Gaussian measures of entanglement versus negativities: Ordering of two-mode Gaussian states

Gerardo Adesso and Fabrizio Illuminati

Phys. Rev. A 72, 032334 – Published 28 September 2005

Abstract

We study the entanglement of general (pure or mixed) two-mode Gaussian states of continuous-variable systems by comparing the two available classes of computable measures of entanglement: entropy-inspired Gaussian convex-roof measures and positive partial transposition–inspired measures (negativity and logarithmic negativity). We first review the formalism of Gaussian measures of entanglement, adopting the framework introduced in M. M. Wolf et al., Phys. Rev. A 69, 052320 (2004), where the Gaussian entanglement of formation was defined. We compute explicitly Gaussian measures of entanglement for two important families of nonsymmetric two-mode Gaussian state: namely, the states of extremal (maximal and minimal) negativities at fixed global and local purities, introduced in G. Adesso et al., Phys. Rev. Lett. 92, 087901 (2004). This analysis allows us to compare the different orderings induced on the set of entangled two-mode Gaussian states by the negativities and by the Gaussian measures of entanglement. We find that in a certain range of values of the global and local purities (characterizing the covariance matrix of the corresponding extremal states), states of minimum negativity can have more Gaussian entanglement of formation than states of maximum negativity. Consequently, Gaussian measures and negativities are definitely inequivalent measures of entanglement on nonsymmetric two-mode Gaussian states, even when restricted to a class of extremal states. On the other hand, the two families of entanglement measures are completely equivalent on symmetric states, for which the Gaussian entanglement of formation coincides with the true entanglement of formation. Finally, we show that the inequivalence between the two families of continuous-variable entanglement measures is somehow limited. Namely, we rigorously prove that, at fixed negativities, the Gaussian measures of entanglement are bounded from below. Moreover, we provide some strong evidence suggesting that they are as well bounded from above.

Received 20 June 2005

DOI:https://doi.org/10.1103/PhysRevA.72.032334

Authors & Affiliations

Gerardo Adesso and Fabrizio Illuminati

Dipartimento di Fisica “E. R. Caianiello,” Università di Salerno; CNR-Coherentia, Gruppo di Salerno; and INFN Sezione di Napoli-Gruppo Collegato di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy

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Vol. 72, Iss. 3 — September 2005

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Figure 1
(Color online) Comparison between the ordering induced by Gaussian EM’s on the classes of states with extremal (maximal and minimal) negativities. This extremal ordering of the set of entangled two-mode Gaussian states is studied in the space of the CM parameters ${s, d, g}$ , related to the global and local purities by the relations $μ_{1} = {(s + d)}^{- 1}$ , $μ_{2} = {(s - d)}^{- 1}$ , and $μ = g^{- 1}$ . The intermediate, meshed surface is constituted by those global and local mixednesses such that the Gaussian EM’s give equal values for the corresponding GMEMS (states of maximal negativities) and GLEMS (states of minimal negativities). Below this surface, the extremal ordering is inverted (GMEMS have fewer Gaussian EM’s than GLEMS). Above it, the extremal ordering is preserved (GMEMS have more Gaussian EM’s than GLEMS). However, it must be noted that this does not exclude that the individual orderings induced by the negativities and by the Gaussian EM’s on a pair of nonextremal states may still be inverted in this region. Above the uppermost, lighter surface, GLEMS are separable states, so that the extremal ordering is trivially preserved. Below the lowermost, darker surface, no physical two-mode Gaussian states can exist. All the quantities plotted are dimensionless.Reuse & Permissions
Figure 2
(Color online) Summary of entanglement properties of two-mode Gaussian states in the projected space of the local mixedness $b = μ_{2}^{- 1}$ of mode 2 and of the global mixedness $g = μ^{- 1}$ , while the local mixedness of mode 1 is kept fixed at a reference value $a = μ_{1}^{- 1} = 5$ . Below the thick curve, obtained imposing the equality in inequality (42), the Gaussian EM’s yield GLEMS more entangled than GMEMS, at fixed purities: the extremal ordering is thus inverted. Above the thick curve, the extremal ordering is preserved. In the coexistence region (see Ref. 19), GMEMS are entangled while GLEMS are separable. The boundaries of this region are given by Eq. (36) (dashed line) and Eq. (35) (dash-dotted line). In the separability region, GMEMS are separable too, so all two-mode Gaussian states whose purities lie in that region are not entangled. The shaded regions cannot contain any phisical two-mode Gaussian state. All the quantities plotted are dimensionless.Reuse & Permissions
Figure 3
(Color online) Comparison between Gaussian EM’s and negativities for two-mode Gaussian states. On the horizontal axis we plot the symplectic eigenvalue ${\tilde{ν}}_{-} (σ)$ of the partially transposed CM $\tilde{σ}$ of a generic two-mode Gaussian state with CM $σ$ . On the vertical axis we plot the symplectic eigenvalue ${\tilde{ν}}_{-} (σ_{opt}^{P})$ of the partially transposed CM ${\tilde{σ}}_{opt}^{P}$ of the optimal pure state with CM $σ_{opt}^{P}$ , which minimizes Eq. (23). The negativities are all monotonically decreasing functions of ${\tilde{ν}}_{-} (σ)$ , while the Gaussian EM’s are all monotonically decreasing functions of ${\tilde{ν}}_{-} (σ_{opt}^{P})$ . The equations of the two boundary curves are obtained from the saturation of inequality (43) (upper bound) and inequality (46) (lower bound), respectively. The dots represent 50 000 randomly generated CM’s of two-mode Gaussian states. Of up to $1 \times 10^{6}$ random CM’s, none has been found to lie below the lower solid-line curve, enforcing the conjecture that it be an absolute boundary for all two-mode Gaussian states. All the quantities plotted are dimensionless.Reuse & Permissions
Figure 4
(Color online) Comparison between the Gaussian entanglement of formation $G_{E_{F}}$ and the logarithmic negativity $E_{N}$ for two-mode Gaussian states. Symmetric states accomodate on the lower boundary (solid line), determined by the saturation of inequality (47). GMEMMS with infinite, average local mixedness, lie on the dashed line, whose defining equation is obtained from the saturation of inequality (48). All GMEMS and GLEMS lie below the dashed line. The latter is conjectured, with strong numerical support, to be the upper boundary for the Gaussian EOF of all two-mode Gaussian states, at fixed negativity. All the quantities plotted are dimensionless.Reuse & Permissions

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