ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
x Contents 2.2.2.2. Symplectic representation of the uncertainty principle 36 2.3. Degree of information encoded in a Gaussian state 37 2.3.1. Purity and generalized entropies 37 2.3.2. Comparison between entropic measures 38 2.4. Standard forms of Gaussian covariance matrices 40 2.4.1. Mixed states 41 2.4.1.1. Standard form of two-mode Gaussian states 42 2.4.2. Pure states 42 2.4.2.1. Phase-space Schmidt decomposition 43 2.4.3. Symmetric states 45 2.4.3.1. Fully symmetric Gaussian states 45 2.4.3.2. Bisymmetric M × N Gaussian states 46 Part II Bipartite entanglement of Gaussian states 49 Chapter 3. Characterizing entanglement of Gaussian states 51 3.1. How to qualify bipartite Gaussian entanglement 51 3.1.1. Separability and distillability: PPT criterion 51 3.1.1.1. Symplectic representation of PPT criterion 52 3.1.2. Additional separability criteria 53 3.2. How to quantify bipartite Gaussian entanglement 53 3.2.1. Negativities 53 3.2.2. Gaussian convex-roof extended measures 54 Chapter 4. Two-mode entanglement 57 4.1. Symplectic parametrization of two-mode Gaussian states 57 4.2. Entanglement and symplectic eigenvalues 58 4.2.1. Partial transposition and negativities 58 4.2.2. Entanglement of formation for symmetric states 59 4.2.3. EPR correlations 59 4.3. Entanglement versus Entropic measures 61 4.3.1. Entanglement vs Information (I) – Maximal negativities at fixed global purity 61 4.3.2. Entanglement vs Information (II) – Maximal negativities at fixed local purities 62 4.3.3. Entanglement vs Information (III) – Maximal and minimal negativities at fixed global and local purities 63 4.3.3.1. GMEMS and GLEMS: Extremally entangled states and purity-based separability criteria 64 4.3.4. Entanglement vs Information (IV) – Maximal and minimal negativities at fixed global and local generalized entropies 67 4.3.4.1. Inversion of extremally entangled states 69 4.3.4.2. Classifying entangled states with generalized entropic measures 72 4.4. Quantifying entanglement via purity measures: the average logarithmic negativity 75 4.4.1. Direct estimate of two-mode entanglement 76 4.5. Gaussian entanglement measures versus Negativities 78 4.5.1. Geometric framework for two-mode Gaussian entanglement measures 78 4.5.2. Gaussian entanglement measures for extremal states 80
Contents xi 4.5.2.1. Gaussian entanglement of minimum-negativity states (GLEMS) 81 4.5.2.2. Gaussian entanglement of maximum-negativity states (GMEMS) 82 4.5.3. Entanglement-induced ordering of two-mode Gaussian states 83 4.5.4. Comparison between Gaussian entanglement measures and negativities 86 4.6. Summary and further remarks 89 Chapter 5. Multimode entanglement under symmetry 93 5.1. Bipartite block entanglement of bisymmetric Gaussian states 94 5.1.1. Symplectic properties of symmetric states 95 5.1.2. Evaluation of block entanglement in terms of symplectic invariants 95 5.1.3. Unitary localization as a reversible multimode/two-mode entanglement switch 97 5.1.3.1. The case of the basset hound 99 5.2. Quantification and scaling of entanglement in fully symmetric states 99 5.2.1. 1 × N entanglement 100 5.2.1.1. Block entanglement hierarchy and signatures of genuine multipartite entanglement 100 5.2.1.2. Entanglement scaling with the number of modes 101 5.2.2. M × N entanglement 102 5.2.2.1. Block entanglement hierarchy and optimal localizable entanglement 102 5.2.2.2. Entanglement scaling with the number of modes 103 5.2.3. Discussion 105 Part III Multipartite entanglement of Gaussian states 107 Chapter 6. Gaussian entanglement sharing 109 6.1. Distributed entanglement in multipartite continuous variable systems 110 6.1.1. The need for a new continuous-variable entanglement monotone 110 6.1.2. Squared negativities as continuous-variable tangles 112 6.1.2.1. Gaussian contangle and Gaussian tangle 112 6.2. Monogamy of distributed entanglement in N-mode Gaussian states 114 6.2.1. General monogamy constraints and residual entanglement 115 6.2.2. Monogamy inequality for fully symmetric states 115 6.2.3. Monogamy inequality for all Gaussian states 116 6.2.3.1. Implications and perspectives 120 Chapter 7. Tripartite entanglement in three-mode Gaussian states 121 7.1. Three-mode Gaussian states 121 7.1.1. Separability properties 122 7.1.2. Pure states: standard form and local entropic triangle inequality 122 7.1.3. Mixed states 126 7.2. Distributed entanglement and genuine tripartite quantum correlations 127 7.2.1. Monogamy of the Gaussian contangle for all three-mode Gaussian states 127 7.2.2. Residual contangle and genuine tripartite entanglement 130 7.2.2.1. The residual Gaussian contangle is a Gaussian entanglement monotone 131
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Contents xi<br />
4.5.2.1. Gaussian entanglement of minimum-negativity states (GLEMS) 81<br />
4.5.2.2. Gaussian entanglement of maximum-negativity states (GMEMS) 82<br />
4.5.3. Entanglement-induced ordering of two-mode Gaussian states 83<br />
4.5.4. Comparison between Gaussian entanglement measures and<br />
negativities 86<br />
4.6. Summary and further remarks 89<br />
Chapter 5. Multimode entanglement under symmetry 93<br />
5.1. Bipartite block entanglement of bisymmetric Gaussian states 94<br />
5.1.1. Symplectic properties of symmetric states 95<br />
5.1.2. Evaluation of block entanglement in terms of symplectic invariants 95<br />
5.1.3. Unitary localization as a reversible multimode/two-mode<br />
entanglement switch 97<br />
5.1.3.1. The case of the basset hound 99<br />
5.2. Quantification and scaling of entanglement in fully symmetric states 99<br />
5.2.1. 1 × N entanglement 100<br />
5.2.1.1. Block entanglement hierarchy and signatures of genuine<br />
multipartite entanglement 100<br />
5.2.1.2. Entanglement scaling with the number of modes 101<br />
5.2.2. M × N entanglement 102<br />
5.2.2.1. Block entanglement hierarchy and optimal localizable<br />
entanglement 102<br />
5.2.2.2. Entanglement scaling with the number of modes 103<br />
5.2.3. Discussion 105<br />
Part III Multipartite entanglement of Gaussian states 107<br />
Chapter 6. Gaussian entanglement sharing 109<br />
6.1. Distributed entanglement in multipartite continuous variable systems 110<br />
6.1.1. The need for a new continuous-variable entanglement monotone 110<br />
6.1.2. Squared negativities as continuous-variable tangles 112<br />
6.1.2.1. Gaussian contangle and Gaussian tangle 112<br />
6.2. Monogamy of distributed entanglement in N-mode Gaussian states 114<br />
6.2.1. General monogamy constraints and residual entanglement 115<br />
6.2.2. Monogamy inequality for fully symmetric states 115<br />
6.2.3. Monogamy inequality for all Gaussian states 116<br />
6.2.3.1. Implications and perspectives 120<br />
Chapter 7. Tripartite entanglement in three-mode Gaussian states 121<br />
7.1. Three-mode Gaussian states 121<br />
7.1.1. Separability properties 122<br />
7.1.2. Pure states: standard form and local entropic triangle inequality 122<br />
7.1.3. Mixed states 126<br />
7.2. Distributed entanglement and genuine tripartite quantum correlations 127<br />
7.2.1. Monogamy of the Gaussian contangle for all three-mode Gaussian<br />
states 127<br />
7.2.2. Residual contangle and genuine tripartite entanglement 130<br />
7.2.2.1. The residual Gaussian contangle is a Gaussian entanglement<br />
monotone 131
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