ROBUST STATIONARY DISTRIBUTED DISCORD IN THE JORDAN-WIGNER FERMION SYSTEM UNDER PERTURBATIONS OF THE INITIAL STATE
E. B. Fel'dman* A. I. Zenchuk**
Institute of Problems of Chemical Physics, Russian Academy of Sciences 142432, Chernogolovka, Moscow Region, Russia
Received April 22, 2014
We investigate the Jordan-Wigner fermion clusters with a stationary distributed quantum pairwise discord. Such clusters appear after the Jordan-Wigner transformation of a spin chain governed by the nearest-neighbor XY Hamiltonian with the particular initial state having one polarized node. We show that the quantum discord stationarity in such systems is not destroyed by the "parasitic" polarization of at least two types. The first type appears because the initial state with a single polarized node is hardly realizable experimentally, and therefore the low polarization of neighboring nodes must be taken into account. The second is the unavoidable additional noise polarization of all nodes. Although the stationarity may not be destroyed by perturbations of the above two types, the parasitic polarizations deform the pairwise discord distribution and may destroy clusters of correlated fermions with equal pairwise discords. Such deformations are studied in this paper.
DOI: 10.7868/S0044451014090077
1. INTRODUCTION
Quantum correlations arc responsible for the effective operation of quantum information devices having the essential advantages in comparison with their classical counterparts fl 14]. According to the current standpoint, the total correlations in a multi-particle system are described by mutual information, and quantum correlations for both pure and mixed states are characterized by the quantum discord [2 5,10 14].
In studing quantum correlations, it is important to choose a proper quantum system possessing the desirable properties and realizable in practice. In this regard, we note the chains of nuclear spins, which are suitable for realization of quantum registers and quantum devices transferring and manipulating quantum information. It is challenging that the multiple quantum (MQ) NMR. methods [15, 16] allow constructing the XY interacting spin chains experimentally. Moreover, using the NMR method, it is possible to create conditions providing the concentration of polarization at a single node of the chain (up to the unavoidable experimental errors) [17]. The dynamics of quantum
* E-mail: efeldman'&icp. ac.ru
E-mail: zenchuk'&itp.ac.ru
correlations in this model was first studied in Ref. [18]. Moreover, it was shown recently [19] that such chains are convenient for studying the dependence of the discord 011 the representation basis of the density matrix describing the quantum system state. The quantum discord calculated for interacting nuclear spins differs from that between the fermions arising after the Jordan Wigner transformation [20] of the density matrix operator [19, 21]. It turned out that the quantum discord between fermions may exhibit very interesting properties [19], which have not been observed in the discord between nuclear spins. The most important property is the stationarity of the pairwise discord in a fermion cluster with the above initial state of a spin-1/2 chain. Besides, if we polarize the proper initial node, then the quantum discord is the same for any fermion pair in the selected fermion cluster. Apparently, this fact is important for the implementation of fermion registers in quantum devices because all fermion nodes are equivalent from the quantum correlation standpoint.
The existence of such clusters motivates the study of their stability with respect to both experimental errors in creating single-node polarization and noise effects. We note that the stability of spin dynamics in the presence of different types of noise is a relevant problem because noise is unavoidable in any quantum process. In particular, the fidelity of the perfect state transfer
(in the absence of noises) under noise perturbations of the coupling constants in the Haniiltonian was considered in Refs. [22 26] for two chains: a completely engineered chain and a chain with remote endnodes. In both cases, the important result is that the noise reduces the fidelity without changing the state transfer time.
In this paper, we study the stability of the discord distribution relative to perturbations of the initial state in a homogeneous spin chain (i.e., the coupling constants in the Haniiltonian are assumed to be stable). We show that the stationarity of the quantum discord in the system with a single initially polarized node may not be destroyed by the additional low polarizations of the neighboring nodes, which unavoidably appear in the experiment. This perturbation just leads to the deformation of the pairwise quantum discord and may eventually destroy the clusters of fermions with the equal pairwise discord. The threshold value of the low polarization is found. We also consider the deformation of the stationary discord distribution caused by the noise polarization appearing in all nodes of the spin chain. It is remarkable that the discord stationarity is not disturbed in both cases.
The paper is organized as follows. The Jordan Wigner transformation of the XY Haniiltonian with nearest-neighbor interactions is briefly discussed in Sec. 2. The stability of the pairwise discord stationarity in the Jordan Wigner fermion system of a spin-1/2 chain with single initially polarized nodes under perturbations of the initial state is demonstrated in Sees. 3 and 4 with numerical simulations of the spin dynamics of a 17-node chain. First, in Sec. 3, the parasitic polarization of two neighboring nodes (with respect to the selected inner polarized node) is considered. Then, in Sec. 4, the noise polarization of all nodes is taken into account using the perturbation method. Deformations of the fermion clusters with equal pairwise discord under the above perturbations are also considered in Sees. 3 and 4. The basic results are discussed in Sec. 5. A formula for calculating the discord in the A'-type density matrix [27] is represented in the Appendix.
2. J ORD AN-WIGNER TRANSFORMATION OF THE XY HAMILTONIAN WITH THE NEAREST-NEIGHBOR INTERACTION
We study quantum correlations in the onc-dimcn-sional open spin-1/2 chain of N nodes governed by the XY Haniiltonian with the nearest-neighbor interactions,
N N-l
H = uo Ih + D (Ii?I(i+1)3- + Iiyl{i+l)y), (1) j=l j=l
where u>o is the Larmor frequency in the external magnetic field, D is the spin spin coupling constant between the nearest neighbors, and Iia (i = 1.....N,
a = x.y.z) is the ¿th spin projection on the a axis.
Following Refs. [18, 19, 21], we diagonalize Haniiltonian (1) using the Jordan Wigner transformation method [20],
H = J2 lh - ^NoJo, £k = D cos (A-) + u;0, (2) k
where the fermion operators Bj are expressed in terms of other fermion operators cj by means of the Fourier transformation
N
k Y^mXDn- (3)
3=1
and the fermion operators cj are defined as [20]
<V (^H '/l:/'i: •••/,; H:/; • (4)
Here,
lllAj) TTf (5)
n = 1,2,..., N.
We can readily express the projection operators Ij- in terms of the fermion operators cj as
<•]<—i- Vj. (6)
Hereafter, diagonal representation (2) of the XY Haniiltonian is used to describe the dynamics of the density matrix associated with the spin-1/2 chain.
3. INITIAL STATE WITH THREE POLARIZED NODES
The dynamics of the Jordan Wigner fermions associated with the spin-1/2 chain with a single initially polarized node j0 has been studied in Refs. [18, 19, 21]. There, the stationarity of the pairwise discord in such systems is demonstrated and fermion clusters with equal pairwise discord are revealed.
We now consider the initial state with an inner initially polarized node j0 (i.e., 1 < j0 < AT) and assume the parasitic low polarization of two neighboring nodes; the initial density matrix is therefore given by
¿0 + 1
Po=Z°XPl bkhr.\ =
^'=¿0-1
= W II fl + 2/,,th|
k=jo-l ¿0 + 1
(7)
¿0 + 1
Z = T1 n
_ 0N
= 2n J] dl
h
\k=j o-l
k=jo-l
whoro bj = huijo/l'T, h is the Planck constant, k is the Boltzniann constant, and T is the temperature of the system.
The motivation for considering this initial state is discussed in the introduction. Namely, an experimental scheme may not provide the ideal single-node polarization. Hence, two neighboring nodes j0 ± 1 also acquire some polarization whenever j0 is an inner node, i.e., 1 < jo < N. This polarization might be called parasitic. As was shown in Rcfs. [19, 21], a fermion cluster with equal pairwise discords (which is our subject in this paper) may be obtained if the polarized node j0 is an inner one. This case is related to the density matrix in Eq. (7) and is discussed below.
The evolution of the initial density matrix (7) in the fermion representation of Hamiltonian (2) is given by
p(t) = exp | -it | Po x
x exp j . (8)
Using the identity
exp (¡1 h.) /?|oxp ¡1 h.) =
= exp(-if),4, V<p, (9) we rewrite density matrix (8) as [18]
¿0+1
n
¿=¿0-1
x ^2™v{-it(£k-£k>)}gk(j)gk>U)iltih> I = k,k' J
= 4° + £ exp {-it(ek - ek,)} filfik, + k,k'
k,k',q,q'
= ÏÏ (l^th| + 2th|x
X + Y ^kqlk'q'I' X
k,k',(?,(?',/,/'
X exp {—it(£k + £q + £i - £k> - £q> - £V )} X
(10)
where
1 J0+1 / / II
4J0
¿=¿0-1
K 2
(ID
x 1 — til
frjo+i A 2
th
h,
'¿0-1
1-th (l-th^l
2 J
1 - th ^zl ) (l-tii^\ x
b:
th 9k (jo )gk> (jo )
hjo
x th^gkUo + Dgk'Uo + D
4 Jo
'^-knklnl —
1
'iqi'q' ~ 11 1 - th "V") th ~f X
b ■
x th (jo )gk> (jo )gq (jo + 1 )gq> (jo + 1 )
1 — th — ^ th !- th Îkïl. x
x gk(jo - l)gk'(jo - 1 )gq(jo + l)gq>(jo + 1)
1 - th
h
'¿o+i A
th
h,
'¿0-1
hi
tii^gk(j0-l) x
2 J 2 "" 2
x gk'(jo - 1 ).9<7 (jo )Sq' (jo )
A3.a
1
■th
"¿0-1
th th
"¿0 + 1
kqlk'q'I'
X 9k (jo -1 )gw (jo -1 )gq (jo )gq> (jo )gi (jo+1 )gv (jo+1 ) •
Finally, using the fermion anticommutation relations [28] and relations among the coefficients A3£ k, , and
^¿0 kqlk'q'I' '
N N N
V = V Aj0 = V Aj0
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