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