Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

T. S. Cubitt

J.I. Cirac

•Motivation and goals

•Time evolution of a chain

•Correlation and entanglement wave-packets

•Engineering the dynamics: solitons etc.

•Fermionic gaussian state formalism

•Conclusions

Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

T. S. Cubitt

J.I. Cirac

Conceptual motivation: new Conceptual motivation: new experimentsexperiments

• …motivate new theoretical studies of non-equilibrium behaviour.

• New experiments…

• Many papers on correlations/entanglement of ground states• Fewer on time-dependent behaviour away from equilibrium

• Many papers on correlations/entanglement of ground states.• Fewer on time-dependent behaviour away from equilibrium

Existing resultsExisting results

• 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.

• In J.Stat.Mech. 0504 (2005) p.010, Calabrese and Cardy investigated the time-evolution of block-entropy in spin chains.

• Bravyi, Hastings and Verstraete (quant-ph/0603121) recently used Lieb-Robinson to prove tighter, linear bound.

Practical motivation: quantum repeatersPractical 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?

•Motivation and goals

•Time evolution of a chain

•Correlation and entanglement wave-packets

•Engineering the dynamics: solitons etc.

•Fermionic gaussian state formalism

•Conclusions

T. S. Cubitt

Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

J.I. Cirac

Time evolution of a spin chain (1)Time evolution of a spin chain (1)

• As an exactly-solvable example, we take the XY model…

anisotropy magnetic field

…and start it in some separable state, e.g. all spins +.

• Fourier:

Time evolution of a spin chain (2)Time evolution of a spin chain (2)

• Solved by a well-known sequence of transformations:

• Bogoliubov:

• Jordan-Wigner:

Time evolution of a spin chain (3)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)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:

corresponds to an orthogonal transformation of the covariance matrix:

Gaussian state stays gaussian under gaussian evolution.

Time evolution of a spin chain (5)Time evolution of a spin chain (5)

• Initial state is a fermionic gaussian state in xl and pl.

• Time-evolution is a fermionic gaussian operation in xk and p

k.

Connected by Fourier and Bogoliubov transformations

• Fourier and Bogoliubov transformations are gaussian:

Time evolution of a spin chain (phew!)Time evolution of a spin chain (phew!)

• Putting everything together:

xk , pkxk p

kxk p

k xk , pkxk , pk xl , pltime-evolveinitial state

•Motivation and goals

•Time evolution of a chain

•Correlation and entanglement wave-packets

•Engineering the dynamics: solitons etc.

•Fermionic gaussian state formalism

•Conclusions

Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

T. S. Cubitt

J.I. Cirac

String correlationsString correlations

• We can get string correlations hanz

lbmi for free…

• Equations are very familiar: wave-packets with envelope S propagating according to dispersion relation .

• Given directly by covariance matrix elements, e.g.:

Two-point correlationsTwo-point correlations

• Two-point connected correlation functions hznz

mi - hznihz

mi can also be obtained from the covariance matrix.

• Again get wave-packets (3 of them) propagating according to dispersion relation k.

• Using Wick’s theorem:

• As with all operationally defined entanglement measures, LE is difficult to calculate in practice.

• Best we can hope for is a lower bound.

What about entanglement?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.

• Popp et al., Phys. Rev. A 71, 042306 (2005) : LE is lower-bounded by any two-point connected correlation function.

• In case you missed it: we’ve just calculated this!

•Motivation and goals

•Time evolution of a chain

•Correlation and entanglement wave-packets

•Engineering the dynamics: solitons etc.

•Fermionic gaussian state formalism

•Conclusions

Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

T. S. Cubitt

J.I. Cirac

• 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

• In other parameter regimes: narrower wave-packets in nearly linear regions of dispersion relation

• ! packets maintain their coherence as they propagate

• In some parameter regimes: broad wave-packets and a highly non-linear dispersion relation

• ! correlations rapidly disperse and disappear

Correlation wave-packetsCorrelation wave-packets

• In some parameter regimes: broad wave-packets and a highly non-linear dispersion relation

• ! correlations rapidly disperse and disappear: (=10, =2)

Correlation wave-packets (1)Correlation wave-packets (1)

• The system parameters and simultaneously control both the dispersion relation and form of the wave-packets.

• In other parameter regimes, 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: (=1.1, =2)

Correlation/entanglement solitonsCorrelation/entanglement solitons

• In general, time-ordering is essential.• But if parameters change slowly, dropping it gives

reasonable approximation.

• 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 (1)Controlling the soliton velocity (1)

• If the parameters are changed with time,

• ! Effective evolution under time-averaged Hamiltonian.

• Starting from =1.1, =2.0 and changing at rate +0.01:

Controlling the soliton velocity (2)Controlling the soliton velocity (2)

• 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.

““Quenching” correlations (1)Quenching” correlations (1)

• What happens if we do the opposite: rapidly change parameters from one regime to another?

• Choose parameters so that “frozen” packets remain relatively coherent, whilst others rapidly disperse.

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

““Quenching” correlations (2)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:

•Motivation and goals

•Time evolution of a chain

•Correlation and entanglement wave-packets

•Engineering the dynamics: solitons etc.

•Fermionic gaussian state formalism

•Conclusions

Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

T. S. Cubitt

J.I. Cirac

What about entanglement? (2)What about entanglement? (2)

• There is another LE bound we can calculate…

• Recall concurrence:

• Not a covariance matrix element because of |*i.

• Localisable concurrence:

• Operators a, ay commute

• States specified by symmetric covariance matrix

• Gaussian operations $ symplectic transformations of

Bosonic case

Fermionic gaussian formalismFermionic gaussian formalism

• Recap of gaussian state formalism…

• States specified by antisymmetric covariance matrix

• Gaussian operations $ orthogonal transformations of

Fermionic case

• Operators c, cy anti-commute

• What’s missing? A fermionic phase-space representation.

Fermionic phase space (1)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() )

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!?

• Define gaussian state to have gaussian characteristic function:

Fermionic phase space (2)Fermionic phase space (2)

• Solution: expand fermionic Fock space algebra to include anti-commuting numbers, or “Grassmann numbers”, n.

• Coherent states and displacement operators now work:cncm|i = -cmcn|i = -n m|i = m n|i = cncm|i

• Grassman algebra:n m = -m n ) n

2=0; for convenience n cm = -cm n

• Grassman calculus:

• Characteristic function for a gaussian state is again gaussian:

Fermionic phase space (3)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:

• Finally,

What about entanglement (3)What about entanglement (3)

• 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.

• 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.

Fermionic phase space (4)Fermionic phase space (4)

• 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)

• Already leading to new theoretical results, e.g.:

•Motivation and goals

•Time evolution of a chain

•Correlation and entanglement wave-packets

•Engineering the dynamics: solitons etc.

•Fermionic gaussian state formalism

•Conclusions

Entanglement flowEntanglement flow in multipartite systems in multipartite systems

T. S. Cubitt

J.I. Cirac

ConclusionsConclusions

• 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 beam-splitters…).

The end!