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Quantum Walk-based Generation of Entanglement Between Two Walkers Salvador E. Venegas-Andraca1 and Sougato Bose2 1Quantum Information Processing Group, Tecnol´ogico de Monterrey Campus Estado de M´exico, Carretera Lago Gpe. Km 3.5, Atizap´an de Zaragoza, Edo. M´exico, 52926, M´exico [email protected], [email protected] 2Department of Physics and Astronomy, University College of London. Gower Street, London WC1E 6BT, United Kingdom [email protected] Quantumwalkscanbeusedeitherastoolsforquantumalgorithmdevelopmentorasentanglement generators, potentially useful to test quantum hardware. We present a novel algorithm based on a discrete Hadamard quantum walk on a line with one coin and two walkers whose purpose is to generate entanglement between walkers. We provide several classical computer simulations of 9 ourquantumalgorithm inwhich weshowthat,although theasymptoticalamount ofentanglement 0 generatedbetweenwalkersdoesnotreachthehighestdegreeofentanglementpossibleateachstepfor 0 eithercoinmeasurementoutcome,theentanglementratio(entanglementgenerated/highestvalueof 2 entanglementpossible,foreachstep)tendstoconverge,andtheactualconvergencevaluedependson n thecoininitialstateandonthecoinmeasurementoutcome. Furthermore,ournumericalsimulations a show that, for the quantum walks used in our algorithm, the value towards which entanglement J ratio converges also depends on the position probability distribution symmetry of a quantum walk 6 computed with one single walker and the same coin initial state employed in the corresponding 2 quantumwalk with two walkers. ] PACSnumbers: PACSnumbers: 03.67.Bg,03.67.Lx,03.67.-a,03.67.Ac h p - I. INTRODUCTION building methods in order to test the “quantumness” of t n emerging technologies for the creation of quantum com- a puters. Thusitisofcrucialimportancetodevelopmeth- u Quantum walks were designed as quantum counter- ods of entanglement generation through quantum walks q parts of classical random walks, a branch of stochastic [ sothatthegenuinequantumnatureofagivenwalkwith processes widely used in algorithm development. Al- given physical systems may be tested. 1 though some authors have selected the name “quantum v random walk” to refer to quantum phenomena [1, 2, 3] Quantum entanglement has been incorporated into 6 and, in fact, in the seminal work by R.P. Feynman [4] quantum walks research either as a result of performing 4 aboutquantummechanicalcomputerswefindaproposal a quantum walk [20, 21, 22, 23, 24, 25] or as a resource 9 thatcouldbeinterpretedasa(continuous-time)quantum to build new kinds of quantum walks [26, 27, 28, 29]. 3 walk[5],itisgenerallyacceptedthatthefirstpaperwith Sinceentanglementisakeycomponentinquantumcom- . 1 quantum walks as its main topic was published in 1993 putation,itisworthkeepinginmindthatquantumwalks 0 by Aharonov et al [6]. Thus, the links between classical canbeusedeitherasentanglementgeneratorsorascom- 9 random walks and quantum walks, as well as the utility putational processes taking advantage of this quantum 0 ofquantumwalksincomputerscience,aretwofreshand mechanical property. : v open areas of research. In this paper we introduce a novel algorithm based i X Since one of the main goals in quantum computing is on a discrete quantum walk on a line with one coin and r the development of quantum algorithms, and given the two walkers whose purpose is to generate entanglement a success of employing classicalrandom walks for comput- between walkers. After evolving the quantum walk for ingsolutionstoNP-completeproblems[7,8,9],therehas a certain number of steps, we perform a measurement been a huge interest in understanding the physical and on the coin state. We then obtain a post-measurement computationalpropertiesofquantumwalksoverthe last quantum state composed by the tensor product of one few years on both experimental [10, 11, 12, 13, 14] and coin state and several walker components. We take the theoretical research communities (see [15] for a review walker components of this post-measurement state and ontheoreticalaspects of quantum walks). In addition to calculate the entanglement between walkers. We per- their usage in computer science, the study of quantum form many quantum walks with the same initial condi- walks is relevant to the modelling of physical phenom- tions and evolution operators, so that we have a quan- ena such as energy transfer in photosynthetic systems tum walk ready to be measured for each time step. In [16]. Moreover, although it has been proved that cer- addition to our algorithm, we provide several simulation tain properties of quantum walks are also reproducible results using different initial conditions for the proposed by classical systems (like variance enhancement with re- quantum walks. While this analysis highlights the po- specttoclassicalrandomwalks[17,18,19]),itisalsotrue tential of a quantum walk to entangle high dimensional that uniquely quantum mechanical properties of quan- quantum systems (the dimension of the space available tum walks, such as entanglement, may be employed to to the walkers grow in each step) it can also at times be 2 practically useful. This will be the case when the walk- 02. While (t n) ≤ ers are systems which do not directly interact with each 03. Apply the evolution operator Uˆt = (Sˆ(Cˆ Iˆ))t to other such as two different electromagnetic field modes. ψ 0. ⊗ Then the coin canbe a commonsystemsuchas anatom 0|4i. Perform a measurement on the coin system. Since which interact with both and can entangle them to a coin 2 there are only two possible outcomes. We high degreewith the degree depending onthe number of l|abelith∈emHα0 and α1. steps possible within the reasonable decoherence time of 05. For outcome α0 then the fields. 06. Compute the post-measurement quantum state ψ c0 | it,pm 07. Quantify entanglement between walkers from quan- II. ALGORITHM FOR ENTANGLEMENT tum state ψ c0 GENERATION | it,pm 08. For outcome α1 then 09. Compute the post-measurement quantum state Inthissectionwepresentouralgorithmforthegenera- ψ c1 | it,pm tionofentanglementina familyofquantumwalksonan 10. Quantify entanglement between walkers from quan- unrestrictedline. Asuccintmathematicalrepresentation tum state ψ c1 of a quantum walk after n steps is 11. Increas|eitt,bpym1 As stated in the introduction, we are interested in ψ n =(Uˆ)n ψ initial, (1) quantifyingtheamountofentanglementbetweenwalkers | i | i for each coin outcome, as well as in studying the impact where ψ initial is the initialtotalstateofthe quantum | i ofdifferentinitialquantumstatesinthisquantificationof walk. In our case, the family of quantum walks we shall entanglement. The following lines shows corresponding employ is composed by the tensor product of one coin results using unrestricted quantum walks on a line. and two walkers A. Entanglement Generation in unrestricted coin walker1,walker2 (2) | i⊗| i Quantum Walks on a Line as total initial state. After several applications of an evolution operator composed of a coin operator and a We shall use Eqs. (4a)-(4e) as total initial states, shift operator, we perform a measurement on the coin where each initial condition has the form ψ 0 = | i state. Theresultofthisoperationisapost-measurement coin 0 position 0,with coin 0 ascoininitialstateand | i ⊗| i | i quantum state composed by the tensor product of one position 0 as walker initial state. | i coin state and several walker components. We take the walker components of this coin post-measurement state and calculate the entanglement between walkers using ψ 0 = 0 c 0,0 p (4a) | i | i ⊗| i the von Neumann entropy ψ 0 = 1 c 0,0 p (4b) | i | i ⊗| i d 2 2 E(|ψi)=S(ρA)=S(ρB)=−Xαi log2(αi). (3) 1 i i=1 ψ 0 =( 0 c+ 1 c) 0,0 p (4c) | i √2| i √2| i ⊗| i where ψ = d α i i is the Schmidt decom- | i Pi=1 i| Ai| Bi position of a bipartite quantum state ψ . We compute | i i 1 n quantum walks using the same initial states and ψ 0 =( 0 c+ 1 c) 0,0 p (4d) evolution operator in order to measure the degree of | i √2| i √2| i ⊗| i entanglement between walkers for each step, so that the final result of this algorithm is a graph with the amount of entanglement available at each step. We summarize |ψi0 =(√0.85|0ic−√0.15|1ic)⊗|0,0ip (4e) this explanation in algorithm 1. where subindex c stands for ‘coin’ and subindex p stands for ‘walker position’. Algorithm 1. Quantification of entanglement. Input: A maximum number of steps n for the quantum Additionally, we use the Hadamard operator as coin walk, and n identically prepared total initial states ψ 0 | i operator with one coin and two walkers. Objective: To quantify the amount of entanglement between walkers for each step of the quantum walk. 1 Hˆ = (0 0 + 0 1 + 1 0 1 1) (5) 01. Set t=1 √2 | ich | | ich | | ich |−| ich | 3 Our shift operator is given by 10 Entanglement available vs Number of steps. |coin〉 = |0〉 1 Entanglement ratio vs number of steps. |coin〉 = |0〉 Sˆent =|0ich0|⊗Xi 1|ic+11,i+X1ipihi,i1|,+i 1 p i,i (6) Entanglement123456789Maximum dfoerg ereaec ho fs etenptanglementE ntanglement available for each step Entanglement available/maximum entanglement000000......456789 | i h |⊗ i | − − i h | (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0 200 4N00umber of ste6p0s0 800 1000 The observable used for coin measurement (step 4 of FIG. 1: After computing a 1000-steps quantum walk algorithm 1) is given by ψ 1000 = [Sˆent(Hˆ Iˆ)]1000 ψ 0 with ψ 0 given by Eq. (4a) | i ⊗ | i | i and Eqs. (5) and (6) as coin (Hˆ) and shift (Sˆ) operators, we perform a coin measurement on ψ 1000 using measurement Mˆ =α0Mˆ0+α1Mˆ1 =α0 0 c 0 +α1 1 c 1 (7) operator Mˆ0 (Eq. (7)). Thethin l|inie of (i) (red color online) | i h | | i h | shows the maximum degree of entanglement between walk- With the purpose of exemplifying the behavior of al- ersattainableinthepost-measurementquantumstate ψ c0 | it,pm gorithm1,we showinthe followinglines three steps ofa (forexample,log 2=1forthesecondstepandlog 3=1.585 2 2 quantumwalkandcorrespondingentanglementmeasure- forthethirdstep),andthethicklineof(i)(bluecoloronline) ment using Eq. (4a) as total initial state, and Eqs. (5) shows the actual entanglement between walkers available at and(6)as correspondingcoinandshift operators. Using eachstep. Wecanseethat,asymptotically,theentanglement availableisabout80%ofthecorrespondingmaximumdegree Eq. (1) we find that of entanglement (plot (ii)). 1 ψ 1 = (0 c 1,1 p+ 1 c 1, 1 p) (8) 10 Entanglement available vs Number of steps. |coin〉 = |1〉 1 Entanglement ratio vs number of steps. |coin〉 = |1〉 |ψi2 = 12|(|0iic|2,√2i2p+||1iic||0,0iip+||0iic||0−,0ip−−|1iic|−2,−(29i)p) Entanglement123456789Maximfour mea dcehg sreteep o f e n t a n g l e mentE ntanglement available for each step Entanglement available/maximum entanglement0000....00006789....55556789 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.550 200 4N00umber of ste6p0s0 800 1000 1 |ψi3 = 2√2(|0ic|3,3ip+|1ic|1,1ip+|0ic|1,1ip FψIG10.002:=A[Sˆfetnetr(HˆcompIˆu)]t1i0n0g0 ψof0awi1t0h00ψ-st0epgsiveqnuabnytuEmq. w(4aalk) | i ⊗ | i | i 1 1, 1 + 0 1,1 + 1 1, 1 and Eqs. (5) and (6) as coin (Hˆ) and shift (Sˆ) operators, c p c p c p −| i |− − i | i | i | i |− − i we perform a coin measurement on ψ 1000 using measure- 0 1, 1 + 1 3, 3 ) (10) | i −| ic|− − ip | ic|− − ip ment operator Mˆ1 (Eq. (7)). The thin curve of (i) (red coloronline)showsthemaximumdegreeofentanglementbe- For ψ 1 (Eq. (8)), the post-measurement quantum | i tween walkers attainable in the post-measurement quantum state after performing a coin measurement with mea- state ψ c1 (forexample,log 2=1forthesecondstepand surement operator Mˆ0 (Eq. (7)) is given by |ψic10,pm = log23|=it1,p.5m85 for the third st2ep), and the thick curve of (i) 0 c 1,1 p,andthedegreeofentanglementbetweenwalk- (blue color online) shows the actual entanglement between | i | i ers is clearly 0. As for coin 1, we perform a coin mea- walkers available at each step. We can see that, for large surement on ψ 1 (Eq. (8)) using measurement operator number of steps, the entanglement available is about 90% of | i Mˆ1 (Eq. (7)), obtaining as post-measurement quantum the corresponding maximum degree of entanglement (graph state ψ c1 = 1 1, 1 . It is also clear that the (ii)). degree| oif1e,pnmtang|lemice|n−t be−tweiepn walkers in ψ c1 is 0. | i1,pm Instep2(Eq. (9)), wehave ψ c0 = 1 0 (2,2 + | i2,pm √2| ic | ip 1 (0 )(1,1 + 3, 3 ), with degree of entangle- 0,0 p) as coin 0 c post-measurement state, and cor- √2 | ic | ip |− − ip | i | i ment between walkers equal to 1. responding entanglement betwen walkers is equal to 1, since 1 (2,2 + 0,0 )is a maximally entangledstate. We show in Figs. (1), (2) and (3), simulation results √2 | ip | ip for a 1000-steps quantum walk performed with Eq. (4a) Along the same lines, the coin 1 post-measurement stateis givenby ψ c1 = 1 (1 |)i(c0,0 + 2, 2 ). as total initial state and Eqs. (5) and (6) as coin and | i2,pm √2 | ic | ip |− − ip shift operators. Since 1 (0,0 + 2, 2 ) is a maximally entangled √2 | ip |− − ip Fig. (1) presents the results of measuring entangle- state, its degree of entanglement is equal to 1. ment between walkers in a coin 0 post-measurement Finally,instep3(Eq. (10)),|ψic30,pm = √16|0ic(|3,3ip+ state|ψict,0pm. InFig. (1.i)weshow| itcwocurves. The thin 21,1 1, 1 ), and corresponding degree of en- curve (red color online) indicates, for each step of the p p | i −|− − i tanglement between walkers is equal to 1.2516 (maxi- quantum walk, the maximum amount of entanglement mum degree of entanglement attainable between walk- between walkers achievable at each time step, while the ers is log 3 = 1.585.) As for coin 1 , ψ c1 = thick curve (blue color online) shows the actual degree 2 | ic | i3,pm 4 oWagiifmr)ee.Ie)oenncuotanFafntnieggnso.letfeaemen(nt2geht)nlaaetnmtw,bgeeelanetspmtwrtaeeehtenstenetanniwatnuvaaamtlbhkilbleeaeerbsirslsaeaoamvfvbaessoiltuatrehtebpsel8seu0mlif%tnosacrxr(aeFeismaaigcsuiehunmsr,seFttde(ihp1gee-... Probability of locating walker at step n000...000000...012000555123 Hadamard walk. Initial condition |0〉c ⊗ |0〉p Entanglement23456789EEntnatnagnlgelmemenetn| c(at|o cCvaionavoi〉inaml aI=〉inl pba =i|altb1 ei|arl〉0 eifls〉 o so)frotn are toaeefca: ehc|0n hs〉t ctaseotnipeng p⊗le m|0e,0n〉t waavlkaeirlsable. 1 (1) but for a coin |1ic post-measurement state |ψict,1pm. (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4N00umber of ste6p0s0 800 1000 Firstofall,we noticethat, asinthe previousparagraph, the degree of entanglement between walkers available in FIG. 3: Plot (i) presents the probability vs location graph ψ c1 (thin line (red color online) of Fig. (2.i)) does of a 1000-step Hadamard quantumwalk with an initial state | it,pm not reach the highest degree of entanglement attainable 0 c 0 p andshiftoperatorprovidedbyEq. (11). Thesym- | i ⊗| i at each time step (thick (blue color online) line in Fig. metryofthiswalk,aboutalinepassingthroughtheoriginand perpediculartothexaxis,isthesameasthatofaHadamard (2.i)). However,it canbe seen by comparingthe asymp- totical behavior shown in Fig. (1.i) and Fig. (2.i) that, quantum walk with initial state given by |ψi = |0ic⊗|0,0ip and shift operator given by Eq. (6). Plot (ii) is a summary if the coin measurement outcome is α1 (Fig. (2.i)) then of Figs. (1.i) and (2.i), and shows that the amount of entan- the amount of entanglement available between walkers glementbetweenwalkersavailableinpost-measurementstate tends to be higher (about 90%, Fig. (2.ii)) than the cor- ψ c1 tends to be higher than the amount of entanglement responding degree of entanglement between walkers for b|eittw,pemenwalkersavailableinpost-measurementstate ψ c0 . a coin measurement outcome α0 (Fig. (1.i)) which is, as | it,pm shown in Fig. (1.ii), about 80%. In Fig. (3.i) we display the probability vs location 10 Entanglement available vs Number of steps. |coin〉 = |0〉 1 Entanglement ratio vs number of steps. |coin〉 = |0〉 gionrniaetpiawhlaoslktfaeatre)1ag0ni0vd0e-nsshtbeifpytH|o0paiecdra⊗amto|a0rridppr(qoivu.eia.dnetodunmbeycwoainlkawndithonalny Entanglement123456789 Maximum dfoerg ereaech o sf teenptanglemeEnnttanglement available for each step Entanglement available / maximum entanglement0000....00006789....55556789 Sˆ=|0ich0|⊗X|i+1iphi|+|1ich1|⊗X|i−1iphi|. (11) (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.550 200 4N00umber of ste6p0s0 800 1000 i i FIG.4: Entanglementvaluesforcoinpost-measurementstate thrTohueghsytmhemoertirgyinoafntdhipserwpaelnkd,icaublaoruttoathlienex paxasiss,inigs [|ψSˆeinctt,0p(mHˆcomIˆ)p]u10t0e0dψfro0mwiath100ψ0-0stgeipvsenqubayntEuqm. w(4ablk),|ψEiq1s0.00(5=) the same as that of a Hadamardquantum walk with ini- and(6)a⊗scoin(Hˆ|)iandshift| (iSˆ)operators,andmeasurement tial state given by |ψi=|0ic⊗|0,0ip and shift operator operator Mˆ0 (Eq. (7)). Thethin line of (i) (red color online) givenbyEq. (6). TheblackcurveofFig. (3.ii)showsthe showsthemaximumdegreeofentanglementbetweenwalkers amountofentanglementavailablebetweenwalkersinthe attainable in the post-measurement quantum state ψ c0 , post-measurement state |ψict,0pm (as in Fig. (1.i)), while and the thick line of (i) (blue color online) shows the| aictt,pumal the gray curve (red color online) shows the correspond- entanglementbetweenwalkersavailableateachstep. Wecan ing degreeof entanglementavailablebetween walkersfor see that, asymptotically, the entanglement available is about post-measurement state ψ c1 (as in Fig. (2.i)). The 90% of the corresponding maximum degree of entanglement purpose of Fig. (3) is to|reilta,ptme the amount of entangle- (plot (ii)). Note that this amount of entanglement available between walkers (90%) is higher than the amount of entan- mentavailableforeachcoinpost-measurementstatewith the symmetry of the quantum walk and, consequently, glement available between walkers (80%) for coin |0ic post- with the total initial state of the quantum walk. We measurement quantum state with initial state |0ic ⊗|0,0ip (Fig. (1)). shall come back to Fig. (3) shortly. We now focus on Figs. (4), (5) and (6), which present the numerical behavior of a quantum walk with initial the symmetry of the probability distribution computed quantum state given by Eq. (4b), and Eqs. (5) and (6) with initial quantum state given by Eq. (4b) (Fig. (6.i)) as coin and shift operators, respectively. seems to have a significant effect on the actual entangle- As in the previous case, Figs. (4) and (5) display ment values for ψ c0 and ψ c1 . the results of measuring entanglement between walkers | it,pm | it,pm in a coin 0 post-measurement state ψ c0 and a coin So, a natural step forward is to compute quantum 1 post-|miecasurement state ψ c1 .|Hiot,wpmever, and in walks with initial states that produce symmetric prob- c|oinctrast to Figs. (1)-(3), in|thiits,pmcase we see that, as abilitydistributions, inorderto seethe asymptoticalbe- the number of steps increases, the entanglement between havior of entanglement. With this thought in mind we walkers for ψ c0 (about 90% with respect to the degree havecomputedthefollowingthreesetsofnumericalsim- of entanglem|eintt,pmattainable in each step, Fig. (4.ii)) is ulations. higherthanthat ofstate ψ c1 (about80% with respect The first set consists of Figs. (7), (8) and (9), in | it,pm tothedegreeofentanglementattainableineachstep,Fig. which we expose the numerical behavior of a quantum (5.ii)). As we can see by comparing Figs. (3) and (6), walk with initial quantum state given by Eq. (4c), i.e. 5 Entanglement10123456789 MaximfouEr mneta adcnehgg lsreetmeep eo n f t e a n v t aa in l ag bl e lem veEsnn tNtaunmgbleemr eonf ts atevpasil.a |bcloei nf〉o =r e|1a〉c.h step Entanglement available / maximum entanglement000000......1456789 Entanglement ratio vs number of steps. |coin〉 = |1〉 Entanglement10123456789 MaximfourE mena tdacenhgg srleeteemp oe fn et na tv a a n i lg a l be lme evnEst nntuamngbleerm oef nstt eapvsa.i la|cbolein 〉f o=r |e0a〉ch step Entanglement available / maximum entanglement0000000.......13456789 Entanglement ratio vs number of steps. |coin〉 = |0〉 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0 200 4N00umber of ste6p0s0 800 1000 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.20 200 4N00umber of ste6p0s0 800 1000 FIG.5: Entanglementvaluesforcoinpost-measurementstate FIG. 7: Entanglement values for coin 0 c post-measurement [|Sψˆeinctt,1p(mHˆc⊗omIˆ)p]u10t0e0d|ψfrio0mwiath10|0ψ0i-0stgeipvsenqubayntEuqm. w(4ablk),|ψEiq1s0.00(5=) s|ψtait1e000|ψi=ct,0pm[Sˆecnot(mHˆpu⊗tedIˆ)]f1r0o0m0|ψai01w00it0|h-sit|eψpis0 q=uan(tu√1m2|0iwcal+k amnednt(6o)pearsatcoorinMˆ(1Hˆ()Eaqn.d(7s)h)i.ftT(hSˆe)tohpineralitnoerso,fa(ni)d(mreedascuolroer- o√pi2e|r1aitco)r⊗s g|0iv,e0nipbgyivEenqs.by(5E)qa.n(d4c()6,)croeisnpe(cHˆti)vealny,dasnhdiftm(eSˆa)- online) shows themaximum degree of entanglement between surement operator Mˆ0 (Eq. (7)). The thin line of (i) (red walkers attainable in the post-measurement quantum state coloronline)showsthemaximumdegreeofentanglementbe- ψ c1 ,andthethicklineof(i)(bluecoloronline)showsthe | it,pm tween walkers attainable in the post-measurement quantum actual entanglement between walkers available at each step. state ψ c0 ,andthethicklineof(i)(bluecoloronline)shows We can see that, asymptotically, the entanglement available | it,pm the actual entanglement between walkers available at each isabout80%ofthecorrespondingmaximumdegreeofentan- step. The asymptotical behavior of entanglement values for glement (plot (ii)). Note that this amount of entanglement this quantum walk is the same as that shown by a quantum availablebetweenwalkers(80%)islessthantheamountofen- walk with total initial state 0 c 0,0 p (Fig. (1)). tanglementavailablebetweenwalkers(90%)forcoin 1 post- | i ⊗| i | i measurement quantum state with initial state 0 c 0,0 p | i ⊗| i (FigProbability of locating walker at step n.000...000000...012000(5551232)).Hadamard walk. Initial condition |1〉c ⊗ |0〉p Entanglement23456789Entanglemen|tc aoCvinao〉im l=aInpb |iatl0eiar〉 ilsf osortn ae toaeEfc: n eh|t1n as〉tncateognipnlge ⊗lme|c me|o0nei,nt0n 〉a〉t w=vaa av|l1kialea〉i rlb sa l be l e f o. r each step (i) Entanglement1001234567890MaximfourE mena tdac2ehn0g g0srleteeemp oe fn et na tv a a n i 4lg aN 0l be 0ulmem evbnEset nrn touafmn sgbtleee6pmr0 so0efn st taevpasi.l a|cbolei8n 〉0f o=0r |e1a〉 ch step1000 (ii) Entanglement available / maximum entanglement00000.....000056789....55555167890 E2n0ta0nglement ra4tN0io0u vmsb neur mofb setre 6op0fs 0steps. |coin〉8 =0 0|1〉 1000 1 (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4N00umber of ste6p0s0 800 1000 FstIaGte. 8:ψEcn1tangcloemmpeunttevdalfureosmforac1o0in00|1-sitceppsostq-umaenatsuumremweanlkt | it,pm FIG. 6: Plot (i) presents the probability vs location graph |ψi1000 = [Sˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√12|0ic + of a 1000-step Hadamard quantumwalk with an initial state √i2|1ic)⊗|0,0ip given by Eq. (4c), coin (Hˆ) and shift (Sˆ) 1 c 0 p andshiftoperatorprovidedbyEq. (11). Thesym- operators given by Eqs. (5) and (6) respectively, and mea- | i ⊗| i metryofthiswalk,aboutalinepassingthroughtheoriginand surement operator Mˆ1 (Eq. (7)). The thin line of (i) (red perpediculartothexaxis,isthesameasthatofaHadamard coloronline)showsthemaximumdegreeofentanglementbe- quantum walk with initial state given by |ψi = |1ic⊗|0,0ip tween walkers attainable in the post-measurement quantum (Eq. (4b)) and shift operator given by Eq. (6). Plot (ii) state ψ c1 ,andthethicklineof(i)(bluecoloronline)shows is a summary of Figs. (4.i) and (5.i), and shows that the | it,pm the actual entanglement between walkers available at each amount of entanglement between walkers available in post- step. The asymptotical behavior of entanglement values for measurementstate ψ c1 tendstobelessthantheamountof | it,pm this quantum walk is the same as that shown by a quantum entanglementbetweenwalkersavailableinpost-measurement state ψ c0 , in stark contrast to the numerical results com- walk with total initial state |0ic⊗|0,0ip (Fig. (2)). | it,pm putedforaquantumwalkwithtotalinitialstate 0 c 0,0 p | i ⊗| i (Figs. (1-3)). (9.ii) shows that the asymptotical behavior of entangle- ment values for a quantum walk with initial state given |ψi0 = (√12|0ic+ √i2|1ic)⊗|0,0ip, and Eqs. (5) and (6) bcoymEpqu.ted(4fao)riasqtuhaenstaummewaaslkthwoisteh iennittaianlgsletamteengtivveanlubeys as coin and shift operators, respectively. Fig. (7) shows Eq. (4c). the results of measuring entanglement between walkers in a coin 0 post-measurementstate ψ c0 , while Fig. Figs. (10) - (12) introduce the asymptotics of en- (8) introd|uicces corresponding results fo|r ait,cpomin 1 post- tanglement values for a quantum walk with initial state c measurement state ψ c1 . | i given by Eq. (4d). Again, although the initial state Although an initi|alitq,pumantum state of the form given |ψi0 =(√i2|0ic+ √12|1ic)⊗|0,0ip produces a symmetri- by Eq. (4c)producesabalancedprobabilitydistribution cal probability distribution (Fig. (12.i)), we notice that (Fig. (9.i)),suchapropertydoesnothaveasignificantef- the asymptotical behavior of entanglement values for a fectonthe degreeofentanglementbetweenwalkers(Fig. coin 0 post-measurement quantum state ψ c0 is dif- | i | it,pm (9.ii)). In fact, comparing plots from Figs. (3.ii) and ferentfromthatofacoin 1 post-measurementquantum | i 6 Probability of locating walker at step n00000000........000000000.000011110246824681 Hadamard walk. Initial condition [1/(2)1/2|0〉c + i/(2)1/2|1〉c] ⊗ |0〉p Entanglement123456789EntangInleitmi|acelo nsintta 〉a t=Cev a:o| 1i[ml(a〉1 pb / a(l e2 r i) sf1 o o/ 2r n) e| E0oa〉nfcc teohain nnst ga+teln ep(gim/|l(ce2eom)ni1nte/ 〉2an )=v|t1 aa|〉0icvla〉oa ib ni ll] ae ⊗ b f lo e | r0. ,e0a〉wchal ksetresp Entanglement10123456789 MaximEunmta ndfogerlg eermeaeceh no tsf ateevnpatialanbgElleen mtvasen nngtulemmbeenrt oafv satielapbsl.e | cfoori ne〉 a=c |h1 〉step Entanglement available / maximum entanglement0000000.......13456789 Entanglement ratio vs number of steps. |coin〉 = |1〉 (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4N00umber of ste6p0s0 800 1000 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.20 200 4N00umber of ste6p0s0 800 1000 FIG. 9: Plot (i) presents the probability vs location graph FIG.11: Entanglementvaluesforcoin 1 c post-measurement of a 1000-step Hadamard quantumwalk with an initial state state ψ c1 computed from a 1000|-siteps quantum walk (√12|0ic+ √i2|1ic)⊗|0ip and shift operator provided by Eq. |ψi1000| i=t,pm[Sˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√i2|0ic + (p1lo1t).(Ti)hiesstyhmemsaemtrye oafstthheatproofbaabiHliatydadmisatrrdibuqtuioanntsuhmowwnailnk √12|1ic)⊗|0,0ip given by Eq. (4d), coin (Hˆ) and shift (Sˆ) wanitdhsihniifttiaolpsetraatteorgigvievnenbbyy|ψEiq0.=(6().√1A2|l0tihco+ug√hi2t|h1eics)y⊗mm|0,e0triyp ompeenrtatoopresrgaitvoernMbˆy1E(qEsq..(5()7)a)n.dT(6h)ertehspinecltiniveeloyf,a(in)d(mreedascuorloer- online) shows the maximum degree of entanglement between of plot (i) is significantly different from that of Fig. (3.i), walkers attainable in the post-measurement quantum state plot(ii)showsthesameasymptoticalbehaviorasthatofFig. ψ c1 ,andthethicklineof(i)(bluecoloronline)showsthe (3.ii). | it,pm actual entanglement between walkers available at each step. The asymptotical behavior of entanglement values for this Entanglement available vs number of steps. |coin〉 = |0〉 Entanglement ratio vs number of steps. |coin〉 = |0〉 quantumwalk is thesame as that shown bya quantumwalk 10 1 Fs(tiIa)Gte.Entanglement10123456789|00ψM:aiximfoEctur me,20a 0dpc0nehg smretetep o af e n nt 4a N0nc 0ug E mgl en ombtaeenlrng motelef mstmee6npp0ts 0avaeuilabnlte foetr8 e0da0cvh staepflr1u00o0emsf(oiair)c1Entanglement available / maximum entanglemento000000i.....000056789....555556789n000|-0sit20c0eppso4Ns00umtqbe-ru omf sate6p0sen0atsuum800remw10e0a0nlkt w(ii)thProbability of locating walker at step n00000000........000000000t−.000011110124682468100o00tHaadalmardi− w5n0a0lk. iInittiali cNoaunmditblioenr0 o[ fsi /s(2tet)1p/s2a|0〉 +t 1/e(25)010/2|1|〉]1 ⊗ |0i,0〉cp10⊗00 |0(,ii0)ipEntanglement(01234567890FEntaingglem|.ceo2Inni0nti 0t〉ai a=(vCl a |so0il5tma〉a b tp el ae :) r [if so ( oi)r4 / (nNe02 a0uo).1cmf/ h2eb)E n se|nt0traet 〉anop +ngf gls(e1lteme/6m(p|e20csen)o01nti/n 2ta 〉)av =|va1 ai|〉l1ai]l a〉b⊗ b le l| e.0 8 , f0 0o 〉 0r p each step1000 |ψi1000 = [Sˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√i2|0ic + √12|1ic)⊗|0,0ip given by Eq. (4d), coin (Hˆ) and shift (Sˆ) FofIGa.11020:0-sPtleoptH(ia)dparmesaerndtsquthaentpurmobwabaliklitwyitvhs aloncaintiiotinalgsrtaapthe operatorsgivenbyEqs. (5)and(6)respectively,andmeasure- ment operator Mˆ0 (Eq. (7)). The thin line of (i) (red color (√i2|0ic+ √12|1ic)⊗|0ip and shift operator provided by Eq. (11). The symmetry of the probability distribution shown in online) shows themaximum degree of entanglement between plot(i)isthesameasthatofaHadamardquantumwalkwith walkers attainable in the post-measurement quantum state |aψctiuct,0apml e,natnadngtlheemtehnitckbleitnweeoefn(iw)a(lbkleurescaovlaoirlaobnlelinaet)esahcohwssttehpe. ionpietiraaltsotragteivgeinvebnybEyq|.ψ(i6=). (A√lit2h|0oiucg+h√t1h2e|1siyc)m⊗m|0e,t0ryipoafnpdlosth(ifi)t The asymptotical behavior of entanglement values for this issignificantlydifferentfromthatofFig. (6.i),plot(ii)shows quantumwalk is thesame as that shown bya quantumwalk thesame asymptotical behavior as that of Fig. (6.ii). with total initial state 1 c 0,0 p (Fig. (4)). | i ⊗| i posed to previous cases in which asymptotical values of state ψ c1 (Fig. (12.ii)). Infact,comparingplotsfrom entanglement between walkers were different for post- tFuigms.s|t(a4ti)et,apnmψdc(010),afonrdapclootinsf|r0oimpoFsitg-sm.e(a5s)uarenmde(n1t1)qfuoarna- mFiegass.u(r1e3m)eanntdst(a1t4e)st|hψaitct,0tphmeaansydm|ψptioct,1tpimcs,owfebocathnesneteanin- coin 1 po|sti-mt,pemasurementquantumstate ψ c1 , shows glementcurvestendtothesameefficiencyof85%approx- that|thie asymptotics of entanglement val|ueist,pfomr initial imately. This tendency can also be seen in Fig. (15.ii) states given by Eqs. (4b) and (4d) are the same. where we show that both entanglement curves overlap. However and in stark contrast to the previous cases, the symmetry properties of the probability distribution ofa quantumwalk withinitial state ψ 0 =(√0.850 c III. CONCLUSIONS | i | i − √0.151 ) 0,0 (Eq. (4e)) does have an effect of the c p | i ⊗| i entanglement between walkers produced from coin post- We have proposed an algorithm to generate entangle- measurement quantum states. ment between walkers, after measuring the coin state, In Figs. (13) and(14)we exhibit the asymptoticalbe- for a Hadamardquantum walk with one (2-dimensional) havior of entanglement values of coin post-measurement coin and two walkers. Our numerical simulations show states ψ c0 and ψ c1 respectively, for a quantum that, asymptotically,the amount of entanglementgener- | it,pm | it,pm walk with initial state given by Eq. (4e). As op- ated between walkers does not reach the highest degree 7 Entanglement10123456789 MaximuEmnt adfonergg elreeamech eo nsf tte eanpvtaainlagblelem vesn tnEunmtabnegrl eomf setnetp as.v a|ciloaibn〉l e= f o|0r〉 each step Entanglement available / maximum entanglement0000....00005678....55556789 Entanglement ratio vs number of steps. |coin〉 = |0〉 Probability of locating walker at step n00000000........000000000.000011110246824681 Hadamard walk. Initial condition [(0.85)1/2|0〉c − (0.15)1/2|1〉c] ⊗ |0〉p Entanglement123456789 EntanglemIneintitaC alo svmtaEaipEnltaea|tncab:rto inla[sei(gnno0 fl〉gn.eo 8l= mreo5 mef|)e 01aene〉/|c2n tcn| hto0vta ai〉sanncltv gu〉e− ale=pe is(l ma0|( 1bob.e1〉lvl ane5 e ct ) r fk 1alo a / v2rc p |aue1 ira〉 lva cc e]b h )⊗l e s . |t 0e 〉p p (gray curve) (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.50 200 4N00umber of ste6p0s0 800 1000 (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4NN00uummbbeerr ooff ssttee6pp0ss0 800 1000 FIG.13: Entanglementvaluesforcoin 0 c post-measurement FIG. 15: Plot (i) presents the probability vs location graph state ψ c0 computed from a 1000|-siteps quantum walk of a 1000-step Hadamard quantumwalk with an initial state √|ψ0i.110500|1=ict),p[mSˆe0n,t(0Hˆp⊗givIˆe)n]10b0y0|Eψqi0. (w4eit)h,c|oψini0(H=ˆ)(a√nd0.8sh5i|f0tic(Sˆ−) b(√y0E.8q5.|0i(c1−1).√0T.1h5e|1iscy)m⊗m|e0tirpy aonfdthsehifptroobpaebrailtitoyr pdrisotvriidbeud- | i ⊗| i tion shown in plot (i) is the same as that of a Hadamard operatorsgivenbyEqs. (5)and(6)respectively,andmeasure- monelninteo)psehroawtosrthMˆe0m(aExqi.mu(7m)).deTgrheeetohfinenltiannegolefm(ie)n(trbedetwcoeleonr q√u0a.n15tu|1mic)w⊗alk|0w,0itihp ainnidtiaslhsifttatoepgerivaetnorbgyiv|eψni=by(E√q0..8(56|0)i.cI−n thiscase,theasymptoticsofentanglementvaluesforbothcoin walkers attainable in the post-measurement quantum state |ψict,0pm, and the thick line of (i) (blue color online) shows pψosct1-me(agsruaryemcuenrvtestoaftpeslo|tψ(iicti,0)p)mte(bnldactkoctuhrevseaomfeplvoatlu(iei)s). and the actual entanglement between walkers available at each | it,pm step. Wecanseethattheasymptoticsofentanglementvalues given in plot (ii) tend to the same values as those shown in Fig. (14), obtained from a coin 1 c post-measurement state interesting result: the actual value towards which the ψ c1 computedfromaquantum| iwalkwiththesameinitial entanglement ratio converges, for each coin measure- | it,pm state (Eq. (4e)). ment outcome, depends on the symmetry of the coin initial state. However, the relationship is not straight- forward, as it is possible to find two coin initial states Entanglement10123456789 MaximfouEr mneta adcnehgg lsreetmeep eo n f t e a n v t aa in l ag bl e lem vesn tnEunmtabnegrl eomf setnetp as.v a|ciloaibn〉l e= f o|1r〉 each step Entanglement available / maximum entanglement0000....000005678.....555556789 Entanglement ratio vs number of steps. |coin〉 = |1〉 sd(asw|uusiψasycrltiehmkr0meipbtr=ehtsunoattat√itso1i,nc,2ndaa|sc0ll,otievhonx+anvopleululy√rogeigr2shoei|nno1betgfiooecattnohtnhtihdneapeinri|rngφsoleaiidetn0mimuatc=ealeennsv√gttbaa,lael0tufmle.oa8erne(.5|nc|bφ0etoGiid0tco−h)apinmnrc√gooa0btiknth.a1oeebms5rit|elt1ewibhtaiyyoe)- (i) 00 200 400 600 800 1000 (ii) 0.450 200 4N00umber of ste6p0s0 800 1000 reveal differences in two quantum walks which are not differentiated easily in the usual case of a single walker. FIG.14: Entanglementvaluesforcoin 1 c post-measurement state ψ c1 computed from a 1000|-siteps quantum walk A noteworthy feature of our algorithm is the high |ψi1000| =it,p[mSˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√0.85|0ic − aermsowuhnitcohfgthroewesntwaintghletmheenntugmenbeerraotefdstbeeptsw.eeOnutrhsecwheamlke- √0.15|1ic)⊗|0,0ip givenbyEq. (4e),coin (Hˆ)andshift (Sˆ) is particularly applicable in physical realizations where operatorsgivenbyEqs. (5)and(6)respectively,andmeasure- the coin is a qubit (such as an atom or a superconduct- ment operator Mˆ1 (Eq. (7)). The thin line of (i) (red color ing qubit) which interacts with a distinct physical sys- online) shows themaximum degree of entanglement between tem (such as an electromagnetic field mode) acting as a walkers attainable in the post-measurement quantum state ψ c1 , and the thick line of (i) (blue color online) shows walker [11, 12, 30]. Taking two such walkers, each being |theit,apmctual entanglement between walkers available at each a distinct system, one can probe how entanglement is step. Wecanseethattheasymptoticsofentanglementvalues generatedbetweenthemthroughouralgorithm. Assuch given in plot (ii) tend to the same values as those shown in walkers do not naturally interact with each other, using Fig. (13), obtained from a coin 0 c post-measurement state thecoin(qubit)systemistheonlywaytoentanglethem. |ψict,0pm computedfromaquantum| iwalkwiththesameinitial A recent circuit QED suggestion for the physical imple- state (Eq. (4e)). mentation of a quantum walk [30] even estimate a walk of a significant number of steps to be carried out within the decoherence times of the relevant physical systems. of entanglement possible at each step (purely from the Simplyenhancingsuchschemestotwowalkerswoulden- dimensionality of the space explored by the walkers in a able observing the entanglement generation mechanism givenstep)foreithercoinmeasurementoutcome. Never- that we have presented and analyzed in this paper. theless, our simulations alsoshow that the entanglement ratio (= entanglement generated/highestvalue of entan- glement possible, for each step) tends to converge to a IV. ACKNOWLEDGEMENTS rather high value (for example, to 0.8 or 0.9), and the actual convergence value seems to depend on the coin S.V-A.gratefullyacknowledgesusefuldiscussionswith initial state and on the coin measurement outcome. Dr J.L. Ball as well as the support of SNI, CONA- Convergence of entanglement ratio leads to a most CyT and Tecnol´ogico de Monterrey Campus Estado de 8 M´exico. S.B. acknowledges the support of the EPSRC, and the Wolfson Foundation. UK, the QIP IRC (GR/S82176 /01), the Royal Society [1] S.Godoy and S.Fujita, A quantumrandom-walk model [17] P.L.Knight,E.Rold´an,andJ.E.Sipe,Quantumwalkon for tunneling diffusion in a 1D lattice, J. Chem Phys. thelineasaninterferencephenomenon,Phys. Rev.A68 97(7) (1992) 5148-5154. (2003) 020301. [2] S.P.Gudder,Quantumprobability(AcademicPressInc., [18] P.L. Knight, E. Rold´an, and J.E. Sipe, Optical cavity 1998) implementations of the quantum walk, Optics Commu- [3] N. Konno, Limit theorems and absorption problems for nications 227 (2003) 147-157. quantum random walks in one dimension, Quantum In- [19] P.L. Knight, E. Rold´an, and J.E. Sipe, Propagating formation and Computation 2 (2002) 578-595. quantum walks: the origin of interference structures, J. [4] R.P. Feynman, Quantum Mechanical Computers, Foun- Mod. Op. 51(12) (2004) 1761-1777. dations of Physics 16(6) (1986) 507-531. [20] I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon, and [5] B. A. Chase and A.J. Landhal, Universal quantum P.L. Knight, Entanglement in coined quantum walks on walks and adiabatic algorithms by 1D Hamiltonians, regular graphs, New J. Phys. 7 (2005) 156. arXiv:0802.1207 (2008). [21] O. Maloyer and V. Kendon, Decoherence vs entangle- [6] Y. Aharonov, L. Davidovich, and N. Zagury, Quantum ment in coined quantum walks, New J. Phys. 9 (2007) random walks, Phys. Rev. A48 (1993) 1687-1690. 87. [7] U. Sch¨oning, A probabilistic algorithm for K-SAT and [22] G.Abal,R.Siri,A.Romanelli,andR.Donangelo,Quan- constraint satisfaction problems, in Proceedings of the tumwalkontheline: entanglementandnon-localinitial 40th Annual Symposium on Foundations of Computer conditions, Phys. Rev. A 73 (2006) 042302. Science (FOCS), (IEEE, 1999), pp. 410-414. [23] G.Abal,R.Donangelo,andH.Fort,Long-timeentangle- [8] K. Iwama and S. Tamaki, Improved upper bounds for mentinthequantumwalk,inAnnalsofthe1stWorkshop 3-SAT, Electronic Colloquium on Computational Com- onQuantumComputationandInformation9-11October plexity 53 (2003). 2006, Pelotas, RS, Brazil (2007) pp.189-200. [9] R. Motwani and P. Raghavan, Randomized Algorithms [24] S.K. Goyal and C.M. Chandrashekar, Spatial entangle- (Cambridge University Press, 1995). ment in many body system using quantum walk, quant- [10] J.Du,H.Li,X.Xu,M.Shi,J.Wu,X.Zhou,andR.Han, ph/Arxiv:0901.0671 (2009). Experimental implementation of the quantum random- [25] M.Annabestani,M.R.Abolhasani,andG.Abal,Asymp- walk algorithm, Phys. Rev. A67 (2003) 042316. totic entanglement in a coherent 2D quantum walk, [11] B.C. Sanders, S.D. Bartlett, B. Tregenna, and P.L. quant-ph/Arxiv:0901.1188 (2009). Knight, Quantum quincunx in cavity quantum electro- [26] Y.Omar,N.Paunkovi´c,L.Sheridan,andS.Bose,Quan- dynamics, Phys. Rev. A 67 (2003) 042305. tumWalkonaLinewithTwoEntangledParticles,Phys. [12] G.S.AgarwalandP.K.Pathak,Quantumrandomwalk Rev. A 74 (2006) 042304. of the field in an externally driven cavity, Phys. Rev. A [27] S.E. Venegas-Andraca, J.L. Ball, K. Burnett, and S. 72 (2005) 033815. Bose, Quantum Walks with Entangled Coins, New J. [13] C.M.Chandrashekar,Implementingtheone-dimensional Phys. 7 (2005) 221. quantum (Hadamard) walk using a Bose-Einstein con- [28] C. M. Chandrashekar, Discrete time quantum walk densate, Phys. Rev. A 74 (2006) 032307. model for single and entangled particles to retain entan- [14] A. Rai, G. S. Agarwal, and J. H. H. Perk, Transport glement in coin space, arXiv:quant-ph/0609113 (2006). andquantumwalkofnonclassicallightincoupledwaveg- [29] C. Liu and N. Petulante, One-dimensional quan- uides, Phys. Rev. A 78 (2008) 042304. tum random walks with two entangled coins, quant- [15] S.E. Venegas-Andraca, Quantum Walks for Computer ph/Arxiv:0807.2263 (2009). 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Quantum Walk-based Generation of Entanglement Between Two Walkers PDF

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Preview Quantum Walk-based Generation of Entanglement Between Two Walkers

Quantum Walk-based Generation of Entanglement Between Two Walkers Salvador E. Venegas-Andraca1 and Sougato Bose2 1Quantum Information Processing Group, Tecnol´ogico de Monterrey Campus Estado de M´exico, Carretera Lago Gpe. Km 3.5, Atizap´an de Zaragoza, Edo. M´exico, 52926, M´exico [email protected], [email protected] 2Department of Physics and Astronomy, University College of London. Gower Street, London WC1E 6BT, United Kingdom [email protected] Quantumwalkscanbeusedeitherastoolsforquantumalgorithmdevelopmentorasentanglement generators, potentially useful to test quantum hardware. We present a novel algorithm based on a discrete Hadamard quantum walk on a line with one coin and two walkers whose purpose is to generate entanglement between walkers. We provide several classical computer simulations of 9 ourquantumalgorithm inwhich weshowthat,although theasymptoticalamount ofentanglement 0 generatedbetweenwalkersdoesnotreachthehighestdegreeofentanglementpossibleateachstepfor 0 eithercoinmeasurementoutcome,theentanglementratio(entanglementgenerated/highestvalueof 2 entanglementpossible,foreachstep)tendstoconverge,andtheactualconvergencevaluedependson n thecoininitialstateandonthecoinmeasurementoutcome. Furthermore,ournumericalsimulations a show that, for the quantum walks used in our algorithm, the value towards which entanglement J ratio converges also depends on the position probability distribution symmetry of a quantum walk 6 computed with one single walker and the same coin initial state employed in the corresponding 2 quantumwalk with two walkers. ] PACSnumbers: PACSnumbers: 03.67.Bg,03.67.Lx,03.67.-a,03.67.Ac h p - I. INTRODUCTION building methods in order to test the “quantumness” of t n emerging technologies for the creation of quantum com- a puters. Thusitisofcrucialimportancetodevelopmeth- u Quantum walks were designed as quantum counter- ods of entanglement generation through quantum walks q parts of classical random walks, a branch of stochastic [ sothatthegenuinequantumnatureofagivenwalkwith processes widely used in algorithm development. Al- given physical systems may be tested. 1 though some authors have selected the name “quantum v random walk” to refer to quantum phenomena [1, 2, 3] Quantum entanglement has been incorporated into 6 and, in fact, in the seminal work by R.P. Feynman [4] quantum walks research either as a result of performing 4 aboutquantummechanicalcomputerswefindaproposal a quantum walk [20, 21, 22, 23, 24, 25] or as a resource 9 thatcouldbeinterpretedasa(continuous-time)quantum to build new kinds of quantum walks [26, 27, 28, 29]. 3 walk[5],itisgenerallyacceptedthatthefirstpaperwith Sinceentanglementisakeycomponentinquantumcom- . 1 quantum walks as its main topic was published in 1993 putation,itisworthkeepinginmindthatquantumwalks 0 by Aharonov et al [6]. Thus, the links between classical canbeusedeitherasentanglementgeneratorsorascom- 9 random walks and quantum walks, as well as the utility putational processes taking advantage of this quantum 0 ofquantumwalksincomputerscience,aretwofreshand mechanical property. : v open areas of research. In this paper we introduce a novel algorithm based i X Since one of the main goals in quantum computing is on a discrete quantum walk on a line with one coin and r the development of quantum algorithms, and given the two walkers whose purpose is to generate entanglement a success of employing classicalrandom walks for comput- between walkers. After evolving the quantum walk for ingsolutionstoNP-completeproblems[7,8,9],therehas a certain number of steps, we perform a measurement been a huge interest in understanding the physical and on the coin state. We then obtain a post-measurement computationalpropertiesofquantumwalksoverthe last quantum state composed by the tensor product of one few years on both experimental [10, 11, 12, 13, 14] and coin state and several walker components. We take the theoretical research communities (see [15] for a review walker components of this post-measurement state and ontheoreticalaspects of quantum walks). In addition to calculate the entanglement between walkers. We per- their usage in computer science, the study of quantum form many quantum walks with the same initial condi- walks is relevant to the modelling of physical phenom- tions and evolution operators, so that we have a quan- ena such as energy transfer in photosynthetic systems tum walk ready to be measured for each time step. In [16]. Moreover, although it has been proved that cer- addition to our algorithm, we provide several simulation tain properties of quantum walks are also reproducible results using different initial conditions for the proposed by classical systems (like variance enhancement with re- quantum walks. While this analysis highlights the po- specttoclassicalrandomwalks[17,18,19]),itisalsotrue tential of a quantum walk to entangle high dimensional that uniquely quantum mechanical properties of quan- quantum systems (the dimension of the space available tum walks, such as entanglement, may be employed to to the walkers grow in each step) it can also at times be 2 practically useful. This will be the case when the walk- 02. While (t n) ≤ ers are systems which do not directly interact with each 03. Apply the evolution operator Uˆt = (Sˆ(Cˆ Iˆ))t to other such as two different electromagnetic field modes. ψ 0. ⊗ Then the coin canbe a commonsystemsuchas anatom 0|4i. Perform a measurement on the coin system. Since which interact with both and can entangle them to a coin 2 there are only two possible outcomes. We high degreewith the degree depending onthe number of l|abelith∈emHα0 and α1. steps possible within the reasonable decoherence time of 05. For outcome α0 then the fields. 06. Compute the post-measurement quantum state ψ c0 | it,pm 07. Quantify entanglement between walkers from quan- II. ALGORITHM FOR ENTANGLEMENT tum state ψ c0 GENERATION | it,pm 08. For outcome α1 then 09. Compute the post-measurement quantum state Inthissectionwepresentouralgorithmforthegenera- ψ c1 | it,pm tionofentanglementina familyofquantumwalksonan 10. Quantify entanglement between walkers from quan- unrestrictedline. Asuccintmathematicalrepresentation tum state ψ c1 of a quantum walk after n steps is 11. Increas|eitt,bpym1 As stated in the introduction, we are interested in ψ n =(Uˆ)n ψ initial, (1) quantifyingtheamountofentanglementbetweenwalkers | i | i for each coin outcome, as well as in studying the impact where ψ initial is the initialtotalstateofthe quantum | i ofdifferentinitialquantumstatesinthisquantificationof walk. In our case, the family of quantum walks we shall entanglement. The following lines shows corresponding employ is composed by the tensor product of one coin results using unrestricted quantum walks on a line. and two walkers A. Entanglement Generation in unrestricted coin walker1,walker2 (2) | i⊗| i Quantum Walks on a Line as total initial state. After several applications of an evolution operator composed of a coin operator and a We shall use Eqs. (4a)-(4e) as total initial states, shift operator, we perform a measurement on the coin where each initial condition has the form ψ 0 = | i state. Theresultofthisoperationisapost-measurement coin 0 position 0,with coin 0 ascoininitialstateand | i ⊗| i | i quantum state composed by the tensor product of one position 0 as walker initial state. | i coin state and several walker components. We take the walker components of this coin post-measurement state and calculate the entanglement between walkers using ψ 0 = 0 c 0,0 p (4a) | i | i ⊗| i the von Neumann entropy ψ 0 = 1 c 0,0 p (4b) | i | i ⊗| i d 2 2 E(|ψi)=S(ρA)=S(ρB)=−Xαi log2(αi). (3) 1 i i=1 ψ 0 =( 0 c+ 1 c) 0,0 p (4c) | i √2| i √2| i ⊗| i where ψ = d α i i is the Schmidt decom- | i Pi=1 i| Ai| Bi position of a bipartite quantum state ψ . We compute | i i 1 n quantum walks using the same initial states and ψ 0 =( 0 c+ 1 c) 0,0 p (4d) evolution operator in order to measure the degree of | i √2| i √2| i ⊗| i entanglement between walkers for each step, so that the final result of this algorithm is a graph with the amount of entanglement available at each step. We summarize |ψi0 =(√0.85|0ic−√0.15|1ic)⊗|0,0ip (4e) this explanation in algorithm 1. where subindex c stands for ‘coin’ and subindex p stands for ‘walker position’. Algorithm 1. Quantification of entanglement. Input: A maximum number of steps n for the quantum Additionally, we use the Hadamard operator as coin walk, and n identically prepared total initial states ψ 0 | i operator with one coin and two walkers. Objective: To quantify the amount of entanglement between walkers for each step of the quantum walk. 1 Hˆ = (0 0 + 0 1 + 1 0 1 1) (5) 01. Set t=1 √2 | ich | | ich | | ich |−| ich | 3 Our shift operator is given by 10 Entanglement available vs Number of steps. |coin〉 = |0〉 1 Entanglement ratio vs number of steps. |coin〉 = |0〉 Sˆent =|0ich0|⊗Xi 1|ic+11,i+X1ipihi,i1|,+i 1 p i,i (6) Entanglement123456789Maximum dfoerg ereaec ho fs etenptanglementE ntanglement available for each step Entanglement available/maximum entanglement000000......456789 | i h |⊗ i | − − i h | (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0 200 4N00umber of ste6p0s0 800 1000 The observable used for coin measurement (step 4 of FIG. 1: After computing a 1000-steps quantum walk algorithm 1) is given by ψ 1000 = [Sˆent(Hˆ Iˆ)]1000 ψ 0 with ψ 0 given by Eq. (4a) | i ⊗ | i | i and Eqs. (5) and (6) as coin (Hˆ) and shift (Sˆ) operators, we perform a coin measurement on ψ 1000 using measurement Mˆ =α0Mˆ0+α1Mˆ1 =α0 0 c 0 +α1 1 c 1 (7) operator Mˆ0 (Eq. (7)). Thethin l|inie of (i) (red color online) | i h | | i h | shows the maximum degree of entanglement between walk- With the purpose of exemplifying the behavior of al- ersattainableinthepost-measurementquantumstate ψ c0 | it,pm gorithm1,we showinthe followinglines three steps ofa (forexample,log 2=1forthesecondstepandlog 3=1.585 2 2 quantumwalkandcorrespondingentanglementmeasure- forthethirdstep),andthethicklineof(i)(bluecoloronline) ment using Eq. (4a) as total initial state, and Eqs. (5) shows the actual entanglement between walkers available at and(6)as correspondingcoinandshift operators. Using eachstep. Wecanseethat,asymptotically,theentanglement availableisabout80%ofthecorrespondingmaximumdegree Eq. (1) we find that of entanglement (plot (ii)). 1 ψ 1 = (0 c 1,1 p+ 1 c 1, 1 p) (8) 10 Entanglement available vs Number of steps. |coin〉 = |1〉 1 Entanglement ratio vs number of steps. |coin〉 = |1〉 |ψi2 = 12|(|0iic|2,√2i2p+||1iic||0,0iip+||0iic||0−,0ip−−|1iic|−2,−(29i)p) Entanglement123456789Maximfour mea dcehg sreteep o f e n t a n g l e mentE ntanglement available for each step Entanglement available/maximum entanglement0000....00006789....55556789 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.550 200 4N00umber of ste6p0s0 800 1000 1 |ψi3 = 2√2(|0ic|3,3ip+|1ic|1,1ip+|0ic|1,1ip FψIG10.002:=A[Sˆfetnetr(HˆcompIˆu)]t1i0n0g0 ψof0awi1t0h00ψ-st0epgsiveqnuabnytuEmq. w(4aalk) | i ⊗ | i | i 1 1, 1 + 0 1,1 + 1 1, 1 and Eqs. (5) and (6) as coin (Hˆ) and shift (Sˆ) operators, c p c p c p −| i |− − i | i | i | i |− − i we perform a coin measurement on ψ 1000 using measure- 0 1, 1 + 1 3, 3 ) (10) | i −| ic|− − ip | ic|− − ip ment operator Mˆ1 (Eq. (7)). The thin curve of (i) (red coloronline)showsthemaximumdegreeofentanglementbe- For ψ 1 (Eq. (8)), the post-measurement quantum | i tween walkers attainable in the post-measurement quantum state after performing a coin measurement with mea- state ψ c1 (forexample,log 2=1forthesecondstepand surement operator Mˆ0 (Eq. (7)) is given by |ψic10,pm = log23|=it1,p.5m85 for the third st2ep), and the thick curve of (i) 0 c 1,1 p,andthedegreeofentanglementbetweenwalk- (blue color online) shows the actual entanglement between | i | i ers is clearly 0. As for coin 1, we perform a coin mea- walkers available at each step. We can see that, for large surement on ψ 1 (Eq. (8)) using measurement operator number of steps, the entanglement available is about 90% of | i Mˆ1 (Eq. (7)), obtaining as post-measurement quantum the corresponding maximum degree of entanglement (graph state ψ c1 = 1 1, 1 . It is also clear that the (ii)). degree| oif1e,pnmtang|lemice|n−t be−tweiepn walkers in ψ c1 is 0. | i1,pm Instep2(Eq. (9)), wehave ψ c0 = 1 0 (2,2 + | i2,pm √2| ic | ip 1 (0 )(1,1 + 3, 3 ), with degree of entangle- 0,0 p) as coin 0 c post-measurement state, and cor- √2 | ic | ip |− − ip | i | i ment between walkers equal to 1. responding entanglement betwen walkers is equal to 1, since 1 (2,2 + 0,0 )is a maximally entangledstate. We show in Figs. (1), (2) and (3), simulation results √2 | ip | ip for a 1000-steps quantum walk performed with Eq. (4a) Along the same lines, the coin 1 post-measurement stateis givenby ψ c1 = 1 (1 |)i(c0,0 + 2, 2 ). as total initial state and Eqs. (5) and (6) as coin and | i2,pm √2 | ic | ip |− − ip shift operators. Since 1 (0,0 + 2, 2 ) is a maximally entangled √2 | ip |− − ip Fig. (1) presents the results of measuring entangle- state, its degree of entanglement is equal to 1. ment between walkers in a coin 0 post-measurement Finally,instep3(Eq. (10)),|ψic30,pm = √16|0ic(|3,3ip+ state|ψict,0pm. InFig. (1.i)weshow| itcwocurves. The thin 21,1 1, 1 ), and corresponding degree of en- curve (red color online) indicates, for each step of the p p | i −|− − i tanglement between walkers is equal to 1.2516 (maxi- quantum walk, the maximum amount of entanglement mum degree of entanglement attainable between walk- between walkers achievable at each time step, while the ers is log 3 = 1.585.) As for coin 1 , ψ c1 = thick curve (blue color online) shows the actual degree 2 | ic | i3,pm 4 oWagiifmr)ee.Ie)oenncuotanFafntnieggnso.letfeaemen(nt2geht)nlaaetnmtw,bgeeelanetspmtwrtaeeehtenstenetanniwatnuvaaamtlbhkilbleeaeerbsirslsaeaoamvfvbaessoiltuatrehtebpsel8seu0mlif%tnosacrxr(aeFeismaaigcsuiehunmsr,seFttde(ihp1gee-... Probability of locating walker at step n000...000000...012000555123 Hadamard walk. Initial condition |0〉c ⊗ |0〉p Entanglement23456789EEntnatnagnlgelmemenetn| c(at|o cCvaionavoi〉inaml aI=〉inl pba =i|altb1 ei|arl〉0 eifls〉 o so)frotn are toaeefca: ehc|0n hs〉t ctaseotnipeng p⊗le m|0e,0n〉t waavlkaeirlsable. 1 (1) but for a coin |1ic post-measurement state |ψict,1pm. (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4N00umber of ste6p0s0 800 1000 Firstofall,we noticethat, asinthe previousparagraph, the degree of entanglement between walkers available in FIG. 3: Plot (i) presents the probability vs location graph ψ c1 (thin line (red color online) of Fig. (2.i)) does of a 1000-step Hadamard quantumwalk with an initial state | it,pm not reach the highest degree of entanglement attainable 0 c 0 p andshiftoperatorprovidedbyEq. (11). Thesym- | i ⊗| i at each time step (thick (blue color online) line in Fig. metryofthiswalk,aboutalinepassingthroughtheoriginand perpediculartothexaxis,isthesameasthatofaHadamard (2.i)). However,it canbe seen by comparingthe asymp- totical behavior shown in Fig. (1.i) and Fig. (2.i) that, quantum walk with initial state given by |ψi = |0ic⊗|0,0ip and shift operator given by Eq. (6). Plot (ii) is a summary if the coin measurement outcome is α1 (Fig. (2.i)) then of Figs. (1.i) and (2.i), and shows that the amount of entan- the amount of entanglement available between walkers glementbetweenwalkersavailableinpost-measurementstate tends to be higher (about 90%, Fig. (2.ii)) than the cor- ψ c1 tends to be higher than the amount of entanglement responding degree of entanglement between walkers for b|eittw,pemenwalkersavailableinpost-measurementstate ψ c0 . a coin measurement outcome α0 (Fig. (1.i)) which is, as | it,pm shown in Fig. (1.ii), about 80%. In Fig. (3.i) we display the probability vs location 10 Entanglement available vs Number of steps. |coin〉 = |0〉 1 Entanglement ratio vs number of steps. |coin〉 = |0〉 gionrniaetpiawhlaoslktfaeatre)1ag0ni0vd0e-nsshtbeifpytH|o0paiecdra⊗amto|a0rridppr(qoivu.eia.dnetodunmbeycwoainlkawndithonalny Entanglement123456789 Maximum dfoerg ereaech o sf teenptanglemeEnnttanglement available for each step Entanglement available / maximum entanglement0000....00006789....55556789 Sˆ=|0ich0|⊗X|i+1iphi|+|1ich1|⊗X|i−1iphi|. (11) (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.550 200 4N00umber of ste6p0s0 800 1000 i i FIG.4: Entanglementvaluesforcoinpost-measurementstate thrTohueghsytmhemoertirgyinoafntdhipserwpaelnkd,icaublaoruttoathlienex paxasiss,inigs [|ψSˆeinctt,0p(mHˆcomIˆ)p]u10t0e0dψfro0mwiath100ψ0-0stgeipvsenqubayntEuqm. w(4ablk),|ψEiq1s0.00(5=) the same as that of a Hadamardquantum walk with ini- and(6)a⊗scoin(Hˆ|)iandshift| (iSˆ)operators,andmeasurement tial state given by |ψi=|0ic⊗|0,0ip and shift operator operator Mˆ0 (Eq. (7)). Thethin line of (i) (red color online) givenbyEq. (6). TheblackcurveofFig. (3.ii)showsthe showsthemaximumdegreeofentanglementbetweenwalkers amountofentanglementavailablebetweenwalkersinthe attainable in the post-measurement quantum state ψ c0 , post-measurement state |ψict,0pm (as in Fig. (1.i)), while and the thick line of (i) (blue color online) shows the| aictt,pumal the gray curve (red color online) shows the correspond- entanglementbetweenwalkersavailableateachstep. Wecan ing degreeof entanglementavailablebetween walkersfor see that, asymptotically, the entanglement available is about post-measurement state ψ c1 (as in Fig. (2.i)). The 90% of the corresponding maximum degree of entanglement purpose of Fig. (3) is to|reilta,ptme the amount of entangle- (plot (ii)). Note that this amount of entanglement available between walkers (90%) is higher than the amount of entan- mentavailableforeachcoinpost-measurementstatewith the symmetry of the quantum walk and, consequently, glement available between walkers (80%) for coin |0ic post- with the total initial state of the quantum walk. We measurement quantum state with initial state |0ic ⊗|0,0ip (Fig. (1)). shall come back to Fig. (3) shortly. We now focus on Figs. (4), (5) and (6), which present the numerical behavior of a quantum walk with initial the symmetry of the probability distribution computed quantum state given by Eq. (4b), and Eqs. (5) and (6) with initial quantum state given by Eq. (4b) (Fig. (6.i)) as coin and shift operators, respectively. seems to have a significant effect on the actual entangle- As in the previous case, Figs. (4) and (5) display ment values for ψ c0 and ψ c1 . the results of measuring entanglement between walkers | it,pm | it,pm in a coin 0 post-measurement state ψ c0 and a coin So, a natural step forward is to compute quantum 1 post-|miecasurement state ψ c1 .|Hiot,wpmever, and in walks with initial states that produce symmetric prob- c|oinctrast to Figs. (1)-(3), in|thiits,pmcase we see that, as abilitydistributions, inorderto seethe asymptoticalbe- the number of steps increases, the entanglement between havior of entanglement. With this thought in mind we walkers for ψ c0 (about 90% with respect to the degree havecomputedthefollowingthreesetsofnumericalsim- of entanglem|eintt,pmattainable in each step, Fig. (4.ii)) is ulations. higherthanthat ofstate ψ c1 (about80% with respect The first set consists of Figs. (7), (8) and (9), in | it,pm tothedegreeofentanglementattainableineachstep,Fig. which we expose the numerical behavior of a quantum (5.ii)). As we can see by comparing Figs. (3) and (6), walk with initial quantum state given by Eq. (4c), i.e. 5 Entanglement10123456789 MaximfouEr mneta adcnehgg lsreetmeep eo n f t e a n v t aa in l ag bl e lem veEsnn tNtaunmgbleemr eonf ts atevpasil.a |bcloei nf〉o =r e|1a〉c.h step Entanglement available / maximum entanglement000000......1456789 Entanglement ratio vs number of steps. |coin〉 = |1〉 Entanglement10123456789 MaximfourE mena tdacenhgg srleeteemp oe fn et na tv a a n i lg a l be lme evnEst nntuamngbleerm oef nstt eapvsa.i la|cbolein 〉f o=r |e0a〉ch step Entanglement available / maximum entanglement0000000.......13456789 Entanglement ratio vs number of steps. |coin〉 = |0〉 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0 200 4N00umber of ste6p0s0 800 1000 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.20 200 4N00umber of ste6p0s0 800 1000 FIG.5: Entanglementvaluesforcoinpost-measurementstate FIG. 7: Entanglement values for coin 0 c post-measurement [|Sψˆeinctt,1p(mHˆc⊗omIˆ)p]u10t0e0d|ψfrio0mwiath10|0ψ0i-0stgeipvsenqubayntEuqm. w(4ablk),|ψEiq1s0.00(5=) s|ψtait1e000|ψi=ct,0pm[Sˆecnot(mHˆpu⊗tedIˆ)]f1r0o0m0|ψai01w00it0|h-sit|eψpis0 q=uan(tu√1m2|0iwcal+k amnednt(6o)pearsatcoorinMˆ(1Hˆ()Eaqn.d(7s)h)i.ftT(hSˆe)tohpineralitnoerso,fa(ni)d(mreedascuolroer- o√pi2e|r1aitco)r⊗s g|0iv,e0nipbgyivEenqs.by(5E)qa.n(d4c()6,)croeisnpe(cHˆti)vealny,dasnhdiftm(eSˆa)- online) shows themaximum degree of entanglement between surement operator Mˆ0 (Eq. (7)). The thin line of (i) (red walkers attainable in the post-measurement quantum state coloronline)showsthemaximumdegreeofentanglementbe- ψ c1 ,andthethicklineof(i)(bluecoloronline)showsthe | it,pm tween walkers attainable in the post-measurement quantum actual entanglement between walkers available at each step. state ψ c0 ,andthethicklineof(i)(bluecoloronline)shows We can see that, asymptotically, the entanglement available | it,pm the actual entanglement between walkers available at each isabout80%ofthecorrespondingmaximumdegreeofentan- step. The asymptotical behavior of entanglement values for glement (plot (ii)). Note that this amount of entanglement this quantum walk is the same as that shown by a quantum availablebetweenwalkers(80%)islessthantheamountofen- walk with total initial state 0 c 0,0 p (Fig. (1)). tanglementavailablebetweenwalkers(90%)forcoin 1 post- | i ⊗| i | i measurement quantum state with initial state 0 c 0,0 p | i ⊗| i (FigProbability of locating walker at step n.000...000000...012000(5551232)).Hadamard walk. Initial condition |1〉c ⊗ |0〉p Entanglement23456789Entanglemen|tc aoCvinao〉im l=aInpb |iatl0eiar〉 ilsf osortn ae toaeEfc: n eh|t1n as〉tncateognipnlge ⊗lme|c me|o0nei,nt0n 〉a〉t w=vaa av|l1kialea〉i rlb sa l be l e f o. r each step (i) Entanglement1001234567890MaximfourE mena tdac2ehn0g g0srleteeemp oe fn et na tv a a n i 4lg aN 0l be 0ulmem evbnEset nrn touafmn sgbtleee6pmr0 so0efn st taevpasi.l a|cbolei8n 〉0f o=0r |e1a〉 ch step1000 (ii) Entanglement available / maximum entanglement00000.....000056789....55555167890 E2n0ta0nglement ra4tN0io0u vmsb neur mofb setre 6op0fs 0steps. |coin〉8 =0 0|1〉 1000 1 (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4N00umber of ste6p0s0 800 1000 FstIaGte. 8:ψEcn1tangcloemmpeunttevdalfureosmforac1o0in00|1-sitceppsostq-umaenatsuumremweanlkt | it,pm FIG. 6: Plot (i) presents the probability vs location graph |ψi1000 = [Sˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√12|0ic + of a 1000-step Hadamard quantumwalk with an initial state √i2|1ic)⊗|0,0ip given by Eq. (4c), coin (Hˆ) and shift (Sˆ) 1 c 0 p andshiftoperatorprovidedbyEq. (11). Thesym- operators given by Eqs. (5) and (6) respectively, and mea- | i ⊗| i metryofthiswalk,aboutalinepassingthroughtheoriginand surement operator Mˆ1 (Eq. (7)). The thin line of (i) (red perpediculartothexaxis,isthesameasthatofaHadamard coloronline)showsthemaximumdegreeofentanglementbe- quantum walk with initial state given by |ψi = |1ic⊗|0,0ip tween walkers attainable in the post-measurement quantum (Eq. (4b)) and shift operator given by Eq. (6). Plot (ii) state ψ c1 ,andthethicklineof(i)(bluecoloronline)shows is a summary of Figs. (4.i) and (5.i), and shows that the | it,pm the actual entanglement between walkers available at each amount of entanglement between walkers available in post- step. The asymptotical behavior of entanglement values for measurementstate ψ c1 tendstobelessthantheamountof | it,pm this quantum walk is the same as that shown by a quantum entanglementbetweenwalkersavailableinpost-measurement state ψ c0 , in stark contrast to the numerical results com- walk with total initial state |0ic⊗|0,0ip (Fig. (2)). | it,pm putedforaquantumwalkwithtotalinitialstate 0 c 0,0 p | i ⊗| i (Figs. (1-3)). (9.ii) shows that the asymptotical behavior of entangle- ment values for a quantum walk with initial state given |ψi0 = (√12|0ic+ √i2|1ic)⊗|0,0ip, and Eqs. (5) and (6) bcoymEpqu.ted(4fao)riasqtuhaenstaummewaaslkthwoisteh iennittaianlgsletamteengtivveanlubeys as coin and shift operators, respectively. Fig. (7) shows Eq. (4c). the results of measuring entanglement between walkers in a coin 0 post-measurementstate ψ c0 , while Fig. Figs. (10) - (12) introduce the asymptotics of en- (8) introd|uicces corresponding results fo|r ait,cpomin 1 post- tanglement values for a quantum walk with initial state c measurement state ψ c1 . | i given by Eq. (4d). Again, although the initial state Although an initi|alitq,pumantum state of the form given |ψi0 =(√i2|0ic+ √12|1ic)⊗|0,0ip produces a symmetri- by Eq. (4c)producesabalancedprobabilitydistribution cal probability distribution (Fig. (12.i)), we notice that (Fig. (9.i)),suchapropertydoesnothaveasignificantef- the asymptotical behavior of entanglement values for a fectonthe degreeofentanglementbetweenwalkers(Fig. coin 0 post-measurement quantum state ψ c0 is dif- | i | it,pm (9.ii)). In fact, comparing plots from Figs. (3.ii) and ferentfromthatofacoin 1 post-measurementquantum | i 6 Probability of locating walker at step n00000000........000000000.000011110246824681 Hadamard walk. Initial condition [1/(2)1/2|0〉c + i/(2)1/2|1〉c] ⊗ |0〉p Entanglement123456789EntangInleitmi|acelo nsintta 〉a t=Cev a:o| 1i[ml(a〉1 pb / a(l e2 r i) sf1 o o/ 2r n) e| E0oa〉nfcc teohain nnst ga+teln ep(gim/|l(ce2eom)ni1nte/ 〉2an )=v|t1 aa|〉0icvla〉oa ib ni ll] ae ⊗ b f lo e | r0. ,e0a〉wchal ksetresp Entanglement10123456789 MaximEunmta ndfogerlg eermeaeceh no tsf ateevnpatialanbgElleen mtvasen nngtulemmbeenrt oafv satielapbsl.e | cfoori ne〉 a=c |h1 〉step Entanglement available / maximum entanglement0000000.......13456789 Entanglement ratio vs number of steps. |coin〉 = |1〉 (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4N00umber of ste6p0s0 800 1000 (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.20 200 4N00umber of ste6p0s0 800 1000 FIG. 9: Plot (i) presents the probability vs location graph FIG.11: Entanglementvaluesforcoin 1 c post-measurement of a 1000-step Hadamard quantumwalk with an initial state state ψ c1 computed from a 1000|-siteps quantum walk (√12|0ic+ √i2|1ic)⊗|0ip and shift operator provided by Eq. |ψi1000| i=t,pm[Sˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√i2|0ic + (p1lo1t).(Ti)hiesstyhmemsaemtrye oafstthheatproofbaabiHliatydadmisatrrdibuqtuioanntsuhmowwnailnk √12|1ic)⊗|0,0ip given by Eq. (4d), coin (Hˆ) and shift (Sˆ) wanitdhsihniifttiaolpsetraatteorgigvievnenbbyy|ψEiq0.=(6().√1A2|l0tihco+ug√hi2t|h1eics)y⊗mm|0,e0triyp ompeenrtatoopresrgaitvoernMbˆy1E(qEsq..(5()7)a)n.dT(6h)ertehspinecltiniveeloyf,a(in)d(mreedascuorloer- online) shows the maximum degree of entanglement between of plot (i) is significantly different from that of Fig. (3.i), walkers attainable in the post-measurement quantum state plot(ii)showsthesameasymptoticalbehaviorasthatofFig. ψ c1 ,andthethicklineof(i)(bluecoloronline)showsthe (3.ii). | it,pm actual entanglement between walkers available at each step. The asymptotical behavior of entanglement values for this Entanglement available vs number of steps. |coin〉 = |0〉 Entanglement ratio vs number of steps. |coin〉 = |0〉 quantumwalk is thesame as that shown bya quantumwalk 10 1 Fs(tiIa)Gte.Entanglement10123456789|00ψM:aiximfoEctur me,20a 0dpc0nehg smretetep o af e n nt 4a N0nc 0ug E mgl en ombtaeenlrng motelef mstmee6npp0ts 0avaeuilabnlte foetr8 e0da0cvh staepflr1u00o0emsf(oiair)c1Entanglement available / maximum entanglemento000000i.....000056789....555556789n000|-0sit20c0eppso4Ns00umtqbe-ru omf sate6p0sen0atsuum800remw10e0a0nlkt w(ii)thProbability of locating walker at step n00000000........000000000t−.000011110124682468100o00tHaadalmardi− w5n0a0lk. iInittiali cNoaunmditblioenr0 o[ fsi /s(2tet)1p/s2a|0〉 +t 1/e(25)010/2|1|〉]1 ⊗ |0i,0〉cp10⊗00 |0(,ii0)ipEntanglement(01234567890FEntaingglem|.ceo2Inni0nti 0t〉ai a=(vCl a |so0il5tma〉a b tp el ae :) r [if so ( oi)r4 / (nNe02 a0uo).1cmf/ h2eb)E n se|nt0traet 〉anop +ngf gls(e1lteme/6m(p|e20csen)o01nti/n 2ta 〉)av =|va1 ai|〉l1ai]l a〉b⊗ b le l| e.0 8 , f0 0o 〉 0r p each step1000 |ψi1000 = [Sˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√i2|0ic + √12|1ic)⊗|0,0ip given by Eq. (4d), coin (Hˆ) and shift (Sˆ) FofIGa.11020:0-sPtleoptH(ia)dparmesaerndtsquthaentpurmobwabaliklitwyitvhs aloncaintiiotinalgsrtaapthe operatorsgivenbyEqs. (5)and(6)respectively,andmeasure- ment operator Mˆ0 (Eq. (7)). The thin line of (i) (red color (√i2|0ic+ √12|1ic)⊗|0ip and shift operator provided by Eq. (11). The symmetry of the probability distribution shown in online) shows themaximum degree of entanglement between plot(i)isthesameasthatofaHadamardquantumwalkwith walkers attainable in the post-measurement quantum state |aψctiuct,0apml e,natnadngtlheemtehnitckbleitnweeoefn(iw)a(lbkleurescaovlaoirlaobnlelinaet)esahcohwssttehpe. ionpietiraaltsotragteivgeinvebnybEyq|.ψ(i6=). (A√lit2h|0oiucg+h√t1h2e|1siyc)m⊗m|0e,t0ryipoafnpdlosth(ifi)t The asymptotical behavior of entanglement values for this issignificantlydifferentfromthatofFig. (6.i),plot(ii)shows quantumwalk is thesame as that shown bya quantumwalk thesame asymptotical behavior as that of Fig. (6.ii). with total initial state 1 c 0,0 p (Fig. (4)). | i ⊗| i posed to previous cases in which asymptotical values of state ψ c1 (Fig. (12.ii)). Infact,comparingplotsfrom entanglement between walkers were different for post- tFuigms.s|t(a4ti)et,apnmψdc(010),afonrdapclootinsf|r0oimpoFsitg-sm.e(a5s)uarenmde(n1t1)qfuoarna- mFiegass.u(r1e3m)eanntdst(a1t4e)st|hψaitct,0tphmeaansydm|ψptioct,1tpimcs,owfebocathnesneteanin- coin 1 po|sti-mt,pemasurementquantumstate ψ c1 , shows glementcurvestendtothesameefficiencyof85%approx- that|thie asymptotics of entanglement val|ueist,pfomr initial imately. This tendency can also be seen in Fig. (15.ii) states given by Eqs. (4b) and (4d) are the same. where we show that both entanglement curves overlap. However and in stark contrast to the previous cases, the symmetry properties of the probability distribution ofa quantumwalk withinitial state ψ 0 =(√0.850 c III. CONCLUSIONS | i | i − √0.151 ) 0,0 (Eq. (4e)) does have an effect of the c p | i ⊗| i entanglement between walkers produced from coin post- We have proposed an algorithm to generate entangle- measurement quantum states. ment between walkers, after measuring the coin state, In Figs. (13) and(14)we exhibit the asymptoticalbe- for a Hadamardquantum walk with one (2-dimensional) havior of entanglement values of coin post-measurement coin and two walkers. Our numerical simulations show states ψ c0 and ψ c1 respectively, for a quantum that, asymptotically,the amount of entanglementgener- | it,pm | it,pm walk with initial state given by Eq. (4e). As op- ated between walkers does not reach the highest degree 7 Entanglement10123456789 MaximuEmnt adfonergg elreeamech eo nsf tte eanpvtaainlagblelem vesn tnEunmtabnegrl eomf setnetp as.v a|ciloaibn〉l e= f o|0r〉 each step Entanglement available / maximum entanglement0000....00005678....55556789 Entanglement ratio vs number of steps. |coin〉 = |0〉 Probability of locating walker at step n00000000........000000000.000011110246824681 Hadamard walk. Initial condition [(0.85)1/2|0〉c − (0.15)1/2|1〉c] ⊗ |0〉p Entanglement123456789 EntanglemIneintitaC alo svmtaEaipEnltaea|tncab:rto inla[sei(gnno0 fl〉gn.eo 8l= mreo5 mef|)e 01aene〉/|c2n tcn| hto0vta ai〉sanncltv gu〉e− ale=pe is(l ma0|( 1bob.e1〉lvl ane5 e ct ) r fk 1alo a / v2rc p |aue1 ira〉 lva cc e]b h )⊗l e s . |t 0e 〉p p (gray curve) (i) 00 200 4N00umber of ste6p0s0 800 1000 (ii) 0.50 200 4N00umber of ste6p0s0 800 1000 (i) −10000 −500 Number0 of steps 500 1000 (ii) 00 200 4NN00uummbbeerr ooff ssttee6pp0ss0 800 1000 FIG.13: Entanglementvaluesforcoin 0 c post-measurement FIG. 15: Plot (i) presents the probability vs location graph state ψ c0 computed from a 1000|-siteps quantum walk of a 1000-step Hadamard quantumwalk with an initial state √|ψ0i.110500|1=ict),p[mSˆe0n,t(0Hˆp⊗givIˆe)n]10b0y0|Eψqi0. (w4eit)h,c|oψini0(H=ˆ)(a√nd0.8sh5i|f0tic(Sˆ−) b(√y0E.8q5.|0i(c1−1).√0T.1h5e|1iscy)m⊗m|e0tirpy aonfdthsehifptroobpaebrailtitoyr pdrisotvriidbeud- | i ⊗| i tion shown in plot (i) is the same as that of a Hadamard operatorsgivenbyEqs. (5)and(6)respectively,andmeasure- monelninteo)psehroawtosrthMˆe0m(aExqi.mu(7m)).deTgrheeetohfinenltiannegolefm(ie)n(trbedetwcoeleonr q√u0a.n15tu|1mic)w⊗alk|0w,0itihp ainnidtiaslhsifttatoepgerivaetnorbgyiv|eψni=by(E√q0..8(56|0)i.cI−n thiscase,theasymptoticsofentanglementvaluesforbothcoin walkers attainable in the post-measurement quantum state |ψict,0pm, and the thick line of (i) (blue color online) shows pψosct1-me(agsruaryemcuenrvtestoaftpeslo|tψ(iicti,0)p)mte(bnldactkoctuhrevseaomfeplvoatlu(iei)s). and the actual entanglement between walkers available at each | it,pm step. Wecanseethattheasymptoticsofentanglementvalues given in plot (ii) tend to the same values as those shown in Fig. (14), obtained from a coin 1 c post-measurement state interesting result: the actual value towards which the ψ c1 computedfromaquantum| iwalkwiththesameinitial entanglement ratio converges, for each coin measure- | it,pm state (Eq. (4e)). ment outcome, depends on the symmetry of the coin initial state. However, the relationship is not straight- forward, as it is possible to find two coin initial states Entanglement10123456789 MaximfouEr mneta adcnehgg lsreetmeep eo n f t e a n v t aa in l ag bl e lem vesn tnEunmtabnegrl eomf setnetp as.v a|ciloaibn〉l e= f o|1r〉 each step Entanglement available / maximum entanglement0000....000005678.....555556789 Entanglement ratio vs number of steps. |coin〉 = |1〉 sd(asw|uusiψasycrltiehmkr0meipbtr=ehtsunoattat√itso1i,nc,2ndaa|sc0ll,otievhonx+anvopleululy√rogeigr2shoei|nno1betgfiooecattnohtnhtihdneapeinri|rngφsoleaiidetn0mimuatc=ealeennsv√gttbaa,lael0tufmle.oa8erne(.5|nc|bφ0etoGiid0tco−h)apinmnrc√gooa0btiknth.a1oeebms5rit|elt1ewibhtaiyyoe)- (i) 00 200 400 600 800 1000 (ii) 0.450 200 4N00umber of ste6p0s0 800 1000 reveal differences in two quantum walks which are not differentiated easily in the usual case of a single walker. FIG.14: Entanglementvaluesforcoin 1 c post-measurement state ψ c1 computed from a 1000|-siteps quantum walk A noteworthy feature of our algorithm is the high |ψi1000| =it,p[mSˆent(Hˆ ⊗ Iˆ)]1000|ψi0 with |ψi0 = (√0.85|0ic − aermsowuhnitcohfgthroewesntwaintghletmheenntugmenbeerraotefdstbeeptsw.eeOnutrhsecwheamlke- √0.15|1ic)⊗|0,0ip givenbyEq. (4e),coin (Hˆ)andshift (Sˆ) is particularly applicable in physical realizations where operatorsgivenbyEqs. (5)and(6)respectively,andmeasure- the coin is a qubit (such as an atom or a superconduct- ment operator Mˆ1 (Eq. (7)). The thin line of (i) (red color ing qubit) which interacts with a distinct physical sys- online) shows themaximum degree of entanglement between tem (such as an electromagnetic field mode) acting as a walkers attainable in the post-measurement quantum state ψ c1 , and the thick line of (i) (blue color online) shows walker [11, 12, 30]. Taking two such walkers, each being |theit,apmctual entanglement between walkers available at each a distinct system, one can probe how entanglement is step. Wecanseethattheasymptoticsofentanglementvalues generatedbetweenthemthroughouralgorithm. Assuch given in plot (ii) tend to the same values as those shown in walkers do not naturally interact with each other, using Fig. (13), obtained from a coin 0 c post-measurement state thecoin(qubit)systemistheonlywaytoentanglethem. |ψict,0pm computedfromaquantum| iwalkwiththesameinitial A recent circuit QED suggestion for the physical imple- state (Eq. (4e)). mentation of a quantum walk [30] even estimate a walk of a significant number of steps to be carried out within the decoherence times of the relevant physical systems. of entanglement possible at each step (purely from the Simplyenhancingsuchschemestotwowalkerswoulden- dimensionality of the space explored by the walkers in a able observing the entanglement generation mechanism givenstep)foreithercoinmeasurementoutcome. Never- that we have presented and analyzed in this paper. theless, our simulations alsoshow that the entanglement ratio (= entanglement generated/highestvalue of entan- glement possible, for each step) tends to converge to a IV. ACKNOWLEDGEMENTS rather high value (for example, to 0.8 or 0.9), and the actual convergence value seems to depend on the coin S.V-A.gratefullyacknowledgesusefuldiscussionswith initial state and on the coin measurement outcome. Dr J.L. Ball as well as the support of SNI, CONA- Convergence of entanglement ratio leads to a most CyT and Tecnol´ogico de Monterrey Campus Estado de 8 M´exico. S.B. acknowledges the support of the EPSRC, and the Wolfson Foundation. UK, the QIP IRC (GR/S82176 /01), the Royal Society [1] S.Godoy and S.Fujita, A quantumrandom-walk model [17] P.L.Knight,E.Rold´an,andJ.E.Sipe,Quantumwalkon for tunneling diffusion in a 1D lattice, J. Chem Phys. thelineasaninterferencephenomenon,Phys. Rev.A68 97(7) (1992) 5148-5154. (2003) 020301. [2] S.P.Gudder,Quantumprobability(AcademicPressInc., [18] P.L. Knight, E. Rold´an, and J.E. Sipe, Optical cavity 1998) implementations of the quantum walk, Optics Commu- [3] N. Konno, Limit theorems and absorption problems for nications 227 (2003) 147-157. quantum random walks in one dimension, Quantum In- [19] P.L. Knight, E. Rold´an, and J.E. Sipe, Propagating formation and Computation 2 (2002) 578-595. quantum walks: the origin of interference structures, J. [4] R.P. 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Rev. A 67 (2003) 042305. tumWalkonaLinewithTwoEntangledParticles,Phys. [12] G.S.AgarwalandP.K.Pathak,Quantumrandomwalk Rev. A 74 (2006) 042304. of the field in an externally driven cavity, Phys. Rev. A [27] S.E. Venegas-Andraca, J.L. Ball, K. Burnett, and S. 72 (2005) 033815. Bose, Quantum Walks with Entangled Coins, New J. [13] C.M.Chandrashekar,Implementingtheone-dimensional Phys. 7 (2005) 221. quantum (Hadamard) walk using a Bose-Einstein con- [28] C. M. Chandrashekar, Discrete time quantum walk densate, Phys. Rev. A 74 (2006) 032307. model for single and entangled particles to retain entan- [14] A. Rai, G. S. Agarwal, and J. H. H. Perk, Transport glement in coin space, arXiv:quant-ph/0609113 (2006). andquantumwalkofnonclassicallightincoupledwaveg- [29] C. Liu and N. Petulante, One-dimensional quan- uides, Phys. Rev. A 78 (2008) 042304. tum random walks with two entangled coins, quant- [15] S.E. Venegas-Andraca, Quantum Walks for Computer ph/Arxiv:0807.2263 (2009). 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