Open quantum random walks: bi-stability and ballistic diffusionpillet.univ-tln.fr/aoqs/www/pdf/Bernard.pdf · 2013. 7. 13. · «open quantum random walks»: Open quantum random walks

with Michel Bauer and Antoine Tilloy

Autrans, July 2013

Open quantum random walks:bi-stability and ballistic diffusion

Open quantum brownian motion

mardi 9 juillet 2013

Different regimes in«open quantum random walks»:

Open quantum random walks describe quantum random motions on the linefor systems with internal and orbital degrees of freedom (alias a gyroscope).

A Brownian like regime. A ballistic like (but diffusive) regime.

On our way, we shall visit the «open quantum Brownian motion»


Outline:

--- What quantum random walks could be?(quantum noise, what objects?)

--- Open quantum random walk trajectories.(a random walk but with an internal quantum gyroscope)

--- Projection on pure states(via sub-martingale theorem, and true more generally)

--- Two regimes and bi-stability (diffusion or ballistics).(gyroscope flips a la Kramer’s)

--- Open quantum brownian motion.--- Continuous limit--- Bi-stability and Q-jumps.

More on...:


What quantum noise and quantum random walk could be?... or not be!

* Classical random walks or stochastic processes.

* A quantum Brownian motion (via open system models)

--- A free (quantum) particle in contact with a (quantum) reservoir.

Feymann-Vernon, Keldish, Caldeira-Leggett, Froelich,....

Usually:Sum, or trace over, or average, over the reservoir degree of freedom, and look for an effective description of the quantum particle dynamics.

Lost of (some) noise related information (and of the filtration)

-- Coins are reset at each time step (i.i.d. variables)

Events: ω = (+,+,−,+, · · · ) = (ε1, ε2, ε2, · · · )

* Discrete (unitary) quantum random walks.

A (quantum) particle on a line (say Z), coupled to a (quantum) coin.

-- The quantum coin can be in two different states, (+ or -).-- The particle move to the left and to the right conditioned on the coin state.-- Use the same quantum coin at each step.

Aharonov, Davidovich, Zagury, ’93, and .......


* Discrete «open» quantum random walks (I).

A (quantum) particle on a line (say Z), with internal degree of freedom,coupled to different quantum coins at each time step (recursive interactions).

Attal, Petruccione, Sabot, Sinayskiy, 2012.

Actually a slight extension of their definition...

The quantum coin can be in two different states, (+ or -).The particle move to the left and to the right conditioned on the coin state:-- if the coin is measured in state +, the particle move one step to the right.-- if the coin is measured in state -, it moves to the left.But the coins act on the internal degrees of freedom (gyroscope),And the probability to go left/right depends on this action.

* Hilbert space of states: p=probes (or coins);c=color (or spin), internal d.o.f ’s;o=position, orbital d.o.f.’s.

Hc = C2, Ho = CZ

Hc ⊗Ho ⊗H⊗∞p H⊗∞p = C2 ⊗ C2 ⊗ · · ·

Coins are reset at each step.

--- Notion of noise (with a measure, the probe state).--- Notion of arrow of time (information on the probes)--- Reservoir, by summing over the probes.


* Discrete «open» quantum random walks (II).

* Interaction (without measurements):

|ψ〉c ⊗ |n〉o ⊗ |φ〉p

(B+|ψc〉 ⊗ |n + 1〉o ⊗ |+〉p + (B−|ψ〉c)⊗ |n− 1〉o ⊗ |−〉p

On state in Hc ⊗Ho ⊗Hp , the unitary evolution

* Measurements and «quantum trajectories»:

Measuring the probes, one may find + or - with probabilities,and then «project» the state on |+> or |->:

Iterating interactions with different probes produces an «entangled» state,sum of states each indexed by a random walk (quantum parallelism).

∝ (B±|ψ〉c)⊗ |n ± 1〉o with probability c〈ψ|B†±B±|ψ〉c

Iterating gives random states indexed by random walks.The events, the output of the probe measurements, are in one-to-one correspondence with random walks.

* The original definition of open QRW was the mean of this process(not keeping track of the measurements).


* «Open» quantum random walks trajectories:

These are classical random processes with values on the line X the internal states.

(I) For pure (internal) state:

(I) For mixed (internal) state: ie. the system is not described by a vector but by a density matrix (a positive hermitian normalised matrix).

If after n step,

The updating is:

with probability:

(ρn, xn)

(p±(n)−1 B±ρnB†±, xn ± 1)

p±(n) = tr(B±ρnB†±)

If after n step, (|ψn〉c, xn)

The updating is: (|ψn+1〉c = p±(n)−1/2 B±|ψn〉c, xn+1 = xn ± 1)

with probability: p±(n) = c〈ψn|B†±B±|ψn〉c

--- The matrices B are the moduli parameters of the walks.--- The random process is now classical (but vector valued).--- The internal system acts as a random (quantum) gyroscope.

The position is slave to the probe measurements. --- Events are random walks (output measurements),

but with probabilities induced by the gyroscope motion.


Open Quantum Brownian motion:

--- Quantum trajectories: Probes are measured (with random outputs),and the system density matrix evolves randomly (according to the measurement outputs).

.... and this can be generalised with many packets and/or entangled states.

--- Quantum dynamical map: Probes (reservoir) are *not* measured but trace out, and the system density matrix evolves with Lindblad equation

ρ̄t :=∫

dx ρ(x, t) ⊗ |x〉o〈x|

∂tρ̄t = − 12 [P, [P, ρ̄t]] + i

(N [P, ρ̄t] + [P, ρ̄t]N†) + i[H, ρ̄t] + LN (ρ̄t)

for the density matrix

--- Quantum Stochastic equation: Probes (reservoir) are *not* measured but *not* trace out, and the *total* state evolves with a quantum-SDE.

dAt = i[P + iN†, At] dξt + i[P − iN, At] dξ†t + L∗(At) dt

for observable A, and dual Linbladian L.,and (quantum) noises: dξt dξ†t = dt, dξ†t dξt = 0

more below.....


Discrete numerical simulations:

A Brownian like regime. A ballistic like (but diffusive) regime.

Unitarity or normalisation of probabilities imposes: B†+B+ + B†

−B− = I

A choice for the simulations: B+ = δ−1(

u rs v

)B− = δ−1

(−v sr −u

)

δ =√

u2 + v2 + r2 + s2

and

with

u = 1.1, v = 1.00 and r = −s = 0.00015u = 1.005, v = 1.00 and r = −s = 0.00015

Kramer’s like transition for a particle in a double well potential (... but not quite).


Projection a.s. on pure states:

As seen from the numerics, the determinant is vanishes (quickly).

--- Claim: «Whatever initial value, the internal density matrix converges to pure states.»

--- This is true for the internal system alone (with or without orbital d.o.f.),i.e. also true for a Q-bit coupled to series of Q-bits.

Proof: ρn+1 = p±(n)−1 B±ρnB†± with probability p±(n) = tr(B±ρnB†

±)

Let ∆n := det ρn be the determinant of the internal density matrix.

E[∆1/2n ] = cn ∆1/2

0 with c := det1/2(M2+) + det1/2(M2

−) < 1,and limn→∞ E[∆1/2

n ] = 0.

Then:

Actually, ∆n is a bounded sub-martingale and as such it converges almost surely and in L1.

limn→∞

∆n = 0

--- Similar to progressive collapse in non-demolition measurement (Haroche’s group experiments) but the target states keep involving randomly.


Scaling limit: «Open quantum Brownian motion».

Both the density matrix and the position have stochastic evolution but the position drift is guided by the internal gyroscope.

Proof: decompose the process as the sum of martingale plus a predictable process, i.e. a Doob decomposition.

Mn =∑n

k=1 πk with πk := ρk − E[ρk|Fk−1] and On := ρn −Mn

(New)As for classical Brownian motion, scale time, distance and moduli simultaneously:

B± = 1√2[I±

√εN + ε(iH± ±M − 1

2N†N) + o(ε)]

with ε a small parameter and H±, M hermitian but not N .

In the scaling limit one gets a time continuous process (with continuous measurement)

dρt =(i[H, ρt] + LN (ρt)

)dt + DN (ρt) dBt,

dXt = UN (ρt) dt + dBt,

A Brownian motion,coding for all probe measurements.

with LN (ρ) := NρN† − 12 (N†Nρ + ρN†N)

DN (ρ) := Nρ + ρN† − ρ UN (ρ) and UN (ρt) := tr(Nρ + ρN†)

The scaling limit is ε→ 0, t = nε fixed, and dx2 ∼ dt.

--- This is for one trajectory, but multiple trajectory are ok (non localised initial wave packets)

--- Averaging over the noise: «open quantum brownian motion» in contact with a reservoir.


Transition between two regimes:Take H := ω0 σ2 and N = a σ3 with σ1,2,3 the usual Pauli matrices.Describe ρt as a pure state: ρt = 1

2 (I+q1σ1+q3σ3) with q1 = sin θ, q3 = cos θ.

dθt = −2(ω0 + a2 sin θt cos θt)dt− 2a sin θt dBt

a2 < ω0 a2 > ω0

Two potential minima, with Kramer’s transition between them.

Trajectories of theta are oriented : they never cross 0 and Pi anti-clockwise.

Eqs. dρt =(i[H, ρt] + LN (ρt)

)dt + DN (ρt) dBt, becomes:

a2 > ω0

θtθtθ∗+

θ∗−


Flips between two pure states:The pure states are those associated to the two potential minima,but need for a change of variable to see it.Let yt := − log | tan θt/2|. It satisfies dyt = 2a dBt − V ′(yt)dt with potential

V (y) = −2(±ω0 sinh y + 2a2 log cosh y)

a2 < ω0a2 > ω0

A cubic like potential but with exponential ramps.

A y-trajectorywith Kramer’s transition across the energy barrier.

No flips.Flips:

E[τflip] ! e∆V/4a2! a2


Ballistically induced diffusion:

dXt = 2a cos θt dt + dBt

2a cos θ∗± ! ∓2awith

Trajectories are ballistic,with seesaw profiles induced by gyroscope flips,and large mean free paths.

But at very large time the position is Gaussian,with large effective diffusion constant.

dXt = UN (ρt) dt + dBt becomes:

E[X2t ] = Deff t , with Deff = 1 + 4a4/ω2

0

Question:Can we find similar phenomena (phenomenological description)producing a large effective diffusion constant (at large time)?


dρt =(i[H, ρt] + LN (ρt)

)dt + DN (ρt) dBt,

dXt = UN (ρt) dt + dBt,

More on the continuous limit:

Proof: decompose the process as the sum of martingale plus a predictable process, i.e. a Doob decomposition.

Mn =∑n

k=1 πk with πk := ρk − E[ρk|Fk−1] and On := ρn −Mn

ρn+1 = ρ(+)n I{sn+1=+} + ρ(−)

n I{sn+1=−}

xn+1 − xn = I{sn+1=+} − I{sn+1=−}

with ρ(±)n := B±ρnB†

±/p±n

p±n := E[I{sn+1=±}|Fn] = tr(B±ρnB†±)

gives: 2πk =(ρ(+)

k − ρ(−)k

)(I{sk+1=+} − p+

k + p−k − I{sk+1=−})

Taylor expanding dMt := Mn+1 − Mn, dρt := ρn+1 − ρn and dXt :=√ε(xn+1 − xn). Identifying ε with dt, we get dMt = DN (ρt) dBt,


More on bi-stability:

-- The system (alias the gyroscope) is a Q-bit with hamiltonian H = ω0 σ2

with a (real) pure state:

-- In absence of measurement the system state oscillates.

ρ = 12

(1 + cos θσ3 + sin θσ1

)

-- Measuring an observable commuting with H : (progressive) collapse.

-- Measuring an observable not commuting with H : Q-jumps.

Generalizations: --

A system (spin half) under continuous measurement.


ρ̄t :=∫

dx ρ(x, t) ⊗ |x〉o〈x|

∂tρ̄t = − 12 [P, [P, ρ̄t]] + i

(N [P, ρ̄t] + [P, ρ̄t]N†) + i[H, ρ̄t] + LN (ρ̄t)

for the density matrix

More on the dynamical map:

Proof:

From trajectory to mean density matrix,

∫dx ρ(x, t) ⊗ |x〉o〈x| := E[ ρt ⊗ |Xt〉o〈Xt|

Then routine application of Ito calculus.

∂tρ(x, t) =12∂2

xρ(x, t)− (N∂xρ(x, t) + ∂xρ(x, t)N†) + i[H, ρ(x, t)] + LN (ρ(x, t)),

Computations are done by considering matrix elements∫

dxf∗(x)ρ(x, t)g(x) andidentifying them with f∗(Xt)g(Xt) ρt with Xt, ρt Q-trajectory.

This is equivalent to eq. above.Generalisable in higher dimension and with in-homogenieties.


dAt = i[P + iN†, At] dξt + i[P − iN, At] dξ†t + L∗(At) dt

for observable A, and dual Linbladian L., and (quantum) noises: dξt dξ†t = dt, dξ†t dξt = 0

More on the SDE (open Brownian motion):

A la physicist:-- Noise dξt :=

∫ t+dtt a(s)ds, [dξ†t , dξt] = dt with [a†(s), a(t)] = δ(s− t)

and a state, i.e. a measure (vacuum, Gibbs,...)

-- Look for quantum SDE: dA = D(A)dξt + D†(A)dξ†t + L∗(A)dt

with L* given (dual Lindbladian) and D(.) derivative.with Ito rule dξtdξ†t = (1 + γ) dt and dξ†t dξt = γ dt for some γ.

-- Consistency condition:

L∗(AB) = L∗(A)B + AL∗(B)−D(A)(1 + γ)D†(B)−D†(A)γD(B)

I.e. L(.) second order differential operator and D(.) first order.This determines D(.) and gamma.(can be generalized with multi-noise).


Thank you.


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Open quantum random walks: bi-stability and ballistic diffusionpillet.univ-tln.fr/aoqs/www/pdf/Bernard.pdf · 2013. 7. 13. · «open quantum random walks»: Open quantum random walks