Quantum Information in Gravitational Fields 1st Edition by Marco Lanzagorta 1627053298 978-1627053297

1 Introduction

1.1 Quantum information

1.2 Quantum communications

1.3 Quantum computing

1.4 Quantum sensors

1.5 Relativistic quantum information

1.6 Summary

2 Special and general relativity

2.1 Special relativity

2.2 Lorentz transformations

2.3 Lagrangian dynamics

2.4 The principle of equivalence

2.4.1 Particle dynamics in a gravitational field

2.4.2 Torsion

2.4.3 Geodesics and geodesic congruences

2.5 The principle of general covariance

2.5.1 Tensor analysis

2.5.2 Covariant derivatives

2.5.3 The coordinate basis

2.5.4 The minimal substitution rule

2.5.5 The energy–momentum tensor

2.5.6 The Euler–Lagrange equations

2.6 The Hamilton–Jacobi equations

2.7 Einstein’s field equations

2.8 Principles of conservation

2.9 Killing vectors

2.10 Tetrad fields

2.11 Spin in general relativity

2.12 The spin–curvature coupling

2.13 Summary

3 Relativistic quantum fields

3.1 The Schrödinger equation

3.2 The Klein–Gordon equation

3.3 Scalar quantum fields

3.4 The quantum Poincare transformations

3.5 Wigner rotations

3.5.1 Massive particles

3.5.2 Massless particles

3.6 The Dirac equation

3.6.1 SO(3) and SU(2)

3.6.2 SL(2,ℂ) and SO+(3,1)

3.6.3 Four-spinors

3.6.4 Particle dynamics

3.6.5 Free particle spinors

3.6.6 Spin and helicity

3.7 Dirac quantum fields

3.8 Group representations in quantum field theory

3.8.1 Non-unitary representations of the Lorentz group and quantum fields

3.8.2 Unitary representations of the Poincare group and quantum states

3.8.3 Unitary/non-unitary representations and wave functions

3.9 Representations of quantum fields with arbitrary spin

3.9.1 Scalar fields

3.9.2 Vector fields

3.9.3 Tensor fields

3.9.4 Dirac fields

3.10 The quantum vacuum in flat spacetime

3.11 Summary

4 Quantum information in inertial frames

4.1 Qubit transformations

4.2 Relativistic dynamics

4.3 Steganographic quantum channel

4.3.1 Relativistic communications

4.3.2 Relativistic fixed points

4.4 The teleportation channel

4.4.1 Relativistic teleportation

4.4.2 Absence of relativistic fixed points

4.5 Spread momentum states

4.6 Summary

5 Quantum fields in curved spacetimes

5.1 Scalar fields in curved spacetime

5.2 Quantum dynamics in general relativity

5.2.1 The plane wave approximation

5.2.2 Hilbert spaces

5.2.3 Scalar orbital angular momentum eigenstates

5.2.4 Scalar four-momentum eigenstates

5.3 The quantum vacuum in a gravitational field

5.4 The spin-statistics connection

5.5 Quantum vector fields in curved spacetime

5.6 Spinors in curved spacetimes

5.7 Covariant derivative for fields of arbitrary spin

5.7.1 General form

5.7.2 Scalar fields

5.7.3 Vector fields

5.7.4 Tetrad fields

5.7.5 Dirac fields

5.8 Spinor dynamics with tetrad fields

5.9 Dirac spinors in curved spacetime

5.9.1 Solution at order ℏ0

5.9.2 Geodesic deviation at order ℏ1

5.10 The spin–curvature coupling

5.11 Summary

6 Qubits in Schwarzschild spacetime

6.1 Metric tensor

6.2 Structure of Schwarzschild spacetime

6.3 Tetrad fields and connection one-forms

6.3.1 Affine connections

6.3.2 Curvature tensor

6.3.3 Tetrad fields

6.3.4 Connection one-forms

6.4 Geodesics

6.5 Quantum dynamics

6.6 Wigner rotations

6.6.1 Equatorial radial fall (θ = π/2, J = 0)

6.6.2 Equatorial circular orbits (θ = π/2, ur = 0)

6.6.3 General equatorial circular paths (θ = π/2, ar ≠ 0)

6.6.4 Geodetic precession of classical gyroscopes

6.7 Radiation damping

6.8 Summary

7 Spin–curvature coupling in Schwarzschild spacetime

7.1 Spinor components

7.2 Constant spinors in spherical coordinates

7.3 The Lemaitre tetrad

7.3.1 Tetrad fields

7.3.2 Connection one-forms

7.3.3 The spinor connection

7.4 Radial fall

7.5 The co-moving tetrad field

7.6 General motion in the equatorial plane

7.7 Circular orbits

7.8 Non-geodetic motion

7.9 Wigner rotations

7.10 Summary

8 Qubits in Kerr spacetime

8.1 The metric tensor

8.2 Structure of Kerr spacetime

8.3 Tetrad fields and connection one-forms

8.3.1 Affine connection

8.3.2 Tetrad fields

8.3.3 Connection one-forms

8.4 Geodesics

8.5 Quantum dynamics

8.6 Wigner rotations

8.6.1 Equatorial radial fall (θ = π/2, J = 0)

8.6.2 Equatorial circular orbits (θ = π/2, ur = 0)

8.7 Summary

9 Quantum information processing in curved spacetimes

9.1 Unitary qubit transformations

9.2 Qubit states

9.3 Quantum communications

9.4 Quantum teleportation

9.5 EPR experiments

9.6 Quantum computation

9.6.1 General iterative algorithm

9.6.2 Grover’s algorithm

9.6.3 Shor’s algorithm

9.7 Gravity-induced entanglement

9.8 Quantum sensing and gravimetry

9.8.1 Interferometric quantum gravimetry

9.8.2 Algorithmic quantum gravimetry

9.9 Summary

10 Conclusions

10.1 Classical information in classical gravitational fields

10.2 Quantum information in classical gravitational fields

10.3 Quantum information in quantum gravitational fields

10.4 Summary