imperfect efficiency, this entanglement transfer can be realized by means of Jaynes-Cummings interactions [176], representing a key step for a reliable physical interface between fields and qubits in a distributed quantum information processing network (see also Refs. [130, 216]). 1 2 3 4 σσσσ
CHAPTER 11 Efficient production of pure N-mode Gaussian states Recently, a great insight into the role of entanglement in the description of quantum systems has been gained through the quantum information perspective, mostly focusing on the usefulness of entanglement, rather than on its meaning. In these years, quantum entanglement has turned from a paradoxical concept into a physical resource allowing for the encoding, manipulation, processing and distribution of information in ways forbidden by the laws of classical physics. In this respect, we have evidenced how CV entanglement between canonically conjugate observables of infinite-dimensional systems, like harmonic oscillators, light modes and atomic ensembles, has emerged as a versatile and powerful resource. In particular, multimode Gaussian states have been proven useful for a wide range of implementations in CV quantum information processing [40], and advances in the characterization of their bipartite and multipartite entanglement have been recently recorded (see the previous Parts of this Dissertation). In experiments, one typically aims at preparing pure states, with the highest possible entanglement, even though unavoidable losses and thermal noises will affect the purity of the engineered resources, and hence the efficiency of the realized protocols (a direct evidence is reported in Sec. 9.2). It is therefore important to understand the structure of correlations in pure Gaussian states of an arbitrary number of modes, and to provide economical schemes to produce such states in the lab with minimal elements, thus reducing the possibility of accumulating errors and unwanted noise. 11.1. Degrees of freedom of pure Gaussian states: practical perspectives In the instance of two- and three-mode Gaussian states, efficient schemes for their optical production with minimal resources have been presented, respectively, in Chapters 9 and 10. In the general case of pure N-mode Gaussian states with N > 3, we know from Eq. (2.56) that the minimal number of degrees of freedom characterizing all their structural and informational properties (up to local unitaries) is (N 2 − 2N) (see also Sec. 2.4.2 and Appendix A). It would be desirable to associate the mathematically clear number (N 2 − 2N) with an operational, physical insight. In other words, it would be useful for experimentalists (working, for instance, in quantum optics) to be provided with a recipe to create pure N-mode Gaussian states with completely general entanglement properties in an economical way, using exactly (N 2 − 2N) optical elements, such as squeezers, beam-splitters and phase shifters. A transparent approach to develop 183
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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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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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