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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 mark.williamson@quantuminfo.org 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