Acknowledgments . . . . vii

Contents

Page

Acknowledgments . . . . vii

Abstract . . . . ix

Résumé . . . . xi

List of publications . . . . xiii

1 Introduction, motivation and scope 1 I BASICS OF QUANTUM INFORMATION IN PHASE SPACE 7 2 Quantum optics in phase space 9 2.1 Quantization of the electromagnetic field . . . . 9

2.2 Wigner function . . . . 11

2.3 Gaussian states . . . . 15

2.4 Gaussian unitaries . . . . 17

2.4.1 Symplectic transformations . . . . 17

2.5 Examples of Gaussian unitaries and Gaussian states . . . . 20

2.5.1 Coherent states and displacement operator . . . . 20

2.5.2 Squeezed states and squeezing operator . . . . 21

2.5.3 Phase-shift operator . . . . 23

2.5.4 Beam splitter . . . . 23

2.5.5 Two-mode squeezer . . . . 24

2.5.6 Thermal states . . . . 25

2.6 Passive states . . . . 26

3 Shannon information theory 27 3.1 Discrete variables . . . . 27

3.1.1 Shannon entropy . . . . 27

3.1.2 Joint entropy . . . . 28

3.1.3 Relative entropy . . . . 28

3.1.4 Mutual information . . . . 29

3.2 Continuous variables . . . . 29

3.2.1 Shannon differential entropy . . . . 29

xv

3.2.2 Joint entropy, relative entropy, mutual information and prop-

erties . . . . 30

3.2.3 Entropy of Gaussian distributions . . . . 31

3.2.4 Entropy power . . . . 32

3.2.5 Rényi entropy . . . . 32

3.2.6 Wehrl entropy . . . . 33

4 Uncertainty relations 35 4.1 Variance-based uncertainty relations . . . . 35

4.1.1 Heisenberg uncertainty relation . . . . 35

4.1.2 Robertson-Schrödinger uncertainty relation . . . . 37

4.1.3 Proof of the Robertson-Schrödinger uncertainty relation . . . . 40

4.1.4 Gaussianity-bounded uncertainty relation . . . . 41

4.1.5 Purity-bounded uncertainty relation . . . . 44

4.2 Entropy-based uncertainty relations . . . . 45

4.2.1 Continuous-variable entropic uncertainty relations . . . . 45

4.2.2 Proof of the original entropic uncertainty relation . . . . 47

4.2.3 Discrete-variable entropic uncertainty relations . . . . 48

5 Separability criteria 51 5.1 Entangled states . . . . 52

5.2 Discrete-variable separability criteria . . . . 53

5.3 Continuous-variable separability criteria . . . . 54

5.3.1 Variance-based separability criterion . . . . 54

5.3.2 Entropy-based separability criterion . . . . 59

5.3.3 Other separability criteria . . . . 60

II CONTINUOUS-VARIABLE UNCERTAINTY RELATIONS 61 6 Tight entropic uncertainty relation for canonically conjugate variables 63 6.1 Entropy-power uncertainty relations . . . . 64

6.2 Extended forms of entropic uncertainty relations . . . . 65

6.2.1 Motivation . . . . 65

6.2.2 Tight uncertainty relation saturated by all pure Gaussian states 66 6.2.3 Numerical evidence of relation (6.14) . . . . 68

6.3 Conditional proof of relation (6.14) . . . . 70

6.3.1 Special case of Gaussian states . . . . 70

6.3.2 General case . . . . 71

6.4 Generalization to n modes . . . . 76

6.4.1 Conditional proof of the n-mode generalization . . . . 79

6.5 Attempts towards a full proof . . . . 82

6.6 Conclusion . . . . 85

xvi

7 Entropic uncertainty relations for arbitrary quadratures 87

7.1 Linear canonical transforms . . . . 88

7.2 Multidimensional uncertainty relations . . . . 91

7.2.1 Entropic uncertainty relation between two linear canonical trans- forms . . . . 91

7.2.2 Entropic uncertainty relation based on a commutators matrix . 94 7.2.3 Extension to Rényi entropies . . . . 98

7.2.4 Corresponding covariance-based uncertainty relation . . . . 99

7.3 Conclusion . . . . 100

8 Tight entropic uncertainty relation for arbitrary quadratures 103 8.1 A general entropic uncertainty relation saturated by all pure Gaussian states . . . . 105

8.2 Explicit entropy calculation for Gaussian states . . . . 106

8.3 Other formulations . . . . 110

8.4 Conditional proof of Eq. (8.11) . . . . 111

8.5 Attempt to define a purity-bounded entropic uncertainty relation . . . 114

8.6 Corresponding covariance-based uncertainty relation . . . . 115

8.7 Conclusion and perspectives . . . . 117

9 Wigner Entropy 119 9.1 A conjecture for positive Wigner functions . . . . 119

9.2 Numerical evidences . . . . 122

9.3 A complex-valued Wigner entropy . . . . 123

9.3.1 Definition . . . . 123

9.3.2 Properties of the complex Wigner entropy . . . . 124

9.3.3 An entropic uncertainty relation? . . . . 126

10 Symplectic-invariant entropic uncertainty relation based on a multi-copy uncertainty observable 131 10.1 2-copy uncertainty observable ˆ L

. . . . 132

10.1.1 Definition of ˆ L

and link with the uncertainty relation . . . . 132

10.1.2 Physical realization . . . . 134

10.1.3 Ladder operators . . . . 136

10.1.4 Alternative definitions . . . . 139

10.1.5 Eigensystem of L

. . . . 140

10.1.6 Symmetry property . . . . 142

10.1.7 Entropic uncertainty relation based on ˆ L

. . . . 143

10.1.8 Invariance of H ( L ^ˆ

)

. . . . 143

10.1.9 Special case of Gaussian states . . . . 144

10.1.10 Example of non-Gaussian states . . . . 146

10.2 3-copy uncertainty observable ˆ L

. . . . 148

10.2.1 Definition of ˆ L

and link with the uncertainty relation . . . . 148

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10.2.2 Alternative definitions . . . . 151

10.2.3 Physical realization . . . . 152

10.2.4 Special case of Gaussian states . . . . 153

10.2.5 Case of non-Gaussian states . . . . 154

III CONTINUOUS-VARIABLE SEPARABILITY CRITERIA 155 11 Improved continuous-variable separability criterion based on the degree of Gaussianity 157 11.1 Improved separability condition . . . . 158

11.2 Detection of non-Gaussian entangled states . . . . 162

11.2.1 Non-Gaussian states generated from Fock states or phase-diffused coherent states . . . . 162

11.2.2 Squeezed single-photon path-entangled state . . . . 169

11.3 Conclusion . . . . 170

12 Improved entropic separability criterion for non-Gaussian states 173 12.1 Derivation of the criterion . . . . 173

12.2 Example of an entangled mixed state . . . . 175

12.3 Discussion . . . . 177

13 Conclusion 179

Bibliography 185

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