Parallel Transport & Entanglement

1. Parallel Transport & Entanglement Mark Williamson1, Vlatko Vedral1 and William Wootters2 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

2. 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

3. Ingredients:Parallel transport

4. Parallel Transport Parallel transport on a sphere

5. Ingredients:Geometric phase An observable resulting from parallel transport of the phase factor of the wavefunction

6. What is the geometric phase? What is the geometric phase? Geometric phase Dynamical phase M. V. Berry (1984), Proc. R. Soc. 392, 45-57. F. Wilczek & A. Zee (1984), Phys. Rev. Lett. 52, 2111.

7. What is the geometric phase? Geometric phase Dynamical phase M. V. Berry (1984), Proc. R. Soc. 392, 45-57. F. Wilczek & A. Zee (1984), Phys. Rev. Lett. 52, 2111.

8. Ingredients:Introduction to entanglement

9. Intro to entanglement • Mutual information of two states: • - Entangled (maximal quantum correlations) • - Separable (maximal classical correlations)

10. Intro to entanglement • Mutual information of two states: • - Entangled (maximal quantum correlations) • - Separable (maximal classical correlations) Entanglement allows systems to be more correlated.

11. Intro to entanglement Entangled (quantum correlations) Separable (classical correlations)

12. Intro to entanglement Entangled (quantum correlations) Separable (classical correlations)

13. Intro to entanglement Entangled (quantum correlations) Separable (classical correlations)

14. Intro to entanglement Entangled (quantum correlations) Separable (classical correlations)

15. Intro to entanglement Entangled (quantum correlations) Separable (classical correlations)

16. Intro to entanglement Entangled (quantum correlations) Separable (classical correlations)

17. Idea/analogy:Nonlocality & geometry Understanding nonlocality from parallel transport

18. Nonlocality & Geometry in QM Aharonov-Bohm Effect Phase shift is the geometric phase Y. Aharonov & D. Bohm, Phys. Rev. 115 485-491 (1959).

19. h/e R should be periodic with period h/2e R. A. Webb et al., Phys. Rev. Lett. 54 (25), 2696 (1985).

20. Aharonov-Bohm topology Same phase picked up no matter what path taken. Only need to encircle tip of cone (topological property)

21. Research:State space curvature due to subsystem correlations Work with Vlatko Vedral

22. Studying the Effect of Entanglement on Geometric Phase Aim: Compare subsystem and composite state geometric phases under fixed entanglement. Keep entanglement fixed by evolving states under local unitaries Composite (pure) Subsystem (mixed) State Geometric phase

23. 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.

24. Effect of Entanglement on Quantum Phase II • Geometrical phase Composite state geometric phase generally not sum of subsystem geometric phases unless state product state:

25. 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.

26. 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

27. 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 <N particles. • W – Very robust to loss of particles. All particles maximally pairwise entangled. No multiparty entanglement.

28. Characterising correction term Difference missing correlations make to the curvature of the state space GHZ states – Classical correlations entirely responsible for the difference. W states – Quantum correlations (aka entanglement) entirely responsible for the difference. See MW & Vedral, quant-ph/0702080 How to determine which correlations are responsible? Use ideas of distance entanglement measures…

29. Entanglement distance measures Vedral et al. (1997) PRL 78, 2275.

30. Research:Subsystem correlations as a rule for parallel transport of an observable Work with Bill Wootters

31. Nonlocal invariants • Properties that are unaltered with local transformations. • 2 qubits: One nonlocal invariant

32. Nonlocal invariants • Properties that are unaltered with local transformations. • 3 qubits: Five nonlocal invariants Kempe invariant

33. Linear combinations of nonlocal invariants A. Sudbery (2001), J. Phys. A 34,643

34. 3 qubit states: GHZ & W states • GHZ – All entanglement in tabc, no pairwise entanglement • W – All entanglement in tab, tac and tbc, no 3 particle entanglement. Inequivalent forms of entanglement under stochastic local operations and classical communication (SLOCC)

35. Basic idea: Entanglement & twisting • correlations provide a notion of twisting (Bill Wootters idea/intuition, also see an earlier paper, ref below) • use the correlations present between subsystems to define your rule for parallel transport between subsystems • parallel transport an arbitrary matrix (could represent an observable) • non-trivial parallel transport in a system of correlated qubits means correlations twisted? W. K. Wootters (2002), J. Math. Phys. 43(9): 4307

36. Nonlocal Invariants From Parallel Transport Transforms observable from one qubit to another (parallel transporter)

37. Nonlocal Invariants From Parallel Transport Transforms observable from one qubit to another (parallel transporter) Kempe invariant = tr R(A’,A) = tr {R(A’,C)R(C,B)R(B,A)} Can be interpreted as the average fidelity between initial and parallel transported observables.

38. General M to parallel transport Transformations (parallel transporters) to move M around the qubits Stretch M into 4 vector Can also write as a linear map acting on M Kempe invariant

39. Interpretation of mathematics/ What does Kempe mean? • Can be interpreted as the fidelity between initial and parallel transported observables averaged over all observables having a fixed length. • Mathematically Initial observable Parallel transported observable • Conditions on matrix, Ma: • Hermitian i.e. an observable • Fixed length

40. Other paths Using spin flip operation on parallel transporters we can obtain ‘conjugate’ basis of invariants, the t quantities.

41. Paths in t basis Spin flip operation performed on white qubit

42. Properties of parallel transporters • Polar decomposition: Hermitian positive semi-definte Unitary Properties of parallel transporters: Paths that enclose no area: (no unitary, just shrinking, P) Paths that enclose area: (shrinking and unitary)

43. Invariants associated to t are just shrinking (only P matrices) • P matrices are hermitian positive semi-definite • Deforms the space of all equal length vectors, a hypersphere, into a hyperellipse • Deformation related to the eigenvalues of P • Kempe invariant also has unitary as well as shrinking • Unitaries can represent rotations and reflections • Kempe invariant has deformation of hypersphere into hyperellipse and a reflection or rotation of it.

44. Form of unitaries • Each link has a unitary associated with it. • Only unique if R invertible • Turns out R only invertible if the subsystems joined by the link are entangled (only W state has unique U for Kempe). • Unitaries associated with links turn out to be orthogonal matrices composed of reflection and rotation.

45. Cyclic product of unitaries • Form cyclic product of SO(2) rotations Entanglement cost of twisting? Total rotation plotted against average residual entanglement

46. Similarities? • Geometric phase (particularly Pancharatnam form) • Wilson loops in lattice gauge field theory

47. Nonlocal invariants • Nonlocal invariants – things about the state that are invariant under local transformations, for example entanglement. • 2 qubits in pure state – 1 nonlocal invariant • 3 qubits in pure state – 5 nonlocal invariants (algebraically independent) • 4 qubits in pure state – 18 nonlocal invariants • 5 qubits in pure state <= 58 nonlocal invariants J. Kempe (1999), PRA 60, 910.

48. 4 and 5 qubits Generalisations to any number of quDits? Can understand why number of nonlocal invariants explodes with number of particles and have an understanding of what they mean using path invariant formalism.

49. Conclusion • Geometric phases appear due to curvature of underlying space. • Geometric phases for entangled systems are not equal to the sum of their parts – correlations change state space curvature. • Only classical correlations change the state space geometry for a GHZ state, only entanglement changes the geometry for a W state. • Nonlocal effects (Ahranov-Bohm) can be interpreted as geometrical phases. • Could entanglement be a(n unmeasurable) gauge dependent quantity like the electromagnetic vector potential? Gauge field theory of entanglement? • Using correlations between subsystems as rule for parallel transport we can understand and obtain nonlocal invariants. • For paths enclosing area there is twisting in correlations very much like Pancharatnam. • Essence of Kempe invariant seems to be how twisted the entanglement is.

50. Mixed state geometric phase Pure states are equivalent if they belong to the same ray (equivalent up to a global phase) Mixed states belong to projective Hilbert space (no global phase) How to define a geometric phase?