Entangling without Entanglement T. Cubitt, F. Verstraete, W. Dür, J. I. Cirac “Separable State can be used to Distribute Entanglement” (to appear in PRL vol. 91, issue 3)

l Send separable ancilla particle l Send entangled ancilla particle l Local Operations & Classical Communication Entangling two distant particles  

C AB Define “separable”? l What does “separable” mean for the messenger? C AB = l Implies separability tracing out one particle: l Choose strongest possible meaning: B  C A l For mixed states: l For pure states:

Don’t Entangle the Messenger l Alice and Bob prepare initial state: l Bob applies CNOT to B and a: l Alice applies CNOT to A and a:

l If we think of rates of entanglement generation as ‘flows’… B a A How does entanglement ‘flow’? l Chain with nearest neighbour interactions BA a ? l …can entanglement ‘flow’ be 0 between & and &, yet be non-zero between and ? A B a a AB B ? l Can and be entangled without entangling the ancilla ? B A a

Physical relevance l Interactions are often mediated by an ‘ancilla’ particle l Ion traps: interactions between ions mediated by phonons l Cavity QED: interactions between atoms mediated by photons in the cavity l Fundamentally, all interactions are mediated by the gauge bosons of particle physics

l Evolution can be discretized by Trotter formula Continuous and discrete cases BA a l Immediately gives a discrete procedure. l Continuous case is stronger than discrete case. >

l So Pure states: impossibility proof l Start with separable state l Condition on separability of ancilla is then l Multiplying by gives l Evolve under for an infinitesimal time-step:Separable

l As achieve effect for e.g. initial state. l Expand in perturbation theory: Don’t entangle the mediator BA aTrivial? l Want ancilla to really be separable, not just arbitrarily small entanglement as

l After evolution ( ): Just add a dash of noise l Use mixed initial state: BA a l Choose  large enough to destroy all entanglement with. (States near maximally mixed state are separable). a Separable in ( )- a ABEntangled in ( )- a AB l Choose  small enough such that mixing does not destroy - entanglement. B A

Entanglement properties of partitions C AB C AB l For mixed states, partitions are independent C AB && A C B l For pure states, entanglement properties of bipartite partitions are inter-dependent C AB  C AB &

C A B C A B Theoretical insight l Alice and Bob prepare initial state: l Bob applies CNOT to B and a: l Alice applies CNOT to A and a: C A B

Conclusions “Wacky but Lovely” – Seth Lloyd l Upsets notions of entanglement flow l Forces us to abandon any intuitive ideas of entanglement being sent through a quantum channel l (At least for general – i.e. mixed – states…) l Separable states can be used to distribute entanglement