Maximal entanglement from quantum random walks

0.28 MB·English

by  B. Alles

Maximal entanglement from quantum random walks
B. All´esa, S. Gu¨ndu¨c¸b, Y. Gu¨ndu¨c¸c
aINFN Sezione di Pisa, Pisa, Italy
bHacettepe Vocational School, Hacettepe University, 06800 Beytepe, Ankara, Turkey
cDepartment of Physics, Hacettepe University, 06800 Beytepe, Ankara, Turkey
The conditions under which entanglement becomes maximal are sought in the general one–
dimensional quantum random walk with two walkers. Moreover, a one–dimensional shift operator
for the two walkers is introduced and its performance in generating entanglement is analyzed as a
function of several free parameters, some of them coming from the shift operator itself and some
others from thecoin operator. To simplify the investigation an averaged entanglement is defined.
2 03.67.Mn; 89.70.-a; 42.50.Dv
1
0
2
I. INTRODUCTION randomwalk steps may yield position states with strong

n entanglement among the two walkers. In particular, our
a
QRW has been modelled in such a way that a further
J Quantumrandomwalks[1–4](QRW)areageneraliza-
positionmeasurementonone walkerfixesthe positionof
1 tionofclassicalrandomwalks. Ateachstepofthe latter
the other, thus rendering the correlation among the two
1 the position of a classical particle (walker) is shifted ac-
walkers a maximum.

cordingtotheresultoftossingacoin. Thus,theposition
]
h of the walker is decided by following a probability dis-
The relevance of entanglement in QRW is crucial.
p tribution over classically well–defined positions. In the
- QRW case the tossing of the coin is substituted by the While this property does not appear in classical walks,
t it becomes the characterizing ingredient of QRW. The
n actionofaunitaryoperator(thatshallbecalledcoinop-
a erator) on a 2–state system (called coin state) and the interest of entanglement lies in the fact that it is of ut-
u mostrelevanceforresearchinquantumcomputationpro-
motion of the walker by a unitary transformation of the
q tocols and quantum information [22–26] (cryptography,
positionstatebyashiftoperator. Thestateofthewalker
[
communication, algorithms,etc.). We want to study the
is in generala quantum superposition ofseveralposition

2 eigenstates. growth of entanglement among the two walkers and, in
v particular, to discover whether maximal entanglement
Manyaspectsofsuchsystemshavebeenstudiedinthe
3 may be achieved. We also pay attention to the quan-
past [5–8]. In [9] the consequences of making one walker
2 tum probability P to obtain such entangled states after
0 move in more than one dimension are analyzed. The in-
the coin measurement.
6 terplay betweendecoherence and entanglementhas been
. unveiled in [10] while in [11] two coins have been used.
1
1 The so–called meeting problem has been studied in [12] As a novel ingredient of the present paper, we will in-
0 fortwowalkers. Weendthisshortreviewofrecentresults troduce a shift operator that to some extent generalizes
1 by recallingRef. [13]where the evolutionof the chirality the ones so far used in literature. Its form contains sev-
: of the coin has been investigated. eral free parameters which, added to the parameters of
v
i QRW also attract interest in some specific research the coin operator, supply enough freedom to easily find
X topics. For example, the efficiency of the energy transfer conditions under which entanglement is maximally en-
r inphotosynthesiscanbe raisedatthelevelofabout99% hanced.
a
if it is modelled by a QRW interacting with the thermal
fluctuations of the environment [14]. Besides, quantum The plan of the paper is as follows. In section II the
transport properties of electron systems and dielectric model for QRW is presented in detail. It includes a dis-
breakdown driven by strong electric fields can also be cussion about the general type of shift operator. An an-
studied by utilizing QRW [15]. In ancilla–based quan- alytic study of the QRW during the first steps in the
tum computation schemes [16] the ancilla plays the rˆole walk is shown in section III. This approach, though ap-
of the coinandthe measurementproduces entanglement plicable to a very limited number of random walk steps,
between the qubits. Moreover, experimental set–ups for turns out to be extremely useful to interpret the numer-
QRW are nowadays available [17–19]. ical results derived in section IV. Indeed, in the latter
Inthepresentpaperweproposeaquantumwalkmade section, thanks to a numerical treatment of the random
ofonespin 1 state(thecoin)coupledtotwoparticles(the walk evolution, we study the entanglement at an arbi-
2
walkers)whosepositionscantakeanyintegervalueonan trarilylargenumberofsteps. Inthissectionanaveraged
infinite line. QRW with two walkers are a valid tool for entanglement will be defined to simplify the search for
obtaining highly entangled states [20,21]. Indeed, per- the conditions of maximal entanglement. Some conclu-
formingmeasurementsonthe2–statecoinsystemaftern sive comments will be given in section V.
1
II. THE SPIN–PARTICLE QRW byadequatelychoosingthereferenceaxes. Suchasimpli-
ficationdoes notentaila loss ofgeneralityfor the reason
Webeginbystatingtheprecisedefinitionofthemodel that the position state and the coin state in (1) are un-
under study. The two walkers can roam on a line of correlated.
discrete spatial positions. The accessible position eigen- A single step in the random walk consists in the con-
states are i,i for i Z (i = 0 is the origin) where the catenation of two operators, Ushift Ucoin, iteratively ap-
firstitemr|eferistothe∈positionofthefirstwalkerandthe plied to φ . Operator Ushift acts o·n position states and
| i
second one of the second walker. Thus the two walkers Ucoin on spin states.
are supposed to stay together during the walk. The coin Let us begin by introducing the shift operator Ushift.
stateisaspin 1 system. Itseigenstatesarethespincom- A quite general form is
2
ponent along the Z–axis. The complete quantum state U = α(n) i+n,i+n i,i
is a sum of terms of the form φ s ψ where the shift |↑ih↑| | ih |
| i ≡ | i⊗| i Xn ↑↑ Xi
firstfactorreferstothespinandthesecondoneindicates
the position state. The symbol emphasizes the tensor + α(n) i+n,i+n i,i
structureofthestatespa⊗ce. Themostgeneralpureposi- Xn ↑↓ |↑ih↓|Xi | ih |
twioitnhstaite|cci|o2n=tem1.pTlahteedeibgyenovuerctmoorsdeolfitsh|eψZi–=coPmipcoin|ie,niit +Xn α↓(n↑)|↓ih↑|Xi |i+n,i+nihi,i|
tohfethZeP–scpoimnparoenednetno+te1d(by1a).nIatrsromwo:st|↑gien(|er↓ail)pstuarnedsstafoter + α(n)|↓ih↓| |i+n,i+nihi,i|, (3)
is d +d with2 d−22+ d 2 =1. A similar model Xn ↓↓ Xi
was↑i|n↑tiroduc↓e|d↓iin [20].| ↑| | ↓| with α(n) complex coefficients. They must satisfy
ss′
The above model is hardly realizable in laboratory several constraints derived from imposing unitarity,
(for a more physical model with two walkers see for ex- UshiftUs†hift = I, (I is the unit operator). After some
ample [27] and references therein). Rather it must be algebra, the l.h.s. of this condition is
viewed as the limiting case where entanglement is ex-
pected to be maximal. Indeed, imagine a more general i+n,i+n i+m,i+m
QRWmodelforwhichthepositionstateofthetwowalk- nX,m,i h |↓ih↑|| ih |
ers after the spin measurement(performed after n QRW
steps)is i,jcij|i,ji(i,j ∈Zand i,j|cij|2 =1)where, ×(cid:16)α↓(n↑)α(↑m↑)∗+α↓(n↓)α(↑m↓)∗(cid:17)
as beforeP, the first (second) item inPthe ket refers to the + i+m,i+m i+n,i+n
|↑ih↓|| ih |
first (second) walker. This state displays large entangle-
ment among the walkers if after a further measurement, ×(cid:16)α↓(n↑)∗α(↑m↑)+α↓(n↓)∗α(↑m↓)(cid:17)
this time on the position state of one of the two walkers, + i+n,i+n i+m,i+m
thepositionoftheotherwalkergetsfixedwithmaximum |↑ih↑|| ih |
probability. Evidently,the limiting caseofthe abovecir- α(n)α(m)∗+α(n)α(m)∗
×(cid:16) ↑↑ ↑↑ ↑↓ ↑↓ (cid:17)
cumstance occurs when j is a function of i, j = f(i)
+ i+n,i+n i+m,i+m
where f is a predetermined known function. As for f |↓ih↓|| ih |
wehavechosentheidentityfunction. Moreover,sincewe α(n)α(m)∗+α(n)α(m)∗ , (4)
have chosen such an idealized QRW model, the precise ×(cid:16) ↓↓ ↓↓ ↓↑ ↓↑ (cid:17)i
which must equal
connection among the spin variable and the walkers will
not be better specified. In the rest of the paper we will I= + i,i i,i. (5)
study the entropythat entanglesthe two particlesin the (cid:16)|↑ih↑| |↓ih↓|(cid:17)Xi | ih |
state c i,i after the coin measurement.
SincPeiwei|wanittostudythegenerationofentanglement, To fulfil the condition, the first two terms in (4) must
throughoutthepaperthefactorcorrespondingtothepo- be zero and note also that the second term is the h.c.
sition of the walkers in the initial state φ of the QRW of the first one. These considerations suggest to define
0
shall be free of entanglement, | i the complex 2–vectors V(n) (α(n),α(n)) and V(n)
1 ≡ 2 ≡
(α(n),α(n)) because in terms of t↑h↑em↑↓the above con-
φ =(d +d ) 0,0 , (1)
| i0 ↑|↑i ↓|↓i ⊗| i dit↓i↑ons ↓b↓ecome V1(n) · V2(m)∗ = 0, V1(n) · V1(m)∗ = 0,
wfichieenretsthde,adboviseansosrummaeldizaatnidonthcoensduibtisocnripretg0arsdtianngdcsofeofr- V2(n) · V2(m)∗ = 0 for all n 6= m and V1(n) · V2(n)∗ = 0,
initial sta↑te (↓0–th step of the QRW). Furthermore, since V1(n)·V1(n)∗ =V2(n)·V2(n)∗ =1foralln. Sinceincomplex
2–spacethereareonlytwoindependentorthonormalvec-
for any spin state s there exists some axis Z such that
| i ′ tors, there is no much freedom to choose V(n) and V(n).
s = ′ ,the statein(1)canbesimplifiedto(dropping 1 2
| i |↑i The most general solution reduces the sum over n and
primes)
m to one single term, n = p, m = q for p,q fixed inte-
φ = 0,0 , (2) gers and V(p) = (α,β), V(q) = ( β ,α ) (for complex
| i0 |↑i⊗| i 1 2 − ∗ ∗
2
numbersα, β) andallthe other V–vectorsequaltozero. Atanymomentthewalkcanbestoppedandthevalue
Moreoverthe two coefficients α,β satisfy α2+ β 2 =1. ofthespinmeasured. Ifthemeasurementwasperformed
| | | |
Thus, the shift operator is after each step in the iteration then a classical random
walk would result. Hence, our interest for QRW consists
U = α +β i+p,i+p i,i inrelegatingthemeasureaftermanystepsbecauseinthis
shift
(cid:16) |↑ih↑| |↑ih↓|(cid:17)Xi | ih | fashion clear entanglement is displayed. Let us consider
ameasurementperformedonlyafternQRWstepsgiving
+ α β i+q,i+q i,i, (6)
∗ ∗ spin s and a position pure state, result of the wavefunc-
(cid:16) |↓ih↓|− |↓ih↑|(cid:17)Xi | ih | tion collapse, denoted by ψ s and equal to
| in
with p,q Z fixed and α,β verifying the above normal-
∈ ψ s c i,i . (11)
ization. | in ≡ i| i
For p = q no entanglement is generated by the QRW Xi
as every term in U would move the couple of parti-
shift If the spin measurement yields up (down) spin we will
cles to the same spatial position. Instead, entanglement
write ψ up (ψ down). The evidententanglementcreated
aimppmeaatresriaaslfsainrcaesdpiff6=ereqn,tthpeairsspepc,iqficarvearlueleasteodf pb,yqrebsecianlg- among| tihne tw|oiwnalkersassociatedwith |ψisn shallbe cal-
culated by the von Neumann entropy
ings on the line of discrete positions where the walkers
wander. During the present study we will stick to the
E c 2log c 2, (12)
values p=+1 and q = 1 all the time. n ≡− | i| 2| i|
− Xi
Without loss of generality, we can take α real in (6).
Indeed, the spin eigenstates , can be redefined where the subscript indicates that the measurement has
| ↑i | ↓i
in such a way to absorb the phase of α. Calling ζ been performed after the n–th step. Certainly E 0
n
such a phase, transformations eiζ/2 and and maximal entanglement is attained when the va≥lues
| ↑i → | ↑i
|la↓rip→rocee−diζu/r2e|a↓bileeltimoiwniatthedritawfrotmhe(p6h)a.sTehoefrβetisoon.oTshimusi-, c|coin|taarinesaNll etqeurmals., (SEon,)imfatxhe=sluomg2Nin.|ψWisne s=haPll uicsui|ai,lliyi
the Ushift operator depends on only two variables: α calculate the normalized entanglement defined as
| |
(henceforth called just α > 0) and arg(β). In literature
the choice α=1, β =0 is generally found. En
. (13)
The coin operator U can be represented by a uni- En ≡ (E )
coin n max
tary 2 2 matrix. The most general such a matrix is an
elemen×t of the U(2) group, This choice of units allows to easily understand when
maximal entanglement has been attained. As the num-
Ucoin =(cid:18) √1 √ρρe−i(θ+η) √1√−ρρe−ei2(iθη−η) (cid:19) eiϕ, balesroNtheofdteenrommsininat|oψrisninin(1g3e)ndeerpaelnddespeonndns.oWnthheensNtep=n1,
− − both E and vanish, the former due to (12) and the
(7) n En
latter by definition.
where 0 θ,η π and 0 ϕ<2π are arbitrary phases
≤ ≤ ≤
and 0 ρ 1. When ϕ = η (7) becomes an element of
≤ ≤ III. ANALYTIC STUDY OF ENTANGLEMENT
the subgroup SU(2). Actually, the overall phase ϕ plays
GENERATION
norˆoleinthefollowingdiscussion,soinpracticewecould
limit ourselves to study SU(2) matrices to represent the
The first few steps of a QRW can be followed analyti-
most general coin operator.
cally. Startingfromstate(2),usingthecoinoperator(7)
ParticularinstancesoftheunitarymatrixU arethe
coin (where the irrelevant global phase eiϕ has been skipped)
Hadamard coin (ρ=1/2, ϕ=0 and η =θ =π/2),
and the shift operator (6) it is easy to see that entangle-
1 +1 +1 ment can only arise after two steps. Indeed, the walkers
UH = √2(cid:18)+1 1(cid:19), (8) plus coin state after the first step (indicated with a sub-
− script 1) is
the Kempe coin [2] (ρ=1/2, ϕ=η =0 and θ =π/2),
φ 1 =(α√ρ β 1 ρe−i(θ+η)) 1,1
| i − − |↑i⊗| i
UK = √12(cid:18)++1i ++1i (cid:19), (9) −(β∗√ρ+αp1−ρep−i(θ+η))|↓i⊗|−1,−1i, (14)
and the wavefunctions after the measurement, ψ up =
| i1
or the Z coin (ρ=1, η =π/2, ϕ=0 and θ arbitrary), 1,1 and ψ down = 1, 1 , whenever they exist, are
| i | i1 |− − i
clearlyfreeofentanglement. Letusassumethatthespin
+1 0 measurement after the second step yielded +1. The re-
U = . (10) 2
Z (cid:18) 0 1(cid:19) sulting entangled position state is
−
3
2
|ψiu2p ∝(cid:16)α√ρ−βp1−ρe−i(θ+η)(cid:17) |2,2i ttehramtwmiasexi(mexacleepnttafonrgalepmoesnsitboleccgulorsbawlhpehnasaer)g,(wβe)=disθco+veηr.
e−2iη α 1 ρ+β√ρe−i(θ+η) 2 0,0 . (15) Ageneralizationoftheseanalyticresultstoniterations
− (cid:12) p − (cid:12) | i ofthe QRW is too complicatedas the length ofthe coef-
The presence of(cid:12)(cid:12)the proportionality symb(cid:12)(cid:12)ol reminds the ficients in |ψinup,down grows exponentially with n. Then,
in the following section we study the above problem nu-
lackofanormalizationconstant. We willoftenwrite the
mericallywitha computerprogram. The programyields
wavefunctionresultingfromthespinmeasurementinthis
exact numerical results but without the corresponding
way purposely because it renders more transparent the
explicit analytical expression.
evaluation of the probability P with which such a state
comes out (P is just the square of the missing normal-
ization). It is easy to verify that maximal entanglement
IV. ENTANGLEMENT AFTER AN ARBITRARY
is achieved when the condition
NUMBER OF STEPS
α√ρ β 1 ρe−i(θ+η)
(cid:12) − p − (cid:12) In this section the entanglementcreatedby measuring
(cid:12) (cid:12)
(cid:12)= α 1 ρ+β√ρe−i(θ+(cid:12)η) (16) the spin of the coin after an arbitrary number of QRW
(cid:12) p − (cid:12) steps having started with the initial state (2) shall be
(cid:12) (cid:12)
(cid:12) (cid:12) studied. To this end we introduce an averagedentangle-
holds. If the Hadamard coin were used, the condition
ment definedoverthefirstnstepsasfollows. Consider
would became α+β = α β which is fulfilled when- n
E
| | | − | n replicas of the QRW andevolve eachofthem indepen-
everαisrealandβ pureimaginary. FortheKempecoin,
dently of the others. In the first replica we measure the
condition(16)reduces to α+iβ = α iβ whichis sat-
| | | − | spin and then the entanglement after the first step ob-
isfied for any α and β as far as both are real. For the Z
taining ,inthesecondoneafterthesecondstepobtain-
coin, condition (16) requires α = β . 1
E
| | | | ing , etc., thus gathering the collection , , , ...,
If the result of the measurement after two steps yields 2 1 2 3
E E E E
spin 1, then the position state is n. On the understanding that eachmeasurementyields
−2 tEhesameresult(i.e.,eitherallmeasurementoutputsgive
up spin or all give down spin), we define
ψ down αβ(1 ρ)e 2i(θ+η)+ β 2 ρ(1 ρ)e i(θ+η)
| i2 ∝(cid:16) − − | | p − − n
1
−α2pρ(1−ρ)e−i(θ+η)−αβ∗ρ(cid:17)|0,0i En ≡ n−1Xa=2Ea. (19)
+ αβ (1 ρ)e 2iη+(β )2 ρ(1 ρ)ei(θ η)
∗ − ∗ −
(cid:16) − p − Thefirststep(a=1)isexcludedfromthesumintheav-
α2 ρ(1 ρ)e−i(θ+η)e−2iη eragebecauseitgenerateszeroentanglementinanycase.
− −
αβpρe 2iη 2, 2 . (17) Notethatin(19)the subscriptndoesnotmeanthe step
∗ −
− (cid:17)|− − i after which the measurementis done, as in (12), but the
number of steps over which the average is performed.
To findthe circumstancesunder whichentanglementbe-
Theconvenienceofusingsuchanaveragedentanglement
comes a maximum we discuss different values of ρ sep-
becomesobviouswhenit is1 because,being the meanof
arately. When either ρ = 0 or 1 then only one term
manypositivenumbersnotlargerthan1,sucharesultfor
survives in each coefficient of (17) and maximal entan-
necessarilyimpliesthatallofthemareprecisely1,i.e.
glement occurs for all θ, η, α and arg(β). For ρ = 1 En
2 it identifies situations where entanglement is constantly
entanglement is a maximum again for all possible values
maximal (therefore the precise value of n used to evalu-
that the parameters can take. To see this we substitute
ate is largely immaterial, as long as it is sufficiently
ρ= 1 in (17) and obtain after some algebra En
2 large). This assertion applies equally well when n = 0,
E
but clearly this case is physically less interesting.
|ψid2own ∝e−i(θ+η)(cid:16)|β|2−α2+2iα|β|sin∆(cid:17)|0,0i abAovec–odmespcuritberedcpordoecewduarsepforerptahreedQRtoWimduprlienmgeantnutmhe-
+e−2iηe−iarg(β) (2β 2 1)cos∆ ber n of steps. It calculates step by step the entangle-
(cid:16) | | −
ment,eitheraveragedornot,aswellastheprobabilityof
isin∆ 2, 2 , (18)
havingupordownspinafterthemeasurementofthecoin
− (cid:17)|− − i
state and also the number of terms N in the collapsed
where ∆ arg(β) θ η. It is easy to verify that both position wavefunction (the latter enables us to compute
≡ − −
coefficients have the same modulus. the exact value of (E ) ). The numerical procedure
n max
Aninstanceofmaximumentanglementcanalsobede- used to extract N was demanding that c in (11) be
i
duced for the general case 0 ρ 1. First note that larger than a pre–fixed threshold, in our c|as|e 10 10. All
−
≤ ≤
everyterminthetwocoefficientsin(17)areequal,apart the results presented in this section have been obtained
from phases. Imposing that the phases are also equal by using this code.
4
The next three subsections contain the study for the QRW is φ = 1,1 . Proceeding with the choice
1
three coin operators of section II, namely the Hadamard α = β =| i1 , a|c↑ailc⊗ul|ationi similar to that exhibited in
√2
UH, the Kempe UK and UZ, respectively. The last sub- sectionIII allowsto showthat φ 2 = 2,2 . By in-
section is dedicated to the general coin case, (expres- duction this can be immediatel|y giener|a↑liiz⊗ed|to ainy step,
sion (7) with ϕ=0). getting φ = n,n . Hence,entanglementvanishes
n
when α|=iβ =|1↑i.⊗| i
√2
Now we show why for step n = 2 the entanglement
of the state resulting from a down spin measurement
1 approaches its maximal value but suddenly drops to
zero at α = β = 1 . Taking α,β in (17) real and
√2
0.8 close to each other leads to the expression ψ down
β=+(1-α2)1/2 (β2 α2)( 2, 2 0,0 )which indeeddisp|layi2s max∝i-
− |− − i−| i
0.6 mal entanglement although it suddenly becomes zero as
ε soon as α exactly equals β. This expression also indi-
200 |ψ>down
0.4 catesthatdespitetheentanglementofthestateresulting
|ψ>up from a down spin measurement converges to the maxi-
0.2 mal value, as α tends to β the probability of obtaining
such a measurement output becomes very small, of or-
der O(β2 α2 2). It seems reasonableto expect that an
0
0 0.2 0.4 α 0.6 0.8 1 analogo|us−mech|anism explains the low probability also
FIG. 1. Averaged entanglement obtained during the first for n>2.
200stepsofaQRWwiththeHadamardcoin(8)andshiftop- InFig.2theratio n(thistimenon–averaged)forthree
E
erator(6)withα,βreal. Theplotsfordown(up)spinoutput different values of α (taking both α and β realand posi-
of the measurements are shown with a continuous (dashed) tive) are shown as a function of n. It is clear that, as α
line. Symbols ψ up,down here and in thenext figures refer to approachesβ,bothtendingto 1 ,theentanglementgets
| i √2
theoutput of the spin measurement. stuck to the maximum for more and more steps. There-
foreweinferfromFig.2thatwhenα=0.71theaveraged
entanglementafter200stepsis 1whileaftermore
200
E ≈
A. Hadamard coin steps, say 800, it is patently less than 1, < 1. In-
800
stead for α = 0.7071, a figure closer to E1 , both
√2 E200
This is the most frequently used coin operator for the and 800 are firmly anchoredto 1. Recall that, although
E
kind of problems at hand. Its expression is given in (8). not shown in this graph, we have proved before that at
Tostudy the entanglementgeneratedbyaQRWevolved exactly α = β = 1 the entanglement drops to zero.
√2
with this coin, we first calculate for n = 200 steps Theplotforothervaluesofαarequalitativelyanalogous
n
E
for real and positive values of α and β = +√1 α2 to the one shown in the jagged line of Fig. 2 (here and
−
in (6). The result is displayed in Fig. 1 as a function throughout the rest of the paper, the number 0.37 will
of α (0 α 1). just represent an arbitrary value of α). Thus, a method
≤ ≤
TheplotsinFig.1showthatasαandβtendtobealike toobtainhighlyentangledstateswouldbeusingrealval-
andreal(thismeansbothapproachthevalue 1 )entan- ues of α and β, very close to each other, and look for a
√2
glement tends to be maximal (minimal) if the measure- down spin measurement output in the first QRW steps.
ment yields down (up) spin. The number of entangled Thedifficultyisthat,asexplainedabove,suchdownspin
termsN inthecollapsedwavefunctionafterthemeasure- outputs are very unlikely.
ment is equal to the step number n. This must be inter- Next we add a phase to β while keeping α real. In
pretedasthatentanglementgetsricherandricherasthe this way we will have covered all relevant (real or com-
QRW goes on. However at the exact values α=β = 1 plex) numerical values for α, β. No dramatic changes in
√2 theentanglementgenerationarefound. Inparticular,no
entanglement suddenly drops to zero for both kinds of
cases with arg(β)=0 exist with maximal entanglement,
measurement outputs. This is how the vertical line in
see Fig. 3 and Fig6. 4. When α = 1 the dependence on
Fig. 1 (which applies to the down spin measurement) 6 √2
mustbe deciphered. Moreover,the probabilityofhaving arg(β) is qualitatively as in Fig. 3. For α = β = 1
− √2
such maximally entangled states (for α β and both the averaged entanglement goes to zero, as Fig. 4 shows
≈
real) after the measurement is extremely small, P 1. manifestly, meaning that the true, non–averaged entan-
≪
The origin of all the above facts can be clarified by glement vanishes at all the QRW steps.
using the analytical expressions obtained in section III. Theuseoftheaveragedentanglement afternQRW
n
First of all we recall that for any values of α and β the steps must not divert our attention awaEy from the fact
first step always gives no entanglement at all. For in- that the real interesting quantity is the normalized en-
stance, taking α = β = 1 the state after one step of tanglement achievedby the quantum state after each
√2 En
5
single measurement. Our computer code allows to dis-
1
coverisolatedcases of maximalentanglement(being iso-
lated, they get lost in the average of ). They occur
En 0.8
mainly in the first few steps of the QRW and are quite
likely since the probability of obtaining the related spin ε 0.6 α=0.7071~~2-1/2
measurement is not very low. n α=0.71 β=+(1-α2)1/2
Letus enumerateallcasesof this kind foundwhen us- 0.4 α=0.37
ing the Hadamard coin. We start by the position states
0.2
originated by measurements that led to down spin: the
entanglement is a maximum at the second step for all α
or β (real or complex) (except for α = β = 1 as we 00 200 400 600 800
√2 n
already know) and at steps n = 3 and 4 when the argu- FIG.2. Entanglementobtainedinthefirst800QRWsteps
mentofβ is arg(β)=π/2,3π/2. Ifinsteadthe outputof after a down spin measurement with the Hadamard opera-
the measurement at n=2 is up spin, then the entangle- tor and α,β real. The continuous line shows the result for a
mentisamaximumforallαrealandarg(β)=π/2,3π/2. value of α very close to 1 = 0.7071067812 . The dashed
√2 ···
When n = 2 these features can be proved by resorting line corresponds to a value of α not so close to 1 and the
to expressions (15) and (17) (but the formalism of sec- √2
lower continuous (jagged) line to a value decidedly different
tionIIIdoesnotsufficetoinvestigatethepresenceorab- from 1 .
senceofmaximalentanglementbeyondthe secondQRW √2
step). Moreover,inallcasestheprobabilityforobtaining
theindicatedresultfromthespinmeasurementisaround
P 0.2 0.5 and the number N of entangled terms in 1
∼ −
the wavefunction collapsed after the measurement coin-
cides with the QRW step number, N = n except when 0.8
n=3forwhichN is2. Thereseemstobenoothersteps
n with n equal to 1. We have checked this assertion up 0.6 |ψ>down
E
to n as large as 1000. ε
200 |ψ>up
0.4
0.2 α=0.37 |β|=(1-α2)1/2
B. Kempe coin 0
0 1 2 3 4 5 6
arg(β)/radians
FIG. 3. Averaged entanglement 200 as a function of the
phase of parameter β for a value ofEα far from 1 with the
√2
When using the Kempe coin (9), the average entan- Hadamard coin.
glement turns out to be independent of α,β as far as
n
E
both coefficients are real andpositive (but depends on n
andontheresult,upordownspin,ofthemeasurement).
1
However, the dependence on the argument of a complex
β displays the rich structure shown in Fig. 5. The first
0.8
aspect of this graph to highlight is that, once fixed the
result of the spin measurement, the averaged entangle-
0.6
mentsatrealandpositiveα,β (therightmostorleftmost ε
edgesinthefigure)areindeedthesameforthetwovalues 2000.4 α=|β|=2-1/2 |ψ>down
of α that have been utilized. The second aspect is that |ψ>up
for down spin measurements and α = β = 1 there is
| | √2 0.2
maximal entanglement as arg(β) tends to become 3π/2
but at this precise value, it suddenly drops to zero (indi-
0
cated by the vertical line at this value of arg(β)), a phe- 0 1 2 3 4 5 6
nomenon similar to the one described for the Hadamard arg(β)/radians
wcoaivnefiunnFctiigo.n1.aftIenr fnacstt,eapts αis =φ √12=and β =n−,n√i2. tAhes phFasIeGo.f4.paAravmereatgeerdβefnotranαg=lem1entwiEt2h00thaesHaafduanmctairodncoofint.he
n √2
| i | ↑i⊗ | i
also happened for real values α β in the Hadamard
≈
coin, the probability of having this output after the spin The behavior near arg(β) = π/2 can be described as
measurement is negligibly small for arg(β) 3π/2. follows: the averaged entanglement becomes = 0.5
200
≈ E
6
for α = β = 1 and arg(β) close to π/2 while it drops π/2 3π/2
| | √2 1
to zero when the phase attains this precise value. This
patterncanbespelledoutbystudyingthenon–averaged
0.8
entanglement: for α β 1 and arg(β) = π/2 it
≈ | | ≈ √2
is maximal for even QRW steps and a down spin mea-
0.6
surement while is zero for odd steps. Instead, for up ε
spin output, the order is reverted: for even steps is 200
0.4
zero and for odd steps is maximal. This explains the
value 0.5 that attains for both spin measurements
E200 0.2
near arg(β) = π/2 in Fig.5 (in other places where
200
E
equals 0.5 the above–described alternance between even
0
and odd steps is not seen). Unfortunately the probabil- 0 1 2 3 4 5 6
ity of having such measurement outputs (down spin for arg(β)/radians
even steps or up spin for odd steps) is negligibly small, FIG. 5. Averaged entanglement 200 as a function of the
E
P 1. Furthermore, in all these cases the number of phaseofparameterβ forvariousvaluesofαwiththeKempe
≪
terms in the collapsed wavefunction is just N = 2 for coin. The thick continuous line refers to down spin measure-
any step n denoting a rather poor entanglement. When ment output with α= β = 1 ; the thick dashed line to up
| | √2
exactly α = β = 1 and arg(β) = π/2, the entan- spinandα= β = 1 ;thethincontinuouslinetodownspin
| | √2 | | √2
glement vanishes at all steps for any measurement out- with α=0.37 and β =+√1 α2; and thethin dashed line
put (actually, the wavefunction after n QRW steps is to up spin with α=| 0|.37 and −β =+√1 α2.
| | −
φ = n, n ). This accounts for the zero of
n
| i inF| i↓gi.⊗5a|t−stric−tlyiarg(β)=π/2andα= β = 1 .
E200 | | √2
Againwestudythenon–averagedentanglementduring 1
the initial steps fo the QRW. If the measurementoutput
is down spin then entanglement is maximal at the steps 0.8
n = 2,3,4 of the QRW for all α,β real and at the step
n=2 for α real and any complex β (the cases for n=2 0.6
follow also from (17)). When instead the measurement ε
gives up spin, the QRW produces a state with maximal 2000.4 β=+(1-α2)1/2 |ψ>down
entanglementonlyatthe secondstepforallαandβ real |ψ>up
(it matches with the discussion after expression (16)).
0.2
All the above events with maximal entanglement have
a significant probability to occur, P 0.2 0.5 and
∼ − 0
the number of terms N in the collapsed position wave- 0 0.2 0.4 α 0.6 0.8 1
functioncoincideswiththeQRWstepnexceptforn=3,
downspinandα,β real,forwhichN is2. Upton=1000 FIG.6. Averagedentanglement E200 asafunctionof αfor
real α,β with theZ coin.
we have found no other maximum non–averaged entan-
glement results for any coin measurement output.
Maximal entanglement can be achieved only in spe-
cific cases. A detailed analysis performed with the non–
averaged entanglement enables us to discover that in-
deed this event happens for all real α and all complex
C. Z coin
β when the measurement output yields down spin at
the step n = 2 of the QRW. The number of entan-
gled terms in the collapsed wavefunction ψ down after
| i2
the spin measurement is N = 2 and the probability is
InFig.6 weshowthe averagedentanglementobtained
P 0.2 0.5, again definitely far from zero. Further-
withthe useofthe Z coin(10)asafunctionofαforreal mo∼re, for−α = β = 1 and arbitrary arg(β) maximal
β. From this plot it is apparent that there are no val- | | √2
ues of these parameters that allow to have maximal or entanglement appears for n = 2,3,4 steps and with a
almost maximal entanglement during the entire QRW. similarlyhighprobability. Inthiscase,though,thenum-
Moreover,havingfixedthe (real)valuesofαand β , the ber N of entangled terms in the resulting wavefunction
plot of as a function of arg(β) (not shown)|c|omes after the spin measurement is equal to the step n only
200
out flat,Eindicating no dependence on this phase. There- for n =2,4 while for n = 3, N is 2. On the other hand,
fore we conclude that the averaged entanglement never if the measurement output is up spin, then the entan-
attains the maximum when the Z coin operator is used, glement will become maximal only when α = β = 1
| | √2
implying that there areno values of α,β that allowmax- and arbitrary arg(β) at the second step with P = 0.5
imal entanglement for all (or almost all) n>2. and N = 2 entangled terms. As α and β tend to
| |
7
be equal, the entanglement grows and the probability state is also completely general because by an adequate
approaches 0.5. All these results for n = 2 can be definition of the axes orientation any spin state can be
easily recovered by using the formalism of section III. viewed as the up spin eigenstate along the Z–axis.
In particular, we have ψ up α2 2,2 + β 2 0,0 and
| i2 ∝ | i | | | i
|ψid2own ∝α∗β∗|−2,−2i−αβ∗|0,0i. We have introduced a shift operator in the QRW with
two walkers. It includes two free parameters that, to-
gether with those present in the coin operator, provides
D. The general coin enough freedom with which a measurement performed
on the coin state may yield maximal entanglement on
the resulting position quantum state. We have studied
We have also run a battery of QRW with the general
the problemboth analyticallyalongthe firststeps ofthe
coin operator (expression (7) with ϕ = 0) in order to
QRW and by a numerical computer code that allows to
look for precise combinations of the parameters in U
coin
probeanarbitrarilylargenumberofQRWsteps. Besides
allowing maximal entanglement. The initial state was
the entanglement, we have measured the probability of
still given by (2).
having such highly entangled states and the quality of
Twokindsofrunsweredone. Inthefirstonethe aver-
their entanglement, given by the number of terms that
aged entanglement was calculated after 200 QRW steps
the collapsed position wavefunction contains. We have
for various values of ρ, θ, η (in U ) and α, arg(β) (in
coin
also devised an averaged measure of the entanglement
U ). The ranges being 0 ρ,α 1, 0 θ,η π and
shift
≤ ≤ ≤ ≤ that simplifies the search for a QRW with all or almost
0 arg(β) < 2π, data were taken by varying the values
≤ all steps presenting high entanglement.
of each parameter by jumps of 0.05. The goal was to
find QRW presenting maximal entanglement during all
the steps. WehaveusedtheHadamard,KempeandZ coinoper-
The second kind of run was designed to find isolated atorsasthey are the mostgenerallyseenthroughoutthe
casesofmaximalentanglementduring the firstfew steps literature. Moreover, in subsection IVD we also tackled
ofthe QRW.Forthis reasonthe QRWwereevolvedonly the general coin operator.
for 10 steps and the non–averaged normalized entangle-
ment was extracted. The same five parametersas before In general maximal entanglement is more frequently
were swept within their ranges. generated when the spin measurement output is oppo-
Nohighlyprobableparameterregionsproducingmaxi- site to the one used as initial state, in our case down
malornearmaximalaveragedentanglement( >0.99 spin.
200
E
and P > 0.15) were found. Instead our computer runs
wereplentyofisolatedcaseswithmaximalentanglement We have found two kinds of situations with maximal
at steps n=2,3,4 of the QRW, ( n =1 and P >0.15). entanglement: (i) near maximal entanglement through-
E
There were so many of them that giving a complete out almost the entire QRW, regardless of the number of
record lies beyond the limits of this short paper. Most steps, but with negligible probability to occur P 1
of those at step n = 2 are summarized in the general (the larger is the number of steps, the lower is the v≪alue
discussion for arbitrary ρ after (17). of P) and (ii) exactly maximal entanglement during a
few steps at the beginning of the QRW with moderately
high probabilities, P 0.2 0.5.
∼ −
V. CONCLUSIONS
Therearenohighlyprobableentangledstatescontain-
Aquantumrandomwalk(QRW)modelwithtwowalk- ing a large number of entangled terms. Only during the
ershasbeendevisedtostudythegenerationofmaximum firststepsoftheQRWonecanachievemaximalentangle-
entanglement among the two walkers by the process of ment while having a reasonably high probability. How-
coinmeasurement. Themodelisratherunphysicalbutit ever, such states contain a limited number of entangled
hasbeenformulatedinthewayitisinordertomaximize terms precisely because they occur at the first steps of
the entanglement among walkers. the QRW. This happens for instance for downspin mea-
Theallowedpositionsofthewalkersarethesetof(pos- surement at the second step for arbitrary α and arg(β)
itive,negativeornull)integersalonganinfiniteline. The with any of the three coins mentioned in section II. The
2–state coin is represented by a spin 1 system. Every largest absolute entanglement with an acceptably high
2
QRW was started with the state 0,0 , that is, the probability and with the above three coins is achieved
|↑i⊗| i
spin in the (Z–component) up eigenstate and the walk- afteradownspinmeasurement,occurringwithprobabil-
ers at the originofthe line. The only considerationused ityP =0.25,atthen=4stepandmanagingtoentangle
to select this choice of the initial state was starting the N = 4 terms in the position wavefunction (thus giving
walk with a position state free of entanglement. With E4 = log24 = 2). They are: (i) Hadamard coin for all
this premise, and by a redefinition of the origin, any ini- α andarg(β)=π/2,3π/2,(ii) Kempe coinwith anyα,β
tialpositioneigenstate is asvalid asany other. The spin real and (iii) Z coin with α= β = 1 and all arg(β).
| | √2
8
VI. ACKNOWLEDGEMENTS ing problem in the quantum walk”, J. Phys. A: Math.
Gen. 39, 14965, (2006).
[13] A. Romanelli, “Distribution of chirality in the quantum
walk: Markov process and entanglement”, Phys. Rev. A
Part of the work was done during a visit of B.A. to
81, 062349, (2010).
the Hacettepe University. B.A. thanks the warm hospi-
[14] M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru–
tality during the stay. B.A. also wishes to thank Emilio
Guzik, “Environment–Assisted Quantum Walks in Pho-
d’EmilioandHans–ThomasElzeforusefulconversations
tosynthetic Energy Transfer”, J. Chem. Phys. 129,
and Vivien Kendon for providing us with some refer-
174106, (2008).
encesonexperimentalrealizationsofQRW.Weacknowl-
[15] T.Oka,N.Konno,R.Arita,andH.Aoki,“Breakdownof
edge the CINECA (Bologna, Italy) ISCRA Award N.
anElectric–Field DrivenSystem: AMappingtoaQuan-
HP10CB7JY6.2010 for the availability of high perfor- tum Walk”, Phys.Rev.Lett. 94, 100602 (2005).
mance computing resources and support. [16] J. Anders, D. K. L. Oi, E. Kashefi, D. E. Browne1 and
E. Andersson, “Ancilla–driven universal quantum com-
putation”, Phys.Rev.A 82, 020301, (2010).
[17] J.Du,H.Li,X.Xu,M.Shi,J.Wu,X.ZhouandR.Han,
“Experimentalimplementationofthequantumrandom–
walk algorithm”, Phys. Rev.A 67, 042316, (2003).
[18] C.A.Ryan,M.Laforest,J.C.BoileauandR.Laflamme,
[1] Y. Aharonov, L. Davidovich and N. Zagury, “Quantum “Experimental implementation of a discrete–time quan-
random walks”, Phys.Rev.A 48, 1687, (1993). tumrandomwalkonanNMRquantum–informationpro-
[2] J. Kempe, “Quantum random walks: an introductory cessor”, Phys. Rev.A 72, 062317, (2005).
overview”, Contemporary Physics 44, 307, (2003). [19] H.B.Perets,Y.Lahini,F.Pozzi,M.Sorel,R.Morandotti
[3] I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon and and Y. Siberberg, “Realization of Quantum Walks with
P. L. Knight, “Entanglement in coined quantum walks Negligible Decoherence in Waveguide Lattices”, Phys.
on regular graphs”, New J. Phys.7, 156, (2005). Rev. Lett.100, 170506 (2008).
[4] V. Kendon,“Decoherence in quantum walks: a review”, [20] S. E. Venegas–Andraca and S. Bose, “Quantum Walk–
Math. Struct.in Comp. Sci 17, 1169, (2006). based Generation of Entanglement Between Two Walk-
[5] N. Konno, “Quantum random walks in one dimension”, ers”, arXiv:0901.3946.
QuantumInformation Processing 1, 345, (2002). [21] S. K. Goyal and C. M. Chandrashekar, “Spatial entan-
[6] G.Abal,R.Siri,A.RomanelliandR.Donangelo,“Quan- glementusingaquantumwalkonamany–bodysystem”,
tumwalk ontheline: Entanglement andnonlocal initial J. Phys.A: Math. Theor. 43, 235303, (2010).
conditions”, Phys.Rev.A 73, 042302, (2006). [22] E. Farhi and S. Gutmann, “Quantum computation and
[7] R. P. Feynman and A. R. Hibbs, “Quantum mechanics decision trees”, Phys.Rev. A 58, 915, (1998).
and path integrals”, McGraw–Hill, (1965). [23] A. Nayak and A. Vishwanath, “Quantum Walk on the
[8] D.Meyer,“Fromquantumcellularautomatatoquantum Line”, quant–ph/0010117.
latticegases”,J.Stat.Phys.85,551,(1996);“Ontheab- [24] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J.
senceofhomogeneous scalar unitarycellular automata”, Watrous“One–dimensionalquantumwalks”,inProc.33
Phys.Lett. A 223, 337, (1996). Annual ACM Symposium on Theory of Computing, pg.
[9] G. Grimmett, S. Janson, and P. F. Scudo, “Weak limits 37–49 (2001).
for quantum random walks”, Phys. Rev. E 69, 026119, [25] J. Watrous, “Quantum simulations of classical random
(2004). walks and undirected graph connectivity”, J. Comput.
[10] O. Maloyer and V. Kendon,“Decoherence versus entan- Syst. Sci. 62, 376, (2001).
glement in coined quantum walks”, New J. Phys. 9, 87, [26] M.A.NielsenandI.L.Chuang,“Quantumcomputation
(2007). andquantuminformation”,CambridgeUniversityPress,
[11] C. Liu and N. Petulante, “One–dimensional quantum (2000).
random walks with two entangled coins”, Phys. Rev. A [27] K. Mayer, M. C. Tichy, F. Mintert, T. Konrad and
79, 032312, (2009). A. Buchleitner, “Counting Statistics of Many–Particle
[12] M.Sˇtefanˇ´ak,T.Kiss,I.JexandB.Mohring,“Themeet- Quantum Walks”, arXiv:1009.5241.
9