Transcript Entanglement Flow in Multipartite Systems

Engineering correlation and
entanglement dynamics in spin chains
T. S. Cubitt
J.I. Cirac
Engineering correlation and
entanglement dynamics in spin chains
T. S. Cubitt
• Motivation
• Time
J.I. Cirac
and goals
evolution of a chain
• Correlation
and entanglement wave-packets
• Engineering
• Fermionic
the dynamics: solitons etc.
gaussian state formalism
• Conclusions
Conceptual motivation: new experiments
•
•
•
Many papers on correlations/entanglement of ground states
Fewer on time-dependent behaviour away from equilibrium
New experiments…
•
…motivate new theoretical studies of non-equilibrium
behaviour.
Existing results
•
•
Many papers on correlations/entanglement of ground states.
Fewer on time-dependent behaviour away from equilibrium
•
•
In Phys. Rev. A 71, 052308 (2005), we used our
entanglement rate equations to bound the time taken to
entangle the ends of a length L chain.
Left open question of whether our pL lower bound is tight.
•
Bravyi, Hastings and Verstraete (quant-ph/0603121)
recently used Lieb-Robinson to prove tighter, linear bound.
•
In J.Stat.Mech. 0504 (2005) p.010, Calabrese and Cardy
investigated the time-evolution of block-entropy in spin
chains.
Practical motivation: quantum repeaters
•
“Traditional” solution to entanglement distribution:
build a quantum repeater.
•
But a real quantum repeater involves particle interactions, e.g.
atoms in cavities.
Alternative (e.g. Popp et al., Phys. Rev. A 71, 042306 (2005)): use
entanglement in ground state:
•
•
•
Getting to the ground state may be unrealistic.
Why not use non-equilibrium dynamics to distribute
entanglement?
Engineering correlation and
entanglement dynamics in spin chains
T. S. Cubitt
• Motivation
• Time
J.I. Cirac
and goals
evolution of a chain
• Correlation
and entanglement wave-packets
• Engineering
• Fermionic
the dynamics: solitons etc.
gaussian state formalism
• Conclusions
Time evolution of a spin chain (1)
•
As an exactly-solvable example, we take the XY model…
…and start it in some separable state, e.g. all spins +.
anisotropy
magnetic field
Time evolution of a spin chain (2)
•
Solved by a well-known sequence of transformations:
•
Jordan-Wigner:
•
Fourier:
•
Bogoliubov:
Time evolution of a spin chain (3)
•
Initial state N|+i…
…is vacuum of the cl=zj-l operators.
•
Wick’s theorem: all correlation functions hxm…pni of the ground
state of a free-fermion theory can be re-expressed in terms of
two-point correlation functions.
•
Our initial state is a fermionic Gaussian state: it is fully specified
by its covariance matrix:
Time evolution of a spin chain (4)
•
Hamiltonian…
…is quadratic in x and p.
•
From Heisenberg equations, can show that time evolution under
any quadratic Hamiltonian:
Gaussian state stays gaussian under
gaussian evolution.
corresponds to an orthogonal transformation of the covariance
matrix:
Time evolution of a spin chain (5)
•
Initial state is a fermionic gaussian state in xl and pl.
Connected by Fourier and
Bogoliubov transformations
•
Time-evolution is a fermionic gaussian operation in xk and pk.
•
Fourier and Bogoliubov transformations are gaussian:
Time evolution of a spin chain (phew!)
•
Putting everything together:
xxxkkktime-evolve
,initial
,,pppk k

state
xxxkkl ,,,pppkl
Engineering correlation and
entanglement dynamics in spin chains
T. S. Cubitt
• Motivation
• Time
J.I. Cirac
and goals
evolution of a chain
• Correlation
and entanglement wave-packets
• Engineering
• Fermionic
the dynamics: solitons etc.
gaussian state formalism
• Conclusions
String correlations
•
We can get string correlations hanzlbmi for free…
•
Given directly by covariance matrix elements, e.g.:
•
Equations are very familiar: wave-packets with envelope S
propagating according to dispersion relation ().
Two-point correlations
•
Two-point connected correlation functions hznzmi - hznihzmi can
also be obtained from the covariance matrix.
•
Using Wick’s theorem:
•
Again get wave-packets (3 of them) propagating according to
dispersion relation (k).
What about entanglement?
•
The relevant measure for entanglement distribution (e.g. in
quantum repeaters) is the localisable entanglement (LE).
•
Definition: maximum entanglement between two sites (spins)
attainable by LOCC on all other sites, averaged over measurement
outcomes.
•
As with all operationally defined entanglement measures, LE is
difficult to calculate in practice.
Best we can hope for is a lower bound.
•
•
Popp et al., Phys. Rev. A 71, 042306 (2005) : LE is lowerbounded by any two-point connected correlation function.
•
In case you missed it: we’ve just calculated this!
Engineering correlation and
entanglement dynamics in spin chains
T. S. Cubitt
• Motivation
• Time
J.I. Cirac
and goals
evolution of a chain
• Correlation
and entanglement wave-packets
• Engineering
• Fermionic
the dynamics: solitons etc.
gaussian state formalism
• Conclusions
Correlation wave-packets
•
•
•
•
•
•
In some parameter regimes: broad wave-packets and a highly
non-linear dispersion relation
! correlations rapidly disperse and disappear
In other parameter regimes: narrower wave-packets in nearly
linear regions of dispersion relation
! packets maintain their coherence as they propagate
In particular, around =1.1, =2.0 all three wave-packets in the
two-point correlation equations are nearly identical
! localised packets propagate at well-defined velocity with
minimal dispersion: “soliton-like” behaviour
Correlation wave-packets (1)
•
The system parameters  and  simultaneously control both the
dispersion relation and form of the wave-packets.
•
In some parameter regimes: broad wave-packets and a highly
non-linear dispersion relation
! correlations rapidly disperse and disappear: (=10, =2)
•
Correlation/entanglement solitons
•
•
In other parameter regimes, all three wave-packets in the twopoint correlation equations are nearly identical
! localised packets propagate at well-defined velocity with
minimal dispersion: “soliton-like” behaviour: (=1.1, =2)
Controlling the soliton velocity (1)
•
If the parameters are changed with time,
•
In general, time-ordering is essential.
But if parameters change slowly, dropping it gives reasonable
approximation.
•
•
•
•
! Effective evolution under time-averaged Hamiltonian.
If we stay within “soliton” regime, adjusting the parameters only
changes gradient of the dispersion relation, without significantly
affecting its curvature.
! Can speed up and slow down the “solitons”.
Controlling the soliton velocity (2)
•
Starting from =1.1, =2.0 and changing at rate +0.01:
“Quenching” correlations (1)
•
What happens if we do the opposite: rapidly change parameters
from one regime to another?
•
Can calculate this analytically using same machinery as before.
Resulting equations are uglier, but still separate into terms
describing multiple wave-packets propagating and interfering.
•
Get four types of behaviour for the wave-packets:
• Evolution according to 1 for t1, then 2
• Evolution according to 1 for t1, then -2
• Evolution according to 1 until t1, no evolution thereafter
• Evolution according to 2 starting at t1
•
Choose parameters so that “frozen” packets remain relatively
coherent, whilst others rapidly disperse.
“Quenching” correlations (2)
•
•
•
Choose parameters so that “frozen” packets remain relatively
coherent, whilst others rapidly disperse.
! can move correlations/entanglement to desired location, then
“quench” system to freeze it there.
E.g. =0.9, =0.5 changed to =0.1, =10.0 at t1=20.0:
Engineering correlation and
entanglement dynamics in spin chains
T. S. Cubitt
• Motivation
• Time
J.I. Cirac
and goals
evolution of a chain
• Correlation
and entanglement wave-packets
• Engineering
• Fermionic
the dynamics: solitons etc.
gaussian state formalism
• Conclusions
What about entanglement? (2)
•
There is another LE bound we can calculate…
•
Recall concurrence:
•
Localisable concurrence:
•
Not a covariance matrix element because of |*i.
Fermionic gaussian formalism
•
Recap of gaussian state formalism…
Bosonic case
Fermionic case
•
Operators a, ay commute
•
Operators c, cy anti-commute
•
States specified by symmetric
covariance matrix
•
States specified by antisymmetric
covariance matrix
•
Gaussian operations $
symplectic transformations of 
•
Gaussian operations $
orthogonal transformations of 
•
What’s missing? A fermionic phase-space representation.
Fermionic phase space (1)
For bosons…
• Eigenstates of an are coherent states: an|i = n|i
• Define displacement operators: D()|vaci = |i
•
Characteristic function for state  is: () = tr(D() )
•
Define gaussian state to have gaussian characteristic function:
For fermions…
• Try to define coherent states: cn|i = n|i…
•
…but hit up against anti-commutation:
cncm|i = m n|i but cncm|i = -cmcn|i = -n m|i
•
Eigenvalues anti-commute!?
Fermionic phase space (2)
•
Solution: expand fermionic Fock space algebra to include anticommuting numbers, or “Grassmann numbers”, n.
•
Coherent states and displacement operators now work:
cncm|i = -cmcn|i = -n m|i = m n|i = cncm|i
•
Characteristic function for a gaussian state is again gaussian:
•
Grassman algebra:
n m = -m n ) n2=0;
•
Grassman calculus:
for convenience n cm = -cm n
Fermionic phase space (3)
•
•
We will need another phase-space representation: the fermionic
analogue of the P-representation.
Essentially, it is a (Grassmannian) Fourier transform of the
characteristic function.
•
Useful because it allows us to write state  in terms of coherent
states:
•
For gaussian states:
What about entanglement (3)
•
Finally,
• Recall bound on localisable entanglement:
•
•
Substituting the P-representation for states  and * :
and expanding xn and pn in terms of cn and cn y, the calculation
becomes simple since coherent states are eigenstates of c.
Not very useful since bound  0 in thermodynamic limit N  1.
Fermionic phase space (4)
•
•
However, experimentalists are starting to build atomic analogues
of quantum optical setups: e.g. atom lasers, atom beam splitters.
Fermionic gaussian state formalism may become important as
fermionic gaussian states and operations move into the lab.
•
Already leading to new theoretical results, e.g.:
•
Michael Wolf has used fermionic gaussian states to prove an area
law for a large class of fermionic systems, in arbitrary
dimensions: Phys. Rev. Lett. 96, 010404 (2006)
Entanglement flow
in multipartite systems
T. S. Cubitt
• Motivation
• Time
J.I. Cirac
and goals
evolution of a chain
• Correlation
and entanglement wave-packets
• Engineering
• Fermionic
the dynamics: solitons etc.
gaussian state formalism
• Conclusions
Conclusions
•
Have shown that correlation and entanglement dynamics in a spin
chain can be described by simple physics: wave-packets.
Correlation dynamics can be engineered:
• Set parameters to produce “soliton-like” behaviour
• Control “soliton” velocity by adjusting parameters slowly in time
• Freeze correlations at desired location by quenching the system
•
Developed fermionic gaussian state formalism, likely to become
more important as experimentalists are starting to do gaussian
operations on atoms in the lab (atom lasers, atomic beamsplitters…).
The end!