Engineering correlation and entanglement dynamics in spin chains
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Transcript of Engineering correlation and entanglement dynamics in spin chains
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Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains
T. S. Cubitt
J.I. Cirac
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•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
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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
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• 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.
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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?
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•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
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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 +.
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• 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:
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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:
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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.
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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:
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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
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•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
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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.:
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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:
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• 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!
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•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
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• 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
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• 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.
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• 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
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• 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.
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• Starting from =1.1, =2.0 and changing at rate +0.01:
Controlling the soliton velocity (2)Controlling the soliton velocity (2)
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• 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
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““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:
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•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
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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:
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• 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.
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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:
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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:
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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:
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• 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.
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• 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.:
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•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
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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…).
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The end!