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On the Eigenvalue Problem in Multipartite Quantum Systems

M Enriquez1'2

1UPIITA, Instituto Politécnico Nacional, Av. Instituto Politécnico Nacional 2580 CP 07340 México D.F., México

2Départamento dé Física, Cinvestav, AP 14-740, 07000 México DF, México E-mail: menriquezf@fis.cinvestav.mx

Abstract. Wé show that thé Hamiltonian of a multiqudit systém can bé diagonalizéd through a séquéncé of unitary transformations writtén in térms of Hubbard opérators. As an application this formalism is appliéd to particular casés of two and thréé-qubit systéms.

1. Introduction

The eigenvalue problem plays a relevant role in quantum mechanics. Indeed, the set of eigenvalues of a linear operator A yields the numerical results that one would obtain when the observable represented by A is measured [1]. In the case where the system of study is integrated by several parts some algebraic subtleties should be taken into account. For instance, consider a system of n subsystems with d levels each, the pure states are represented by vectors in the Hilbert space Hfn, here 0 denotes the Kronecker (tensor or direct) product [2,3], and Hd stands for the Hilbert space of a single d-level system (qudit). An operator Ak acting on the k-th subsystem is promoted to act on the entire space in the following way

Afc o Id 0 Id Id ®Afc ®I d Id, (1)

(k-1)—times (n-k)-times

where Id is the identity operator in Hd. It is not difficult to realize that the matrix representation of (1) has at most d"+1 < d2n entries different from zero. This fact suggests to look for a convenient framework to handle with the involved cumbersome calculations. To approach the tensor product algebra in a more convenient way we proposed the use of Hubbard operators [2-4]. Indeed, these are also refered as X-operators and satisfy the following properties

i) XijXkm = 5jkXim (multiplication rule)

ii) £ Xk'k = I (completeness)

iii) (Xij )t = Xj,i (non-hermiticity)

Because of property ii) any linear operator A acting on the whole system can be expressed in terms of the X-operators. For instance, the Hamiltonian of the system in this representation

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of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

H = E hln_1....il0j3n_1....j30 Xin-1,jn-1 Xi1,j1 <g> Xi0,j0, (2)

¿n-iV" ¿0

Of particular interest is the diagonalization of Hamiltonians with nearest-neighbour spin interaction. Among them are the bipartite three dimensional anisotropic Ising model [5], the XY Heisenberg model for spin-1/2, which can be exactly diagonalized using the Jordan-Wigner transformation [6], etc. For higher spins the diagonalization of some Heisenberg models have been accomplished numerically [7]. In the present work we use the tensor product properties of the Hubbard operators to diagonalize some spin interacting Hamiltonians. Indeed, it has been shown in [2,4] that the tensor products in equation (2) can be written as an X-operator depending on just two indices. Such a representation allows us to generalize the Unitary Transformation Method (UTM) introduced by Ovchinnikov [8] to diagonalize a Hermitian operator in an easy manner.

This contribution is organized as follows. In Section 2 we present a short review of the UTM for a single d-level system. The formalism of Hubbard operators for systems of several qudits is discussed in Section 3. It is shown that cumbersome calculations of the tensor product can be turned into simple relations of indices. Moreover, we generalize the UTM for hermitian operators acting on a multiqudit space. In Sec. 4 this formalism is applied to specific cases of two and three-qubit Hamiltonians. Finally, some remarks are given in Sec. 5.

2. A review of the unitary transformation method

Let B = {|p) : p = 0,1,..., d — 1} be an orthonormal basis of Hd. Thus, a vector e H can be written as a normalized linear combination of the basis elements

d-1 d-1 №) = E Cp|p), Cp e C, E |cpl2 = 1. (3)

p=0 p=0

On the other hand, the simplest representation of Hubbard operators acting on Hd is constructed as the following outer products

Xp,q := |p)<q|, p,q = 0,1,...,d — 1, (4)

where the sub-index d makes reference to the dimension of the space. It is not difficult to prove that the operator so defined fulfill the properties i) - iii). Furthermore, the matrix representation of (4) has 1 in the entry (i + 1, j + 1) and zero elsewhere. The action of the operator XP'q on any basis vector follows immediately

Xpq |r) = 5q,r (5)

Thus, the Hamiltonian H of the system can be written as a linear combination of Hubbard operators

d-1 d-1

H = E £pXP'P + E Vp,qXP'q, £d e R, Vp,q e C, Vqp = Vp*q, (6)

p=0 p=q

the conditions on the coefficients follow from the fact Ht = H. We restrict ourselves to the case where all coefficients are real, hence Vq,p = Vp,q. In order to solve the eigenvalue equation

H E) = EpE), p = 0,1,...,d — 1, (7)

we look for an unitary operator U so that the transformed Hamiltonian

H' = UHU t = £ EvXpv. (8)

is indeed diagonal. Note that in this representation the states |p) are eigenstates of (8) with eigenvalue Ep. Remark that degeneracy is not been taking into account. Moreover, the states fulfilling (7) are obtained as |Ep) = Ut|p), for p = 0,1,..., d — 1. In order to find U we first consider the d(d — 1)/2 unitary operators defined as

Uk/(ak/) = exp[ak/(X,¡'1 — Xed'k)], k < i, ak/ e R. (9)

For fixed indices k and i and using the Baker-Campell-Hausdorff formula it is easy to show that the operator (9) changes H to

H(1) = UM (ak,i)HUJl/ (aM) = £ e^X™ + £ V^X™, (10)

p=0 p=q

e^ = 2(ek + ee) + 2(ek — e£) cos2ak,e + Vk/ sin2ak/, e^P = 1 (ek + ee) — 2 (ek — ee)cos2ak,e — VM sin2ak,e,

Vk(1) = —1 (ek — ee) sin2ak,e + Vk,ecos2ak,e, (11)

Vk(,p) = Vk,p cos ak,e + Ve,p sin o^e, V^ = Vp/ cos ak,e — Vp,k sin o^e, p = k, i, ep^ = ep, Vp,q) = Vp,q, p,q = k, i.

The former procedure can be repeated iteratively to get H(r) = VHVt where the transformation V = V (akr ,er,..., ®ki ,ei) = Ukr ,er ••• Uk1/1, with r < n(n — 1)/2. To erase the off-diagonal elements we require V^^ = Vs^ = ... = V^^ = 0. These conditions constitue a system of r non-linear equations in the parameters akl,ei,..., akr,er, whose solution allows to find the operator V. Clearly, the eigenvalues of both H( r) and H are the numbers ePr). Moreover, the eigenvectors of H are computed as |Ep) = Vt|p). According to Ovchinnikov [8] we should solve d(d — 1)/2 non-linear equations, however it will be shown that in some cases the number of equations and parameters can be reduced.

3. Multiquidit systems

The Hilbert space Hf" of a system of n qudits is constructed as follows

Hf" = Span{|i"-i) ® ■ ■ ■ ® |ii) ® |io), ie = 0,1,..., d — 1}. (12)

According to Ref. [4] the basis elements can be labeled using a single-index notation identifying |in-i)®-■ |i1 )®|i0) = |in-1 ■ ■ ■ i0) = |i), where i = (in-1 ■ ■ ■ i0)d. The last expression indicates that the numbers ie are the coefficients of the expansion of i in base d and they are related with the physical state of the (n — i)-th subsystem. Moreover, any vector e Hf" is a linear combination of the form

dn-1 dn-1

=^2 ci|i) =^2 Ci|i"-1 i0), i = (in-1 ••• i0)d. (13)

i=0 i=0

On the other hand, it was shown in [4] that any linear operator A acting on the Hilbert space of a multipartite system can be written in terms of Hubbard operators using a single-index for both rows and columns, for instance the Hamiltonian (2) reads

H = E hin-i,...,io;in-i,...jo, i = (in-1 ■ ■ ■ j = (jn-1 ■ ■ ■ j0)d. (14)

This representation is convenient to deal with some algebraic calculations. For instance, the action of H on a linear combination (13) follows immediately from equation (5). Note that if a n-qudit Hamiltonian is written in this form then can be diagonalized by the UTM. This observation generalizes the method for multiqudit systems. In the following Section two particular cases are studied.

4. Applications

An operator defined as follows

H2 = eraXn + emXd , + V^mjX/ + Xd , }, m > n, em > e„. (15)

is known as a two-level form. In particular, when d = 2 the well-known two-level Hamiltonian is recovered. We now consider the transformation (10) on H2 with k = n and £ = m. To take the operator (15) to a diagonal form we require V^m = 0, which yields the condition

) = 2Vn,m cos(2an,m), (16)

hence it is obtained

sin(2an,m) = [(¡n-^m-2M^^mm2, cos(2an,m) = j^-^m^kmm2. (17)

Substituting the last equations in the expressions for ei1- and in (11) the two eigenvalues of H2 are immediately computed

En — ^ — 2(en + em) — 1 [(en — em)2 + 4(Vn,m)2] ^

Em — em — 2 em ) + 2 |_(era em)

The eigenvectors are calculated as |Ep) = U»t,m(a:ra,m)|p), for p = n, m. Hence, we obtain

|En) = cosa„,m|n) + sinan,m|m), |Em) = cos |n) - sina„,m|m), (19)

cos an,m

As an example consider the Hamiltonian for the two-qubit anisotropic Ising model proposed by Delgado [5]

H2 = - E Xiai ® + 6з(! ® ^a) + сз(^з ® 1), (21)

where a are the Pauli matrices and Aj,63 and c3 are real parameters. Using equation (14) and identifying A1 = A, A2 = x and A3 = P this operator can be written as H2 = h1 + h2, where

h1 = (x - A){X40,3 + X43,0} + (b3 + C3 - ^)X40,0 - (63 + C3 + ^)X43,3

h2 = (x + A){X41,2 + X42,1} + (-63 + C3 + P)X41,1 - (63 - C3 + P)X42,2.

are two-level forms. In order to diagonalize H2 we apply a sequence of two transformations (9)

as follows H22) = U1,2U0,3H2Uq 3Uj2. Note that the action of U0,3 (U1,2) leaves h2(h1) invariant.

(2) (2)

So, the conditions V0 3 = V 2 = 0 yield the parameters a0,3 and a1,2. Moreover, the eigenvalues are computed using equations (19)

E = -P - [(63 + C3)2 + (x - A)2]1/2 , E1 = p - [(63 - C3)2 + (x + A)2]1/2 , E = P + [(63 - C3)2 + (x + A)2]1/2 , E3 = -P + [(63 + C3)2 + (x - A)2]1/2 .

The corresponding eigenvectors in the single-index and computational basis read

|£b) = cos a0,3|0) + sin «0,313) = cos a0,3|00) + sin 0:0,3! 11), |E1) = cos a1,2|1) + sin a1,2|2) = cos a1,2|01) + sin a1,2|10), |E2) = cosa1,2|1) - sina1,2|2) = cos a1,2|01) - sina1,2|10), |E3) = cosa0,3|0) - sina0,3|3) = cos a0,3|00) - sina0,3|11),

cos Oo 3

E3 — eo

, and cos a12 =

E3 - E0

We now consider a particular case of a four-level form given by H4 = epXp,p + eq (Xq,q + Xd,r + X,T)

E2 — ei E2 — Ei

+ {Y + XT + X?^) + 5 + Xf + X7) + h.c.}, Y, 5 G R.

In order to diagonalize H4 we take a sequence of transformations (9). The entries of the transformed operator h41- = Uq,SH4Uq,S read

¡q1- = eq + 5 sin(2aq,s), eS1- = eq - 5 sin(2ag,s), V^S = 5 cos(2ag,s),

V^ = Y cos aq,s + Y sin aq,s, Vq(1 = 5 cos aq,s + 5 sin aq,s, (27)

Vm = Y cos aq,s - Y sin aq,s, Vr(S) = 5 cos aq,s - 5 sin aq,s.

Remark that V^fe- = V^ and the other entries remain the same. As a second step we make the transformation H^2) = Uq,rH^1)Uq',r, where

eq2) = 1 (eq1- + e^) + 1 (e^ - ei-1-) cos(2aq,r) + sin(2aq,r), el2) = 1 (eq1- + e^) - 1 (eq1- - e^) cos(2aq,r) - sin(2aq,r)

Vq(2) = -2(eq1- - e^) sin(2aq,r) + cos(2aq,r), (28)

Vq(,S- = Vq(1 cos aq,r + V^S- sin aq,r, Vq(2 = Vq(1 cos aq,r + V-^ sin ar,p, Vp2 = Vp1 cos aq,r - Vp1 sin aq,r, Vs(2) = Vi1 cos aq,r - vS1 sin ar,p.

Quantum Fest 2015 IOP Publishing

Journal of Physics: Conference Series 698 (2016) 012021 doi:10.1088/1742-6596/698/1/012021

(2) (2)

To erase the off-diagonal elements we require Vqs = Vgr = 0. The first condition fulfils if Vq1 = Vr(,S) = 0 and hence aq,s = n/4, the latter yields cos2(aq,r) = 1/9 and reduces to a two-form

fff = epXp,p + (€q + 25)X*,q + (eq - 5)(X^ + Xs/) + V37{Xp,q + X^}. (29) This can be diagonalized using the previous method. The eigenvalues of H read

Ep = 2 (ep + eq + 25) - 2 [(ep - eq - 25)2 + 1272]1/2 , Eq = 2(ep + eq + 25) - 2 [(ep - eq - 25)2 + 1272]1/2 , Er = Es = eq - 5.

The normalized eigenvectors are obtained as |Ek) = Uq,sUq,rUp,q|k), for k = p, q,r, s. We find

|Ep) = cos &p,q |p) + ^^sin Qp,q (|q) + |r) + |s}) ,

|Eq) = - sin ap,q|p) + ~13 cos ap,q (|q) + |r) + |s)),

1 2 1 (31)

1 r E - e 11/2

|Es) = ^=(|s)-|q)), where cos ap,q = q p

Eq — Epi

The four-level form (26) is useful in the study of a three-qubit system with pairwise interaction given by (21). In such a case it can be shown that the corresponding Hamiltonian is written as the sum of two four-forms (26) with d = 23, and its eigenvalues and eigenvectors are obtained using the method formerly presented [9]. More general problems can be approached using this formalism. For instance, one can consider the eigenvalue problem of the hierarchy of n-qubit Hamiltonians studied in [10,11]. In the case of higher spin, the Hamiltonian (21) can be generalized by changing ^ Ji, where J is the i-th generator of SU(2) with arbitrary j. In such a case the corresponding eigenvalues and eigenvectors can be found by diagonalizing two 22j-level forms. These results will be reported elsewhere.

5. Conclusions

The unitary transformation method was originally proposed to diagonalize a single d-level system Hamiltonian. However, we have shown that the method can be extended to the case of n qudits by means the tensor product algebra of Hubbard operators. The developed formalism was applied to particular cases with d = 2 and n = 2,3, still the general case can be treated once the Hamiltonian is written as a sum of appropriate k-level forms.

Acknowledgments

The author is in indebted with the organizing committee members of Quantum Fest 2015 for their kind invitation. The financial support of CONACyT is acknowledged.

References

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