Fundamentals of Quantum Entanglement

F. J. Duarte, Fundamentals of Quantum Entanglement (Institute of Physics, Bristol, 2019)

ISBN: 978-0-7503-2226-3

29 chapters, 10 appendices, 71 figures, and approximately 700 equations, in 240 pages.

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"(|x, y> - |y, x>)... was my first lesson in quantum mechanics, and in a very real sense my last, since all the rest is mere technique, which can be learnt from books" (Ward 2004).

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What's important to highlight is that the derivation of the quantum entanglement probability amplitude, à la Dirac, flows naturally; it is transparent and straightforward.

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1. Introduction

1.1 Introduction

1.2 A few words on quantum mechanics

1.2.1 The photon from a quantum perspective

1.3 Ward’s observation

1.4 History of quantum entanglement

1.4.1 The philosophical path

1.4.2 The physics path

1.5 The field of quantum entanglement

1.6 Fundamentals of quantum entanglement

1.7 Intent


2. Dirac’s contribution

2.1 Introduction

2.2 Dirac’s pair theory

2.3 Dirac’s notation

2.4 Dirac’s notation in N-slit interferometers

2.5 Semi coherent interference

2.6 From quantum probabilities to measurable intensities

2.7 Dirac’s identities


3. The Einstein Podosky Rosen (EPR) paper

3.1 Introduction

3.2 EPR doubt’s on quantum mechanics

3.3 EPR’s landmark definition of a correct theory


4. The Schrödinger papers

4.1 Introduction

4.2 The first Schrödinger paper

4.3 The second Schrödinger paper


5. Wheeler’s paper

5.1 Introduction

5.2 Wheeler’s paper significance to quantum theory

5.3 Wheeler’s paper significance to quantum experiments


6. The probability amplitude for quantum entanglement

6.1 Introduction

6.2 The Pryce-Ward paper

6.2.1 Theoretical legacy of the Pryce-Ward paper

6.2.2 Experimental legacy of the Pryce-Ward paper

6.3 Ward’s doctoral thesis

6.4 Summary


7. The quantum entanglement experiment

7.1 Introduction

7.2 The quantum entanglement experiment

7.3 Historical notes


8. The annihilation quantum entanglement experiments

8.1 Introduction

8.2 The first three quantum entanglement experiments

8.3 Further significance of the annihilation experiments


9. The Bohm and Aharonov paper

9.1 Introduction

9.2 Significance to the development of quantum entanglement research

9.3 Philosophy and physics


10. Bell’s theorem

10.1 Introduction

10.2 von Neumann’s

10.3 Bell’s theorem or Bell’s inequalities

10.4 An additional perspective on Bell’s theorem

10.5 Example

10.6 More philosophy and physics


11. Feynman’s Hamiltonians

11.1 Introduction

11.2 Probability amplitudes via Hamiltonians à la Feynman

11.3 Arrival to quantum entanglement probability amplitudes

11.4 Discussion


12. The second Wu quantum entanglement experiment

12.1 Introduction

12.2 Salient features

12.3 Bell’s theorem and hidden variables


13. The hidden variable theory experiments

13.1 Introduction

13.2 Testing for local hidden variable theories

13.3 Early optical experiment

13.3 Observations and discussion


14. The optical quantum entanglement experiments

14.1 Introduction

14.2 The Aspect experiments

14.2.1 The first Aspect experiment

14.2.2 The second Aspect experiment

14.2.3 The third Aspect experiment

14.3 Observations and discussion


15. The quantum entanglement probability amplitude 1947-1992

15.1 Introduction

15.2 The quantum entanglement probability amplitude 1947-1992

15.3 Observations and discussion


16. The GHZ probability amplitudes

16.1 Introduction

16.2 The GHZ probability amplitude

16.3 Observations and discussion


17. The interferometric derivation of the quantum entanglement probability amplitude for n = N = 2

17.1 Introduction

17.2 The meaning of the Dirac-Feynman probability amplitude

17.3 The derivation of the quantum entanglement probability amplitude

17.4 Identical states of polarization

17.5 Discussion


18. The interferometric derivation of the quantum entanglement probability amplitude for n = N = 21 22, 23, 24… 2r

18.1 Introduction

18.2 The quantum entanglement probability amplitude for n = N = 4

18.3 The quantum entanglement probability amplitude for n = N = 8

18.4 The quantum entanglement probability amplitude for n = N = 16

18.5 The quantum entanglement probability amplitude for n = N = 21 22, 23… 2r

18.6 Discussion


19. The interferometric derivation of the quantum entanglement probability amplitudes for n = N = 3, 6

19.1 Introduction

19.2 The quantum entanglement probability amplitude for n = N = 3

19.3 The quantum entanglement probability amplitude for n = N = 6

19.4 Discussion


20. What happens with the entanglement at n = 1 and N = 2 ?

20.1 Introduction

20.2 Reversibility: from entanglement to interference

20.3 Schematics

20.4 Experimental and theoretical perspectives

20.4.1 Experimental perspective

20.4.2 Theoretical perspective

20.5 Interference for N slits and n = 1


21. Quantum entanglement probability amplitudes and Bell’s theorem

21.1 Introduction

21.2 Probability amplitudes

21.3 Quantum polarization

21.4 Quantum probabilities and Bell’s theorem

21.5 Example

21.6 Discussion


22. Cryptography via quantum entanglement

22.1 Introduction

22.2 Measurement protocol

22.3 Experimental


23. Quantum entanglement and teleportation

23.1 Introduction

23.2 The mechanics of teleportation

23.3 Technology


24. Quantum entanglement and quantum computing

24.1 Introduction

24.2 Entropy

24.3 Qbits

24.4 Quantum entanglement and Pauli matrices

24.5 Pauli matrices and quantum entanglement

24.6 Quantum gates

24.6.1 Pauli gates

24.6.2 The Hadamard gate

24.7 The Hadamard matrix and quantum entanglement

24.8 Multiple entangled states

24.9 Technology


25. Space-to-space and space-to-Earth communications via quantum entanglement

25.1 Introduction

25.2 Free-space configurations

25.3 The space-to-Earth experiment

25.4 Further horizons


26. Space-to-space quantum interferometric communications: an alternative to quantum entanglement communications?

26.1 Introduction

26.2 The generalized N-slit quantum interference equations

26.3 The generation and transmission of interferometric characters

26.4 The inherent quantum security mechanism

26.5 Discussion References

27. Quanta pair sources for quantum entanglement experiments

27.1 Introduction

27.2 Positron-electron annihilation

27.3 Atomic Ca emission

27.4 Type I parametric down conversion

27.5 Type II spontaneous parametric down conversion

27.6 Further horizons


28. More on quantum entanglement

28.1 Introduction

28.2 Consequences of the EPR paper

28.3 Hidden variable theories

28.4 The perspectives of EPR and Schördinger

28.5 Indistinguishability and Dirac’s identities

28.6 Photon non-locality

28.7 Discussion


29. On the interpretation of quantum mechanics

29.1 Introduction

29.2 Quantum critical

29.2.1 On “The moral aspects of quantum mechanics”

29.2.2 On “Against ‘measurement’’

29.3 Pragmatic perspective

29.4 Fundamental principles

29.5 The Dirac-Feynman-Lamb doctrine

29.6 The importance of the probability amplitude

29.7 The best interpretation of quantum mechanics

29.8 Discussion



A. Revisiting the Einstein Podosky Rosen (EPR) paper

A.1 Introduction

A.2 EPR and the Uncertainty Principle

A.3 Conclusion


B. Revisiting the Pryce-Ward probability amplitude

B.1 Introduction

B.2 Exciting times and extreme succinctness

B.3 Conclusion


C. Classical and quantum interference

C.1 Introduction

C.2 The classical interference equation

C.3 The N-slit quantum interference equations

C.4 The difference between classical and quantum interference


D. Interferometers and their probability amplitudes

D.1 Introduction

D.2 Interferometers

D.2.1 The Mach-Zehnder interferometer

D.2.2 The Michelson interferometer

D.2.3 The Sagnac interferometer

D.2.4 The N-slit interferometer

D.3 Beam splitter matrices


E. Polarization rotators

E.1 Introduction

E.2 Wave plates

E.3 Rhomboids and prismatic rotators


F. Vector products in quantum notation

F.1 Introduction

F.2 Vector products

F.2.1 Dot product

F.2.2 Cross product

F.2.3 Density matrix

F.2.4 Vector direct product

F.2.5 Vector outer product

F.2.6 Kronecker product or tensor product

F.3 Equivalence in vector notation for entangled polarizations


G. Trigonometric identities

G.1 Trigonometric identities

H. More on quantum notation

H.1 Introduction

H.2 Certainly not classical

H.3 Multiplication of probability amplitudes


I. From quantum principles to classical optics

I.1 Introduction

I.2 From quantum interference to generalized diffraction

I.3 From generalized diffraction to generalized refraction

I.4 From generalized refraction to reflection

I.5 From quantum interference to Heisenberg’s uncertainty principle

I.7 The cavity linewidth equation

I.6 Generalized multiple-prism dispersion

I.8 Discussion


J. Introduction to Hamilton’s quaternions

J.1 Introduction

J.2 Basic quaternion identities References

Fundamentals of Quantum Entanglement: corrigendum

Page published on the 12th of October, 2019

Updated on the 25th of November, 2019