MM-PARTICLE QUANTUM WALKS WITH INTERACTION
CLEMENT AMPADU
31 Carrolton Road Boston, Massachusetts, 02132 U.S.A. e-mail: [email protected]
Abstract
We consider directional correlations between M particles on a line. For non-interacting particles we find analytic asymptotic expressions. When -interaction is introduced in the model we study the Fourier analysis and obtain general analytic formula for the wave function of the walk in the case M for the transformation C , which can be considered an unfactorized version of the Hadamard walk in two-dimensions. KEY WORDS: quantum random walk, interacting particles, particle correlation, entanglement
PACS: Introduction
Quantum walk on the line with a single particle possess a classical analogue, involving more walkers opens up the possibility to study collective quantum effect, such as many particle correlations. In this context, entangled initial states and indistinguishability of the particles play a role [4]. In this paper we investigate the non-classical effects in the M particle discrete-time quantum walk on the line. Starting in Section 2 we recall some facts about the quantum walk of a single particle on a line, in Section 3 we extend the ideas of Section 2 to include M particles, and give the probability )( tP sameside of finding the M particles on the positive or negative side of the line. In Section 4 we obtain the asymptotic limit of )( tP sameside . In Section 5 entangled initial states are considered, there we we consider two approaches in analyzing )( tP sameside for coin states that do not factorize. In Section 5.1 we analyze the case of maximally entangled Bell-type basis , and in Section 5.2 employ the equivalence etween the M -particle walk on the line and single particle walk on an M dimensional lattice. This equivalence allows to invoke the weak limit theorem of Grimmet et.al [2] which have been successfully used by a number of authors [1,6]. In Section 6 we consider the probability )( tP sameside for indistinguishable particles, and show that for a particular choice of M bosons or fermions the problem reduces to the case of distinguishable particles with maximally entangled coins. Stefanak et.al [4] noted in their work involving directional correlations in quantum walk with two particles, that entanglement in two-particle non-interacting quantum walks cannot break the limit of probabilities they found for separable particles, and posed the following question: What happens if we consider interacting particles? This motivated them to introduce the concept of two-particle quantum walks with interaction to the solution of their question. The authors found out that by introducing a interaction one can exceed the limit derived for non-interacting particles. In Section 7 we commence the study of this new model focusing on the Fourier analysis. Section 8 is devoted to the conclusions.
2. Quantum walk on the line with one particle
The Hilbert space of the quantum walk is given by the tensor product CP HHH , where P H is the position space of the particle , and C H is the coin space of the particle. The position space is spanned by the set of orthonormal states }:{ Zii , and the coins space is spanned by RL , . To define the movement of the walker in one dimension, we first consider what happens on one step in the quantum walk. We first make superposition on the coin space with the coin operator C U and move the particle according to the coin state with the translation operator S as follows CW UISU , where I is the identity operator in the particle’s position space, W U is the coin operator on the position space, and the translation operator S is given by x RRxxLLxxS . The evolution of the quantum walk is then defined y )()1( tUt W , which by induction on t , can be written in terms of the initial state of the particle as )0()1( Wt Ut . If we take * HU C , where ),( * qpH is a one- dimensional generalization of the Hadamard walk, for example, Ampadu [1], we get the most studied case of the Hadamard coin. We can express the wave vector )( t as the spinor xtR tLt Zx xx )( )()( , where the quibit T RL ),( has an upper (lower) component associated to the left (right) chirality, and the states x are the eigenstates of the position operator corresponding to the site x on the line, from which it follows that probability distribution generated by the quantum walk is given by )()(),( tRtLtxp xx .
3. Quantum Walk on the line with M particles.
We assume that the M particles are distinguishable. Let M HH ,, be the Hilbert spaces of the M particles, and let M UU ,, be the coin operator on the position space of the M particles respectively, then the extension of the formalism above for an M particle walk is as follows. For the M distinguishable particles, the Hilbert space of the composite system can be written as MMcomposite
HHHHHH . Assuming further that the time evolution of the particles is independent, then the coin operator on the position space of the composite system can be written as MMcomposite
UUUUUU . We suppose further that the all the particles start from the origin then the initial state of the M particle quantum walk is given by CtimesM , where C is the initial coin state of the M particles. Suppose that the initial coin state is factorizable. Since the time evolution of the particles are independent, entanglement is neglible and the factorization of the M -particle state remains unaffected, thus the joint probability istribution ),,,,( tmmmp M of finding the M particles at sites M mmm ,,, , respectively at time t can be written as the product of the single particle distribution to give Mi iM tmptmmmp ),(),,,,( , where ),( tmp i is the probability distribution of a single-particle quantum walk given that the initial coin state was i . If the initial coin state is not factorizable, then Mi iM tmptmmmp ),(),,,,( , however we can map the M -particle walk on a line to a quantum walk of a single particle on lattice whose dimension is the same as the number particles in the M -particle walk on the line, namely M itself, then it follows that we can write coin operator on the position space of the composite system as timesM CCcompositecompositecomposite UUISU , where composite I is the identity operator on the position space of the composite system, and Mcomposite
SSS , and each i S in the tensor product represents the translation operator of the individual particles. Henceforth we will take * HU C , where ),( * qpH is a one-dimensional generalization of the Hadamard walk, for example, Ampadu [1], so timesM CCcompositecompositecomposite UUISU , the coin operator on the position space of the composite system can be interpreted as the coin operator of a single particle walk on an M dimensional lattice with the coin given by the tensor product of M Hadamard operators. The directional correlation between particles can be quantified in terms of the probability that the particles are found after t steps of the quantum walk on the same side of the line Stefanak et.al [4]. In the case of the M particle quantum walk, if all the particles are distinguishable, letting )( tP sidesame denote the probability that the particles are found after t steps of the quantum walk on the same side of the line, we have tm tm tm tm Mtm tm tm tm Msidesame M MM M tmmmptmmmptP ),,,,(),,,,()(
If the particles are indistinguishable, then the probabilities ),,,,( tmmmp M and ),,,,( tmmmp MM are equivalent, hence )( tP sidesame have to be restricted over an ordered pair ),,,( M mmm with ii mm for all Mi . In particular we have Mi tm tmm Mtm mm Msameside i iii ii tmmmptmmmptP ),,,,(),,,,()( .
4. Separable initial states
For the M particles assuming the initial coin state C can be factorized as MC , then we recall from the previous section that the joint probability distribution ),,,,( tmmmp M factorizes, so we can write the probability )( tP sameside as Mi iMi isameside tPtPtP )()()( , where )( tP i denotes the probability that the particle has started the quantum walk with the coin state i is on the positive or negative side of the line after t steps, in particular, tm ii i tmptp ),()( and ),()( tm ii i tmptp . Let us consider a general separable coin state of the form RbLaRbLaRbLa
MMMMC . The asymptotic distribution for a single particle due to Konno [3] can be written as, Stefanak et.al [4], txtxt bbaabatxbatxp iiiiiiii (4.1) he probability that the single particle is on the negative or positive side of the line has been calculated by Stefanak et.al [4] and are given as iiiiiit iiiii bbaabadxbatxpbap (4.2) iiiiiit iiiii bbaabadxbatxpbap (4.3) Now substituting (4.2) and (4.3) into Mi iMi isameside tPtPtP )()()( we get Mi Mi iiiiiiiiiiiiMsameside bbaababbaabatP (4.4) Now we recast (4.4) in a suitable form that one can use for example to determine when the particles are most likely to be on the same side or opposite side of the half-axis by looking into the direction of maximal bias in the probability distribution. Now we consider the basis formed by the eigenstates of the Hadamard coin, Stefanak et.al [4], have shown that * H has the following expression in the standard basis, RL (4.5) Further they have shown that by decomposing the coin state of one particle in the Hadamard basis as iii hh , the relation between the coefficients in the standard and the Hadamard basis is given by iii hha and iii hhb , from which it follows that equation (4.4) can be written as Mi Mi iiiiMsameside hhhhtP (4.6) where the parameters i h are given by the by the overlap of the coin state i with the eigenstate .
5. Entangled initial states 5.1 On initial coin states of the Bell-type
If the coin state is not factorizable, then the joint probability distribution is no longer a product of the single-particle distribution. However, we can decompose the M particle state in terms of single- particle amplitudes. In this way, we decompose the joint probability distribution into single-particle distributions plus an interefence term [4]. We then can use the results of the previous section to find the asymptotic value of the probability )( tP sameside , say. We first consider the following Bell-type Basis for the M particle quantum walk LRRLRLLRRLRLLRLR , RRRLLL . Note that we have used the term Bell-type basis , because the M particle quantum walk is not a bipartite system, but the basis has the same form as the original Bell basis, and its self-explanatory for the M particle quantum walk. Let ),( tm Li denote the amplitude of a single particle being after t steps at the position m with the coin state i , RLi , , provided that the initial coin state was L . Similarly, let ),( tm Ri denote the amplitude of a single particle being after t steps at the position m with the coin state i , RLi , , provided that the initial coin state was R . Let us agree to define ),(),(),(),(,,,, tmtmtmtmtmmm MRMLRkLkMLRLR kkk M (5.1) ),(),(),(),(,,,, tmtmtmtmtmmm MLMRLkRkMRLRL kkk M (5.2) ),(),(),(),(,,,, tmtmtmtmtmmm MLMRRkLkMRLLR kkk M (5.3) ),(),(),(),(,,,, tmtmtmtmtmmm MRMLLkRkMLRRL kkk M (5.4) ),(),(),(),(,,,, tmtmtmtmtmmm MLMLLkLkMLLLL kkk M (5.5) ),(),(),(),(,,,, tmtmtmtmtmmm MRMRRkRkMRRRR kkk M (5.6) The joint probability distribution generated by the quantum walk of the M particle with the initially entangled coins described by the Bell-type basis, upon using (5.1)-(5.6) is given by ,,,,,,,, ,,,,,,,,21),,,,( RLkk MLRRL kkkMRLLR kkk MRLRL kkkMLRLR kkkM
M MM MM tmmmtmmm tmmmtmmmtmmmp and ,,,,,,,,21),,,,( RLkk MRRRR kkkMLLLL kkkM
M MM tmmmtmmmtmmmp where the superscript in both the formulas ),,,,( tmmmp M and ),,,,( tmmmp M indicates the initial coin state. Since * H and the Bell-type initial states contain only real entries, we can drop the absolute value in the formulas for ),,,,( tmmmp M and ),,,,( tmmmp M , and expand the joint probability distribution in the form Mi iMLRRLMRLLR MRLRLMLRLRM tmtmmmptmmmp tmmmptmmmptmmmp ),(),,,,(),,,,( ),,,,(),,,,(21),,,,( (5.7) and Mi iMLLLLMLLLLM tmtmmmptmmmptmmmp ),(),,,,(),,,,(21),,,,( (5.8) ote that in (5.7) and (5.8) terms of the form ),,,,( tmmmp MLRLR , for example, are defined as ),(),(),(),(),,,,( tmptmptmptmptmmmp MRMLRLMLRLR (5.9) and the terms in the product Mi i tm ),( are defined for example in (5.8) for ),( tm as ),(),(),( tmtmtm LRLRRLRLRL , where in this definition ),( tm LRLRL for example is understood as ),(),(),(),(),( tmtmtmtmtm MRLMLLRLLLLRLRL . Recall for distinguishable particles, tm tm tm tm Mtm tm tm tm Msidesame M MM M tmmmptmmmptP ),,,,(),,,,()( (5.10) If we insert either (5.7) and (5.8) into (5.10) we find we can write the result as )()()( tItptp
LRLRsamesidesamside and )()()( tItptp LLLLsamesidesamside , where )( tI is in the interference term, and is given by )()()( tttI , where ),()( tm tmt and tm tmt ),()( . On the M dimensional lattice the time evolution of the Hadamard walk is determined by the generator MMM kUkUkUkkkU )()(),,,( , where )( r kU denotes the generator of a single-particle walk on the line and this is given by * HeeDkU ikikr . Due to the tensor product decomposition of ),,,( M kkkU we can write its eigenvalues as the product of the eigenvalues )( r kU and we can also write the eigenvectors of ),,,( M kkkU as the tensor product of the eigenvectors of )( r kU . The eigenvalues of ),,,( M kkkU are given by Mt ttxM kwiMxxx ekkkk (321 ,,,, , M xxx , where )( kiw x e are the eigenvalues of the atrix )( r kU , and )( kw x is determined by kkw , )()( kwkw . Similarly, the eigenvectors of ),,,( M kkkU are given by the tensor product )()()(),,,( MxxxMxxx kvkvkvkkkv MM of the eigenvectors of the matrices )( r kU which are given by Tikkiwik eeeknkv )(11 , Tikkiwik eeeknkv )(22 , where kkkkn cos1cos)(cos12)( and kkkkn cos1cos)(cos12)( . Now invoking the weak limit theorem of Grimmett et.al, the cumulative distribution function is then given by )~,,~,~,~( M Mxx MM xx Iw xxxxxM dttttF , where we have let ttt ii ~ , and M tttI ~,~,~, . The probability measure is determined by Mz zCMxxxxxxxx dkkkkv MM . Note that the vector C which corresponds to the initial state of the coin C consists of M components. From the explicit form of the eigenvectors ),,,( Mxxx kkkv M , the probability measure M xxxx can be seen to equal Mi iMi Mj ji ji ji j ivv viiixxixxxxiiiixMxxxx dkkSkCxkSSkCCkSSkCC jjji iiM
11 2 1 1 1 11 where kkkC cos1 )cos()( , kkkS cos1 )sin()( , and the coefficients XSC ,, are determined from the initial state of the coin C . To obtain the cumulative distribution function we have need the domain of integration, and these are obtained from Mxxxx kkkw M ,,, , that is, the gradients of the phases Mxxxx kkkw M ,,, of the eigenvalues of ),,( M kkU . From the explicit form of the eigenvalues of ),,( M kkU , the gradients are MMM xxxxxxMxxxx
CCCkkkw )1(,,)1(,)1(,,, . Using the above results and he substitution iiii qkkkC cos1 cos)( , ii ii qq dqdk , the cumulative distribution function takes the form Mi Mj ji ixxxiiMi t ii iMM qCqCqq dqtttF ji . Recall that differentiation of the cumulative distribution function gives the probability density function, in particular the relation MMM ggg Fggggp ),,,,( , implies that Mi Mj ji jixxxxiiMi iiMM ttCttCttttttttp j , where we have let ttq ii . Now using ),,,,( M ttttp and replacing the sums in tm tm tm tm Mtm tm tm tm Msidesame M MM M tmmmptmmmptP ),,,,(),,,,()( by integrals we get
Mi t iMMi t iMsameside dtttttpdtttttptP ),,,,(),,,,()( , where ),,,,( M ttttp is given as above. Whilst we have not been able to obtain a closed form formula, we should note that there is a simplification of the formula above which will give the same formula in the case of seperable states, that is, for seperable states Mi Mi iiiiM Mi t iMMi t iMsameside hhhh dtttttpdtttttptP This implies that the bounds obtained by both formulas is the same.
6. Asymptotic Probability for Indistinguishable particles.
It is natural in this case to use the so-called second quantization formalism [5]. To give the second quantization formalism for the single particle dynamics. Let us denote the bosonic creation operators by , ˆ rmT a , and the fermionic creation operators similarly. Note that , ˆ rmT a creates one bosonic particle at position m with the internal state r , RLr , ; similarly for the fermionic operator. The dynamics of the quantum walk with indistinguishable particles on a one particle level, for bosonic particles is given by the following transformation of the creation operators (which is also similar for fermionic particles) RmTLmTLmT aaa ,1,1, ˆˆ21ˆ RmTLmTRmT aaa ,1,1, ˆˆ21ˆ The difference between the bosonic and fermionic operators is that the bosonic operators fulfill the following commutation relations ,, rmrm aa and ,, ˆ,ˆ rrmmrmTrm aa whilst the fermionic operators satisfy the anti-commutor relations ),(),( rmrm bb and ,, ˆ, rrmmrmTrm bb otice that since the dynamics is defined for a single particle, we can describe the state of the M indistinguishable particles after t steps of the quantum walk in terms of the single-particle amplitudes. As the initial state of the coin we choose LRLR ,00 , that is, the M particles are initially at the origin with alternating coin states starting with L for the first particle, R for the second particle, and so on. Now recalling the amplitudes Li and Ri for the single particle performing the quantum walk with the initial coin state L , R respectively, the state of the M bosons (and similarly for fermions) can be stated in the following form vacatmmt M M MMM mm RLrr rrmmTMLRLRrrBosons ,, ,,, ,;,1)( ˆ),,,()( , where vac denotes the vacuum state, and we have defined ),(),(),(),,,( tmtmtmtmm MRrMLrLrMLRLRrr
MMM and similarly for MM rrmmT a ,;, ˆ . Note that the summation indices M mm ,, run over all possible sites. Now we consider the joint probability );,,,,( tmmmmp M that after t steps we detect a particle at sites M mmmm ,,, ,321 simultaneously with ii mm . If ii mm , then, ),,,,(),,,,(),,,,(),,,,( )(1);,,,,( RLrr MLRRL rrMRLLR rrMRLRL rrMLRLR rr FermionsBosonsrrrmmmMFermionsBosons
M MMMM MM tmmmtmmmtmmmtmmm ttmmmmp where the plus sign on the right corresponds to bosonic, and the negative sign corresponds to fermionic. If we compare these expressions to those for the typeBell states considered previously we get the following relations );,,(2);,,,( tmmptmmmp MMBosons and );,,(2);,,,( tmmptmmmp MMFermions . Now we suppose that M mmm , then for bosons one has );,,,();,,,( );,,,();,,,( );,,,(4);,,,(4);,,( tmmmtmmm tmmmtmmm tmmmtmmmtmmp LRLRLRRLLRLRRLLR LRLRRLRLLRLRRRRR LRLRRRRRLRLRLLLLBosons
Similarly for fermions one has );,,,();,,,( );,,,();,,,();,,,( tmmmtmmm tmmmtmmmtmmmp
LRLRLRRLLRLRRLLR LRLRRLRLLRLRLRLRFermions
If we compare the last two expressions for those obtained in the case of the Bell-type states we also get for the case MM mmmm , the following );,,,();,,,( tmmmptmmmp Bosons and );,,,();,,,( tmmmptmmmp Fermions . Recall for indistinguishable particles we had Mi tm tmm Mtm mm Msameside i iii ii tmmmptmmmptP ),,,,(),,,,()( , where the summation is restricted to ordered pairs ),,,( M mmm with ii mm . However using the results );,,(2);,,,( tmmptmmmp MMBosons , );,,(2);,,,( tmmptmmmp MMFermions , );,,,();,,,( tmmmptmmmp Bosons , );,,,();,,,( tmmmptmmmp Fermions , we can replace );,,,( tmmmp MFermionsBosons in Mi tm tmm Mtm mm Msameside i iii ii tmmmptmmmptP ),,,,(),,,,()( by ),,,( tmmp M and extend the summation over M mm ,, to conclude that )()( tPtP samsideBosonssameside and )()( tPtP samesideFermionssameside , which shows that results obtained are the same for distinguishable particles starting the quantum walk with entangled coin states. . On the notion of interaction: Fourier Analysis of C We begin by defining the evolution operator for quantum walks with interaction. To define this, we change the factorized time evolution on the position space of the composite system given by MMcomposite
UUUUUU . Recall that in the original time evolution the coin was the same factorized coin in all lattice points ),,,( M mmm , in the interaction we change the coin to a non-factorized one C , when the particles are at the same lattice point M mmmm . We then define the unitary time evolution operator for the quantum walk with M interacting particles on a line as CPSUUPSU compositetimesM CCcomposite , where P is the projector on the joint position state of the M particles, and is defined by timesMm timesM mmmmmmP , and PIP composite . In the context of the previous paragraph it should be noted that C has dimension MM , however here we will restrict ourselves to the case M , and consider C , this coin can be considered the unfactorized version of the Hadamard coin in two dimensions, in that it is similar to the Hadamard coin in two dimensions, however, ** HHC . Morever, recall that the 2-particle walk on the line is equivalent to a single particle walk in two dimensions. Now the spatial Fourier transform ),,( tkk yx for ,, yx kk of the wave function ),,( tyx over Z is given y ),( )( ),,(),,( Zkk ykxkiyx yx yx etyxtkk . Let ),,( ),,( ),,( ),,(),,( tyx tyx tyx tyxtyx UDRL , where the subscripts ,,,
DRL and U refer to the “left”, “right”, “down” and “up” chirality states, that is, ),,( ),,( ),,( ),,(),,( tyx tyx tyx tyxtyx UDRL is the four component vector of amplitudes of the particle being at point ),( yx at time t . Define L M , R M , D M , U M . The dynamics of the C walk is given by ),1,(),1,(),,1(),,1()1,,( tyxMtyxMtyxMtyxMtyx DULR . From the dynamics of we may deduce the following about : ),,(),1,( ),1,(),,1(),,1( ),1,(),1,(),,1(),,1()1,,(ˆ , ))1(( , ))1((, ))1((, ))1(( )(, tkkMeMeMeMe etyxMe etyxMeetyxMeetyxMe etyxMtyxMtyxMtyxMtkk yxDikUikLikRik yx ykxkiDik yx ykxkiUikyx ykxkiLikyx ykxkiRik ykxkiyx DULRyx yyxx yxy yxyyxxyxx yx his implies that we have ),,(ˆ)1,,(ˆ , tkkMtkk yxkkyx yx , where yyyy yyyy xxxx xxxxyyxxyx ikikikik ikikikik ikikikik ikikikikDikUikLikRikkk eeee eeee eeee eeeeMeMeMeMeM , Note that we can write CeeeeDM yyxxyx ikikikikkk ),,,( , , where ),,,( yyxx ikikikik eeeeD is the diagonal matrix with entries as shown and C is the unitary matrix mentioned earlier that acts on the chirality state of the particle. Hence, yx kk M , is also a unitary matrix. In terms of the initial spatial Fourier transform we can write )0,,(),,(ˆ , yxt kkyx kkMtkk yx . A popular method to calculate t kk yx M , is to diagonalize the unitary matrix yx kk M , , this method has been used successfully by a number of authors including Ampadu, Watabe et.al. Now we notice if yx kk M , has normalized eigenvectors ,,, yxyxyxyxyxyxyxyx kkkkkkkkkkkkkkkk NNNN , where i kk yx N , are appropriate normalization constants, and corresponding eigenvalues ,,,, ,,, ykxkykxkykxkykxk , then the evolution matrix t kk yx M , can be written as i kki kki kkti i kkt kk yxyxyxyxyx NM ,,2,41 ,, . It follows that the Fourier transform at time t is given by
41 ,,2,, )0,,(~),,(~ i i kkyxi kki kkti kkyx yxyxyxyx kkNtkk .
8. Concluding Remarks