Entanglement flow in multipartite systems

1. T. S. Cubitt F. Verstraete J.I. Cirac • 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

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

3. T. S. Cubitt F. Verstraete J.I. Cirac • 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

4. A One particle

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

6. T. S. Cubitt F. Verstraete J.I. Cirac • 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

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

8. Aside: entangling without entanglementT. S. Cubittet. al.,PRL 91, 037902 (2003)

9. For pure states A Hab Hbc • General • Qubits! B C Three particles: flow through

10. T. S. Cubitt F. Verstraete J.I. Cirac • 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

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

12. !Rate equations: set of coupled differential equations. HO O- C H3C H OH HO • Rate at which products are produced depends on the amounts of its immediate precursors that are present: H3C H3C C O C OH H3C H3C H3C …which in turn depend on the amounts of their precursors: etc. Remedial chemistry HO- -OH

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

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

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

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

17. Putting all this together, we arrive at: Entanglement rate equations (4) • 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 Hij that cross the boundary of A or B ):

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

19. T. S. Cubitt F. Verstraete J.I. Cirac • 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

20. 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? … Entanglement generation in chains Fbn/2c -1

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

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

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

24. T. S. Cubitt F. Verstraete J.I. Cirac • 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

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

26. The end!