Almost Perfect State Transfer in Quantum Spin Chains

arXiv:1205.4680v3 [quant-ph] 23 Oct 2012 et ilntb e.Ntoal,w sueta the require- that assume PST we precise Notionally, the met. that be and not perfect will ments be not will complementary a following the in approach. sim- adopt randomly shall also We ro- are ulated. more which transfer (which imperfections make against chains [4] to bust modulated states In boundary [23]), PST. to and coupled with [17] are couplings to chain similar the a methods of to uses mod- fields perturbations are magnetic device random and the adding the of by [8]) [7], production eled also the quan- (see the in [9] In to , imperfections measurements. imputable here [6] and errors shall wires them We the tum circumvent studies. to various or particularly of correct object relate the to been are how has transfer [20] state and of etc. [22] fidelity the biases [16], on influences systematic their fabrication interactions, What operations, additional read-out and defects, input imperfect or specifications. theoretical the experimental from and in deviations manufacturing properties unavoidable transfer the ideal of these view of robustness the of that couplings chains [11]. the spin [1], modulating in sites achieved the and between be engineering can properly PST shown, by one been that so, has is particular, It this (PST). in when transfer finding state one; perfect be of of output probability speaks the the as that state wished initial the is it Ideally, high. chain. the through for of realized need dynamics is the ad- intrinsic key transfer minimizes the A the it as that time. interventions some is external after method transfered end this is other chain of the the vantage of at the end qubit one in the at to systems qubit devices, the these basic of In state possible the channels. quantum as such have [1] of chains construction [3], spin years, proposed quantum recent been many In to relevance protocols. of information is locations distant tween etk o ie httemnfcuigo h chains the of manufacturing the that given for take We nonsynchronous errors: of sources many are There is importance practical of question a context, this In be must wires quantum of reliability and efficiency The be- states quantum transfer to models of design The PT n oevracmoae h nvial imperfect co unavoidable restrictive the less accommodates much moreover under and (PST) corollar happens a APST as obtained is approximation. result energi This 1-excitation numbers. the rational over of when help systems, the mirror-symmetric with in wires quantum efficient of modelling h aua oino lotpretsaetase AS)i (APST) transfer state perfect almost of notion natural The .INTRODUCTION 1. nttt o hsc n ehooy .uebr t.72, str. 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PST nor APST are possible. In a significant model, the where J are the constants coupling the sites l 1 and l l − times for which APST is attained can be estimated show- and Bl are the strengths of the magnetic field at the sites x y z ing in this instance that a good approximation to PST l (l =0, 1,...,N). The symbols σl , σl , σl stand for the is obtained in finite time independently of the size of the Pauli matrices which act on the l-th spin. chain. It is immediate to see that Let us finally draw attention to recent related and com- 1 N plementary publications that have made use of the same [H, (σz +1)]= 0 2 l tools from number theory in the case of spin chains with Xl=0 uniform couplings. In his thesis [5], using the Kronecker theorem, Burgarth has shown that the Heisenberg XX which implies that the eigenstates of H split in subspaces spin chains with prime lengths have arbitrary good fi- labeled by the number of spins over the chain that are delity. In [12], Godsil introduced the notion of pretty up. In order to characterize the chains with APST, it good state transfer on graphs which is equivalent to our will suffice to restrict H to the subspace spanned by the definition of APST. Our terminology is dictated by the states witch contain only one excitation. A natural basis fact that in these instances the transition amplitudes are for that subspace is given by the vectors almost periodic functions. Recently, the characterization e = (0, 0,..., 1,..., 0), n =0, 1, 2,...,N, of the uniform XX Heisenberg chains which admit this | ni pretty good state transfer was analyzed using again num- where the only ”1” occupies the n-th position. The re- ber theoretic methods [13]. striction J of H to the 1-excitation subspace acts as fol- In dealing with the Heisenberg XX spin chain with lows uniform coupling and zero magnetic fields, the mirror- symmetry of the one-excitation Hamiltonian is de facto J en = Jn+1 en+1 + Bn en + Jn en−1 . (2.2) | i | i | i | i ensured as this operator coincides with the adjacency matrix of the (N + 1)-path. As will be seen, this symme- Note that try requirement comes into play in relation with APST, J = J = 0 (2.3) when more general couplings and magnetic fields are con- 0 N+1 sidered. It is remarked in [12] that in order to get a good is assumed. approximation to perfect state transfer, the waiting time Consider the polynomials χn(x) obeying the recurrence must be very large. Numerical results presented in [13], relation indicate that for a given level of fidelity, or approximation, the required times grow linearly with N, the num- Jn+1χn+1(x)+ Bnχn(x)+ Jnχn−1(x)= xχn(x) (2.4) ner of sites in the chain minus one. (We shall comment with on these observations in Section 5, using exact results on almost perfect return [14] as applied to the uniform χ−1 =0, χ0 =1. (2.5) XX chain.) Thus Godsil [12] expressed the view that pretty good state transfer or APST will not be a satis- They satisfy the orthogonality relations factory substitute for perfect state transfer in practice. N We here wish to stress in this connection that when con- sidering non-uniform couplings, there are situations, as wsχn(xs)χm(xs)= δnm, (2.6) shown by an aforementioned example in Section 5, where Xs=0 high fidelity can be achieved in finite time irrespectively where ws are the discrete weights taken to satisfy of the chain length. This suggests that the class of spin chains with APST, a much larger one than the PST fam- N ily, should not be so radically deemed of impractical use ws =1. (2.7) at this point. Xs=0

In what follows we shall take the eigenvalues xs in increasing order 2. 1-EXCITATION STATE TRANSFER IN XX x < x < x < < x . (2.8) SPIN CHAINS 0 1 2 ··· N Let

We shall consider XX spin chains with nearest- N neighbor interactions. Their Hamiltonians H are of the xs = √wsχn(xs) en . (2.9) form | i | i nX=0 N−1 N It is easily seen that these vectors s are eigenstates of 1 x x y y 1 z | i H = Jl+1(σ σ + σ σ )+ Bl(σ + 1), J with eigenvalues x 2 l l+1 l l+1 2 l s Xl=0 Xl=0 (2.1) J x = x x . (2.10) | si s| si 3

Since both bases s and en are orthonormal this means that there then exists a time t for which the {| i} {| i} iφ(t) state e0(t) coincides (up to a phase factor e ) with ′ ′ | i en em = δnm, xs xs = δss , the state e . h | i h | i N In the case| i of almost perfect state transfer, there is no we also have time t for which condition (3.18) is verified. Neverthe- N less, it is possible to approach the state e with any | N i e = √w χ (x ) x . (2.11) prescribed degree of accuracy. From a practical point of | ni s n s | si Xs=0 view, there is no essential difference between perfect and almost perfect state transfer, owing to inevitable tech- Let PN+1(x) be the characteristic polynomial of J nological and measurement errors. However, the APST conditions are much weaker than the PST ones. APST P (x) = (x x )(x x ) ... (x x ). (2.12) N+1 − 0 − 1 − N therefore widens the possibilities for constructing efficient The discrete weights are expressed as [10] quantum wires. Let us first derive the necessary condition for APST. hN Taking into account expansion (2.11), we have ws = ′ , s =0, 1,...,N (2.13) PN (xs)P (xs) N+1 N N − e (t) = √w e ixst x , e = √w χ (x ) x with h = J 2J 2 ...J 2 and P (x)= h1/2χ (x). | 0 i s | si | N i s N s | si N 1 2 N N N N Xs=0 Xs=0 (3.19) and hence 3. NECESSARY AND SUFFICIENT 2 CONDITIONS FOR ALMOST PERFECT −iφ(t) e e0(t) eN = QUANTUM STATE TRANSFER | i − | i N 2 −iφ(t)−itxs By ”almost perfect state transfer” (APST) we mean ws e χN (xs) . (3.20) − the following. Xs=0

Assume that the spin chain is prepared at time t = 0 It is easily seen that in order to fulﬁll condition (3.15) we in the pure state e0 . This means that that at site n =0 need the spin is up while| i all other spins are down. For arbitrary times t, this state will evolve into the χN (xs) =1. (3.21) state | |

− To convince oneself of this fact, suppose that (3.21) does e (t) = e itJ e . (3.14) | 0 i | 0i not hold and assume that for some s = 0, 1,...,N, we have χ (x ) = a with a = 1. In this case, using the We demand that for any small ǫ> 0, there exists a value N s | | 6 of time t such that reverse triangle inequality we have

2 2 iφ(t) −iφ(t)−itxs 2 e0(t) e eN < ǫ, (3.15) e χN (xs) ( a 1) > 0 | i− | i − ≥ | |− where φ(t) is a real parameter which can depend on t. and as a consequence the r.h.s. of (3.15) cannot be made This means that the state e0(t) can be as close to the arbitrarily small. state e as desired and that| e i has thus undergone at Since χ (x) has only real coeﬃcients (3.21) implies | N i | 0i N time t an almost perfect transfer. that χN (xs) = 1. From general properties of orthog- The notation η ξ 2 for two vectors ξ and η stands as onal polynomials± [10], [23] it follows, using (2.12), that usual for | − | (3.21) amounts to

N χ (x ) = ( 1)N+s. (3.22) ξ η 2 = ξ η 2 (3.16) N s − | − | | k − k| Xk=0 As shown in [23], (3.22) implies that the matrix J is persymmetric, or mirror-symmetric i.e. where ξk are the expansion coeﬃcients of the vector ξ over a basis, say e : | ki RJR = J, (3.23) N ξ = ξ e . (3.17) where R is the reﬂection matrix k| ki Xk=0 0 0 ... 0 1 0 0 ... 1 0 Recall that the condition for 1-excitation PST reads R =   ...... − e (t) = e iφ(t) e ; (3.18)  1 0 ... 0 0  | 0 i | N i   4

We have thus obtained a necessary condition for APST For every δ > 0 there exist a real parameter φ and a namely, that the Jacobi matrix J corresponding to the value of time t such that spin XX Hamiltonian should be mirror symmetric. This x t πs + φ <δ (mod 2π). (3.31) coincides with one of the necessary conditions for PST | s − | [17]. The other condition for PST requires that In more details, (3.31) can be rewritten in the form π x x = M , (3.24) δ < xst πs + φ +2πMs < δ, (3.32) s+1 − s t s − − where M are integers which may depend on s. where M are arbitrary positive odd numbers. s s This obviously amounts to a condition on the spectrum We shall now obtain the conditions on the spectrum of of J for APST to occur. The question is: what are the J for APST. properties that the eigenvalues x must possess to ensure The matrix J is Hermitian and mirror symmetric with s that it is possible to find a time t and integers M so J > 0 and hence all its eigenvalues x are real and dis- s i s that (3.32) is satisfied and hence APST realized. In other tinct. words, given the set of real numbers a = φ πs, this is Let f (t) be the amplitude for finding the system in s 0n asking: when is it possible to find values of−t for which the state n at time t if was in the state e at time 0 x t is approximated in terms of integers by a 2πM t = 0, | i | i s s s with any prescribed accuracy? − −iJt The solution of this Diophantine approximation prob- f0n(t)= en e e0 . (3.25) h | | i lem (3.32) is given remarkably by the Kronecker theorem It is easily seen that [23] [15], [19]. In order to state it, let us introduce the following def- N inition [15], [19]: A set αi,i =1, 2,... of real numbers is −ixst f0N (t)= wse χN (xs). (3.26) called linearly independent if for any n the only rational Xs=0 values of r1,...,rn satisfying Taking into account (3.22) we have r α + r α + + r α =0 1 1 2 2 ··· n n N − are r1 = r2 = = rn = 0. f (t)= w e ixst( 1)N+s. (3.27) ··· 0N s − The Kronecker theorem can be formulated in two ver- Xs=0 sions, one a special case of the other. The first version is Mindful of the normalization (2.7), it is seen that con- attributed to Kronecker himself. dition f (t) 1 is equivalent to the condition that | 0N |≈ Version 1 Assume that the real numbers xs, s = − 0, 1,...N are linearly independent over the field of ra- e ixst( 1)N+s eiφ (3.28) − ≈ tional numbers. Let a0,a1,...,aN be fixed arbitrary real for a fixed value t, where means ”approximately equal numbers. The Kronecker theorem states that for every with any prescribed accuracy”.≈ To paraphrase (3.28), δ > 0 there exist a real t and integers M0,M1,...MN there thus should be a value t of time for which the l.h.s. such that the inequalities is as close as desired to a phase independent of s. xst as 2πMs <δ, s =0, 1, 2,...,N (3.33) This means that f0N (t) is an almost-periodic func- | − − | tion. | | hold. Recall that any almost periodic function f(t) is a for- mal trigonometric series [2] We see how this theorem directly applies to the APST problem. Indeed, provided that the eigenvalues are lin- ∞ early independent over the rational numbers, we may iωnt f(t)= ane , (3.29) conclude from the Kronecker theorem that for every pa- −∞ n=X rameter φ it is possible to find a time t so that f0N (t) is as close to 1 as desired. | | where ω are real parameters. For periodic functions n In many cases however, the eigenvalues x are not lin- f(t + T ) = f(t), one has ω = 2πn and (3.29) is the s n T early independent. This means that there can exist L ordinary Fourier series. For almost periodic functions independent relations of the type there exist the so-called almost periods. This means that for every ε > 0 there exists a real parameter T = T (ε) (i) (i) (i) r0 x0 + r1 x1 + + rN xN =0, i =1, 2,...,L N such that the inequality ··· (3.34)≤ (i) f(t + T ) f(t) <ε (3.30) where rs are integers such that for every i =1, 2,...,L | − | at least one of them is nonzero. (The use of integers is holds for all t. equivalent to that of rationals and more practical.) In turn, condition (3.28) is tantamount to an inequality When additional relations such as (3.34) are present, involving the exponents that can be stated as follows: the Kronecker theorem can be formulated as follows [19]. 5

Version 2 Assume that the real parameters xs, s = occurs. Indeed, since (3.22) (which implies mirror- 0, 1,...,N are all distinct and moreover that there symmetry) specifies χN (x) at N + 1 distinct points, this are L relations of the type (3.34) with nontrivial sets polynomial can be obtained by Lagrange interpolation. (i) (i) Two orthogonal polynomials associated to the 3-diagonal r0 ,...rN of integers. { Then, the} approximation condition (3.33) holds for ev- matrix J are therefore known: χN (x) and the charac- N ery δ > 0 if and only if the real quantities a satisfy the teristic polynomial (x xs) of degree N +1. As i s=0 − conditions shown in [23], all theQ orthogonal polynomials can then be found by the Euclidean algorithm thereby providing (i) (i) (i) r a0+r a1+ +r aN 0 (mod 2π), i =1, 2,...,L their 3-term recurrence relation explicitly, i.e. giving all 0 1 ··· N ≡ (3.35) the couplings Ji and magnetic fields Bi. (i) with the same integers ri as in (3.34). The difference in the conditions for PST and APST arises only in the restrictions on the spectrum or the For the proof of this statement see [19]. eigenvalues of J. As is observed, these conditions are Using this (generalized) version of the Kronecker the- much more stringent for PST than for APST. orem we can formulate the necessary and sufficient conditions for APST. 4. SPECTRAL SURGERY AND APST General result. Let x0, x1,...,xN be N +1 distinct eigenvalues of the Jacobi matrix J corresponding to the XX spin chain (2.1). Assume that there are L N rela- A spectral surgery procedure was proposed in [23], as tions of the type (3.34) with nonzero integer parameters≤ a way to construct XX spin chains with PST from such (i) systems already known to possess the PST property. rs . Given the initial spectrum X = x , x ,...,x , un- Then, the following conditions are necessary and suf- N { 0 1 N } der spectral surgery, one or several levels x 1 , x 2 ,...,x ficient for APST: i i ij (i) the Jacobi matrix J is mirror-symmetric, i.e. are removed from the set XN . The reduced set is X˜ − = X R , where R = N j N \ j j Bs = BN−s, Js = JN+1−s, s =0, 1,...N; (3.36) xi1 , xi2 ,...,xij is the set of levels that have been removed.{ The} spectral levels determine the (mirror- (ii) the L linear relations symmetric) Jacobi matrix J (and hence the Hamiltonian of the XX spin chain) uniquely. Thus the new spectral N set X˜ − generates a new Jacobi matrix J˜ of dimen- r(i)(πs φ)=0 (mod 2π), i =1, 2,...L (3.37) N j s − sion (N + 1 j) (N + 1 j). Certain restrictions Xs=0 − × − need to be imposed on the possible choices of sets Rj . are compatible. Namely, Rj can contain any number of levels starting (0) An obvious special case of this statement occurs when from the first one, say R = x , x ,...,x 1− or M1 { 0 1 M 1} all eigenvalues xs are linearly independent over the field similarly, any number of levels from the last level, say (N) of rational numbers. This means that L = 0, i.e. that − − RM2 = xN , xN 1,...,xn M2+1 . The only restriction there are no additional relations such as (3.37) and that is that there{ should be no gaps in} the above sequences. version 1 of the Kronecker theorem suffices to conclude to Apart from these two elementary surgeries, levels can the occurrence of APST . In such instances , we see that also be extracted from the middle of the spectral set the mirror symmetry of J and the linear independence XN ; in this case the requirement is the following: all of the eigenvalues represent the necessary and sufficient such level sets should consist in the union of subsets conditions for APST. (i) RL = xi, xi+1,...,xi+L−1 of even length L and with- Let us now compare the conditions for APST with the { (i) } out gaps within R . Equivalently, one can say that only conditions for PST. It is known that the PST conditions L pairs x , x of neighbor levels can be removed from the are [17]: i i+1 middle of the set X . (i) the Jacobi matrix J is mirror-symmetric; N Remarkably, the Jacobi matrix J˜ corresponding to a (ii) Assuming that the eigenvalues xs are ordered so surgered set X˜ − , is obtained from the initial Jacobi that x < x < < x , the differences ∆ = x x N j 0 1 N s s+1 s matrix by j Christoffel transforms (see [23] for details). are proportional··· to odd numbers: − That is, after j Christoffel transforms, we obtain a new ˜ ∆s = κ(2js + 1), (3.38) Jacobi matrix J which is mirror-symmetric and satisfy the PST property. This observation allows to generate where js are integers and κ is a constant not depending new explicit examples with PST without the need to per- on s. form the inverse spectral problem algorithm. This is ad- We see that mirror-symmetry is necessary for both vantageous since the formulas of the Christoffel transform PST and APST. This implies that the matrix J and are rather simple. hence the Hamiltonian can be uniquely reconstructed One such example was already given in [23]. Let from the eigenvalues x0, x1,...xN when either process N be odd. We start with the uniform grid XN = 6

N/2, N/2+1,...,N/2 and then remove 2L levels parameters ; it will furthermore be seen that the wait- {− − } symmetrically from the middle of the set XN ing times can be estimated and found to be finite independently of the size of the chain for arbitrarily good X − = N/2, N/2+1, L 3/2, L 1/2, N 2L {− − ···− − − − fidelities in some special APST situations. L +1/2,L +3/2,...,N/2 1,N/2 . (4.1) i. Absence of magnetic fields. − } Consider the special situation corresponding to zero Such a spectral set corresponds to the model of the XX magnetic fields Bs = 0. In this case the Jacobi matrix J spin chain with PST proposed in [21]. has only two nonzero diagonals. The corresponding or- Consider what is the effect of spectral surgery with thogonal polynomials are symmetric Pn( x) = Pn(x) respect to APST. and hence the eigenvalues satisfy the properties− − It is almost obvious that under an arbitrary admissible surgery procedure the APST property survives. Indeed, xs + xN−s =0, s =0, 1,...,N (5.1) if all eigenvalues xi were linearly independent over the where it is assumed that x < x <...x . field of rational numbers then any reduced set XÑ−j will 0 1 N obviously satisfy the same property. Consider first the case N odd. Then one has (N + Assume that there exists a set of linear relations (3.34) 1)/2 independent relations (5.1) for s = 0, 1,..., (N 1)/2. They coincide with relations (3.34) where only two− on the spectral levels xs. APST is possible iff the set of linear relation (3.37) is compatible (it is assumed a priori integers are nonzero and given by rs = rN−s = 1. that the matrix J is mirror-symmetric). Relations (3.37) then reduce to the single condition Under spectral surgery some levels of the set X are N πN 2φ =0 (mod2π) (5.2) removed. This means that some of relations (3.34) will − disappear (or remain the same). This implies that the from which we find new Jacobi matrix J˜ will also possess APST. π/2 if N =4m +3 Interestingly, this method can sometimes generate a φ = − (mod 2π). (5.3) chain with APST, even if the initial chain lacks this prop- π/2 if N =4m +1 erty. Indeed, the absence of APST for a given mirror- symmetric XX chain means that relations (3.37) are in- If one assumes that the eigenvalues x0, x1,...,x(N−1)/2 compatible. When we remove some of the levels x the are linearly independent, then the Kronecker theorem i guarantees with φ = π/2 that the corresponding spin number of relations (3.34) can be reduced which in prin- ± ciple can lead to the compatibility of the reduced set of chain will exhibit APST. relations (3.37). Of course one should seek to realize this If N is even, we shall have the N/2 independent relations (5.1) for i =0, 1,...,N/2 1 and in addition in concrete examples. We plan to investigate this prob- − lem in the future. xN/2 =0. (5.4) Conditions (3.37) are compatible iff 5. EXAMPLES AND SPECIAL CASES 0 if N/2 is even φ = (mod 2π). (5.5) We shall present a number of examples and special π if N/2 is odd cases in this section: Again, if the eigenvalues x , x ,...x − are linearly i) we shall indicate how the central result specializes 0 1 N/2 1 independent, the APST property is present. in the absence of magnetic fields; ii. The uniform XX chain with zero magnetic ii) we shall record the circumstances for APST in the fields. uniform XX chain and shall discuss the relation between The APST (or pretty good state transfer) of the XX the waiting time and the length of the chain for this par- chain with uniform couplings and no magnetic fields, i.e. ticular system, using results on almost perfect return (to J = 1 and B = 0 for all s, have recently been sorted the point of origin); s S out in [13]. It has been found there that a chain of length iii) we shall show, not too surprisingly, that the con- N+1 admits APST if and only if N = p 2or2p 2 where ditions for PST are a special case of the conditions for p is prime or iff N =2m 2. As a matter− of fact,− the proof APST; of that result given in [13],− effectively proceeds from the iv) we shall consider in details the case N = 4 corre- application of version 2 of the Kronecker theorem and sponding to spin chains with 5 sites. In this instance, the use trigonometric relations between the eigenvalues it is possible to give a very explicit analysis of APST and to estimate waiting times using standard methods of xs = 2 cos(πs/(N + 2)), s =1, 2,...,N + 1 (5.6) Diophantine approximation. The conditions for all the possible cases (PST, APST and neither) are derived. which are given by the roots of the Chebyshev polyno- v) we shall introduce a remarkable model based on mials of second kind for this model. the para-Krawtchouk polynomials that can exhibit PST, In the same reference [13] (see also [12]), it is pointed APST or neither depending on the value of one of its out, on the basis of numerical analysis, that the waiting 7 times for APST will grow with the size of the chain. This chosen using the dilation factor α.) This freedom relates has cast doubts on the practical use of APST. Our next to the values that the phase φ takes modulo 2π. For a examples will hopefully dissipate this negative view by given spectrum, the APST conditions might restrict φ, showing that there are models exhibiting APST where however shifts in the ground state energy will allow φ to the waiting times stay finite as the size is grown. take arbitrary values. Before we leave this uniform XX chain, we would like iv. Spin chains with 5 sites and without mag- to offer the following comment on the waiting times as netic field the chain is taken to have very large length. Consider the the XX spin chain without magnetic The quantity fields (B0 = B1 = = BN = 0) that contains 5 sites. In this case it is possible··· to express the APST conditions −itJ f00(t)= e0 e e0 (5.7) explicitly and to describe 3 possible scenarii: PST, APST h | | i and neither. represents the return amplitude, i.e. the amplitude to This corresponds to N = 4. There are only two dis- return to the initial (input) state after a time t. Since tinct exchange constants: J1 and J2. Indeed, due to the −itJ iφ itJ −iφ mirror-symmetry condition (3.36) we have (for N = 4) e e0 e eN = e eN e e0 , | i− | i | i− | i that J4 = J1, J3 = J2. The spectrum of the corresponding Jacobi matrix J if we have APST from e0 to eN , we also have APST | i | i can easily be found: from eN to e0 . As a consequence, in a chain with APST,| wei must| observei that there is almost perfect re- x0 = a, x1 = b, x2 =0, x3 = b, x4 = a, (5.11) turn, that is that there are times tn, n =0, 1, 2,... such − − that 2 2 where a = J1 +2J2 , b = J1 are two positive parameters. Clearlyp a>b so the eigenvalues xs in (5.11) are lim f00(tn) =1. (5.8) n→∞ | | given in increasing order. This is a special case of the situation considered in i. The conditions for almost perfect return have been an- The sufficient condition for APST is that the parameters alyzed in [14] and can be simply stated: An XX spin a and b be linearly independent over the rationals. This chain will exhibit almost perfect return if and only if is equivalent to the statement that the ratio a/b is an its one-excitation Hamiltonian J has a pure point spec- irrational number. We thus see that if J 2 +2J 2/J is trum. As is readily seen, the spectrum of the uniform 1 2 1 an irrational number, then the spin chainp with 5 nodes XX chain becomes continuous as N, the number of sites has the APST property. minus one, becomes infinite (see [14] for more details.) It Suppose now that the ratio a/b = n/m is a rational follows that almost perfect return does not occur in the number where n,m are co-prime integers. This means case of a semi-infinite Heisenberg XX chain and hence that there are two conditions on the eigenvalues not surprisingly an infinite time is required for APST in this model. mx0 nx1 = mx4 nx3 = 0 (5.12) iii. PST as a special case of APST. − − If a set x0, x1,...xN of eigenvalues satisfy the APST Then the relations (3.37) which are required for APST conditions,{ it is clear that} the affine transformed eigen- (with φ = 0) reduce to the only condition that n be even valuesx ˜s = αxs + β will also satisfy the APST condi- and m be odd. tions. In particular, we can always assume that x0 = 0 But this is equivalent to the PST condition. Indeed, and x1 = 1. The PST condition (3.38) then implies that we have x2s = K2s are even integers while x2s+1 = K2s+1 are odd a b n − = 1= M/m, (5.13) integers. We have N linear relations b m −

Ksx1 = xs, s =0, 2, 3,...,N. (5.9) where M and m are both odd. Observe that the differences between the eigenvalues are ∆1 = x1 x0 = With φ = 0 we see that the conditions (3.37) become a b, ∆ = x x = b, ∆ = x x = b,−∆ = − 2 2 − 1 3 3 − 2 4 x4 x3 = a b. Hence (5.13) coincides with the PST Ks = s(mod 2). (5.10) condition− for− a chain with 5 sites. In all other cases, i.e. when n is odd and m is odd, or This is equivalent to the condition that the parity of the when both n and m are odd, the APST condition is not integers K coincides with the parity of s, i.e. with the s valid and the XX spin chain does demonstrate neither PST condition. We thus see that the PST condition is a PST nor APST. special case of the APST condition, as expected. In the APST situation (i.e. when a/b is irrational), it A note is in order here. In general the translation is interesting to estimate the waiting times. It is easy to x x + β of the energy spectrum has no physical s s calculate the amplitude f (t) directly from (3.27): meaning:→ it is always possible to redeﬁne the value of 0N the ground state energy and for instance to put it equal a2 b2 b2 1 f (t)= − + cos at cos bt (5.14) to zero as noted. (Similarly the scale can be arbitrarily 0N 2a2 2a2 − 2 8

This is a simple example of almost periodic functions: two number with opposite parities. This means that for two harmonic oscillations with frequencies ω1 = a and any irrational number z = a/b there exists an infinite se- ω2 = b do not constitute a pure periodic function. But quence of generalized convergents cn = un/vn such that the Kronecker theorem guarantees that there exists a se- lim →∞ c z and u is even while v is odd. More- n n → n n quence tn such that f0N (tn) 1 when n . over, the best approximation property (5.17) holds for all There are several simple algorithms→ to→ estimate ∞ the such convergents. quantities tn. Given this sequence, let us put One of them proceeds through the expansion of a/b πv into a continued fraction [15] t = n . (5.18) n b 1 z = a/b = ζ + , (5.15) The amplitude f0N (tn) then becomes 0 1 ζ1 + 2 2 2 ζ2 + ... a b b a 1 f0N (tn)= −2 + 2 cos πvn + . (5.19) 2a 2a b 2 where the quotients ζ0, ζ1,... are integers. The corre- This can easily be simplified to sponding convergents pn/qn of the continued fraction are rational numbers which provide the best approximation b2 of the irrational number a/b [15]. In our case this method 2 f0N (tn)=1 2 sin (πvnεn/2), (5.20) works well if all (or at least infinitely many) convergents − a have the property that pn is even and qn is odd. Oth- where erwise one can apply the method of Farey fractions [15] a un or, equivalently, the method of intermediate convergents εn = (5.21) [18]. b − vn This method can described as follows. Starting with is the accuracy of the rational approximation of the irra- the convergents p /q ,p /q ,... we compute the so- { 0 0 1 1 } tional number a/b. called mediants (or intermediate convergents) by the for- By property (5.17) we have that mula [18] π2b2 p˜ p p f0N 1 . (5.22) n = n+1 ± n . (5.16) ≈ − a2v2 q˜ q q n n n+1 ± n It is seen from this formula that f0N (tn) converges to 1 The convergents pn/qn together with the intermediate when n . Moreover, this formula allows to estimate convergentsp ñ/qñ form a set of generalized convergents the accuracy→ ∞ of this approximation if the (generalized) wn = un/vn,n = 0, 1, 2,... , where un = pn or un =p ñ convergents un/vn are known explicitly. (similarly v = q or v =q ˜ ). From the elementary n n n n It is seen that f0N (tn) 1 when tn . This means properties of the convergents it follows that all fractions that the sequence t is→ associated to→ APST. ∞ { n} un/vn are simple (i.e. un and vn are co-prime). More- Consider the special case J1 = J2 = 1 so that all cou- over, it is possible to show [18] that the generalized con- plings are now equal. For the homogeneous Heisenberg vergents provide the general solution to the best approx- chain (i.e. with Ji = 1 for all i = 0, 1, 2,...,N) it is imation problem: they satisfy the inequality known [11] that PST only occurs when there are 2 or 3 − sites. We are thus in a situation of APST for a uniform z u /v < v 2 (5.17) | − n n| n chain with 5 nodes. In this case a = √3, b = 1. The continued fraction It is easy to show that there are infinitely many general- representation is ized convergents un/vn with the desired property: un is even while vn is odd. 1 √3=1+ . (5.23) Indeed, assume that there is only a finite number of 1 the convergents p /q with the desired property. This 1+ n n 1 means that starting with some n = M the numerators 2+ 1+ ... pn are all odd; the corresponding denominators qn may be either even or odd. Necessarily, for any neighbor pair The first convergents are pn/qn and pn+1/qn+1, the denominators qn and qn+1 should be of opposite parity. Indeed, if one assumes pn 5 7 19 26 71 that q and q have the same parity then their me- = 1, 2, , , , , ,... . (5.24) n n+1 qn 3 4 11 15 41 diant (pn + pn+1)/(qn + qn+1) will be reducible (with both numerator and denominator even) which is impos- It is seen that the first two convergents with the desired sible. Thus the mediantsp ñ/qñ will have the desired property (i.e. an even numerator and an odd denomi- property, becausep ñ = pn+1 pn is even as a sum of nator) are 2/1 and 26/15. Computing the mediants ac- two even numbers andq ˜ = q ± q is odd as a sum of cording to (5.16), we obtain the additional intermediate n n+1 ± n 9 convergent 12/7. Hence, the first 3 desired (intermedi- Indeed, we can always choose an infinite set of (inter- ate) convergents are mediate) convergents u /v ,u /v ,...,u /v ,... with { 1 1 2 2 n n } the property that the numerators un are all odd. Intro- u 2 12 26 n = , , (5.25) duce the value vn 1 7 15 un εn = γ (5.31) The first fraction, 2/1, yields an approximation to the − vn difference between f0N (t) and 1 with an accuracy of 3 10−1; the second fraction| | 12/7 yields an approximation which describes the accuracy of the Diophantine approx- with· accuracy 2 10−2. imation of the parameter γ. We already saw that · So, already in the case of a spin chain with 5 sites we − ε < v 2. (5.32) see that all 3 possibilities: PST, APST or neither can | n| n occur. APST is generic, i.e. APST happens for almost Choose all possible values of J1 and J2. We shall consider next a XX spin chain with an ar- t = πv . (5.33) bitrary (even) number of spins which demonstrates the n n similar property. Substituting this in (3.27) and taking into account that v. Spin chain corresponding to the para- u is odd we have Krawtchouk polynomials. n − − Assume that N is odd and that the spectrum xs is (N 1)/2 (N 1)/2 − such that (up to an affine transformation) f (t )= w + e iπγvn w = 0N n − 2s 2s+1 Xs=0 Xs=0 x =2s, x =2s+γ, s =0, 1,..., (N 1)/2, (5.26) 2s 2s+1 − (N−1)/2 (N−1)/2 −iπvnεn where γ is a real parameter such that 0 <γ< 2. w2s e w2s+1. (5.34) − − This situation can be realized in spin chains that are Xs=0 Xs=0 obtained from the para-Krawtchouk polynomials with In [24] it was shown that for the para-Krawtchouk poly- the recurrence coefficients [24] nomials the identities γ + N 1 B = − , (N−1)/2 (N−1)/2 n 2 w = w =1/2 (5.35) n(N +1 n)((N +1 2n)2 γ2) 2s 2s+1 J 2 = − − − . (5.27) Xs=0 Xs=0 n 4(N 2n)(N 2n + 2) − − hold. Hence The corresponding Jacobi matrix is mirror symmetric. When γ = 1 the grid is uniform and the Jacobi matrix J 1 − f (t )= 1+ e iπvnεn . (5.36) which is associated to the ordinary Krawtchouk polyno- 0N n −2 mials generates PST [1]. It is clear that all eigenvalues can be expressed as linear Now combinations of x and x . Indeed, 1 2 π2 f0N (tn) = cos(πvnεn/2) 1 2 (5.37) x2s = sx2, x2s+1 = x1 + sx2, s =0, 1, 2,... (N 1)/2 | | ≈ − 8vn −(5.28) Hence, according to our central result (3.37), in order to where we have used (5.32). have APST, the following conditions must be realized: Formula (5.37) gives a good approximation of the amplitude f0N if vn is sufficiently large (depending on the a = sa , a = a + sa , (mod 2π) (5.29) | | 2s 2 2s+1 1 2 physical requirements on the accuracy). E.g. for vn > 10 we get an accuracy of 1%. where as = πs φ. These relations are compatible iff φ =0 (mod2π−). Consider, e.g. the value γ = √3. If the parameter γ is irrational then there are no addi- From (5.23) we find the first appropriate (i.e. with all numerators un odd) convergents tional relations for the eigenvalues x1, x2 and thus provided (5.29) is verified, APST will occur. u 5 7 19 45 71 Equivalently, this means that there exists a sequence n = 1, , , , , ,..., . (5.38) t , n =1, 2,... such that vn 3 4 11 26 41 { n } lim f0N (tn) = 1 (5.30) Already the third convergent 7/4 gives an accuracy of n→∞ | | about 1%. This is because the actual accuracy ε3 As in the previous case it is possible to determine the 0.0179 of the third convergent is smaller than what the≈ times tn and estimate the corresponding rate of conver- rhs of formula (5.32) gives. Hence the first waiting time gence in (5.30) using standard Diophantine approxima- corresponding to an accuracy of 1% in the amplitude is tion methods. . t3 =4π. 10

Consider now the case where γ = p/q is a rational theorem in Diophantine approximation proved essential number, with p and q coprime integers. In this case there in this investigation. The necessary and sufficient condi- is an additional linear relation between x1 and x2: tions that have been found for a chain to admit APST bear on the 1-excitation restriction J (a tri-diagonal ma- 2qx1 = px2 (5.39) trix) of the chain Hamiltonian. Not unlike what is needed for PST, the requirements for APST entail a symme- and hence a similar relation should hold for a1 and a2: try condition on J as well as spectral restrictions. As it turns out, like for PST, J must be mirror-symmetric that 2qa1 = pa2 (mod 2π) (5.40) is, invariant against reflection with respect to its anti- diagonal. The conditions on the eigenvalues of J turn where a = π, a = 2π. If the numerator p is odd then 1 2 out (in general) to be much less demanding for APST (5.40) holds for any pair of coprime integers p, q and than for PST. They amount roughly to the conditions hence the APST condition is satisfied. In fact we are for which the Kronecker theorem applies and imply that then in the situation where not only APST but even PST at least a subset of these eigenvalues be linearly indepen- occurs. Indeed, this corresponds to the model with PST dent over the field of the rational numbers. Spin chains that we derived and discussed in [24]. with APST therefore much enlarge, in principle, the class If the numerator p = 2j is even for some integer j of such systems that can be exploited as quantum wires (in this case, necessarily the denominator q is odd) then since their fidelity can be as good as is technically rele- relation (5.39) becomes qx = jx and hence (5.40) reads 1 2 vant. It has also been demonstrated how chains with APST qπ 2jπ =0 (mod2π). (5.41) − can be constructed from a chain that already admits (or Obviously, (5.41) cannot hold when q is odd and there is possibly not) APST, by the removal of single excitation no PST nor APST in this case. energy levels from the parent system via Christoffel trans- In summary the picture is as follows: forms. This offers a constructive method to broaden the (i) when γ is an irrational number, APST is observed; catalog of systems with APST. (ii) when γ = p/q is rational with p odd there is PST; A number of examples and special cases interesting in (iii) when γ = p/q is rational with p even neither PST their own right have also been presented and discussed. nor APST happens. They illustrate situations where PST or APST occur or Let us make the following remark in connection with where neither can happen. One of the models introduced the last two examples. As mentioned in subsection iii, in have provided counter-examples to the view borne out in the discussion of the uniform XX chain, almost perfect particular from the examination of the Heisenberg XX return occurs in chains of the type we have been con- chain with homogeneous couplings, that APST requires sidering, this is so whenever the spectrum of the Jacobi excessively long times in extended wires. matrix J is discrete. Clearly, this will be the case for We trust this report provides interesting information any XX spin chain with nearest-neighbor interactions on the transfer of state in XX spin chains with nearest- that has a finite number of sites. It should be stressed neighbor couplings and suggests that APST requires fur- that finite length does not similarly imply APST; as we ther analysis. have seen, there are examples of systems with finite 1- excitation Hamiltonian J that do not possess APST. Acknowledgments

6. CONCLUSIONS AZ thanks Centre de Recherches Mathématiques (Uni- versité de Montréal) for hospitality. The authors Let us recapitulate the essential elements of our anal- would like to thank M.Christandl, M.Derevyagin and ysis of almost perfect state transfer in spin chains. It A.Filippov for stimulating discussions. They also ac- builds on the knowledge that XX spin chains with prop- knowledge communications from M.Bruderer, D. Bur- erly engineered couplings and magnetic fields can effect garth and S. Severini and are also very grateful to a the transport of states from one end to the other with referee for a very constructive and stimulating review. probability 1 over certain times - these are chains with The research of LV is supported in part by a research perfect state transfer (PST). In view of the fact that there grant from the Natural Sciences and Engineering Re- are always manufacturing or measurement imprecisions, search Council (NSERC) of Canada. our study aimed to characterize the models where al- though not perfect, state transfer could be realized with a probability very close to 1 - such chains have been said to show almost perfect state transfer (APST). We have thus undertaken to categorize all XX spin chains with arbitrary nearest-neighbor couplings and magnetic fields that would exhibit APST. The Kronecker 11

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