1. Motivation and goals One particle, two particles: previous work Three particles: flow through particles Many particles: flow along networks Application: entanglement generation in chains Conclusions and open questions Entanglement flow in multipartite systems T. S. CubittF. VerstraeteJ.I. Cirac

2. Entanglement flow: motivation How do the entanglement dynamics depend on the entanglement in the system? Doesn’t help us understand entanglement dynamics. …entanglement rate ( ) is non-zero. If certain particles are entangled… If nothing is entangled… …entanglement rate ( ) is 0. H SWAP

3. Motivation and goals One particle, two particles: previous work Three particles: flow through particles Many particles: flow along networks Application: entanglement generation in chains Conclusions and open questions Entanglement flow in multipartite systems T. S. CubittF. VerstraeteJ.I. Cirac

4. One particle A

5. Two qubits Entanglement rate neatly splits into separate entanglement- and interaction-dependent parts: f only involves entanglement-related quantities, with interaction details absorbed into coefficient h. W. Dür et. al., PRL 87, 137901 (2001) AB H Entanglement flow Entanglement capability of interactions

6. Motivation and goals One particle, two particles: previous work Three particles: flow through particles Many particles: flow along networks Application: entanglement generation in chains Conclusions and open questions Entanglement flow in multipartite systems T. S. CubittF. VerstraeteJ.I. Cirac

7. Three particles: flow through Two particles: only dynamics is entanglement creation. Tripartite systems already hold more possibilities: C AB H ab H bc How does entanglement flow through … B …to get from to ? A C Entanglement doesn’t have to flow through at all! B Starting from a completely separable mixed state, and can become highly entangled without itself ever becoming entangled. B C A Is there such thing as “flow” of entanglement through ? B

8. Aside: entangling without entanglement Aside: entangling without entanglement T. S. Cubitt et. al., PRL 91, 037902 (2003)

9. General Three particles: flow through C AB H ab H bc Qubits! For pure states

10. Motivation and goals One particle, two particles: previous work Three particles: flow through particles Many particles: flow along networks Application: entanglement generation in chains Conclusions and open questions Entanglement flow in multipartite systems T. S. CubittF. VerstraeteJ.I. Cirac

11. Many particles: flow along A B C Interesting dynamics hidden inside subsystems

12. Remedial chemistry C H3CH3C OH H3CH3C HO Rate at which products are produced depends on the amounts of its immediate precursors that are present: C H3CH3C O-O- H3CH3C HO - OH OHH HO - C H3CH3C O H3CH3C …which in turn depend on the amounts of their precursors: etc. ! Rate equations: set of coupled differential equations.

13. Many particles: flow along Can we derive something similar for entanglement? Maybe rate of entanglement generation between two particles… depends on the entanglement between particles further back along the network. And the rate for those … would depend on the entanglement between particles still further back along the network. B’ A’ B A

14. Entanglement rate equations (1) Uhlmann’s theorem: Density matrix evolves as: Use it to re-express F AB (t) : Use Uhlmann again to re-express F AB (t+  t) :

15. Entanglement rate equations (2) Same relations show that only Hamiltonians “crossing the boundary” of A or B give first-order contributions. Unitaries and state maximizing the expressions don’t change to first-order in  t : First expression for time derivative:

16. Entanglement rate equations (3) Need to re-express terms of singlet fractions. Prove linear algebra Lemma: Using this, with, we have and where if i is in A, we define A’ i =A [ i and B’ i =B, etc.

17. Entanglement rate equations (4) Putting all this together, we arrive at: This is actually a slightly stronger result than stated before, since (from A’ i 2 A’ etc.) Thus we arrive at the stated result (recall that the sum is only over those interactions H ij that cross the boundary of A or B ):

18. Many particles: flow along l Entanglement flow along any network is equivalent to entanglement flow along a chain. l If interaction strengths in chain are set appropriately, we get the same entanglement flow equations. B’ A’ B A b a

19. Motivation and goals One particle, two particles: previous work Three particles: flow through particles Many particles: flow along networks Application: entanglement generation in chains Conclusions and open questions Entanglement flow in multipartite systems T. S. CubittF. VerstraeteJ.I. Cirac

20. Entanglement generation in chains As an example application, look at entanglement generation in qubit chains. How long does it take to entangle end qubits? In particular, how does this time scale with the length of the chain? … F b n/2 c -1

21. Entanglement generation in chains What do the curves F k (t) that saturate the rate equations look like? Time t Generalized singlet fractions F k (t)

22. Entanglement generation in chains End qubits in a chain of length n are maximally entangled when … n

23. Entanglement generation in chains Can’t solve rate equations analytically, but can bound their solutions: Chain length n Time to entangle ends T ent

24. Motivation and goals One particle, two particles: previous work Three particles: flow through particles Many particles: flow along networks Application: entanglement generation in chains Conclusions and open questions Entanglement flow in multipartite systems T. S. CubittF. VerstraeteJ.I. Cirac

25. Conclusions and open questions We have established a quantitative concept of entanglement flow: flow through individual particles flow along general networks of interacting particles As an example application, derived a square-root lower bound on entanglement generation. Open questions: How tight are the inequalities in the entanglement rate equations? Can the square-root bound be saturated? Easily extended to higher dimensions and multipartite entanglement.

26. The end!