ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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44 2. Gaussian states: structural properties respect to an arbitrary A|B bipartition, therefore, the CM σp of any pure N-mode Gaussian state is locally equivalent to the form σ p S = (SA ⊕ SB)σp (SA ⊕ SB) T , with ⎛ ⎞ σ p S = ⎜ ⎝ NA C1 ⋄ ⋄ ⋄ ⋄ C2 ⋄ ⋄ ⋄ ⋄ . .. ⋄ ⋄ ⋄ ⋄ CNA S1 ⋄ ⋄ ⋄ ⋄ S2 ⋄ ⋄ ⋄ ⋄ . .. ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ SNA ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ NB S1 ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ S2 ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ . .. ⋄ ⋄ ⋄ ⋄ ⋄ C1 ⋄ ⋄ ⋄ ⋄ SNA ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ C2 ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ . .. ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ CNA ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ . . . ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⎟ . (2.57) ⎟ ⎠ Here each element denotes a 2 × 2 submatrix, in particular the diamonds (⋄) correspond to null matrices, to the identity matrix, and νk 0 ν2 Ck = , Sk = k − 1 0 0 νk 0 − ν2 . k − 1 The matrices Ck contain the symplectic eigenvalues νk = 1 of both reduced CMs. By expressing them in terms of hyperbolic functions, νk = cosh(2rk) and by comparison with Eq. (2.22), one finds that each two-mode CM Ck Sk , Sk Ck encoding correlations between a single mode from SA and a single mode from SB, is a two-mode squeezed state with squeezing rk. Therefore, the Schmidt form of a pure N-mode Gaussian state with respect to a NA ×NB bipartition (with N = NA +NB, NB ≥ NA) is that of a direct sum [116, 92] σ p S = NA i=1 σ sq i,j (ri) N k=2NA+1 σ 0 k , (2.58) where mode i ∈ SA, mode j ≡ i + NA ∈ SB, and σ 0 k = 2 is the CM of the vacuum state of mode k ∈ SB. This corresponds, on the Hilbert space level, to the product of two-mode squeezed states, tensor additional uncorrelated vacuum modes in the higher-dimensional subsystem (SB in our notation) [29]. The phase-space Schmidt decomposition is a very useful tool both for the understanding of the structural features of Gaussian states in the CM formalism, and for the evaluation of their entanglement properties. Notice that the validity of such a decomposition can be extended to mixed states with fully degenerate symplectic spectrum, i.e. Williamson normal form proportional to the identity [29, 92]. As a straightforward consequence of Eq. (2.58), any pure two-mode Gaussian state is equivalent, up to local unitary operations, to a two-mode squeezed state of the form Eq. (2.22), therefore the minimum number of local-unitary degrees of freedom for pure Gaussian states with N = 2 is just one (the squeezing degree), in
2.4. Standard forms of Gaussian covariance matrices 45 accordance with Eq. (2.56), and as explained by alternative arguments in Appendix A.2.1. In other words, according to the notation of Eq. (2.54), any pure twomode Gaussian state is symmetric (b = a) and its standard form elements fulfill c± = ± √ a2 − 1. Notice also that the phase-space decomposition discussed here is special to Gaussian states and is independent from the general Schmidt decomposition at the Hilbert space level, Eq. (1.20), which can be obtained for any pure state. For CV systems, it will contain in principle infinite terms, as the local bases are infinitedimensional. To give an example, the two-mode squeezed state, whose CM in its “phase-space Schmidt decomposition” is of the form Eq. (2.22), admits the following Hilbert-space Schmidt decomposition [16] |ψ sq 〉i,j = 1 cosh r ∞ tanh n r |n〉 i |n〉 j , (2.59) n=0 with local Schmidt bases given by the number states in the Fock space of each mode. We will now show that for (generally mixed) Gaussian states with some local symmetry constraints, a similar phase-space reduction is available, such that multimode properties (like entanglement) can be unitarily reduced to two-mode ones [GA4, GA5]. 2.4.3. Symmetric states Very often in quantum information, and in particular in the theory of entanglement, peculiar roles are played by symmetric states, that is, states that are either invariant under a particular group of transformations — like Werner states of qudits [264] — or under permutation of two or more parties in a multipartite system, like ground and thermal states of some translationally invariant Hamiltonians (e.g. of harmonic lattices) [11]. Here we will introduce classes of Gaussian states invariant under all permutation of the modes (fully symmetric states) or exhibiting such permutationinvariance locally in each of the two subsystems across a global bipartition of the modes (bisymmetric states). For both we will provide a standard form based on the special properties of their symplectic spectrum. We will limit ourself to a collection of results, which will be useful for the computation and exploitation of entanglement in the corresponding states. All the proofs can be found in Ref. [GA5]. Unless explicitly stated, we are dealing with generally mixed states. 2.4.3.1. Fully symmetric Gaussian states. We shall say that a multimode Gaussian state ϱ is “fully symmetric” if it is invariant under the exchange of any two modes. In the following, we will consider the fully symmetric M-mode and N-mode Gaussian states ϱ α M and ϱ β N , with CMs σ α M and σ β N . Due to symmetry, we have that the CM, Eq. (2.20), of such states reduces to ⎛ ⎞ α ε · · · ε ⎜ σαM = ⎜ ⎝ ε . . α ε ε . .. . . ε ⎟ ⎠ ε · · · ε α , σ ⎛ β ζ · · · ζ ⎞ ⎜ βN = ⎜ ⎝ ζ . . β ζ ζ . .. . . ζ ⎟ , ⎠ (2.60) ζ · · · ζ β where α, ε, β and ζ are 2 × 2 real symmetric submatrices (the symmetry of ε and ζ stems again from the symmetry under the exchange of any two modes).
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ENTANGLEMENT OF GAUSSIAN STATES Ger
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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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Page 43 and 44: CHAPTER 2 Gaussian states: structur
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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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