Spin-chain-star systems: entangling multiple chains of spin qubits

R. Grimaudo,1 A. S. M. de Castro,2 A. Messina,3 and D. Valenti1

1Dipartimento di Fisica e Chimica “Emilio Segre”, Universita degli Studi di Palermo,viale delle Scienze, Ed. 18, I-90128, Palermo, Italy

2Universidade Estadual de Ponta Grossa, Departamento de Fısica, CEP 84030-900, Ponta Grossa, PR, Brazil3Dipartimento di Matematica ed Informatica, Universita degli Studi di Palermo, Via Archirafi 34, I-90123 Palermo, Italy

(Dated: March 15, 2022)

We consider spin-chain-star systems characterized by N-wise many-body interactions between thespins in each chain and the central one. We show that such systems can be exactly mapped intostandard spin-star systems through unitary transformations. Such an approach allows the solutionof the dynamic problem of an XX spin-chain-star model and transparently shows the emergence ofquantum correlations in the system, based on the idea of entanglement between chains.

I. INTRODUCTION

Superposition and entanglement are at the basis of theremarkable advantages of using quantum mechanics overclassical physics in quantum information science [1–3].Such quantum resources are exploited in several tech-nological applications ranging from quantum simulation[4, 5], quantum metrology [6–10] and quantum cryptog-raphy [11, 12] to quantum computing algorithms [13, 14].Besides entanglement, other quantities have been discov-ered and proposed as quantum resources over the lastyears, such as, for example, discord [15, 16], coherence[17], steering [18], and contextuality [19].

Generating entanglement and superposition states oflarge systems is a target which has kindled a growing in-terest in physical scenarios characterized by small quan-tum systems through which it is possible to control andcoherently manipulate mesoscopic environments [20–22].In this sense, in recent decades, a great attention hasbeen payed in studying spin systems which have beensuccessfully applied in quantum information [23–25].Thesimplest imaginable setting consists in a single spin-qubit(either just a qubit, or a spin-1/2 or, more generally, atwo-level system) that controls other N spin-1/2’s homo-geneously distributed in a circle centred on it. This sys-tem is commonly known as central spin system [26] andthe Hamiltonian models used to describe it are calledspin-star models. In such a star-shaped system the N‘environmental’ spins do not interact with each other di-rectly, but only with the central one which, hence, playsthe role of a bridge through which quantum correlationsbetween the spins surrounding it can arise.

The interest towards central spin models has remark-ably grown thanks to: I) their suitability in describingthe hyperfine interaction in quantum dots [27] and theinteractions between nuclear spins and nitrogen-vacancycenters in diamond [28, 29]; II) their broad applicabilityin different fields like quantum information [30], quan-tum metrology and sensing [31, 32], quantum thermo-dynamics [33] and fundamental aspects [34].Further, alots of works have been developed to investigate thequantum correlation and thermal entanglement arisingamong spins in the star framework [35–39], as well as the

Markovian and non-Markovian dynamics induced by asurrounding bath interacting with the spin-star system[40–42].

Recently, the idea of spin-chain-star system has beenproposed [43, 44] where the control spin occupies the cen-ter of a star of M rays (chains), arranged at angular dis-tance 2π/M, each of which hosts the same sequence ofN spins, generally equidistant. There is no interactionbetween different chains and the spins in a given radiusmay not even be the same [45]. The spin-spin interactionswithin each chain and with the control spin are describedby Hamiltonian terms strictly related to the physical sce-nario to be studied. In [46], for example, the same sys-tem has been analysed to study the effects related to thequantum darwinism in such a structured environment.

In this work we consider particular spin-chain-starsystems characterized by the peculiarity that the spinsin each chain are all the same and interact throughN-wise interactions among them and with the centralspin, Fig. 1a. Such a kind of interactions can be im-plemented via quantum simulation apparatus based oneither trapped ions [47, 48] or superconducting circuitsmade of transmon qubits [49]. Moreover, these exoticcouplings have been demonstrated to be exploitable togenerate superposition states and then quantum correla-tions in large spin-chains under the experimental controlof magnetic fields applied on the spin-system [50, 51].

We analytically demonstrate that our spin-chain-starsystems, under appropriate conditions, can be unitar-ily reduced to standard spin-1/2-star systems, Fig. 1b.This result implies the possibility of applying to such alarge system some of the results already achieved for thestandard spin-star systems, as, for example, the eigen-spectrum, the eigenvalues and the quantum dynamics[52]. We show, indeed, that, by appropriately ‘trans-lating’ these results in the spin-chain-star language, it ispossible to bring to light how to to generate entangledstates of the different chains. Further, we show as wellhow to produce different classes of entangled states de-pending on the specific topological configuration chosenfor the spin-chain-star system.

The paper is organized as follows. In Sec. II differ-ent types of N-wise spin-chain star models together with

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their unitarily equivalent standard spin-star models areintroduced and their possible integrability is analysed.The quantum dynamics and the possibility of generatingdifferent classes of entangled states (depending on theconfiguration) for an XX spin-chain-star model are dis-cussed in detail in Sec. III. Finally, concluding remarksare reported in Sec. IV.

II. SPIN-CHAIN-STAR SYSTEMS

In this section we propose a new class of spin-chainstar systems characterized by the presence of only many-body interactions of maximum order. We show that, byconsidering the many-body N-wise interactions for eachchain, such systems are unitarily equivalent to standardspin-chain star systems, which are remarkably consideredand studied for their applications in several fields [30–34, 53].

A. The X Model

Consider the following model:

H = H1 + H2, (1)

with

H1 =hωa

2σ

za + γ1σ

xa ⊗

[M1⊗

i

σx1i

], (2a)

H2 =hωa

2σ

za + γ2σ

xa ⊗

[M2⊗

j

σx2 j

]. (2b)

The physical system described by this Hamiltonian modelcan be thought of as a central spin-1/2, to which we referas ancilla, coupled to two spin-1/2 chains (the subscriptsof the two terms in the Hamiltonian (1) refer to the twodifferent chains). The coupling which characterizes thetwo spin-chains (including the ancilla) consists in N-wiseinteraction terms, that is, a type of interaction involvingall the spins at one time.

It has been demonstrated [50] that the M-wise spin op-erator

⊗Mi=1 σ x

k can be unitarily reduced as (see AppendixA)

M⊗i=1

σxi → σ

x1 , (3)

where σ x1 is intended to be a 2M-dimensional operator.

From a physical and mathematical point of view, it meansthat the M-spin chain effectively behaves as a single two-level system and then can be formally treated as a singlequbit. The origin of such a lucky circumstance can betraced back to the existence of a set of constants of mo-tion which generate an equally numbered set of dynam-ically invariant two-dimensional Hilbert subspaces [50].

This implies that, within each of these subspaces, theM-spin dynamics can be mapped into that of a singletwo-level system [50]. A particularly interesting subdy-namics is that characterized by the Hilbert space spannedby the two states |↓〉⊗M and |↑〉⊗M (with σ z|↓〉=−|↓〉 andσ z|↑〉= +|↑〉).

Therefore, the two Mk-spin operators⊗M1

i σ x1i and⊗M2

j σ x2 j in H1 and H2, can be unitarily transformed into

σ x11 and σ x

21, respectively. In this way, the model in Eq.(1), after the appropriate unitary transformations, canbe written as

H = hωaσza + γ1σ

xa ⊗ σ

x1 + γ2σ

xa ⊗ σ

x2 , (4)

where the second index of the spin operator in the lasttwo terms, indicating the first spin of each chain, hasbeen omitted.

Let us consider now a more general spin-chain systemas the following one

Hx = hωaσza +

N

∑k

γkσxa ⊗

[Mk⊗

j

σxk j

]. (5)

We can call such a physical system a spin-chain-star sys-tem since we can imagine the different N chains to bedisposed in a star-shaped configuration, each coupled tothe same central spin. It is easy to convince oneself thatalso this model, analogously to the two-chain case, canbe transformed through unitary transformations into thefollowing simpler spin model:

Hx = hωaσza +

N

∑k

γkσxa ⊗ σ

xk . (6)

This result shows that a star-shaped spin-chain systemcan be formally described and mathematically treated asa standard spin-star system, that is, a system consistingof N mutually uncoupled spin-1/2’s, each interacting witha unique central spin-1/2. This possibility stems fromthe fact that each N-wise-interacting spin-chain can beeffectively reduced to a single two-level system.

B. The XY Model

It is interesting to point out that also the N-wise spin-operator

⊗Mj=1 σ

yj can be unitarily reduced to [50] (see

Appendix A)

M⊗j=1

σyj →

[(−1)

M−12 γy

(M−1)/2

∏j=1

σz2 j+1

]σ

y1 , (7)

provided that the number M of spins is odd [50], as shownin Appendix A. The operator realizing such a transfor-mation is the same accomplishing that in Eq. (3). Inthe above expressions σ

z2 j+1 are constants of motion and

their possible values can be +1 and -1. The specific values

3

(a) (b)

Figure 1: (a) A spin-chain-star system composed by eight five-spin chains. The central white circle represents theancilla, while the ellipses embracing the spins of each chain and the ancilla represent the N-wise interactions. Theoverlap of the eight ellipses is byproduct of the cartoon and has no physical meaning. (b) The standard spin-starsystem unitarily equivalent to the system shown in panel (a).

assigned to these integrals of motion identify a precise dy-namically invariant subspace [50]. The Hilbert subspacespanned by |↓〉⊗M and |↑〉⊗M, for example, is character-ized by all the constants of motion equal to 1, namelyσ

z2 j+1 = 1, ∀ j. Therefore, the following XY spin-chain-

star system

Hxy = hωaσza +

N

∑k

γxk σ

xa ⊗

[Mk⊗

j

σxk j

]+

N

∑k

γyk σ

ya ⊗

[Mk⊗

j

σyk j

],

(8)can be mapped into the standard XY spin-star system,which reads

Hxy = hωaσza +

N

∑k

γxk σ

xa ⊗ σ

xk +

N

∑k

γyk σ

ya ⊗ σ

yk , (9)

if the number Mk of spins in the k-th chain is chosen sothat (Mk−1)/2 is even, and if all the chains coupled to thecentral ancilla are initially prepared in either |↓〉⊗Mk or|↑〉⊗Mk or any arbitrary superposition of these two states(such as a GHZ-like state).

C. The XY Z Model

It is possible to demonstrate that the same set of uni-tary operators realizing the transformations in Eqs. (3)and (7) realizes also the following transformation [50] (seeAppendix A)

M⊗l=1

σzl →

[γz

(M−1)/2

∏l=1

σz2l+1

]σ

z1, (10)

where M is odd.

Also in this case, if the involved two-level subdynam-ics of each chain is that characterized by the two states|↓〉⊗Mk and |↑〉⊗Mk , then the XY Z spin-chain-star system

Hxyz =hωaσza +

N

∑k

γxk σ

xa ⊗

[Mk⊗

j

σxk j

]+

N

∑k

γyk σ

ya ⊗

[Mk⊗

j

σyk j

]+

N

∑k

γzk σ

za⊗

[Mk⊗

j

σzk j

],

(11)

is unitarily equivalent to the standard XY Z spin-starmodel

Hxyz =hωaσza +

N

∑k

γxk σ

xa ⊗ σ

xk +

N

∑k

γyk σ

ya ⊗ σ

yk +

N

∑k

γzk σ

za⊗ σ

zk .

(12)

A qualitative representation of a star-shaped system com-posed by eight five-spin chains is shown in Fig. 1a. Itsunitarily equivalent standard spin-star system is shownin Fig. 1b.

Finally, it is worth pointing out that the effectivemathematical description, basing on the unitary trans-formation procedure, is not affected by a possible time-dependence of the Hamiltonian parameters. This prop-erty stems from the fact that the unitary operators, whichtransform the Hamiltonians, does not depend on theHamiltonian parameters and, more in general, on time.

D. In presence of fields

In this subsection we see that our analysis and then theunitary reduction of a spin-chain-star model to a stan-dard spin-star model keeps its validity also when fields

4

applied to the entire chains are considered. Let us sup-pose each entire chain in the system to be subject to auniform magnetic field. The models in Eqs. (5), (8) and(11) are then enriched of the term

N

∑k

hωk0

Mk

∑j

σzk j. (13)

It is possible to convince oneself [50] that the unitarytransformations acting as expressed in Eqs. (3), (7) and(10) convert the operators in Eq. (13) into (see AppendixA)

N

∑k

[1 +

Mk

∑j=2

j

∏i=2

σzk j

]hω

k0 σ

zk1, (14)

where σzk j are constants of motions as before. Within

the subspace (for each chain) we are interested in, thatis the one spanned by the two states |↓〉⊗Mk and |↑〉⊗Mk ,the integrals of motions are all equal to 1. We can thenwrite the following effective two-level operators

N

∑k

Mk hωk0 σ

zk , (15)

each of which accounts for the magnetic field applied ona chain, whose dynamics is equivalent to that of a singlespin-qubit system.

Therefore, in this physical scenario, the unitarily trans-formed effective models in Eqs. (6), (9) and (12) aremodified by simply introducing such terms in the Hamil-tonians. It is important to underline that the introduc-tion of the fields uniformly acting upon each chain doesnot alter the symmetry properties of the original Hamil-tonians. In this way, the possibility to perform the sameunitary operations on the Hamiltonian interaction termsresults to be not affected.

As a final remark, we wish to emphasize the im-portance of the symmetry properties possessed by theHamiltonian(s). For the case analysed in this paper,the study of the Hamiltonian symmetries is fundamentalfor the exact solution of the dynamical problem. Morein general, the symmetries, besides allowing to analyti-cally treat some models, have profound physical impli-cations. Indeed, the disclosure of symmetry-protected(sub-)dynamics in different physical systems [54, 55] canlead to discover physical effects which turn out to beuseful and applicable in different fields such as quantummetrology [56, 57].

E. Integrability

It is worth pointing out that the fully isotropic XXXspin-star model is integrable; precisely, it belongs tothe class of XXX Richardson-Gaudin integrable models[52, 58]. The condition of integrability stems from theexistence of an appropriate set of integrals of motion,

allowing to obtain all eigenstates and related eigenval-ues through the use of Bethe ansatz techniques[52]. Thiscircumstance, joined with the fact that the XXX spin-star model well describes systems with spherical sym-metry (such as quantum dots in semiconductors with s-type conduction bands [59]), has spurred several studiesfocused on the equilibrium and dynamical properties ofsuch a model [60, 61].

Very recently, it has been demonstrated that alsothe XX model is integrable [52]. This model naturallyemerges in resonant dipolar spin systems in rotatingframes [62, 63] and its eigenstates are divided into twoclasses: dark and bright states. The former are productstates of the ancilla and of all the other environmentalspins, so that the central spin is thus disentangled fromthe spin-bath. The latter can be written as a combinationof dark states and then exhibit entanglement between theancilla and the other spins [52].

On this basis we therefore claim that, when an XXspin-chain star model can be unitarily reduced to astandard spin-star one, we can derive the exact expres-sions of the eigenvalues and eigenvectors of the spin-chain-star system. It is important to stress that, inthis case, fully integrability cannot be invoked since thespin-chain-star model is exactly solvable only within thespecific subspaces where the mapping to an integrablespin-star model is possible. As previously said, one ofthese subspaces is that spanned by the pair of states{|↑〉⊗M, |↓〉⊗M}. Within other subspaces, instead, al-though the reduction to standard spin-star models is al-ways possible, the effective unitarily equivalent modelspresent inhomogeneities (XY Z) which affect the integra-bility.

III. EXACT SOLUTION FOR XX SPIN-CHAIN-STARSYSTEMS

A. W -like and GHZ-like states of spin-chains

In this section we specialize the XX spin-chain-star sys-tem considered so far by setting γx

k = γyk , ∀ k in Eq. (8),

namely

Hxx = hωaσza +

N

∑k

γk

{σ

xa ⊗

[M⊗j

σxk j

]+ σ

ya ⊗

[M⊗j

σyk j

]}.

(16)Moreover, we suppose a number N of spin-1/2-chains,each consisting of M spin-qubits and satisfying the con-straint that (M − 1)/2 is an even number. Throughthe appropriate unitary transformations previously dis-cussed, we get thus an effective standard XX spin-starsystem (Eq. (9) with γx

k = γyk , ∀ k)

Hxx = hωaσza +

N

∑k

γk[σ

xa ⊗ σ

xk + σ

ya ⊗ σ

yk

]. (17)

5

We consider all the N spin-chains initialized in the M-spin state |↓〉⊗M. As said in the previous section, thesymmetries of the Hamiltonian lead to a two-dimensionaldynamically invariant subspace spanned by |↓〉⊗M and|↑〉⊗M. It means that the k-th chain (k = 1 . . .N) can beeffectively represented in terms of dynamical variablesσ x

k , σyk , σ

zk of a fictitious qubit. The following mapping

(valid for each chain)

|↓〉⊗M ⇐⇒ |−〉, |↑〉⊗M ⇐⇒ |+〉. (18)

(with σ z|±〉 = ±|±〉) enables to fix unambiguously theinitial state of the fictitious standard spin-star system.We suppose the ancilla and the effective standard spin-star system initially prepared in the following state

|ψ(0)〉= |↑a〉|−〉⊗N , (19)

which in terms of spin-chain states is written as

|ψ(0)〉= |↑a〉|↓1〉⊗M . . . |↓k〉⊗M . . . |↓N〉⊗M. (20)

It is known that the exact time evolution of the initialcondition taken into account for the XX spin-star systemis [64, 65]

|ψ(t)〉= α(t)|↑a〉|−〉⊗N + |↓a〉N

∑k

βk(t)|−1 · · ·+k · · ·−N〉,

(21)with

α(t) = cos(ω t)+ iωa

ωsin(ω t), (22a)

βk(t) =−iγk/h

ωsin(ω t), (22b)

where

ω =√

∑k

(γk/h)2 + ω2a . (23)

We underline that, for the initial condition underscrutiny, only one effective frequency, namely ω, char-acterizes the time evolution of the system. Thus, the XXspin-star system comes back to its initial condition witha period T = 2π/ω and behaves as if its dynamics weregoverned by an effective Hamiltonian describing N differ-ent spins homogeneously coupled to the central one, thatis with the same effective coupling constant. Precisely,such an effective model can be obtained by substitutingin Eq. (8) γx

k = γyk = γe f f with γe f f = hω, which is inde-

pendent of k.Moreover, it is interesting to note that, for γk = γ, ∀ k

and ωa = 0, when t = nπ/ω (so that α(t) = 0), we get thestate

|W 〉= |↓a〉

[1√N

N

∑k|−1 · · ·+k · · ·−N〉

], (24)

up to a global factor exp{−iπ/2}. For this scenario, thetime behaviours of the probabilities |α(t)|2 and |βk(t)|2 =|β (t)|2, ∀ k, are shown in Fig. 2(a) in the case of N = 9.

We highlight that the N-two-level system results in aW -like state and that, therefore, each of these N involvedtwo-level systems represents one of the N M-spin chains.Thus, the ancilla-mediated (quantum) correlations whicharise between the effective N spin-1/2’s can be interpretedas correlations get established between the N spin-chains.It means that such a W -state is a ‘macro-state’ consist-ing in a superposition of states involving all the M-spin-chains in the systems. We can then write the W -like statein terms of the spin-chains as

|W 〉= |↓a〉

[1√N

N

∑k|↓1〉⊗M . . . |↑k〉⊗M . . . |↓N〉⊗M

]. (25)

Therefore, we claim that the XX spin-chain-star model[Eq. (16)] allows for the generation of ‘macro’ W -likestates of the chains and then for the creation of a large-scale entanglement between all the subsystems in thespin-chain-star physical scenario.

0.0

0.5

1.0(a)

| ( t)|2| k( t)|2

0 1 2 3 4 5t

0.00

0.05

0.10

|k(

t)|2

(b)/ = 0/ = 1/ = 5/ = 10

Figure 2: (a) Time dependencies of |α(t)|2 (blue dashedline) and |βk(t)|2 = |β (t)|2, ∀ k (red solid line) [see Eq.(22)] in case of N = 9 number of M-spin chains, forγk = γ, ∀ k and ωa = 0. (b) Effects of the detuning onthe time dependence of |βk(t)|2 = |β (t)|2, ∀ k, in case ofN = 9 and γk = γ, ∀ k; different values of the ratio h∆/γ

are considered: 0 (red dotted line), 1 (green dot-dashedline), 5 (blue dashed line) and 10 (black solid line).

B. Effects of Detuning

Let us suppose to apply a uniform magnetic field to allthe identical M-spin-chains of the system, namely

hω0

M

N

∑k

M

∑j

σzk j. (26)

In this case the unitarily equivalent spin-star model reads

H ′xx = hωaσza + hω0

N

∑k

σzk +

N

∑k

γk[σ

xa ⊗ σ

xk + σ

ya ⊗ σ

yk

]. (27)

6

It is easy to check that, for such a physical scenario,the expression of the evolved state in Eq. (21), obtainedwhen the spin-chain-star system is initially prepared inthe state (19), remains formally identical, as well as thatof βk(t) in Eq. (22b) [64, 65]. The only slight variationis found in the mathematical expressions of α(t) and ω

which become now respectively [64, 65]

α(t) = cos(ω t)− i∆

ωsin(ω t), (28a)

ω =√

∑k

(γk/h)2 + ∆2, (28b)

where the detuning is defined as follows:

∆ = ω0−ωa. (29)

The effects of the detuning on the probabilities |βk(t)|2are shown in Fig. 2(b), for γk = γ, ∀ k. We see, as ex-pected, that the greater is the detuning, the lower arethe probabilities |βk(t)|2 (and then the higher the com-plementary probability |α(t)|2), meaning that the systemtends to remain in its initial condition for high values ofthe detuning.

This result, thus, demonstrates also that, although anon-vanishing field ωa is present on the ancilla spin, itis still possible generating the W -state in Eq. (24), pro-vided that a further magnetic field of magnitude ωa/M isuniformly applied to all the M-spin chains (so that ∆ = 0).

C. Entanglement

In this section we investigate the possible quantum cor-relations emerging in the spin-chain-star system. To thisend, we consider the case of N = 2 chains, each madeof M spin-qubits, and γk = γ, ∀ k. It is interesting tonote that in this case, through the procedure previouslyexposed and for ∆ = 0, if we get −1 by measuring theancilla dynamical variable σ z

a at t = π/ω, Eq. (21) fore-sees that the resulting spin-state of the two chains hasthe following form

|GHZ〉=|↑〉⊗M|↓〉⊗M + |↓〉⊗M|↑〉⊗M

√2

. (30)

Since in this state the concurrence [66] between twogeneric spin-1/2’s (in the same or different chains) van-ishes and the measure of the collective z-component ofthe whole spin-chain can give only the two values ±N, itis legitimate to call this multi-spin state a GHZ-like state.The legitimacy of such a denomination of the state (30)can be convincingly strengthened simply observing that,exploiting our mapping (18), the same state (30) can bewritten as the Bell state

|GHZ〉 → |Ψ+〉=|+〉|−〉+ |−〉|+〉√

2, (31)

which is characterized by the maximum level of concur-rence (C = 1). We stress that, as a result of our analysis,

the entanglement should be intended as the signature ofthe existence of quantum correlations between the spin-chains. We wish to emphasize the added value of the rep-resentation in terms of fictitious two-level systems, sinceit provides information about quantum correlations getestablished between the spin-chains. According to thisinterpretation, it becomes relevant to calculate the timedependence of the concurrence exhibited by two chainsin the state (21):

|ψ(t)〉=α(t)|↑a〉|−〉|−〉+ |↓a〉 [β1(t)|+〉|−〉+ β2(t)|−〉|+〉] =

=α(t)|↑a〉|↓〉⊗M|↓〉⊗M

+ |↓a〉[β1(t)|↑〉⊗M|↓〉⊗M + β2(t)|↓〉⊗M|↑〉⊗M] .

(32)It is possible to verify that the concurrence for the densitymatrix of the two effective spin-1/2’s is

C(t) = 2|β1(t)||β2(t)|. (33)

We are therefore able to write the time-dependence of theentanglement emerging between the two chains. The lat-ter results to be maximum for β1(t) = β2(t) = 1/

√2, cor-

responding to the GHZ-like state previously examined.

Analogously, in the case of N = 3 M-spin chains, bymeasuring σ z

a = 1 at t = π/ω, the state reached by thespin system is

|W 〉=|+−−〉+ |−+−〉+ |−−+〉√

3=

|↑〉⊗M|↓〉⊗M|↓〉⊗M + |↓〉⊗M|↑〉⊗M|↓〉⊗M + |↓〉⊗M|↓〉⊗M|↑〉⊗M√

3,

(34)which can be interpreted as a maximally entangled W -like state of the three M-spin chains. Also in this case wemay infer the entanglement between the three spin chainswith the help of the effective description involving threeinteracting two-level systems as well. It is easy to verifythat, for the state (21) in the case of N = 3, each pairi- j of chains exhibits a non-vanishing concurrence equalto 2|βi||β j|. Each pair of two generic true spin-1/2’s is,instead, disentangled as in the case of two chains.

This result can be extended to the case of N spin-chains. In this instance, it is possible to check, in-deed, that the entanglement between two generic truespins vanishes, while the concurrence for a generic pairof chains i- j results to be 2|βi||β j|.

This means that the spin-chain-star system here dis-cussed, besides generating quantum correlations in alarge spin-system, is suitable to generate different typesof entangled states of the chains assumed in the modelunder scrutiny. The origin of such differences can betraced back to specific topological and structural prop-erties of the spin-chain-star system, as for example thenumber of chains and the number of spins per chain.

7

IV. CONCLUSIONS

In this work we have considered a special class of spin-chain-star systems. Precisely, we have focused our atten-tion on spin chains characterized by many body N-wiseinteractions, that is interaction terms involving all thespins in a chain at once. We have taken into accountseveral types of spin-chain-star models including differ-ent types of N-wise interactions as well as the presenceof local magnetic fields on both the ancilla and the spinsin the chains.

We have shown that each model we have consideredcan be analytically treated through unitary transforma-tions and that each chain, for specific initial conditions,effectively behaves and can be thought of as a two-levelsystem. This implies that a spin-chain-star system be-longing to the class under scrutiny is unitarily equiva-lent to a standard spin-star system. Therefore, we canexploit the knowledge and the results obtained for thestandard spin-star models and interpret them in termsof multiple-chain states. For example, we have demon-strated that, by starting from a disentangled state of thespin-chain-star system, our model and scheme allow forthe generation of a ‘macro’-entangled state of all the spinchains which form the system. Therefore, we can speakof quantum correlations arising between the actual spinsand between the spin chains globally described as two-level systems. In case of N = 2 and N = 3 spin chains,thanks to the mapping into a spin-1/2-star model, wecan affirm that the system evolves and reaches a maxi-mally entangled superposition at appropriate instants oftime. Moreover, we are also able to quantify the entan-glement get established between two spin chains throughthe calculation of the concurrence (since the two chainseffectively behave as two spin-qubits).

Finally, we have analysed the effects on the probabil-ity related to the generation of such a state stemmingfrom the presence of magnetic fields acting on the an-cilla and (homogeneously) on the chains. In this way, wehave found the optimal experimental work condition nec-essary to get a macro-entangled state. Our result, thus,paves the way to the possibility of generating large-scaleentanglement in spin systems made of several spins.

Further investigations could concern quantum oscilla-tor baths(s) interacting with the spin chains and/or theancilla. In this case, likely, a numerical approach to solvethe dynamics is necessary. Several approaches could beused to deal with this scenario, from the standard Lind-blad theory [67, 68] to the partial Wigner transform [69–75] and the non-Hermitian formalism [76–85].

ACKNOWLEDGEMENTS

RG and DV acknowledge financial support from thePRIN Project PRJ-0232 - Impact of Climate Changeon the biogeochemistry of Contaminants in the Mediter-ranean sea (ICCC).

APPENDICES

Appendix A: Symmetry-based unitary transformations

Let us consider the following N-spin model

H = γx

N⊗k=1

σxk + γy

N⊗k=1

σyk + γz

N⊗k=1

σzk , (A1)

where σ x, σ y and σ z are the standard Pauli matrices.The N distinguishable spins are coupled only through N-wise interaction terms, that is interaction terms whichinvolve all the N-spins at once.

To exactly diagonalize the model, it is useful to beginwith the easiest case of two interacting spin-1/2’s [86–89]:

H2 = γxσx1 σ

x2 + γyσ

y1 σ

y2 + γzσ

z1σ

z2, (A2)

for which σz1σ

z2 is an integral of motion, [H2, σ

z1σ

z2] = 0. It

is possible to verify that the following unitary and Her-mitian operator (1 is the identity operator in the fourdimensional Hilbert subspace)

T12 =12

[1+ σz1 + σ

x2 − σ

z1σ

x2 ] , (A3)

transforms H2 into

T†12H2T12 = H2 = γxσ

x1 − γyσ

z2σ

x1 + γzσ

z2. (A4)

Since σz2 is a constant of motion for H, it can be treated

as a parameter (=±1), rewriting

Hσz2

= (γx− γyσz2) σ

x1 + γzσ

z2. (A5)

The existence of such a symmetry (giving rise to the con-stant of motion) implies the existence of two dynamicallyinvariant Hilbert subspaces, each of which related to aneigenvalue of σ

z2. Each single-spin-1/2 Hamiltonian, ob-

tainable by assigning a value to σz2, governs the two-spin

dynamics in one of the two dynamically invariant Hilbertsubspaces. Therefore, in this way, we have reduced thetwo-interacting-spin problem to two independent single-spin-1/2 problems.

Let us consider now the analogous three-spin model

H3 = γxσx1 σ

x2 σ

x3 + γyσ

y1 σ

y2 σ

y3 + γzσ

z1σ

z2σ

z3. (A6)

This time the new constant of motion σz2σ

z3 appears. By

applying the analogous procedure used for the two-spincase, we get the following new Hamiltonian

T†23H3T23 =H3 = γxσ

x1 σ

x2 − γyσ

z3σ

y1 σ

x2 + γzσ

z3σ

z1. (A7)

In this case, the unitary transformation involves the sec-ond and the third spin, and the operator accomplishingsuch a transformation, inspired by T12, reads

T23 =12

[1+ σz2 + σ

x3 − σ

z2σ

x3 ] . (A8)

8

Since σz1σ

z2 is a constant of motion for H3, we can exploit

the operator written in Eq. (A3) and apply one moretime the same procedure to H3. So, we get

T†12H3T12 = T†

123H3T123 = ˜H3

= γxσx1 − γyσ

z3σ

y1 + γzσ

z3σ

z1,

(A9)

where we put T123 = T23T12. In this case we have fourdynamically invariant subspaces related to the four pairsof eigenvalues of the two constants of motion σ

z1σ

z2 and

σz2σ

z3. So, we have four two-level Hamiltonians govern-

ing the three-spin dynamics in each subspace. Therefore,we have reduced the initial dynamical problem of threeinteracting spins to four independent single-spin-1/2 dy-namical problems.

Basing on this last result, it is easy to understand thatwe can apply the same sequence of unitary transforma-tions also in the case of N spins. More precisely, wecan iterate the transformation procedure until the ini-tial N-spin Hamiltonian is completely reduced to a setof 2N−1 two-level Hamiltonians. Each of these effectivesingle-spin-1/2 Hamiltonians governs the N-spin dynam-ics within an invariant subspace identified by the specificvalues of the 2N−1 constants of motion. The total unitaryoperator accomplishing the chain of unitary transforma-tions can be written as

T =1

2N−1

N−2

∏k=0

[1+ σ

zN−(k−1) + σ

xN−k− σ

zN−(k+1)σ

xN−k

],

(A10)where each piece of the product acts on a ‘spin-triplet’,say [i, j,k]. As shown in the two- and three-spin cases,each three-spin transformation leaves the Hamiltoniandependent on the dynamical variables of the first spinof the triplet (i-th spin) and on those of all other spinsnot affected by the transformation. The spins j and kappears only with the z-component, i.e. σ

zj and σ

zk which

are constants of motion. They can be therefore treatedas parameters and substituted with their eigenvalues inthe expression of the transformed Hamiltonian.

The effects on the Hamiltonian after the spin-triplettransform are:

• a -1 factor appearing in the interaction term in γy;

• the appearance of the σ z operator (parameter) ofthe last spin of the triplet in the interaction termsin γy and γz;

• the Pauli spin operators (σ x, σ y and σ z) of thefirst spin of the triplet under consideration remainunchanged in each relative interaction term (γx, γyand γz).

The final set of parametric single-spin-1/2 Hamiltoni-ans is then

H = γxσx1 +

[(−1)

N−12 γy

(N−1)/2

∏k=1

σz2k+1

]σ

y1

+

[γz

(N−1)/2

∏k=1

σz2k+1

]σ

z1,

(A11)

in the case of an odd number of spins, and

H = γxσx1 +

[(−1)

N2 γy

N/2

∏k=1

σz2k

]σ

x1 + γz

N/2

∏k=1

σz2k. (A12)

when N is an even number. For the sake of clearness, wepoint out that (N−1)/2 and N/2, appearing respectivelyin Eq. (A11) and (A12), are the numbers of transforma-tions necessary to get the final set of two-level Hamilto-nians from the original N-spin Hamiltonian [Eq. (A1)].

Finally, we can consider local magnetic fields appliedto the spins of the chain, that is a further term in theHamiltonian in Eq. (A1) of the following type

N

∑k=1

hωkσzk . (A13)

In the case of two and three spins (k = 2 and k = 3, re-spectively), the above operator, subject to the unitarytransformation based on the operators (A3) and (A8),acquires the forms

h(ω1 + σz2ω2)σ

z1, (A14a)

h(ω1 + σz2ω2 + σ

z2σ

z3ω3)σ

z1, (A14b)

respectively. Then, in the case of N spins, the generalform of the factor multiplying σ

z1 and depending on the

ωk parameters reads

h

[ω1 +

N

∑k=2

k

∏k′=2

σzk′ωk

]. (A15)

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