Long-time Entanglement in the Quantum Walk

Long-time Entanglementin the Quantum Walk

Gonzalo Abal, Raul Donangelo∗, Hugo Fort

Instituto de Fısica, Universidad de la Republica, Montevideo∗ Instituto de Fısica, UFRJ, Rio de Janeiro

other collaborators at Montevideo:

A. Romanelli, R. Siri and A. Auyuanet (now PhD student at UFRJ)

talk at WECIQ06, UCPel, RS, october 2006. WECIQ06 – p. 1/21

Why look at the Quantum Walk?

some of the best classical algorithms for “hard” problems

(NP-complete) are based on random walks

quantum walks spread faster than random walks

* exponentially faster hitting time

in a hypercube(J. Kempe, quant-ph/0205083)

experimentally at the “proof of principle” stage:

* linear optics(B. Do et al., J. Opt. Soc. Am.B22, 400,2005)

* liquid-state NMR processor(C. Ryan et al, quant-ph/0507267)

version of Grover’s oracle search algorithm on QW:N → N 1/2

Shenvi et al., quant-ph/0210064WECIQ06 – p. 2/21

Quantum Walk on a line

(Y. Aharonov, N. Zagury, L. Davidovich, PRA 48, p.1687, 1993)

|R>|L>

xx− 1 x + 1

the quantum walk uses an additional qubit,

to model the“coin” state

|χ〉 = a|L〉+ b|R〉

WECIQ06 – p. 3/21

quantum walk on the line

discrete position space,Hx, spanned by integersx

{. . . , | − 1〉, |0〉, |1〉, . . .}

coin space,Hc, spanned by{|L〉, |R〉}joint spaceH = Hx ⊗Hc,

quantum state of the walker is described by

|Ψ〉 =∞∑

x=−∞(ax|L〉+ bx|R〉)⊗ |x〉

WECIQ06 – p. 4/21

evolution operator

one-step:U = S · (Ix ⊗ UC)

︸︷︷︸

analog to coin tossing

unitary coin evolution (i.e.UC = H, a Hadamard coin)

followed by aconditional shift operationin H

S =∑

x

|x− 1〉〈x| ⊗ |L〉〈L|+ |x + 1〉〈x| ⊗ |R〉〈R|

aftert steps,|Ψ(t)〉 = U t|Ψ(0)〉 ←− discrete-time quantum walk

WECIQ06 – p. 5/21

coherent evolution

(symmetric initial condition atx = 0, aftert = 100 steps)

position probability distribution variance vs. time

-100 -50 0 50 100position, n

0

0,02

0,04

0,06

0,08

P(n

)

Hadamard WalkSymmetric initial conditions

0 20 40 60 80 100time

0

500

1000

1500

2000

2500

3000

<n2 >

Hadamard walk on the linesymmetric initial state

P (x, t) = |ax|2 + |bx|2 σ2(t) ∝ t2

Classically, the random walk hasσ2clas ∝ t

Pclas(x, t) satisfies a diffusion equationWECIQ06 – p. 6/21

coin-position entanglement

the conditional shift operation is non-separable,∑

x

|x− 1〉〈x| ⊗ |L〉〈L|+ |x + 1〉〈x| ⊗ |R〉〈R|

it entangles the coin and positionof the walker

How do we quantify this entanglement?

WECIQ06 – p. 7/21


for a pure state,ρ = |Ψ〉〈Ψ|, bipartite entanglement may be

quantified using the entropy of thereduced density operator,

ρc = tracex(ρ) =

∑

x |ax|2∑

x axb∗x

∑

x a∗xbx

∑

x |bx|2

WECIQ06 – p. 7/21


for a pure state,ρ = |Ψ〉〈Ψ|, bipartite entanglement may be

quantified using the entropy of thereduced density operator,

ρc = tracex(ρ) =

∑

x |ax|2∑

x axb∗x

∑

x a∗xbx

∑

x |bx|2

for eigenvaluesr and1− r theentropy of entanglementis

SE ≡ −trace(ρc log ρc) = − [r log2 r + (1− r) log2(1− r)]

(SE = 0 for a product state,SE = 1 for full entanglement)

WECIQ06 – p. 7/21

entanglement vs. time

3 initial conditions (same coin state):

|Ψ0〉 = |χc〉 ⊗ |0〉

|Ψ±〉 = |χc〉 ⊗ (| − 1〉 ± | 1〉)/√

2

asymptotic(t≫ 1) values

initial SE

|Ψ+〉 0.98 . . .

|Ψ0〉 0.87 . . .

|Ψ−〉 0.66 . . .

0 10 20 30 40 50 60t

0

0,2

0,4

0,6

0,8

1SE

0 1 2 3 4 5t

0

0,5

1SE

blue: |Ψ+〉; black: |Ψ0〉; red: |Ψ−〉fixed coin:|χc〉 ≡ (|R〉 + i|L〉)/

√2

WECIQ06 – p. 8/21

SE can be obtained analytically

in the Fourier representation|k〉 =∑

x eikx|x〉,

state vector−→ |Φ〉 =

∫ π

π

dk

2π|k〉 ⊗ (ak|R〉+ bkL)

the evolution operator

is diagonal ink space Uk =1√2

e−ik e−ik

eik −eik

|Φk(t)〉 = U tk|Φk(0)〉 andρc can be explicitly calculated fort≫ 1

exact asymptotict≫ 1 expressions for the eigenvalues(r, 1− r) of

ρc are obtained...

details:

G. Abal, R. Siri, A. Romanelli and R. Donangelo, PRA73, 042302, 069905(E)(2006);

quant-ph/0507264WECIQ06 – p. 9/21

eigenvalues ofρc (outline)

write r as

r(t) =1

2

[

1 +√

1− 4∆(t)]

where∆ = AC − |B|2 is the determinant ofρc,

A(t) ≡∑

x

|ax|2 =

∫ π

−π

dk

2π|ak|2

B(t) ≡∑

x

axb∗x =

∫ π

−π

dk

2πakb

∗k (1)

C(t) ≡∑

x

|bx|2 =

∫ π

−π

dk

2π|bk|2.

Normalization requires trace(ρ) = A + C = 1.

WECIQ06 – p. 10/21

dependence on initial coin

Local initial state|Ψ(0)〉 = |χc〉 ⊗ |0〉 with arbitrary coin state

|χc〉 ≡ cos α|L〉+ eiβ sin α|R〉

WECIQ06 – p. 11/21




after some “straightforward calculations”,

limt≫1

∆ = ∆0 − 2b21 cos β sin(4α)

with ∆0 = (√

2− 1)/2 andb1 = (2−√

2)/4.

WECIQ06 – p. 11/21




extreme values:

0.74 . SE . 1

−π/2 −π/4 0 π/4 π/2 −π/8 −3π/8 π/8 3π/4 α

0.7

0.8

0.9

1SE

β=0β = π/2β=π/4

WECIQ06 – p. 11/21

fixed coin, non-local initial conditions

position subspace spanned by| ± 1〉 with |χc〉 = (|R〉+ i|L〉)/√

2

|Ψ(0)〉 =(cos θ| − 1〉+ sin θ e−iϕ| 1〉

)⊗ |χc〉

the same analytical method givesSE exactly

–10

1

theta

–2

0

2

phi

0.7

0.8

0.9

Sc

+

+

+

-

-

-0

0

−π/2

−π/2

π/2

π/2−π/4 π/4π

−π

θ

ϕ

WECIQ06 – p. 12/21

non-local i.c., fixed relative phaseϕ

position subspace spanned by{| − 1〉, | + 1〉}

|Ψ(0)〉 =(cos θ| − 1〉+ sin θ e−iϕ| 1〉

)⊗ |χc〉

-π/2 -π/4 0 π/4 π/2θ

0,6

0,7

0,8

0,9

1S

E

0.872

0.661

0.979ϕ= ±π

WECIQ06 – p. 13/21

Effect of more non-locality in the i.c.

local initial states:SE & 0.76

initial states in subspace spanned by| ± 1〉: SE & 0.66

gaussian initial state with spread∼ σx ≫ 1

SE ∼ log2 σ/4σ4 → 0

with enough non-locality in the initial state

all entanglement levels may be (asymptotically) obtained

WECIQ06 – p. 14/21

Two quantum walkers

U = S · (I ⊗ Uc)

the conditional shift now involves

two particles

S =∑

x,y

|x + 1, y + 1〉〈x, y| ⊗ |00〉〈00| +

|x− 1, y + 1〉〈x, y| ⊗ |01〉〈01| +|x + 1, y − 1〉〈x, y| ⊗ |10〉〈10| +|x− 1, y − 1〉〈x, y| ⊗ |11〉〈11|

x

y

-1 1

-1

1

|00〉

|01〉

|10〉

|11〉

x→ particle 1

y → particle 2

the coin operationUc now acts on two qubits

WECIQ06 – p. 15/21

separable coin operations

entanglement between 1 and 2 is conserved

for UC = H ⊗H (Hadamard walk)

joint probability distribution aftert = 100 steps

-150-100

-50 0

50 100

150-150-100

-50 0

50 100

150

0

0.001

0.002

0.003

0.004

0.005

0.006

X

Y

product state initial coin

|χ1〉 = 12(|0〉+ i|1〉)⊗2

-150-100

-50 0

50 100

150-150-100

-50 0

50 100

0 0.002 0.004 0.006 0.008 0.01

0.012 0.014 0.016 0.018

X

Y

product state initial coin

|χ2〉 = 12(|0〉 − |1〉)⊗2

WECIQ06 – p. 16/21

Non-separable coin operations

Grover’s coin

G =1

2

−1 1 1 1

1 −1 1 1

1 1 −1 1

1 1 1 −1

-150-100

-50 0

50 100

150-150-100

-50 0

50 100

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

X

Y

|χ2〉 =1

2(|00〉−|01〉−|10〉+|11〉)

WECIQ06 – p. 17/21


Grover’s coin

G =1

2

−1 1 1 1

1 −1 1 1

1 1 −1 1

1 1 1 −1

-150-100

-50 0

50 100

150-150-100

-50 0

50 100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X

Y

|χ1〉 =1

2(|00〉+i|01〉+i|10〉−|11〉)

initial coin state makes all the difference!WECIQ06 – p. 17/21


RP coin (of interest for iterated

Quantum Games)A

B

H

UC = CNOT · (H ⊗ I)

=1√2

1 0 1 0

0 1 0 1

0 1 0 −1

1 0 −1 0

-150-100

-50 0

50 100

-150

-100

-50

0

50

100 0

0.002 0.004 0.006 0.008

0.01 0.012 0.014 0.016 0.018

RPA coin

X

Y

|χ1〉 =1

2(|00〉+i|01〉+i|10〉−|11

WECIQ06 – p. 18/21

Entanglement between both walkers

1 10 100t

0

1

2

3

4

5

6

SE

Gr χ1Gr χ2RP χ1RP χ2

for the cases considered the entanglement increases logarithmically

SE(t) ∼ log2 tc with c ≃ 0.9

except for Grover’s coin with|χ1〉, for whichc ≃ 0.6WECIQ06 – p. 19/21

In sum – To do list

inter-particle entanglement generated by Grover and RP coins

increases at a well defined logarithmic rate

the localized distribution from Grover’s coin has the lowest

rate...

other important non-separable coin operations need to be

investigated (FFT?)

analytical methods (as in 1D case) may clarify the dependence

on initial conditions.

given that entanglement can be generated byU at alog t rate,

what happens in the presence of weak noise?

WECIQ06 – p. 20/21

Next: Quantum Games!

WECIQ06 – p. 21/21

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Long-time Entanglement in the Quantum Walk