Entanglement flow
in multipartite systems
T. S. Cubitt
• Motivation
• One
F. Verstraete
J.I. Cirac
and goals
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: motivation
•
If nothing
certain particles
is entangled…
are entangled…
HSWAP
•
•
…entanglement
rate ( on
) isthe
0.
non-zero.
How do the entanglement
dynamics depend
Doesn’t help us understand entanglement dynamics.
entanglement in the system?
Entanglement flow
in multipartite systems
T. S. Cubitt
• Motivation
• One
F. Verstraete
J.I. Cirac
and goals
particle, two particles: previous work
• Three
particles: flow through particles
• Many
particles: flow along networks
• Application:
entanglement generation in chains
• Conclusions
and open questions
One particle
A
Two qubits
W. Dür et. al., PRL 87, 137901 (2001)
H
A
B
•
Entanglement
capability of
interactions
•
Entanglement flow
•
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
in multipartite systems
T. S. Cubitt
• Motivation
• One
F. Verstraete
J.I. Cirac
and goals
particle, two particles: previous work
• Three
particles: flow through particles
• Many
particles: flow along networks
• Application:
entanglement generation in chains
• Conclusions
and open questions
Three particles: flow through
•
Two
particles: doesn’t
only dynamics
entanglement
Entanglement
have toisflow
through Bcreation.
at all!
Tripartite systems already hold more possibilities:
Starting from a completely separable mixed state,
and
C can become
How A
does
entanglement
flowhighly entangled without B itself ever
becoming
entangled.
through
B …
•
Hab
A
•
Hbc
B
C
Is there such thing as “flow” of
entanglement through B ?
…to get from A to C ?
Aside: entangling without entanglement
T. S. Cubitt et. al., PRL 91, 037902 (2003)
Three particles: flow through
•
Qubits!
General
Hab
A
Hbc
B
•
For pure states
C
Entanglement flow
in multipartite systems
T. S. Cubitt
• Motivation
• One
F. Verstraete
J.I. Cirac
and goals
particle, two particles: previous work
• Three
particles: flow through particles
• Many
particles: flow along networks
• Application:
entanglement generation in chains
• Conclusions
and open questions
Many particles: flow along
A
C
B
•
Interesting dynamics hidden inside subsystems
Remedial chemistry
HOH3C
•
HO
C
H3C
H
O
H3C
H3C
C
OH
O-
HO
H3C
C
H3 C
Rate at which products are produced depends on the amounts of
its immediate precursors that are present:
…which in turn depend on the amounts of their precursors:
etc.
•
-OH
! Rate equations: set of coupled differential equations.
OH
Many particles: flow along
•
Can we derive something similar for entanglement?
B
A
A’
B’
•
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.
Entanglement rate equations (1)
•
Uhlmann’s theorem:
•
Use it to re-express FAB(t) :
•
Density matrix evolves as:
•
Use Uhlmann again to re-express FAB(t+ t) :
Entanglement rate equations (2)
•
Unitaries and state maximizing the expressions don’t change to
first-order in t :
•
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 (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.
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 Hij that cross the boundary of A or B ):
Many particles: flow along
a
A


B’
b
B
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.
Entanglement flow
in multipartite systems
T. S. Cubitt
• Motivation
• One
F. Verstraete
J.I. Cirac
and goals
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 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?
…
Fbn/2c -1
Entanglement generation in chains
What do the curves Fk(t) that saturate the rate equations look
like?
Generalized singlet fractions Fk(t)
•
Time t
Time t
Entanglement generation in chains
•
End qubits in a chain of length n are maximally entangled
when
n
…
Entanglement generation in chains
Can’t solve rate equations analytically, but can bound their
solutions:
Time to entangle ends Tent
•
Chain length n
Entanglement flow
in multipartite systems
T. S. Cubitt
• Motivation
• One
F. Verstraete
J.I. Cirac
and goals
particle, two particles: previous work
• Three
particles: flow through particles
• Many
particles: flow along networks
• Application:
entanglement generation in chains
• Conclusions
and open questions
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.
•
Easily extended to higher dimensions and multipartite
entanglement.
Open questions:
• How tight are the inequalities in the entanglement rate
equations?
• Can the square-root bound be saturated?
The end!