ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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36 2. Gaussian states: structural properties equilibrium at a temperature Tk, characterized by an average number of thermal photons ¯nk which obeys Bose-Einstein statistics, ¯nk = νk − 1 2 = 1 . (2.32) exp − 1 ωk kBTk The N quantities νk’s form the symplectic spectrum of the CM σ, and are invariant under the action of global symplectic transformations on the matrix σ. The symplectic eigenvalues can be computed as the orthogonal eigenvalues of the matrix |iΩσ| [207] and are thus determined by N invariants of the characteristic polynomial of such a matrix [208]. One global symplectic invariant is simply the determinant of the CM (whose invariance is a consequence of the fact that Det S = 1 ∀S ∈ Sp (2N,)), which once computed in the Williamson diagonal form reads Det σ = N k=1 ν 2 k . (2.33) Another important invariant under global symplectic operations is the so-called seralian ∆ [GA6], defined as the sum of the determinants of all 2 × 2 submatrices of a CM σ, Eq. (2.20), which can be readily computed in terms of its symplectic eigenvalues as N ∆(σ) = ν 2 k . (2.34) The invariance of ∆σ in the multimode case [208] follows from its invariance in the case of two-mode states, proved in Ref. [211], and from the fact that any symplectic transformation can be decomposed as the product of two-mode transformations [123]. 2.2.2.2. Symplectic representation of the uncertainty principle. The symplectic eigenvalues νk encode essential information on the Gaussian state σ and provide powerful, simple ways to express its fundamental properties [211]. For instance, let us consider the uncertainty relation (2.19). Since the inverse of a symplectic operation is itself symplectic, one has from Eq. (2.23), S −1T ΩS −1 = Ω, so that Ineq. (2.19) is equivalent to ν + iΩ ≥ 0. In terms of the symplectic eigenvalues νk the uncertainty relation then simply reads νk ≥ 1 . (2.35) Inequality (2.35) is completely equivalent to the uncertainty relation (2.19) provided that the CM σ satisfies σ ≥ 0. We can, without loss of generality, rearrange the modes of a N-mode state such that the corresponding symplectic eigenvalues are sorted in ascending order k=1 ν− ≡ ν1 ≤ ν2 ≤ . . . ≤ νN−1 ≤ νN ≡ ν+ . With this notation, the uncertainty relation reduces to ν− ≥ 1. We remark that the full saturation of the uncertainty principle can only be achieved by pure N-mode Gaussian states, for which νi = 1 ∀i = 1, . . . , N , meaning that the Williamson normal form of any pure Gaussian state is the vacuum |0〉 of the N-mode Hilbert space H . Instead, mixed states such that νi≤k = 1 and νi>k > 1, with 1 ≤ k ≤ N, only partially saturate the uncertainty principle, with
2.3. Degree of information encoded in a Gaussian state 37 Hilbert space H Phase space Γ dimension structure ∞ 2N description ϱ σ bona fide ϱ ≥ 0 σ + iΩ ≥ 0 operations U : U † U = ϱ↦→UϱU † S : STΩS = Ω σ↦→SσS T spectra UϱU † = diag{λk} 0≤λk≤1 SσS T = diag{νk} 1≤νk
Page 1 and 2: ENTANGLEMENT OF GAUSSIAN STATES Ger
Page 3: Life’s Entanglement. CR Studio In
Page 7: Abstract This Dissertation collects
Page 10 and 11: x Contents 2.2.2.2. Symplectic repr
Page 12 and 13: xii Contents 7.2.3. Tripartite enta
Page 14 and 15: xiv Contents Chapter 13. Entangleme
Page 17 and 18: Introduction About eighty years aft
Page 19 and 20: Introduction 5 Gaussian states. Imp
Page 21: Introduction 7 The companion Part V
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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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CHAPTER 13 Entanglement in Gaussian
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13.1. Gaussian valence bond states
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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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