ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

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78 4. Two-mode entanglement 4.5. Gaussian entanglement measures versus Negativities In this Section, based on Ref. [GA7], we add a further piece of knowledge on the quantification of entanglement in two-mode Gaussian states. We compute the Gaussian entanglement of formation and, in general, the family of Gaussian entanglement measures (see Sec. 3.2.2), for two special classes of two-mode Gaussian states, namely the states of extremal, maximal and minimal, negativities at fixed global and local purities (GMEMS and GLEMS, introduced in Sec. 4.3.3 [GA2, GA3]). We find that the two families of entanglement measures (negativities and Gaussian entanglement measures) are not equivalent for nonsymmetric twomode states. Remarkably, they may induce a completely different ordering on the set of entangled two-mode Gaussian states: a nonsymmetric state ϱA can be more entangled than another state ϱB, with respect to negativities, and less entangled than the same state ϱB, with respect to Gaussian entanglement measures. However, the inequivalence between the two families of measures is somehow bounded: we show that, at fixed negativities, the Gaussian entanglement measures are rigorously bounded from below. Moreover, we provide strong evidence hinting that they should be bounded from above as well. 4.5.1. Geometric framework for two-mode Gaussian entanglement measures The problem of evaluating Gaussian entanglement measures (Gaussian EMs) for a generic two-mode Gaussian state has been solved in Ref. [270]. However, the explicit result contains so “cumbersome” expressions (involving the solutions of a fourth-order algebraic equation), that they were judged of no particular insight to be reported explicitly in Ref. [270]. We recall here the computation procedure [GA7] that we will need in the following. For any two-mode Gaussian state with CM σ ≡ σsf in standard form Eq. (4.1), a generic Gaussian EM GE is given by the entanglement E of the least entangled pure state with CM σ p ≤ σ, see Eq. (3.11). Denoting by γq (respectively γp) the 2 × 2 submatrix obtained from σ by canceling the even (resp. odd) rows and columns, we have γq = a c+ c+ b , γp = a c− c− b . (4.60) All the covariances relative to the “position” operators of the two modes are grouped in γq, and analogously for the “momentum” operators in γp. The total CM can then be written as a direct sum σ = γq ⊕ γp. Similarly, the CM of a generic pure twomode Gaussian state in block-diagonal form (it has been proven that the CM of the optimal pure state has to be with all diagonal 2 × 2 submatrices as well [270]) can be written as σ p = γ p q ⊕ γ p p, where the global purity of the state imposes (γ p p) −1 = γ p q ≡ Γ (see Appendix A.2.1). The pure states involved in the definition of the Gaussian EM must thus fulfill the condition γ −1 p ≤ Γ ≤ γq . (4.61) This problem is endowed with a nice geometric description [270]. Writing the matrix Γ in the basis constituted by the identity matrix and the three Pauli matrices, x0 + x3 Γ = x1 , (4.62) x1 x0 − x3
4.5. Gaussian entanglement measures versus Negativities 79 the expansion coefficients (x0, x1, x3) play the role of space-time coordinates in a three-dimensional Minkowski space. In this picture, for example, the rightmost inequality in Eq. (4.61) is satisfied by matrices Γ lying on a cone, which is equivalent to the (backwards) light cone of γq in the Minkowski space; and similarly for the leftmost inequality. Indeed, one can show that, for the optimal pure state σ p opt realizing the minimum in Eq. (3.11), the two inequalities in Eq. (4.61) have to be simultaneously saturated [270]. From a geometrical point of view, the optimal Γ has then to be found on the rim of the intersection of the forward and the backward cones of γ−1 p and γq, respectively. This is an ellipse, and one is left with the task of minimizing the entanglement E of σp = Γ ⊕ Γ−1 [see Eq. (3.10)] for Γ lying on this ellipse. 10 At this point, let us pause to briefly recall that any pure two-mode Gaussian state σp is locally equivalent to a two-mode squeezed state with squeezing parameter r, described by the CM of Eq. (2.22). The following statements are then equivalent: (i) E is a monotonically increasing function of the entropy of entanglement; (ii) E is a monotonically increasing function of the single-mode determinant m2 ≡ Det α ≡ Det β [see Eq. (2.53)]; (iii) E is a monotonically decreasing function of the local purity µi ≡ µ1 ≡ µ2 [see Eq. (2.37)]; (iv) E is a monotonically decreasing function of the smallest symplectic eigenvalue ˜ν p − of the partially transposed CM ˜σ p ; (v) E is a monotonically increasing function of the squeezing parameter r. This chain of equivalences is immediately proven by simply recalling that a pure state is completely specified by its single-mode marginals, and that for a single-mode Gaussian state there is a unique symplectic invariant (the determinant), so that all conceivable entropic quantities are monotonically increasing functions of this invariant, as shown in Sec. 2.3 [GA3]. In particular, statement (ii) allows us to minimize directly the single-mode determinant over the ellipse, m 2 = 1 + x1 , (4.63) Det Γ with Γ given by Eq. (4.62). To simplify the calculations, one can move to the plane of the ellipse with a Lorentz boost which preserves the relations between all the cones; one can then choose the transformation so that the ellipse degenerates into a circle (with fixed radius), and introduce polar coordinates on this circle. The calculation of the Gaussian EM for any two-mode Gaussian state is thus finally reduced to the minimization of m2 from Eq. (4.63), at given standard form covariances of σ, as a function of the polar angle θ on the circle [135]. This technique had been applied to the computation of the Gaussian entanglement of formation by minimizing Eq. (4.63) numerically [270] (see also [56]). In addition to that, as already mentioned, the Gaussian entanglement of formation has been exactly computed for symmetric states, and it has been proven that in this case the Gaussian entanglement of formation is the true entanglement of formation [95]. Here we are going to present new analytical calculations, first obtained in [GA7], of the Gaussian EMs for two relevant classes of nonsymmetric two-mode Gaussian states: the states of extremal negativities at fixed global and local purities [GA2, 10 The geometric picture describing the optimal two-mode state which enters in the determination of the Gaussian EMs is introduced in [270]. A more detailed discussion, including the explicit expression of the Lorentz boost needed to move into the plane of the ellipse, can be found in [135].
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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
Page 43 and 44: CHAPTER 2 Gaussian states: structur
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Page 47 and 48: 2.2. Mathematical description of Ga
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128 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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