Classical correlation and quantum entanglement in the mixed-spin Ising-XY model with Dzyaloshinskii-Moriya interaction
MMarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb
International Journal of Modern Physics Bc (cid:13)
World Scientific Publishing Company
CLASSICAL CORRELATION AND QUANTUM ENTANGLEMENTIN THE MIXED-SPIN ISING-XY MODEL WITHDZYALOSHINSKII-MORIYA INTERACTION
HAMID ARIAN ZAD
Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran ∗ [email protected] HOSSEIN MOVAHHEDIAN
Department of Physics, Shahrood University of Technology, 3619995161, Shahrood, Iran
In the present work, initially a mixed-three-spin (1/2,1,1/2) cell of a mixed- N -spin chainwith Ising-XY model is introduced, for which pair spins (1,1/2) have Ising-type interac-tion and pair spins (1/2,1/2) have both XY-type and Dzyaloshinskii-Moriya(DM) inter-actions together. An external homogeneous magnetic field B is considered for the systemin thermal equilibrium. Integer-spins have a single-ion anisotropy property with coeffi-cient ζ . Then, we investigate the quantum entanglement between half-spins (1/2,1/2), bymeans of the concurrence. Classical correlation(CC) for this pair of spins is investigatedas well as the concurrence and some interesting the temperature, the magnetic field andthe DM interaction properties are expressed. Moreover, single-ion anisotropy effects onthe correlation between half-spins is verified. According to the verifications based onthe communication channels category by D. Rossini, V. Giovannetti and R. Fazio ,we theoretically consider such tripartite spin model as an ideal quantum channel, thencalculate its information transmission rate and express some differences in behaviour be-tween this suggested model and introduced simple models in the previous works(chainswithout spin integer and DM interaction) from information transferring protocol pointof view. Keywords :quantum entanglement; classical correlation; channel capacity; Dzyaloshinskii-Moriyainteraction.
1. Preliminaries
Heretofore, there are a lot of interests to investigate the various correlations(whether quantum or classical) for an ideal system , , , , . If we would like to ver-ify the quantum correlation between parts of a system then may be bound us toinvestigate the entanglement. Quantum entanglement is a special property whichcan exist only in the quantum systems , , , . Thereby, most of researchers confinethemselves to verify the entanglement to understand the behaviour of such systemsin the various situations , , , , , , . In this way, spin models are ideal can-didates for generating and manipulating of entangled states and for studying theentanglement , , , , , , by verification some stimulating quantities such as, a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN the concurrence , , , negativity , , , quantum discord , , , , , , quan-tum disorder , correlation functions , and von Neumann entropies , , , .Somewhere, spin models have been studied with the DM interaction , , , , , ,that such interaction arises naturally in the perturbation theory due to the spin-orbit coupling in magnetic systems.Straightforward researches have been caried out to investigate interaction be-tween the next-nearest-neighbour sites of a Heisenberg spin model in Refs. , , , .Such interaction may has an essential role to generate a Heisenberg model with di-amond chain topology by organizing mixture of particles that have different spins.Motivated by this issue, several studies have been done on the mixture of differentspins with various models and many interesting results have been reported , , .Diamond chains as attractive structures among these spin models were exactly in-vestigated from quantum entanglement, quantum correlation, phase transitions etc.view points , , , , , .The motivation for the study of a diamond chain with the Ising-XXZmodel is that it can describes real materials such as natural mineral azurite Cu ( CO ) ( OH ) , where according to experimental results, theoretical calculationsare interestingly reasonable in this case (another polymeric coordination com-pounds such as M ( OH ) with spin-1 Heisenberg diamond chain were investigatedin the literature ). Another quantum spin models consisting of diamond-shapedcells can be theoretically suggested and solved. In this regard, we here are interestedto introduce a few body diamond chain with specific model and verify its bipartiteCC and also entanglement in the some physical situations.In our previous works , , we analyzed bipartite quantum entanglement inthe mixed-three-spin system (1/2,1,1/2) with two different ‘XXX Heisenberg’and‘Ising-XY’models in the vicinity of an external homogeneous magnetic field. Thispaper has been devoted to verify CC and the quantum entanglement between half-spins (1/2,1/2) of same as the second model for which an additional DM interactionis considered between half-spins. Some interesting temperature,the magnetic field,the DM interaction and the other applied coefficients properties especially single-ionanisotropy related to the integer spin are expressed. The main purpose of this workis to provide the exact solution for the generalized version of the mixed spin-1/2and spin-1 Ising-XY diamond chain, which should bring a deep insight into howthe thermal and the magnetic properties depend on the spins-1/2 and spin-1 of themodel, in order words, we are going to understand the spin-1 existence has howmuch physical effects on the correlation between spins-1/2.Forthermore, the suggested model is considered as a memoryless communicationchannel between hypothetical sender Alice and the receiver Bob, then quantum in-formation transmission rate R , , is numerically verified. The ratio R describesthe maximum number of qubits one can transfer through the channel per unit oftime. Before, we proved that similar mixed-spin model can be considered as a com-munication channel for transferring qutrits .arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model The paper is organized as follows. In the next section, we first characterize theconcurrence as a measure of entanglement and CC between the spins (1/2,1/2), alsowe have a quick look at the model as a communication channel(Sec. 3). In Sec. 4 wedefine our favorite model with an analytical Hamiltonian and get its eigenvectors andeigenvalues. Then, we extract density matrix of the bipartite spins (1/2,1/2) fromdensity matrix of the mixed-three-spin system in representation of the basis states.In Sec. 5, we show the numerical calculations and simulations of the concurrenceand CC between the spins (1/2,1/2), with respect to the temperature, the magneticfield, the coupling constant J , the single-ion anisotropy ζ , the DM interaction D and the anisotropy parameter γ associated to the XY interaction. Also, informationtransmission rate of the mixed-three-spin chain channel is theoretically investigated.Section 6 is devoted to discussions and a summary of conclusions.
2. Introduction to the Concurrence and the ClassicalCorrelation(CC)2.1.
Concurrence
The concurrence that is a measure of entanglement, can be defined for bipartitespin systems as C ( ρ ) = max { , λ − (cid:88) i =1 λ i } , (1)where λ = max { λ , λ , λ , λ } and λ i are square roots of the eigenvalues of theinner product R = ρ ˜ ρ, (2)with ˜ ρ = ( σ y ⊗ σ y ) ρ † ( σ y ⊗ σ y ) , (3)where in the basis states {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} , the density matrix of a quantumsystem with Hamiltonian H in thermal equilibrium is defined as ρ eq = exp( − βH ) T r [exp( − βH )] , (4)where β = 1 /T (we set k B = 1) in which T is the temperature and Z = T r [exp( − βH )] is the partition function of the system. ρ † denotes the complex con-jugation of the density matrix ρ , and σ y = (cid:18) − ii (cid:19) . (5)Hitherto, the concurrence was explicitly calculated and simulated in terms ofthe temperature, the magnetic field, the DM interaction etc. for the various spinmodels. In the some of references cited here and references therein, it has beenarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN mentioned that C ( ρ ) behaves as sudden death at striking critical points, which iscalled “entanglement sudden death”(also review Refs. , ). In the following, weinvestigate the concurrence changes with respect to the temperature, the magneticfield, the anisotropy coefficient γ , the single-ion anisotropy ζ and the DM interaction D . Moreover, we would like know, that what is the effect of spin-1 existence in thesystem on the temperature, the magnetic field and the DM interaction dependencesof the concurrence corresponding to the spins (1/2,1/2), consequently we extractsome interesting outcomes. Classical correlation
We here recall the concept of CC for the spins (1/2,1/2) briefly. Total correlationin a bipartite system formed by (sub)systems A and B in a composite Hilbert space H bi = H A ⊗ H B is quantified by the quantum mutual information , , as I ( ρ A : ρ B ) = S ( ρ A ) + S ( ρ B ) − S ( ρ A B ) , (6)where, S ( ρ ) = − T r [ ρ log ( ρ )] is the von Neumann entropy in which ρ A ( B ) = T r A ( B ) ( ρ ). The quantum mutual information includes quantum information andclassical one (see Refs. , ). After a measurement on one of the (sub)systems suchas A , the amount of information obtained about the another (sub)system B isdefined as CC. CC can be defined in terms of POVM measurement , , . Let usconsider a set of projective measurements { B κ } performed locally only on part B then, the probability of measurement outcome κ is defined as p κ = T r
A B [( I A ⊗ B κ ) ρ A B ( I A ⊗ B κ )] , (7)where I A denotes the identity operator for the (sub)system A . After this measure-ment, state of the subsystem A is described by the conditional density operator ρ κ = 1 p κ [( I A ⊗ B κ ) ρ A B ( I A ⊗ B κ )] . (8)The projectors B κ can be characterized as B κ = V Π κ V † where, Π κ = | κ (cid:105)(cid:104) κ | atwhich κ = { , } . We parametrize the matrix V as V = (cid:18) cos ( θ ) e − iφ sin ( θ ) e iφ sin ( θ ) − cos ( θ ) (cid:19) , (9)where V ∈ U (2), 0 ≤ θ ≤ π and 0 ≤ φ ≤ π . We define the suprimumof the difference between the von Neumann entropy S ( ρ A ) and the based-on-measurement(POVM) quantum conditional entropy S ( ρ A B |{ B κ } ) = (cid:80) κ p κ S ( ρ κ )of the subsystem A as CC ( ρ A B ) = sup { B κ } { S ( ρ A ) − S ( ρ A B |{ B κ } ) } , (10)where S min ( ρ A B ) = min { B κ } S ( ρ A B |{ B κ } ). CC has been precisely verified in Ref. .arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model
3. Channel Capacity
Study of the classical information and communication channels was first char-acterized by Shannon and its quantum analogous was promoted by von Neu-mann and more studies have been devoted to these regimes in the last decades , , , , , , , . The information sent via quantum communication channels iscarried by quantum states(qubits), classical information(bits) can also be transmit-ted through quantum channels, namely, any channel that is able to transmit quan-tum information can be likewise used for transmitting classical information. Onecan find profound detections about quantum communication channels category inRefs. , .Recently, it was proposed for using simple spin chains with specific models ascommunication channels , , , . Some interesting suggested models as commu-nication channel are included Pauli channels , , depolarizing channels, dephasingchannels , spin chains channels , , electromagnetic channels and Gaussianchannels , , . Channel capacity is the maximum rate of a communication chan-nel which information can be reliably carried. Hence, researchers are interested tostudy on the capacity of a channel with memory or memoryless for transmitting orstoring unknown quantum states .When an arbitrary state ρ is propagated through a communication channel withcapacity, it can be wholly characterized by designing mapping protocol as the bellowform M : ρ i −→ ρ f = M [ ρ i ] , (11)where ρ i is the initial state and ρ f is the mapped state through channel. Suchmapping is performed by unitary transformation operator related to the feature ofthe channel as M [ ρ ] = U ρU † . (12)Caricature of this protocol is shown in Fig. 1. Fig. 1. Caricature of the transferring information through a mixed-spin chain as a quantumcommunication channel.
This paper puts substantial limits on the amount of information that can betransmitted reliably along a mixed-three-spin chain memoryless channel. The actionarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN of a generic quantum channel denoted on a single system as E can be defined as E N = E ⊗ N , where N represents the number of channel uses. The quantum capacity Q measured in qubits per channel use, is defined as Q = max N →∞ Q N N , (13)where Q N = max ρ (cid:2) S ( E N ( ρ )) − S e ( ρ, E N ) (cid:3) , which denotes the maximum coherentinformation. S represents the von Neumann entropy, and S e is the entropy exchange,namely S e ( ρ, E ) = S (cid:0) ( H ⊗ E )( | Ψ ρ (cid:105)(cid:104) Ψ ρ | ) (cid:1) , and state | Ψ ρ (cid:105) is any purification of ρ by means of a reference quantum system H , i.e. ρ = T r H (cid:2) | Ψ ρ (cid:105)(cid:104) Ψ ρ | ) (cid:3) .The transfer protocol can be as the follows: (I) sender (Alice) applies a SWAPoperation S A for the set of unknown states | ψ n (cid:105) A = | ψ n , · · · ψ , ψ , ψ (cid:105) , and firstpart of the spin chain C A (see Fig. 2). Here, it is assumed that the spin chainis initially in a separable state | ψ (cid:105) C = | ↓ , (cid:9) , ↓(cid:105) which, here states of spins-halfand spin-integer of the tripartite system are set up in the S z and J z down states | (cid:105) = | − / (cid:105) = | ↓(cid:105) and | − (cid:105) = | (cid:9) (cid:105) respectively, (II) after time t = τ , the receiver(Bob) tries to recover the information sent from Alice by means of some decodingprotocols, namely, Bob will extract information sent by applying a SWAP operator S B . This operator couples Bob , s memory with C B which corresponds to the reduceddensity operator for the N -th qubit. In fact, after time t = τ , this protocol can bedescribed as a completely positive trace preserving (CPTP) mapping from inputstate density matrices to output state density matrices, in the form ρ A −→ M [ ρ A ] = T r T B [ U ( ρ A ⊗ ρ C ) U † ] , (14)where M represents the mapping scenario of a real memoryless channel and U is the Fig. 2. Schematic representation of the mixed-three-spin chain as a quantum communicationchannel. unitary transformation which describes joint evolution of the composite Hamiltonian H AC = H A ⊗ H C , T r T B denotes to trace over all spins except receiver B.arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model In the present work, it is considered a scenario where the sender and the re-ceiver use their spins belong to the spin chain for encoding and decoding informa-tion(classical or quantum). From communication theory point of view, this consid-eration is not enough persuasive, but on the one hand, the consequences can betreated analytically.For doing transmission protocol, suppose that Alice has a memory with state | Ψ (cid:105) A = · · · | ψ (cid:105) ⊗ | ψ (cid:105) ⊗ | ψ (cid:105) , where | ψ i (cid:105) = α (cid:48) i | ↓(cid:105) + β (cid:48) i | ↑(cid:105) ( i = { , , , · · · } and | α (cid:48) i | + | β (cid:48) i | = 1). By starting the transmission protocol( t = 0) by Alice via couplingfirst memory element | ψ (cid:105) with state of the first chain spin C A through SWAP gate S A (1), memory element | ψ (cid:105) will replace with the state of C A . We here assumedthat the mixed-three-spin chain be initially in state | ↓ (cid:9) ↓(cid:105) . This procedure can becharacterized as (cid:0) · · · | ψ (cid:105) ⊗ | ψ (cid:105) ⊗ | ψ (cid:105) (cid:1) A ⊗ | ↓ (cid:9) ↓(cid:105) C ⊗ (cid:0) | ↓(cid:105) ⊗ | ↓(cid:105) ⊗ | ↓(cid:105) · · · (cid:1) B S A (1) −−−−→ (cid:0) · · · | ψ ψ ↓(cid:105) (cid:1) A (cid:0) α (cid:48) | ↓ (cid:9) ↓(cid:105) + β (cid:48) | ↑ (cid:9) ↓(cid:105) (cid:1) C (cid:0) | ↓(cid:105) ⊗ | ↓(cid:105) ⊗ | ↓(cid:105) · · · (cid:1) B . (15)After time evolution τ , the first memory element | ψ (cid:105) embedded in C A after firstSWAP S A (1), spreads along the mixed-three-spin chain, thereby, the total state(15) becomes (cid:0) · · · | ψ ψ ↓(cid:105) (cid:1) A ⊗ (cid:0)(cid:0) α (cid:48) | ↓ (cid:9) ↓(cid:105) + β (cid:48) Υ ( τ ) | ↓ (cid:9) ↑(cid:105) (cid:1) C ⊗ (cid:0) | ↓↓↓ · · · (cid:105) (cid:1) B , (16)where Υ ( τ ) = C (cid:104)↓ (cid:9) ↓ | e − iHτ (cid:126) | ↓ (cid:9) ↑(cid:105) C (17)which is the probability amplitude of finding the spin up ( | ↑(cid:105) ) in the 3-th partof the mixed-three-spin chain( C B presented in Fig. 2). In Sec. 5, we will use thesestatements to verify the transmission rate of the channel numerically.
4. Suggested Spin Model and Theoretical Background
We introduce Hamiltonian of the mixed- N -spin system with Ising-XY model(mixed-spin chain shown in Fig. 1) which is in an external homogeneous magneticfield B , as the follows H = M (cid:80) i =1 (cid:0) (1 + γ ) S xi, S xi, + (1 − γ ) S yi, S yi, (cid:1) + M (cid:80) i =1 (cid:126)B · ( (cid:126)S i, + (cid:126)S i, ) + M (cid:80) i =1 (cid:126)D · ( (cid:126)S i, × (cid:126)S i, )+ M (cid:80) i =1 J ( S zi, J zi, + J zi, S zi, ) + M (cid:80) i =1 ζ ( J zi, ) + M (cid:80) i =1 (cid:126)B · (cid:126)J i, , (18)where i denotes the number of mixed-three-spin cell in the chain, γ is anisotropyparameter, J is the Ising coupling between the spins (1,1/2), D is the DM interactionbetween spins-half of the cell and ζ is the single-ion anisotropy parameter consideredarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN for spins-integer. (cid:126)S = { S x , S y , S z } and (cid:126)J = { J x , J y , J z } are spin operators(with (cid:126) = 1) which, are introduced as the following matrices S x = (cid:18) (cid:19) , S y = (cid:18) − ii (cid:19) , S z = (cid:18) − (cid:19) , (19) J x = √ , J y = √ − i i − i i , J z = − . (20)We here consider M = 1 and (cid:126)B = B z and (cid:126)D = D z , which are homogeneousmagnetic field and DM interaction in the z -direction. Note that here all of in-troduced parameters are considered dimensionless parameters. Eigenvectors of theHamiltonian of the mixed-three-spin (1/2,1,1/2) chain are given by | φ (cid:105) = a | ↑ , (cid:13) , ↓(cid:105) + | ↓ , (cid:13) , ↑(cid:105) , | φ (cid:105) = b | ↑ , (cid:13) , ↓(cid:105) + | ↓ , (cid:13) , ↑(cid:105) , | φ (cid:105) = e | ↑ , (cid:8) , ↑(cid:105) + | ↓ , (cid:8) , ↓(cid:105) , | φ (cid:105) = f | ↑ , (cid:8) , ↑(cid:105) + | ↓ , (cid:8) , ↓(cid:105) , | φ (cid:105) = g | ↑ , (cid:13) , ↑(cid:105) + | ↓ , (cid:13) , ↓(cid:105) , | φ (cid:105) = h | ↑ , (cid:13) , ↑(cid:105) + | ↓ , (cid:13) , ↓(cid:105) , | φ (cid:105) = j | ↑ , (cid:9) , ↓(cid:105) + | ↓ , (cid:9) , ↑(cid:105) , | φ (cid:105) = k | ↑ , (cid:9) , ↓(cid:105) + | ↓ , (cid:9) , ↑(cid:105) , | φ (cid:105) = l | ↑ , (cid:9) , ↑(cid:105) + | ↓ , (cid:9) , ↓(cid:105) , | φ (cid:105) = m | ↑ , (cid:9) , ↑(cid:105) + | ↓ , (cid:9) , ↓(cid:105) , | φ (cid:105) = n | ↑ , (cid:8) , ↓(cid:105) + | ↓ , (cid:8) , ↑(cid:105) , | φ (cid:105) = o | ↑ , (cid:8) , ↓(cid:105) + | ↓ , (cid:8) , ↑(cid:105) , (21)where a = − i ( i − D ) √ D , b = i ( i − D ) √ D ,e = − γB + ζ − √ ( ζ +2 B ) +4 J ( ζ + J +2 B )+ γ + J ,f = − γB + ζ + √ ( ζ +2 B ) +4 J ( ζ + J +2 B )+ γ + J ,g = − γ − √ γ +4 B + B , h = − γ √ γ +4 B + B ,j = − i ( i − D ) − ζ + √ ζ + D , k = − i ( i − D ) − ζ − √ ζ + D ,l = γ ζ − B + √ ( ζ − B ) +4 J ( ζ + J − B )+ γ + J ,m = γ ζ − B − √ ( ζ − B ) +4 J ( ζ + J − B )+ γ + J ,n = i ( i − D ) − ζ − √ ζ + D , o = i ( i − D ) − ζ + √ ζ + D , (22)and the corresponding eigenvalues are E = ζ + √ D , E = ζ − √ D E = B + ζ + (cid:112) ( ζ + 2 B ) + 4 J ( ζ + J + 2 B ) + γ ,E = B + ζ − (cid:112) ( ζ + 2 B ) + 4 J ( ζ + J + 2 B ) + γ ,E = ζ + (cid:112) γ + 4 B , E = ζ − (cid:112) γ + 4 B E = ζ − B + (cid:112) ζ + D + 1 , E = ζ − B − (cid:112) ζ + D + 1 E = ζ − B + (cid:112) ( ζ − B ) + 4 J ( ζ + J − B ) + γ ,E = ζ − B − (cid:112) ( ζ − B ) + 4 J ( ζ + J − B ) + γ ,E = ζ + B + (cid:112) ζ + D + 1 , E = ζ + B − (cid:112) ζ + D + 1 (23)arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model In the basis states representation, the total density matrix of the considered tripar-tite system which is a thermal equilibrium state can be characterized by using latestequations. Hence, the density matrix of the pair spins (1/2,1/2) can be expressedas ρ T = 1 Z δ ς P ξ ξ ∗ Q ς ∗ χ , (24)where T is partial trace over second spin(note that the matrix is symmetric). { δ, ς, P, Q, ξ, χ } are functions of T , B , D , γ , J and ζ , and also mixture of com-ponents of the total density matrix.
5. Exact Numerical Solution
In order to provide a detailed analytical and numerical simulation, here, we usethe concurrence as a measure of entanglement for the bipartite (sub)system, alsoCC is investigated and simulated numerically as well as the concurrence.
Correlation functions
With regard to the geometric of correlation functions, we will calculate the con-currence and CC for the particular case bipartite spins (1/2,1/2) whose the densityoperator is presented in the form (24). Arrays of this matrix can be characterizedas the following equations δ = (1 + G iz + G jz + G ijzz ) , P = (1 + G iz − G jz − G ijzz ) ,Q = (1 − G iz + G jz − G ijzz ) , χ = (1 − G iz − G jz + G ijzz ) ,ς = ( G ijxx − G ijyy ) , ξ = ( G ijxx + G ijyy ) , (25)where G kz = (cid:104) σ kz (cid:105) with k = { i, j } , is the magnetization density at site k and G ijµν = (cid:104) σ iµ σ jν (cid:105) with µ, ν = { x, y, z } denote spin-spin correlation functions at sites i and j . Note that the expectation value can be defined as T r [ ρ G ]. If we introduce theelements E = ξ + ξ ∗ + ς + ς ∗ , E = ξ + ξ ∗ − ς + ς ∗ , E = δ + χ − P − Q, E = δ − χ − P + Q, E = δ − χ + P − Q, (26)in accordance with the reconstructed density matrix (24) as ρ T = Z (cid:0) I ⊗ I + (cid:80) i =1 E i ( σ i ⊗ σ i ) + E ( I ⊗ σ ) + E ( σ ⊗ I ) (cid:1) , (27)then, we obtain a simple equation for CC of the bipartite system represented in Eq.(10) as CC ( ρ A B ) = (1 −E )2 log (1 − E ) + (1+ E )2 log (1 + E ) , (28)in which E = max {|E | , |E | , |E |} .arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN
Concurrence
The reduced density matrix presented in Eq. (24) has whatever is needed aboutthe bipartite spins (1/2,1/2), hence the concurrence can be readily obtained as C ( ρ A B ) = 2 max { max (cid:0) , | ςZ | − P (cid:1) , | ξZ | − ( δχZ ) } . (29)The concurrence (29) as function of the temperature T and the magnetic field B at fixed values of the anisotropy, the single-ion anisotropy and DM interactionparameters is shown in Fig. 3. As illustrated in this figure, the concurrence at lowtemperature and weak magnetic field is maximum ( C ( ρ ) = 1), on the other hand,this quantity is minimum ( C ( ρ ) = 0) at high temperature and strong magneticfield. This essential property of the concurrence has been studied for various spinmodels in the previous works , , and here it is true for our favorite bipartite(sub)system and is compatible with the previous works.In the some of used references, authors gained a critical temperature at which theconcurrence vanishes, but here we see that with changes of the magnetic field, theconcurrence vanishes at different critical temperatures. Indeed, by inspecting Fig.4, one can observe that for the various magnetic fields, the concurrence diagramsvanish in the various critical temperatures and that is because, the concurrence ofthe bipartite (sub)system strongly depends on the some extra parameters exceptthe temperature and the magnetic field, such as the single-ion anisotropy and DMinteraction parameters. Also, it is explicitly seen by increasing the anisotropy γ , Fig. 3. Concurrence of the spins (1/2,1/2), with respect to the temperature and the magneticfield at fixed values of ζ = J and D = 5 J , for: (a) γ = 0 . J ; (b) γ = 0 . J . the concurrence vanishes at higher critical temperatures for the fixed values ofthe magnetic field(blue dash line and green dot-dashed line). Meanwhile, at weakmagnetic field, the concurrence not entirely be zero in the higher temperatures.In the stronger anisotropy γ , this phenomenon becomes more clear(Figs. 3(b) and4(b)). As a result, because of verifying the entanglement for such quantum systemat very low temperatures (near T = 0) is practically difficult, it can be easier byincreasing of the anisotropy. Indeed, one can use the anisotropy as an entanglementcontroller in higher temperatures for such model.arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model ζ = J and D = 5 J in the various magnetic fields for: (a) γ = 0 . J ; (b) γ = 0 . J .Fig. 5. Concurrence of the spins (1/2,1/2), with respect to the single-ion anisotropy and DMinteraction parameters at low temperature T = 0 . J and fixed B = J for: (a) γ = 0 . J ; (b) γ = 0 . J . Figure 5 represents the concurrence of the spins (1/2,1/2) with respect to thesingle-ion anisotropy and the DM interaction parameters at low temperature( T =0 . J ) and fixed homogeneous magnetic field B = J . As illustrated in this figure,for the fixed values of the single-ion anisotropy, with decrease of the DM interactionfrom its high values, the concurrence decreases until reaches a minimum in whicha sudden change occurs in a special critical DM interaction. Here, state of thebipartite (sub)system will change, indeed a phase transition occurs in this criticalpoint. with increase of ζ this critical point tends to the stronger DM interaction.If we consider a line that connects these critical points, we see that the concur-rence increases with increase of the DM interaction for the region upper than thisline, namely, the concurrence is proportional to the DM interaction in this region.On the other hand, for the region lower than the connection line, the concurrencedecreases with increase of the DM interaction.This function decreases with increase of the single-ion anisotropy ζ at fixedarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN values of D until reaches that minimum in which phase transition occurs as wementioned before. Hence, the concurrence reaches its maximum value in the strongDM interaction and small single-ion anisotropy. Shape of the concurrence digram ischanged by increasing the anisotropy γ . These properties are obviously presentedin Figs. 6 and 7. Fig. 6. Concurrence of the spins(1/2,1/2) as function of the DM interaction parameter D at lowtemperature T = 0 . J at fixed values of the single-ion anisotropy and B = J for: (a) γ = 0 . J ;(b) γ = 0 . J . As shown in Fig. 6(a), maximum amount of the concurrence decreases with in-crease of the single-ion anisotropy ζ . Moreover, in this figure it is obviously visible,that those critical points in which the concurrence reaches its minimum at low tem-perature tend to the stronger DM interaction, where ζ > γ are consideredfixed. For the strong single-ion anisotropy property ζ > J , we found that the mini-mum amount of the concurrence becomes zero, namely, in this condition state of thebipartite (sub)system is a separable state. With increase of the anisotropy param-eter γ (from 0.2 to 0.8), the critical points shift to the weaker DM interaction(Fig.6(b)).In the present paper, we obliged ourselves to investigate the pairwise entangle-ment of spins (1/2,1/2) from the single-ion anisotropy point of view, which is merelyconsidered for integer-spins in the total spin chain. Fig. 7 depicts the concurrence asfunction of the single-ion anisotropy ζ at fixed values of D at low temperature. Asillustrated in this figure, one can see that the pairwise entanglement sorely dependson the single-ion anisotropy ζ . With increase of the single-ion anisotropy in thetripartite system, the concurrence of the bipartite (sub)system decreases for fixedvalues of D .In the almost weak DM interaction(red solid line and green dot-dashed line), theconcurrence decreases by increasing the single-ion anisotropy from zero and reachesa minimum in which phase transition occurs. With further increase of the single-ion anisotropy, a sudden change happens in the concurrence behaviour at a specialarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model ζ atlow temperature T = 0 . J at fixed values of the DM interaction and B = J for: (a) γ = 0 . J ;(b) γ = 0 . J . critical single-ion anisotropy ζ c and for ζ > ζ c this quantity decreases smoothly. Byincreasing the anisotropy γ , this sudden change(the concurrence minimum point)occurs at almost weaker the single-ion anisotropy for D = 0(red solid line in Fig.7), while for D > γ with increase of the single-ionanisotropy parameter ζ (black dot line).For better understanding the behaviour of the concurrence with respect to theanisotropy γ and DM interaction, we knew interest that depict the anisotropy andthe DM interaction dependences of the concurrence at low temperatures at fixedvalues of the magnetic field, in the form of contour plots as illustrated in Fig. 8.This figure shows that in this circumstances, the concurrence is symmetric versusthe anisotropy and the DM interaction parameters for various magnetic fields. Also,it can be readily seen that in the strong DM interaction i.e., D > | J | for 0 ≤ B ≤ J (Figs. 8(a) − D > | BJ | for 2 J < B ≤ J (Figs. 8(d) − B > J theconcurrence behaviour dramatically alters. We explain this exotic behaviour in thedifferent intervals of the magnetic field in the following.If one follows Fig. 8 step by step then realizes that the thermal concurrenceas a measure of pairwise entanglement has an exotic behaviour versus increasingthe magnetic field. This quantity for interval 0 ≤ B ≤ J behaves different frominterval 2 J < B ≤ J , also its behaviour for B > J is generally different fromformer intervals. Namely, for the first interval, with increase of the magnetic fieldfrom zero to 2 J the concurrence arises at the weaker DM interaction(the blue regiongradually decreases) as far as the concurrence does not vanish even at D = 0and γ ≈
0. While, at the second interval, with increase of the magnetic field, thearch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN
Fig. 8. Contour plots of the concurrence as function of the anisotropy γ and the DM interaction D at fixed values of ζ = J at low temperature T = 0 . J for: (a) B = 0; (b) B = J ; (c) B = 2 J ;(d) B = 3 J ; (e) B = 4 J ; (f) B = 5 J , for spins (1/2,1/2). Colour bars represent changes of theconcurrence from C = 0(black regions) to C = 1(red regions). concurrence arises at stronger the DM interaction( D ≈ | BJ | ), which width changesof the blue region can present this exotic behaviour. Consequently, for the strongarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model magnetic field, the field influence on the concurrence behaviour will overcome theDM interaction. Finally, with regard to Fig. 8(f) for B > J the maximum amountof the concurrence is entirely limited to the DM interaction interval D < | BJ | andstrong anisotropy γ ≈ | J | . Fig. 9. Surfaces of constant geometric concurrence of state ρ ( AB ) defined in Eq. (24) at lowtemperature T = 0 .
15 for: (a) ζ = J and C ( ρ ) = 0 .
03, (b) ζ = J and C ( ρ ) = 0 .
3, (c) ζ = J and C ( ρ ) = 0 .
8. It is clear that the geometry of the concurrence is symmetric versus absolute valuesof the DM interaction and the anisotropy, while there is not any symmetry versus the magneticfield axes.
In Fig. 9, we plot level surfaces of geometric concurrence of the state ρ ( AB )defined in Eq. (24) at low temperature and fixed value of ζ = J . Using this figure,one can recognize that in what regions seeks existence of the entangled states forthe bipartite (sub)system. Obviously, surfaces of the constant geometric concurrenceare symmetric versus DM interaction and anisotropy γ , while we can not find anysymmetry versus the magnetic field axes. Selecting of this special model may bereson of this symmetry breaking and one can choose a model for which symmetryalways be established.5.1.2. Classical correlation
With regard to Sec. 2 and references therein, we start to explain the configurationof the CC between the spins (1/2,1/2) as well as its concurrence, for realizing someextra physical behaviours of the (sub)system which are rare in the other investigatedspin models in the previous works. We here verify this quantity which can be existin the both classical and quantum systems as function of the various parametersthen, compare it with the concurrence. Finally, we get some interesting outcomes.CC as function of the temperature and the magnetic field at fixed values ofthe single-ion anisotropy and the DM interaction parameters is shown in Fig. 10.Let us divide this figure to four segments: (i) at low temperatures( T (cid:28) J ) andinterval 0 . J (cid:46) B (cid:46) . J , CC is maximum and in the outside of this interval thisquantity vanishes. Also here, by increasing the temperature from zero this quantitygradually vanishes; (ii) at interval 0 ≤ B (cid:46)
1, by increasing the temperature fromarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN
Fig. 10. CC of the spins (1/2,1/2) as function of the temperature and the magnetic field at fixedvalues of D = 5 J and ζ = J for: (a) γ = 0 . J ; (b) γ = 0 . J . zero CC arises, and with further increase of the temperature until T ≈ J , thisquantity reaches a maximum value then vanishes; (iii) at B (cid:38) J , by increasingthe temperature from T ≈ . J , CC arises as far as at high temperatures thisquantity reaches another maximum then vanishes again; (iv) for T (cid:38) J at interval0 . J (cid:46) B (cid:46) . J , with increase of the temperature, this quantity arises againand reaches a maximum smaller that other. It is clear that, with increase of theanisotropy the maximum value of CC decreases in the latest three region (ii), (iii)and (iv) but it increases in the first region (i).If we look smartly at Fig. 3 from the top perspective, we realize that for lowtemperatures by increasing the anisotropy, the CC behaviour will almost becomessimilar to the quantum entanglement for the spins (1/2,1/2). The subject matter isthe behaviour of both functions at low temperature which almost become the samewith increasing the anisotropy. Indeed, the anisotropy has the unification ability onthe concurrence and CC functions at low temperature.CC as function of the single-ion anisotropy and the DM interaction at low tem-peratures and fixed value of the magnetic field is shown in Fig. 11. As shown inthis figure, in the strong DM interaction( D (cid:38) J ), CC is maximum independent ofthe single-ion anisotropy changes. But for the weaker DM interaction, with increaseof the single-ion anisotropy, this quantity vanishes at a special region, then arisesagain and reaches another maximum. Such region that presents a critical DM in-teraction domain versus the single-ion anisotropy parameter is depicted in Fig. 12.This figure shows that by increasing the single-ion anisotropy at low temperaturesand fixed magnetic field B = J , CC vanishes at higher critical DM interaction forboth γ = 0 . J and γ = 0 . J where, for the case of γ = 0 . J these critical points areabove line D = 0 . ζ (dashed line), but for the case of γ = 0 . J , they are below thisline. This is means that, for Fig. 11(a) we have 0 . ζ < D c < ζ while, for Fig. 11(b)arch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model T = 0 . J and fixed B = J for: (a) γ = 0 . J ; (b) γ = 0 . J .Fig. 12. The critical DM interaction D c versus single-ion anisotropy ζ at low temperatures T = 0 . J and fixed B = J for: (a) γ = 0 . J (boxes); (b) γ = 0 . J (circles). Note that the criticalDM interaction points for case γ = 0 . J is above of the dashed line which represents linear equation D = 0 . ζ , while they are below of it for γ = 0 . J . inequality 0 . ζ < D c < . ζ is established(also here, there are some points at whichCC is maximum at weak DM interaction and small the single-ion anisotropy). Bycomparing Fig. 11(a) with Fig. 6(a) one can gain some likenesses and differences inbehaviour of the concurrence and CC. For example, at fixed values of ζ > ζ the critical point tend to the stronger the DM interaction. Unlike theconcurrence, both CC maxima before and after the critical point are equivalent for ζ ≥ J . Another achievements can be obtained by comparing all represented figuresarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN corresponding to the concurrence and CC.
Transmission rate
Quantum transmission rate for the introduced protocol in Fig. 2 can be obtainedwith regard to the quantum channel capacity of the memoryless channel map M [ ρ ].This capacity that was defined as the maximum amount of the quantum informationreliably transmitted per use of the channel M (see Ref. ), was used for a spin chainchannel in Ref. and is given by Q ( η ) = max p ∈ [0 , { H ( ηp ) − H ((1 − η ) p ) } , (30)where, H ( X ) = − X log ( X ) − (1 − X ) log (1 − X ) is the dyadic Shannon entropy.The rate of this channel for ε uses in the time interval T = ετ can be defined as R ≡ lim ε →∞ ε Q ( η ) ετ = Q ( η ) τ . (31)Assume a simple spin-1/2 Heisenberg model with general Hamiltonian H G . Theoperator of the total z -component of the spin, given by σ ztotal = (cid:80) i σ zi is conserved,namely (cid:2) σ ztotal , H G (cid:3) = 0. Hence, the Hilbert space H G decomposes into invariantsubspaces, each of which is a distinct orthogonal eigenvector of the operator σ ztotal .Here, the transfer amplitude η in Eq. (30) is obtained using confined Hamiltonian H G . So, perfect and efficient quantum state transfer is happened for the such quan-tum chain with R = 1. But, for our suggested model with Hamiltonian (18) as acommunication channel, we obtain the transfer amplitude η using general Hamilto-nian H G of the system, and prove that the general Hamiltonian of the such systemwith few body is a capable operator to investigate the transmission protocol. Hence,we focus on the maximum and minimum amount of the rate R .For our model, the transfer amplitude η is a sinusoidal function of τ just thesame one in Ref. , but with changeable period π/ − iD ), where i = √− η = | Υ ( τ ) | = | sin(2(1 − iD ) τ | . (32)By setting Eq. (32) in Eq. (30), we can get the quantum transmission rate (31).Figure 13 depicts this rate versus time evolution τ numerically for various fixedvalues of the DM interaction. With regard to this figure, information transferringrate roughly depends on the DM interaction, i.e., for D = 0 . < τ < D from 0.05 to 0.5 the biggest peak of the rateoccurs with the pass of less time and its intensity increases almost ten times(from R = 0 .
25 to R = 2 . Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model τ for: (a) D = 0 .
05; (b) D = 0 .
1; (c) D = 0 . D = 0 . (1/2,1/2) have both XY and DM interactions together, without considering spin-integer. Namely, the transmission rate is independent of the Ising coupling betweenspins (1,1/2) and single-ion anisotropy parameter related to the spin-integer. Butas noted, it is roughly dependent on the DM interaction. If one looks carefully atthe Fig. 13 and follows limited time interval 0 < τ <
6. Summary and conclusions
We have introduced a mixed- N -spin Ising-XY model, then focused on a mixed-three-spin (1/2,1,1/2) cell of it to investigate CC and quantum entanglement be-tween spins (1/2,1/2) in the vicinity of a homogeneous magnetic field. Here, botharch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb H. ARIAN ZAD AND H. MOVAHHEDIAN
XY and DM interactions between spins (1/2,1/2) also Ising interaction betweenspins (1,1/2) and a single-ion anisotropy for spin-integer have been considered. Inthis paper, we restricted ourselves to a finite chain with mixed-three-spin (1/2,1,1/2)of a larger mixed- N -spin chain, and because it is generally difficult to study a spinchain with large length, small-size systems can be good options for obtaining someinformation about large-size systems. Fortunately, small-size cells of a large-size spinchain can also depict well properties of the large-size chain such as the tempera-ture, the magnetic(here the DM interaction and the single-ion anisotropy for thesuggested triangular cell) and the quantum correlation properties.In conclusion, we found that in the fixed values of ζ = J and D = 5 J , the con-currence is maximum at low temperature and weak magnetic field, but with increaseof the temperature this quantity decreases until vanishes at a critical temperature.This critical temperature is changed with the magnetic field changes, namely, forvarious magnetic fields we have different critical temperatures. With increase of theanisotropy, these critical points shift to the higher temperatures.Also, at low temperature and fixed value of B = J , the concurrence is maximumat strong DM interaction and zero single-ion anisotropy, and by increasing thesingle-ion anisotropy also decreasing the DM interaction this quantity decreasesuntil reaches a minimum at which phase transition occurs. This is means that theconcurrence as a measure of entanglement associated to the half-spins, is roughlydependent on the single-ion anisotropy related to the spin-1. Some critical pints ofthe DM interaction and the single-ion anisotropy have been numerically presentedin which the concurrence suddenly changes.Moreover, at low temperatures, we investigated the concurrence as function ofthe DM interaction and anisotropy at various fixed values of the magnetic field, andrealized that this quantity is symmetric in the DM interaction and the anisotropyframework and will extremely change with increase of the magnetic field. As aresult, the concurrence has a different behaviour for field intervals 0 ≤ B ≤ J ,2 J < B ≤ J and B > J .Surfaces of constant geometric concurrence of the (sub)system (1/2,1/2) withstate ρ ( AB ) at low temperature and fixed ζ = J have been presented in this work.We concluded that these surfaces depict the position of regions contained the en-tangled states properly. By using these surfaces we can easier detect circumstancesin which the (sub)system state is entangled.In that follows, we verified CC for the bipartite spins (1/2,1/2), and understoodthat at fixed values of ζ = J and D = 5 J this term has a exotic behaviour inthe temperature and the magnetic field framework. As an interesting outcome, weshowed that at low temperatures, by increasing the anisotropy, the behaviour ofCC will almost be same as the concurrence. Also, this term is investigated at lowtemperature and fixed magnetic field B = J with respect to the DM interactionand the single-ion anisotropy. Hence, we gained a set of the critical DM interactionin which CC is vanished, and by increasing the anisotropy γ range of this set ofarch 15, 2018 0:19 WSPC/INSTRUCTION FILE ws-ijmpb Classical correlation and Quantum Entanglement in the Mixed-Spin Ising-XY Model the critical points is changed in the DM interaction and the single-ion anisotropyframework(Fig. 12).Finally, we theoretically considered the tripartite mixed-spins (1/2,1,1/2) as acommunication channel with capacity, then analyzed quantum information trans-mission rate of it. To perform the transmission protocol, we assumed that an un-known qubit can be reliably transferred through this channel. Here, among theintroduced parameters for the favorite system, the transmission rate is just depen-dent on the DM interaction and that is because of the special Ising-XY consideredmodel for the spin chain which is not mentioned in the previous works. As anotherinteresting result, we showed that by increasing the DM interaction between spins(1/2,1/2), maximum transmission rate occurs at the less time interval and thischannel without considering the spin-1, can reliably transfer quantum information.One can compute another properties of the channel such as amplitude dampingchannel, entanglement-assisted classical capacity of the channel etc. for our sug-gested model and obtains some interesting outcomes. References
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