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)
Entangling two distant particles

Local Operations & Classical Communication 6

Send entangled ancilla particle 4

4
Send separable ancilla particle 6
Define “separable”?

What does “separable” mean for the messenger?

Choose strongest possible meaning:
C
A

=
B

For pure states:

For mixed states:
Implies separability tracing out one particle:
C
A
B

C
A
Don’t Entangle the Messenger



Alice and Bob
prepare
initial state:
Alice applies
CNOT to A
and a:
Bob applies
CNOT to B
and a:
1 3
 k ,   k ,0  k ,   k ,0

6 k 0
1
1
j , j ,1 j , j ,1
j 0 6

1
 GHZ  GHZ
1
3
  ijk  i, j , k
i , j ,k 0
i, j , k
 001   010  1011  1101 
1 
   0 a 0
3
2
 1AB  1 a 1
3
1
6
How does entanglement ‘flow’?


Chain with nearest
neighbour interactions
A
a
H Ba
B
Can A and B be entangled without entangling
the ancilla a ?
A

a
B

H Aa
If we think of rates of
entanglement
generation as ‘flows’…
…can entanglement ‘flow’ be 0 between A & a
and B & a , yet be non-zero between A and B ?
Physical relevance


Interactions are often mediated by an ‘ancilla’
particle
Ion traps: interactions between ions
mediated by phonons


Cavity QED: interactions between
atoms mediated by photons in the
cavity
Fundamentally, all interactions
are mediated by the gauge
bosons of particle physics
Continuous and discrete cases
>
H Aa
A
a
H Ba
B

Continuous case is stronger than discrete case.

Evolution can be discretized by Trotter formula
U  t   e  Aa Ba 
 itH Aa / 2 n  itH Ba / n  itH Aa / 2 n
 lim e
e
e
 it H
n
H


 U AaU BaU AaU Ba

Immediately gives a discrete procedure.
n
Pure states: impossibility proof



Start with separable state A a B
Evolve under H  H Aa  H Ba for an infinitesimal
time-step: U  t   1  i t  H Aa  H aB   O( t 2 )
Condition on separability of ancilla is then
1ABa  t H Aa  H aB  A a B
  A B  t  AB   a  t  a

Multiplying by



So  AB  A B   t A B   t A B

A B 
 A   t A
gives

A B   AB  0
B   t B


Don’t entangle the mediator
2
1
0
H Aa
1
0



A
a

 
H  aˆ  aˆ  1AB   aˆ  aˆ    xA   xB
H Ba
B
1
0
Expand in perturbation theory:
H  H 0   2 H AB  1a  O   4 
As   0 achieve effect for e.g. initial state
0 A 0 a 0 B.
Want ancilla to really be separable, not just
arbitrarily small entanglement as   0  t  

Just add a dash of noise


Use mixed initial state:
  00 00 1a   1ABa

A
a
H Ba
B
After evolution (H  H 0   2 H AB  1a  O   4 ):
  t    AB  AB  1a  O( 4t   )     1ABa
Separable in ( A B )- a

H Aa
Entangled in ( A B )- a
Choose  large enough to destroy all
entanglement with a . (States near maximally
mixed state are separable).
Choose  small enough such that mixing does
not destroy A - B entanglement.
Entanglement properties of partitions

For pure states, entanglement properties of
bipartite partitions are inter-dependent
C
A

C
&
B
A
B

C
A
B
For mixed states, partitions are independent
C
A
C
&
B
A
C
&
B
A
B
Theoretical insight



Alice and Bob
prepare
initial state:
Alice applies
CNOT to A
and a:
Bob applies
CNOT to B
and a:
C
A
B
C
A
B
C
B
A
Conclusions


“Wacky but Lovely” – Seth Lloyd
Separable states can be used to distribute
entanglement
Forces us to abandon any
intuitive ideas of entanglement
being sent through a quantum
channel


Upsets notions of
entanglement flow
(At least for general – i.e. mixed – states…)