Fundamentals of Quantum Entanglement

F. J. Duarte, Fundamentals of Quantum Entanglement, 2nd Edn (Institute of Physics, Bristol, 2022)

Print ISBN: 978-0-7503-5265-9

Online ISBN: 978-0-7503-5269-7


30 chapters, 10 appendices, 92 figures, 1113 equations, and 119 problems, in 308 pages.

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An absolutely exact measurement of momentum p is physically impossible... In this regard, the 'all values' spread in the coordinate, as feared by EPR, is not allowed... Hence, the EPR conclusion that 'the quantum mechanical description of physical reality... is not complete' can be dismissed.

Page 3-4

As noted previously Bell's theorem is completely disconnected from the physics leading to |ψ>+ , |ψ>- , |ψ>+ , and |ψ>- .

Page 10-6

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. The are no 'paradoxes' in the physics of quantum entanglement.

Page 17-5

Nature appears to be very efficient and there is indeed an optimum finesse determining the smallness necessary to gain full and complete representation of the interferometric reality.

Page 29-15

A theory that does not observe Born's rule is not a quantum mechanical theory... Thus, it is logical to conclude that the hidden variable theory introduced by Bohm is not a quantum mechanical theory.

Page 30-3

Contemporaneous criticisms of quantum mechanics fail to mention that, in praxis, there is no measurement problem in quantum mechanics... quantum measurements in the interferometric and polarization domains, can be described without resorting to the concept of the collapse of the wave function or the collapse of the probability amplitude.

Page 30-7

The main impact of Bell's theorem is from a philosophical-historical perspective: it reinforces, outside the physics of quantum entanglement, the incompatibility of hidden variable theories with quantum mechanics.

Page 30-8

From an interferometric perspective, there are no mysteries... and no paradoxes... in the physics of quantum entanglement.

Page 30-14

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Fundamentals of Quantum Entanglement, 2nd Edn: corrigenda

CONTENTS

1. Introduction

1.1 Introduction

1.2 Foundations of quantum mechanics

1.2.1 The mathematical bases of quantum mechanics

1.2.2 The photon from a quantum perspective

1.3 Ward’s observations

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

Problems

References

2. Dirac’s physics

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 Expanded series of N-slit quantum interference probabilities

2.6 The interferometric probability in 2D and 3D

2.7 Semi-coherent interference

2.8 From quantum probabilities to measurable intensities

2.9 Interferometric calculations and quantum coherence

2.10 Dirac’s identities

2.10.1 Indistinguishability identities

2.10.2 Extending the emission identities

Problems

References

3. The Einstein–Podolsky–Rosen (EPR) paper

3.1 Introduction

3.2 EPR’s doubts on quantum mechanics

3.2.1 EPR’s definition of a correct theory

3.3 Transparent resolution of the EPR ‘paradox’

3.3.1 EPR and the uncertainty principle

Problems

References

4. The Schrödinger papers

4.1 Introduction

4.2 The first Schrödinger paper

4.3 The second Schrödinger paper

References

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 5.4 A theoretical opportunity

References

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

Problems

References

7. The quantum entanglement experiment

7.1 Introduction

7.2 The quantum entanglement experiment

7.3 Historical notes

Problems

References

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

Problems

References

9. The Bohm and Aharonov paper

9.1 Introduction

9.2 Significance to the development of quantum entanglement research

9.3 Philosophy and physics

Problems

References

10. Bell’s theorem

10.1 Introduction

10.2 von Neumann’s

10.3 Bell’s theorem or Bell’s inequalities

10.4 Example

10.5 An additional perspective on Bell’s theorem

10.6 More philosophy and physics

Problems

References

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 Hyperfine splitting

11.5 Discussion

Problems

References

12. The second Wu quantum entanglement experiment

12.1 Introduction

12.2 Salient features

12.3 Bell’s theorem and hidden variables

References

13. The hidden variable theory experiments

13.1 Introduction

13.2 Testing for local hidden variable theories

13.3 Early optical experiment

13.4 Observations and discussion

References

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

Problems

References

15. The quantum entanglement probability amplitude 1947–1992

15.1 Introduction

15.2 The quantum entanglement probability amplitude 1947–1992

15.2.1 1947–1949

15.2.2 1948

15.2.3 1951

15.2.4 1957

15.2.5 1965

15.2.6 1975

15.2.7 1990

15.2.8 1992

15.3 Observations and discussion

Problems

References

16. The GHZ probability amplitudes

16.1 Introduction

16.2 The GHZ probability amplitudes

16.3 Observations and discussion

References

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 Beyond single quanta-pair quantum entanglement

17.6 Discussion

Problems

References

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, 24, … 2r

18.6 Discussion

Problems

References

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

Problems

References

20. Quantum 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

20.4.3 Derivation of the Dirac–Feynman principle

20.5 Interference for N slits and n = 1

Problems

References

21. Quantum entanglement probability amplitudes applied to Bell’s theorem

21.1 Introduction

21.2 Probability amplitudes

21.3 Quantum polarization

21.4 Quantum probabilities and Bell’s theorem

21.5 Application to Bell’s theorem

21.6 All-quantum approach

21.7 Discussion

Problems

References

22. Quantum entanglement via matrix notation

22.1 Introduction

22.2 The probability amplitudes of quantum entanglement

22.3 Dirac’s ket vectors and Pauli matrices

22.4 Quantum entanglement in Pauli matrix notation

22.4.1 Mechanics of Pauli matrices

22.5 Quantum entanglement and the Hadamard gate

22.6 Complete set of matrices derived from the probability amplitudes of quantum entanglement

22.7 Polarization rotators for quantum entanglement

22.8 Quantum mathematics with polarization rotators

22.9 Quantum mathematics with the Hadamard gate

22.10 Interconnectivity in quantum mechanics

Problems

References

23. Cryptography via quantum entanglement

23.1 Introduction

23.2 Measurement protocol based on Bell’s theorem

23.2.1 Experiments

23.3 All-quantum measurement protocol

Problems

References

24. Quantum entanglement and teleportation

24.1 Introduction

24.2 The mechanics of teleportation

24.3 Technology

Problems

References

25. Quantum entanglement and quantum computing

25.1 Introduction

25.2 Entropy

25.3 Qbits

25.4 Quantum entanglement and Pauli matrices

25.5 Pauli matrices and quantum entanglement

25.6 Quantum gates

25.6.1 Pauli gates

25.6.2 The Hadamard gate

25.7 The Hadamard matrix and quantum entanglement

25.8 Multiple entangled states

25.9 Technology

Problems

References

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

26.1 Introduction

26.2 Space-to-space configurations

26.3 Experiments

26.3.1 The space-to-earth experiment

26.3.2 The International Space Station experiment

26.4 Further horizons

Problems

References

27. Space-to-space quantum interferometric communications

27.1 Introduction

27.2 The generalized N-slit quantum interference equations

27.3 The generation and transmission of interferometric characters

27.4 The inherent quantum security mechanism

27.5 Discussion

Problems

References

28. Quanta pair sources for quantum entanglement experiments

28.1 Introduction

28.2 Positron–electron annihilation

28.3 Atomic Ca emission

28.4 Type I spontaneous parametric down-conversion

28.5 Type II spontaneous parametric down-conversion

28.6 Quantum description of parametric down-conversion

28.7 Alternative quantum pair sources

28.8 Further horizons

Problems

References

29. Quantum interferometric principles

29.1 Introduction

29.2 Fundamental principles of quantum mechanics

29.3 Nonlocality of the photon

29.4 Indistinguishability and Dirac’s identities

29.5 Quantum measurements

29.5.1 Probability amplitudes

29.5.2 Quantum probabilities

29.5.3 Quantum entanglement measurements

29.5.4 Quantum time and entropy

29.5.5 The quantum measurer

29.6 Quantum entanglement at the foundations of quantum mechanics

29.7 On the origin of the Dirac–Feynman principle

29.7.1 Optimum finesse

29.7.2 Further refinements

29.8 Discussion

Problems

References

30. On the interpretation of quantum mechanics

30.1 Introduction

30.2 Philosophical aspects of quantum entanglement

30.2.1 The perspectives of EPR and Schrödinger on quantum entanglement

30.2.2 Hidden variable theories

30.3 Quantum critical

30.3.1 On ‘The moral aspects of quantum mechanics’

30.3.2 On ‘Against measurement’

30.3.3 On Bell’s criticisms of quantum mechanics

30.4 Conceptual ‘problems’ in quantum mechanics

30.4.1 The ‘measurement problem’

30.4.2 Particle–wave duality

30.4.3 Quantum reality

30.4.4 Unnecessary concerns

30.5 Quantum luminaries

30.6 The pragmatic perspective

30.7 The Dirac–Feynman–Lamb doctrine

30.8 The all-important probability amplitude

30.9 The quantumness derived from the nonlocality of the photon

30.10 The best interpretation of quantum mechanics

30.11 Discussion

Problems

References

Appendices

A. Revisiting the Pryce–Ward probability amplitude for quantum entanglement

A.1 Introduction

A.2 Exciting times and extreme succinctness

A.3 Conclusion

References

B. Classical and quantum interference

B.1 Introduction

B.2 The classical interference equation

B.3 The N-slit quantum interference equations

B.4 From quantum interference to classical interference

Problems

References

C. Interferometers and their probability amplitudes

C.1 Introduction

C.2 Interferometers

C.2.1 The Mach-Zehnder interferometer

C.2.2 The Michelson interferometer

C.2.3 The Sagnac interferometer

C.2.4 The N-slit interferometer

C.3 Beam splitter matrices and Dirac’s bra ket notation

C.3.1 The beam splitter and the Hadamard gate

C.3.2 The Hanbury Brown-Twist interferometer

C.3.3 The HOM interferometer

Problems

References

D. Polarization rotators for quantum entanglement

D.1 Introduction

D.2 Wave plates

D.3 Rhomboid polarization rotators

D.4 Multiple-prism collinear polarization rotators

Problems

References

E. Vectors, vector products, matrices, and tensors for quantum entanglement

E.1 Introduction

E.2 Vector basics

E.3 Vector products

E.3.1 Dot product

E.3.2 Cross product

E.3.3 The ket bra product

E.3.4 Vector direct product

E.3.5 Vector outer product

E.4 Matrix algebra

E.4.1 The identity matrix

E.4.2 The inverse matrix

E.4.3 Matrix determinant and trace

E.4.4 Eigenvalues and eigenvectors

E.5 Unitary matrices

E.6 The tensor product

E.7 Equivalence in vector notation for entangled polarizations

Problems

References

F. Trigonometric identities

F.1 Trigonometric identities

Problems

References

G. More on quantum notation

G.1 Introduction

G.2 Certainly not classical

G.3 Multiplication of probability amplitudes

G.4 On the Dirac identity for quantum entanglement

References

H. From quantum principles to classical optics

H.1 Introduction

H.2 From quantum interference to generalized diffraction

H.3 From generalized diffraction to generalized refraction

H.4 From generalized refraction to reflection

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

H.6 The cavity linewidth equation

H.7 Generalized multiple-prism dispersion

H.8 Discussion

Problems

References

I. Introduction to complex conjugates and Hamilton’s quaternions

I.1 Introduction

I.2 Complex conjugates

I.3 Basic quaternion identities

Problems

References

J. Some open ended quantum questions

J.1 Introduction

J.2 Planck’s constant

J.3 The fine structure constant

J.4 The extreme weakness of gravity in the quantum domain

J.5 Quantum consciousness

J.6 Fundamental physics constants

Problems

References

Index



F. J. Duarte, Fundamentals of Quantum Entanglement, 1st Edn (Institute of Physics, Bristol, 2019)

ISBN: 978-0-7503-2226-3


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

Fundamentals of Quantum Entanglement (Institute of Physics, Bristol, 2019) at Amazon

Fundamentals of Quantum Entanglement (Institute of Physics, Bristol, 2019) at IoP

"(|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).

Page 1-3

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.

Page 17-5


Watch talk on Fundamentals of Quantum Entanglement in youtube


CONTENTS

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

References

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

References

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

References

4. The Schrödinger papers

4.1 Introduction

4.2 The first Schrödinger paper

4.3 The second Schrödinger paper

References

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

References

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

References

7. The quantum entanglement experiment

7.1 Introduction

7.2 The quantum entanglement experiment

7.3 Historical notes

References

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

References

9. The Bohm and Aharonov paper

9.1 Introduction

9.2 Significance to the development of quantum entanglement research

9.3 Philosophy and physics

References

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

References

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

References

12. The second Wu quantum entanglement experiment

12.1 Introduction

12.2 Salient features

12.3 Bell’s theorem and hidden variables

References

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

References

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

References

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

References

16. The GHZ probability amplitudes

16.1 Introduction

16.2 The GHZ probability amplitude

16.3 Observations and discussion

References

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

References

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

References

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

References

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

References

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

References

22. Cryptography via quantum entanglement

22.1 Introduction

22.2 Measurement protocol

22.3 Experimental

References

23. Quantum entanglement and teleportation

23.1 Introduction

23.2 The mechanics of teleportation

23.3 Technology

References

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

References

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

References

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

References

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

References

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

References

Appendices

A. Revisiting the Einstein Podosky Rosen (EPR) paper

A.1 Introduction

A.2 EPR and the Uncertainty Principle

A.3 Conclusion

References

B. Revisiting the Pryce-Ward probability amplitude

B.1 Introduction

B.2 Exciting times and extreme succinctness

B.3 Conclusion

References

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

References

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

References

E. Polarization rotators

E.1 Introduction

E.2 Wave plates

E.3 Rhomboids and prismatic rotators

References

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

References

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

References

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

References

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 1st of September, 2023