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

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

School of Physics & Astronomy FACULTY OF MATHEMATICAL & PHYSICAL SCIENCE Ingredients: Parallel transport

Parallel Transport Parallel transport on a sphere

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

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

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

School of Physics & Astronomy FACULTY OF MATHEMATICAL & PHYSICAL SCIENCE Ingredients: Introduction to entanglement

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

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

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

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

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

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

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

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

School of Physics & Astronomy FACULTY OF MATHEMATICAL & PHYSICAL SCIENCE Idea/analogy: Nonlocality & geometry Understanding nonlocality from parallel transport

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

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

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

School of Physics & Astronomy FACULTY OF MATHEMATICAL & PHYSICAL SCIENCE Research: State space curvature due to subsystem correlations Work with Vlatko Vedral

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

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.

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

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.

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

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.

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/ How to determine which correlations are responsible? Use ideas of distance entanglement measures…

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

School of Physics & Astronomy FACULTY OF MATHEMATICAL & PHYSICAL SCIENCE Research: Subsystem correlations as a rule for parallel transport of an observable Work with Bill Wootters

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

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

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

3 qubit states: GHZ & W states GHZ – All entanglement in t abc, no pairwise entanglement W – All entanglement in t ab, t ac and t bc, no 3 particle entanglement. Inequivalent forms of entanglement under stochastic local operations and classical communication (SLOCC)

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

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

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

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

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 Conditions on matrix, M a : Hermitian i.e. an observable Fixed length Initial observableParallel transported observable

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

Paths in t basis Spin flip operation performed on white qubit

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)

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.

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.

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

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

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.

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

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.

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?

Measuring geometric phase Acts on (pure) path state Acts on (mixed) internal state internal state path state

Measuring geometric phase Acts on (pure) path state Acts on (mixed) internal state internal state path state

What is Geometric Phase? Mixed state geometric phase E. Sjoqvist et al. (2000), Phys. Rev. Lett. 85, 2845.