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Entanglement and Decoherence inCoined Quantum Walks
Peter Knight, Viv Kendon, Ben Tregenna, Ivens Carneiro, MathieuGirerd, Meng Loo, Xibai Xu (IC) & Barry Sanders, Steve Bartlett
(Macquarie), Eugenio Roldan (Valencia) &John Sipe (Toronto)• Random walk as a quincunx• early mention of a quincunx: Caesar's De Bello Gallico, Book VII:
http://www.romansonline.com/sources/dbg/Ch07_73.asp (I’m impressedthat the Roman Empire set up its own web-site)
• Asterix: for the other side of the coin, translation to Englishhttp://www.aunet.org/~thaths/asterix/books/asterix_the_legionary.html
Barry Sanders Viv KendonBen Tregenna Steve Bartlett
UK Australia/Canada
Why are these walks interesting? Motivation:quantum interferencesrelate to algorithmic speed-up, can engineer coin decoherenceand study transition to classical, ….
Also John Sipe, Eugenio Roldan, I Carneiro, Mathieu Girerd, Meng Loo &Xibai Xu
Francis Galton & the Quincunx• Galton’s quincunx demonstrates
random walk.• Each peg is like the coin, and the
ball goes L or R.• Distribution of balls follows
normal distribution withvariance proportional to numberof rows of pegs.
• How to design a quantumquincunx to show quantumwalk?
Reminder of usual 1D classical random walk• Particle confined to motion along a regular lattice in 1D, with points
indexed by integers i (integer time)• Decide on whether to go forward or backward according to whether
an unbiased coin reveals heads (+ ) or tails (-).• The distribution P(i) is binomial and approaches a Gaussian
distribution after many steps.• The average distance from the origin increases according to
ts ∝Sqrt(time) Dependence
0
200
400
600
800
1000
1200
1
30
59
88
117
146
175
204
233
262
291
320
349
378
407
436
465
494
523
552
581
610
639
668
697
726
755
784
813
842
871
900
929
958
987
Step
(St.
Dev
.)^
2
(St.Dev.)^2
Also Julia Kempe, Dorit Aharanov, and many peoplehere have contributed to math description of walks….Julia Kempe review: Contemp Phys 2003/ quant-ph/0303081.-but realisations?-Decoherence?
Quantum coin for Quantum Quincunx• Coin replaced by quantum two-level system:
Quantum coin toss - Hadamard transformation:
Quantum Quincunx
• Head for L, Tail for R:
Superposition of headsand tails goes tosuperposition of left andright: interferences!
Chose symmetric initialstate…
Quantum superpositions with coin flip:
Conditional Translation of Position• The particle’s translation depends on the state of the
coin, and the entire quantum random walk isdeterministic (in a wave sense) over the joint particle-coin Hilbert space. Quantum walk is not a RandomWalk..
• Is it always “quantum?”-- see later
Bouwmeester, Schleich et al- already observed quincunx in classicaloptical experiment. Described in terms of many Landau-Zener crossings
Ions: Travaglione and Milburn;Lattices: Kendon and Briegel etc
Its not random. Is it quantum?• Knight, Roldan & Sipe (quant-
ph 0305165/Opt Comm inpress) optical implementationof walk for polarization orposition “cebits”
• Bouwmeester et al alreadyimplemented in their GaltonBoard experiment
• But look at resources: cancompute entanglement interms of entropy usingSchmidt basis - 4-component=4 propagating spinwaves…entanglement!
The proposed setup
T2T2
T1
T2
T2T2 T2T2 T2
T2 T2
T2 T2T2T2
T2j=1
j=0
k=+1
k=0 k=-1
D0
D-2
D-4
D+4
D+2
dynamiclines
node
Scheme of Paternostro et al
When the walker is a single-photon state: 4th step
T2T2
T1
T2
T2T2 T2T2 T2
T2 T2
T2 T2T2T2
T2
D0
D-2
D-4
D+4
D+2
s1
-4-2024Position0.050.10.150.20.250.30.35Np-4-2024Position0.050.10.150.20.250.30.35Np classical distribution quantum distribution
-40-2002040Position0.020.040.060.080.1Np50th step
Simulating QRW with any state of light! (1)The proposed optical set-up realizes the following unitary evolutionof the state of the walker after N steps
00ˆˆˆˆˆ Φ≡Φ=Φ ====NQW0)1(j1)T(j2)T(jN)T(jN UTUU...U
NNN 22211
ˆ
0 ... αχαχαχα =Φ→=ΦNQWU
For an input coherent state, this is easily explicitly calculated:
ααχαχαχαχαχαχα
ρρααααρ
2
ˆ
...)(
ˆˆ)(
N2NN222111
NQW 0
N QW N
U2 0
P
U UP NQW
d
d
∫∫
⊗⊗⊗=
=→=+
with χi that depend on the structure of the set-up.
For an arbitrary state, in the Glauber diagonal P-representation
Introducing randomness: decoherence in QRW
Let’s suppose we introduce an additional phase shift between twosuccessive operations T2 in the proposed set-up. We shift the state of the
walker, before each T2 , by 2πl, where l is randomly chosen from aGaussian distribution centred at 1 and with standard deviation σps. In
practice, the random phase shift (RPS) is as follows:
T2
T1 T2T2 T2T2
T2
T2 l,l,l are taken from
σps
1
Equal random shift 2πl
Equal random shift 2πl
Equal random shift 2πl
α
Quantum Quincunx in a Cavity:Phys Rev A67 (2003) 042305
• A two-level atom is the coin: a conditional Stark term shifts thephase of the cavity field clockwise or counterclockwise dependingon the state of the atom.
• Sequence of atoms for repeated random phase shifts: classical walk.• The cavity field initially has a “sharp” phase distribution (NB
domain is a circle, not a line).• variance grows linearly with number of atoms: phase diffusion
Recycling the Coin: quantum walk• The quantum ‘coin’ corresponds to the two levels of the atom, and this coin can be
recycled to give a quantum walk.• The cavity evolution is interrupted by periodically spaced Hadamard
transformations, which ‘flip’ the coin: F(ϕ) operates on the atom-cavity system inbetween these Hadamard coin flips.
• It is conceivable to apply (FH)15 within the timescale of an experiment (Paris groupparameters: J M Raimond, private communication)
• . For small ϕ, the results are similar to 1D random walk, but large ϕ is also possible:random walk on a circle.
Quantum Quincunx• Use a single atom (recycle the
coin)• Use π/2 pulse to implement
quantum coin flip• Quantum phase diffusion =
quantum quincunx• Phase spreads quadratically faster• Need open cavity (Haroche)
Conditional phase shifts?• Conditional phase-shift operator• Cavity prepared in a coherent state and the atom in
either the + or – state:• The phase of the cavity coherent state undergoes a
random walk in discrete steps of ϕ.• Ideally consider the (un-normalized) phase state
( ) ( )zaaiF σϕ=ϕ +exp
( ) ±⊗α=±⊗αϕ ϕ± 2/ieF
( ) ∑∞
=
φ∝φ±⊗ϕ±φ=±⊗φϕ0
for ,n
in neF
Wigner functions, α=3, phase step 2π/6?
T=4 T=3
T=2T=1
Note fringes as well as displacements-see via quadraturemeasurements
conclusions• Quantum version of Galton’s quincunx shows differences from
classical walks: interferences, quantum spreading• Feasible: Haroche group developed new cavity allowing Hadamards
during transit of atom (Yamaguchi quant ph-02)• Quadratic enhancement of phase diffusion: quantum speed up;
quantum algorithms• Can engineer decoherence: observe transition from quantum to
classical, suppression of interferences• Funding:
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