School of Physics & Astronomy FACULTY OF MATHEMATICAL & PHYSICAL SCIENCE Parallel Transport & Entanglement Mark Williamson 1 , Vlatko Vedral 1 and William Wootters 2 1 School of Physics & Astronomy, University of Leeds, UK 2 Department of Physics, Williams College, USA [email protected]www.qi.leeds.ac.uk
Transcript
Slide 1
School of Physics & Astronomy FACULTY OF MATHEMATICAL &
PHYSICAL SCIENCE Parallel Transport & Entanglement Mark
Williamson 1, Vlatko Vedral 1 and William Wootters 2 1 School of
Physics & Astronomy, University of Leeds, UK 2 Department of
Physics, Williams College, USA [email protected]
www.qi.leeds.ac.uk
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Overview Ingredients: Parallel transport Geometric phase
Entanglement Idea/Analogy: Nonlocality and geometry Research :
State space curvature due to subsystem correlations Subsystem
correlations as a rule for parallel transport of observables
Conclusion
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School of Physics & Astronomy FACULTY OF MATHEMATICAL &
PHYSICAL SCIENCE Ingredients: Parallel transport
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Parallel Transport Parallel transport on a sphere
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School of Physics & Astronomy FACULTY OF MATHEMATICAL &
PHYSICAL SCIENCE Ingredients: Geometric phase An observable
resulting from parallel transport of the phase factor of the
wavefunction
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What is the geometric phase? M. V. Berry (1984), Proc. R. Soc.
392, 45-57. F. Wilczek & A. Zee (1984), Phys. Rev. Lett. 52,
2111. Geometric phase Dynamical phase
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What is the geometric phase? M. V. Berry (1984), Proc. R. Soc.
392, 45-57. F. Wilczek & A. Zee (1984), Phys. Rev. Lett. 52,
2111. Geometric phase Dynamical phase
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School of Physics & Astronomy FACULTY OF MATHEMATICAL &
PHYSICAL SCIENCE Ingredients: Introduction to entanglement
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Intro to entanglement Mutual information of two states: -
Entangled (maximal quantum correlations) - Separable (maximal
classical correlations)
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Intro to entanglement Mutual information of two states: -
Entangled (maximal quantum correlations) - Separable (maximal
classical correlations) Entanglement allows systems to be more
correlated.
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Intro to entanglement Entangled (quantum correlations)
Separable (classical correlations)
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Intro to entanglement Entangled (quantum correlations)
Separable (classical correlations)
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Intro to entanglement Entangled (quantum correlations)
Separable (classical correlations)
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Intro to entanglement Entangled (quantum correlations)
Separable (classical correlations)
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Intro to entanglement Entangled (quantum correlations)
Separable (classical correlations)
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Intro to entanglement Entangled (quantum correlations)
Separable (classical correlations)
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School of Physics & Astronomy FACULTY OF MATHEMATICAL &
PHYSICAL SCIENCE Idea/analogy: Nonlocality & geometry
Understanding nonlocality from parallel transport
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Nonlocality & Geometry in QM Aharonov-Bohm Effect Phase
shift is the geometric phase Y. Aharonov & D. Bohm, Phys. Rev.
115 485-491 (1959).
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R should be periodic with period R. A. Webb et al., Phys. Rev.
Lett. 54 (25), 2696 (1985). h/e h/2e
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Aharonov-Bohm topology Same phase picked up no matter what path
taken. Only need to encircle tip of cone (topological
property)
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School of Physics & Astronomy FACULTY OF MATHEMATICAL &
PHYSICAL SCIENCE Research: State space curvature due to subsystem
correlations Work with Vlatko Vedral
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Studying the Effect of Entanglement on Geometric Phase Aim:
Compare subsystem and composite state geometric phases under fixed
entanglement. Composite (pure) Subsystem (mixed) StateGeometric
phase Keep entanglement fixed by evolving states under local
unitaries
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Effect of Entanglement on Quantum Phase I Dynamical phase If
Dynamical phase of composite state always sum of subsystem
dynamical phases even if state entangled or not.
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Effect of Entanglement on Quantum Phase II Geometrical phase
Composite state geometric phase generally not sum of subsystem
geometric phases unless state product state:
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Effect of Entanglement on Quantum Phase II Geometrical phase Is
this pointing to a geometrical interpretation of correlations
(entanglement)? Difference missing correlations (classical and
quantum) make to GP and the curvature of the state space.
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GHZ & W States N qubit GHZ state GHZ example N=3 N qubit W
state W example N=3, k=1 State of each of N subsystems (labelled by
n) given by
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Properties of GHZ & W states GHZ If you loose just one
particle, state unentangled but still classically correlated. All N
particles are entangled, no entanglement between