Interference of the signal from a local dynamical process with the quantum state propagation in spin chains
IInterference of the signal from a local dynamicalprocess with the quantum state propagation in spinchains
Saikat Sur and V. Subrahmanyam Department of Physics, Indian Institute Of Technology, Kanpur-208016, IndiaE-mail: [email protected] and [email protected] Abstract.
The effect of a local instantaneous quantum dynamical process (QDP),either unitary or non-unitary, on the quantum state transfer through a unitaryHamiltonian evolution is investigated for both integrable and non-integrable dynamics.There are interference effects of the quantum state propagation and the QDP signalpropagation. The state transfer fidelity is small for further sites, from the site wherethe information is coded, indicating a finite speed for the propagation of the quantumcorrelation. There is a small change in the state transfer fidelity for the case of non-unitary QDP intervening the background unitary dynamics. In the case of unitaryQDP, the change is more pronounced, with a substantial increase in the fidelity forappropriate sites and times. For the non-integrable case, viz. a kicked Harper model,the state transfer fidelity is quite large for further sites for short times, indicating afinite speed for the propagation of the quantum correlation cannot be defined.PACS: 03.65.Ud 03.67.Bg 03.67.Hk 75.10.Pq a r X i v : . [ qu a n t - ph ] O c t nterference of the signal of a local process with the quantum state propagation
1. Introduction
Quantum spin chains have been investigated, over the last few years, from the viewpointof quantum information and communication. A quantum spin chain serves as a possiblechannel for the quantum state transfer[1, 2, 3, 4]. These systems have been studiedextensively, as exactly-solvable condensed matter physics systems, for studying spinordering of novel spin states, and quantum critical behaviour.[5, 6, 7]. These studiesdeal with the quantum dynamics of initial many-body states through the Schroedingertime evolution[8], and the statistical mechanics of the phase transitions[5, 8].The spin chain dynamics can be treated viewing the spin chain as a closed system,with a given initial state with a distribution of correlations. Through the time evolutionthe quantum correlations can be redistributed and the spin chain can exhibit a varietyof multi-party quantum correlation and entanglement structures. The wide spectrum ofthe dynamics investigation of spin chains includes the study of magnon bound states andscattering[9, 10], spin current and relativistic density wave dynamics[11, 12], quantumcorrelations after a quench[13] and light-cone in entanglement spreading[14]. The effectof unitary evolution of the quantum spin chain can be viewed as, in the quantuminformation language, various global and local multi-qubit gate operations acting onthe many-qubit system. The system will undergo a redistribution of entanglement andquantum correlations[15, 16, 17, 18, 19] through the unitary evolution. However, amulti-party qubit system in general can experience non unitary or incoherent processes[20] acting on some qubits locally or globally, that can lead to decoherence. Investigatingthe effect of local decohering process on the quantum correlations and the entanglementstructure is difficult in general, as traditional approximation tools like the time-dependent perturbation theory are not applicable for these systems. In a simple exampleof local decohering process intervening the Schroedinger evolution of quantum states isinvestigated[21], where a the Hamiltonian unitary evolution of a many-qubit quantumstate is interrupted by a local instantaneous quantum dynamical process (QDP) on acertain qubit at a certain epoch of time. The QDP can be a non-unitary decoheringoperation or a unitary operation different from the back ground evolution. The QDPsignal, the effect of the occurrence of the process at a given time and location, andits propagation have been investigated using both magnetisation conserving and non-conserving dynamics for initial states with and without entanglement for various modelHamiltonian dynamics [21]. The signal propagation speed in general depends on thetype of interaction of the qubits, the strength of the interaction, the spatial range of theinteraction and the initial state.The quantum state transfer protocol[1, 2, 3, 22] studied for the spin chains relies onthe Hamiltonian evolution for a faithful communication and detection of the state froman initial qubit location to a target qubit location. The average fidelity, averaged overall possible input qubit state, of the state transfer, is a figure of merit of the protocol.The fidelity depends on the interaction parameters of the Hamiltonian, and in general itis oscillatory as a function of the time of evolution as a result of quantum interference. nterference of the signal of a local process with the quantum state propagation
2. Quantum State transfer through Unitary Dynamics
Faithful and faultless transmission of quantum states between two distant locations isthe main challenge for quantum communication and information processing protocols.The quantum spin chains as channels of quantum state transfer was proposed andinvestigated[1, 2, 22]. The spin of the electrons are basic examples of qubits,viz. two-level quantum systems, and the spin chains can be realised physically inmagnetic systems. One-dimensional spin models have been studied extensively usingthe Heisenberg-XY or transverse-field Ising model Hamiltonians, due to their exactsolvability using the Bethe ansatz technique or Jordan-Wigner fermion techniquerespectively. In this paper, we will be using the Heisenberg dynamics for the mostpart, and will study a variant of transverse-XY model in the last section.In a standard state transfer protocol, a direct-product many-qubit initial state isprepared, at time t = 0, with the first spin in a desired state (or the information is codedinto the state of the first spin). The many-qubit state is evolved through time using theHamiltonian Schroedinger dynamics to a final state at time t . Then the desired state is tobe recovered from a certain target spin, the probability of the state transfer is measuredby the fidelity, normalised to unity, computed by taking the overlap of the target statewith the desired state. A large value of the state transfer fidelity for a spin at a particularlocation at a particular time implies the efficacy of the state transfer protocol. In thispaper, we will deviate from the standard protocol, and intervene the smooth Heisenberg-XY dynamics by a local quantum dynamical process (QDP) at a given location and time,viz. an instantaneous quantum operation is carried out on the many-qubit state at atime t = t , between the coding time t = 0 of the quantum information at the first spin,and the recovery or readout time t of the quantum information from a different spin. In nterference of the signal of a local process with the quantum state propagation | φ (cid:105) = α | (cid:105) + β | (cid:105) . The first qubit is initialised to this state at one end, the manyqubit initial state is | ψ (0) (cid:105) = | φ ... (cid:105) , a direct product of the state | φ (cid:105) for the firstqubit and the state | (cid:105) for all other qubit. The unitary time evolution will transformthe many qubit state to | ψ ( t ) (cid:105) using the Hamiltonian dynamics. The desired qubit | φ (cid:105) can spread out to other qubits during the evolution, i.e., the desired qubit state cantravel to the other locations with a some speed and probability. The speed associatedwith such transfer is inversely related to the interaction strength between the sites. Foran efficient state transfer, the target state ρ l , is the reduced density matrix of the l ’thqubit, calculated by taking a trace over all other qubits from the many qubit state, ρ l = tr (cid:48) | ψ ( t (cid:105)(cid:104) ψ ( t ) | , should be the same as the desired state ρ φ = | φ (cid:105)(cid:104) φ | . The statetransfer fidelity gives a probabilistic measure of the state transferred from one site toanother and is quantified by taking overlap of the target and the desired states, givenby F l,φ ≡ T rρ φ ρ l = (cid:104) φ | ρ l | φ (cid:105) . (1)Now, the fidelity shown above depends on the initial state φ , apart from the targetspin location and time, and the interaction parameters of the Hamiltonian dynamics.A figure of merit measure for the efficacy of the protocol is to average the above overall possible initial qubit states, i.e., over the surface of the Bloch sphere representing allqubit pure states.The first exactly-solvable and integrable non trivial models of interacting quantumspins is a one-dimensional chain of spins interacting with their nearest neighbourHeisenberg exchange interaction, known as the Heisenberg model. We will use thePauli operator (cid:126)σ l to represent the different components of the spin operator (cid:126)S i at i ’thsite. Let us consider a one-dimensional chain of N spins interacting through the nearest-neighbour anisotropic Heisenberg model. The Hamiltonian is given by, H = − J (cid:88) i ( σ xi σ xi +1 + σ yi σ yi +1 + ∆ σ zi σ zi +1 ) . (2)where J is the exchange interaction strength for the nearest-neighbour spins, and ∆ isthe anisotropy strength. As all the three Pauli spin matrices appear in the Hamiltonian,an exchange interaction of neighbouring spin is implied in all three spin dimensions.The model exhibits ferromagnetic (antiferromagnetic) behaviour in the ground state for∆ > < i ’th spin as | (cid:105) (up-spin nterference of the signal of a local process with the quantum state propagation | (cid:105) (down-spin) states, denoting the eigenstates σ zi with eigenvalues +1 and -1respectively. The basis states for the many-qubit system can be chosen to be the directproducts of the basis states of each spin. The z-component of the total spin Σ σ zi is aconstant of motion, which implies that the eigenstates will have a definite number ofdown spins. The many-qubit basis states with l down spins can also be labeled by thelocations ( x , x ..x l ) of the l down spins, where the set is an ordered set with x < x and so on. An eigenstate with l down spins, a l − magnon state, can be written as asuperposition of the basis states as, | ψ (cid:105) = (cid:88) x ,x ..x l ψ ( x , x ..x l ) | x , x ..x l (cid:105) (3)where the eigenfunction ψ ( x , x ..x l ) denotes the wave function amplitude for thecorresponding basis state. The eigenfunction is given by the Bethe Ansatz[27], labeledby the the set of momenta ( p , p ..p l ) of the down spins, which are determined by solvingalgebraic Bethe ansatz equations, with periodic boundary conditions. There is only onezero-magnon state | F (cid:105) = | .. (cid:105) , which is just a ferromagnetic ground state with all thespins polarized along one direction. It is straightforward to see that it an eigenstate ofthe above Hamiltonian with energy (cid:15) = − N J for periodic boundary conditions (closedchain), and (cid:15) = − J ( N −
1) for open boundary conditions (open chain). Starting from | F (cid:105) , one-magnon excitations can be created by turning any one of the spins, giving N localised one-magnon states, which can be labeled by the location of the down spin. One-magnon eigenstates are labeled by the momentum of the down spin, the eigenfunctionis given by, ψ xp = (cid:115) N e ipx ; p = 2 πIN , for a closed chain ψ xp = (cid:115) N + 1 sin( px ); p = πIN + 1 , for an open chain , (4)where the momentum p is determined by an integer I = 1 , , ..N for both cases. Theone-magnon eigenvalue is given by (cid:15) ( p ) = (cid:15) − J cos p . The interaction strength J determines the hopping of the down spins to neighbouring sites, and the interaction ofthe two down spins is determined by ∆. The one-magnon eigen energies are independentof ∆ as the states carry only one down spin. The two-magnon (l=2) and other eigenstates(l >
2) include both scattering states and bound states of the down spins. Theeigenfunction for the two-magnon eigenstate is labeled by two momenta p and p ,is given by ψ x ,x p ,p = A ( p , p )( e ip x e ip x + e iθ ( p ,p ) e ip x e ip x ) , (5)where A ( p , p ) is a normalisation factor. The two terms in the wave function differ bya permutation of the positions x and x . The phase factor θ ( p , p ) depends on themoment and the interaction strengths, and is given bytan θ ( p , p )2 = ∆ sin[( p − p ) / p + p ) / − ∆ cos[( p − p ) / . (6) nterference of the signal of a local process with the quantum state propagation Figure 1.
The evolution of the initial state ρ (0) in three steps: The stateevolves to ρ ( t − ) through a unitary process using the Heisenberg Hamiltoniantill t = t − , the state become ˜ ρ ( t +0 ) through an instantaneous local QDP at t = t , and finally it evolves to ˜ ρ ( t ) through the same unitary process. The two momenta are determined by the Bethe ansatz equations, p N = 2 πI + θ ( p , p ) , p N = 2 πI − θ ( p , p ) , (7)where I , I are any integers. The two-magnon eigenvalues are given by (cid:15) ( p , p ) = (cid:15) + 2(∆ − cos p ) + 2(∆ − cos p ). The two-magnon scattering states have been studiedfor the spin-independent scattering[28]. The detail of solving the above Bethe ansatzequations for the two-magnon scattering and bound states is discussed in the AppendixA, along with the computation of the time-dependent Green’s functions.We will see no dynamics effects if the initial state is this eigenstate of theHamiltonian. To study the dynamics of an arbitrary initial state, we will consider alinear combination of zero-magnon and one-magnon states that are not eigenstates ofthe Hamiltonian, in conjunction with the QDP dynamics below. Let us consider a spinchain with open boundary condition and the initial state given by, | ψ (0) (cid:105) = α | ... (cid:105) + β | .. (cid:105) = α | F (cid:105) + β | x = 1 (cid:105) , (8)where x denotes location of the down spin. The above state is a linear superpositionof an eigenstate of the Hamiltonian with eigenvalue (cid:15) and a one magnon state whichcan be written as a linear combination of one-magnon momentum eigenstates | p (cid:105) of theHamiltonian with eigenvalues (cid:15) ( p ), we have | ψ (0) (cid:105) = α | F (cid:105) + β (cid:88) p ψ p (1) | p (cid:105) , (9)Now, the time evolution of the system will transport the single down spin from the firstsite to other sites and the state after a time t becomes, | ψ ( t ) (cid:105) = αe − i(cid:15) t | F (cid:105) + β (cid:88) y G y ( t ) | y (cid:105) , (10)where the time-dependent function G x (cid:48) x ( t ) [21] is given in terms of the wave functionsdefined above as, G x (cid:48) x ( t ) = (cid:88) p ψ xp ψ x (cid:48) ∗ p e − it(cid:15) ( p ) . (11)Using the one-magnon eigenfunctions, shown in Eq.4, we can express the Green’sfunction as, G x (cid:48) x ( t ) = N π (cid:90) π ψ xp ψ x (cid:48) ∗ p e − it(cid:15) ( p ) dp = ( − i ) ( x − x (cid:48) ) J x − x (cid:48) (2 t ) − ( − i ) x + x (cid:48) J x + x (cid:48) (2 t ) , (12) nterference of the signal of a local process with the quantum state propagation G x (cid:48) x ( t ) = ( − i ) x − x (cid:48) J x − x (cid:48) (2 t ) , (13)for a closed chain. J x ( y ) is the Bessel function of integer order x and argument y . The reduced density matrix (RDM)of the l th qubit is defined by taking trace ofthe whole quantum state over all qubits except the l th one, as ρ l = T r (cid:48) ρ ( t ), where ρ ( t ) = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | , is the time-evolved many-qubit state. Using the basis | (cid:105) , | (cid:105) for the l ’th qubit, the reduced density matrix is given by, ρ l ( t ) = (cid:32) − x l ( t ) y l ( t ) y ∗ l ( t ) x l ( t ) (cid:33) , (14)where the elements of the RDM are time dependent functions that can be calculatedfrom Eq.10 as, x l ( t ) = (cid:104) | ρ l ( t ) | (cid:105) = | β | | G l ( t ) | ,y l ( t ) = (cid:104) | ρ l ( t ) | (cid:105) = αβ ∗ e − i(cid:15) t G ∗ l ( t ) . (15)The state transfer fidelity for the l th site as a function of time is given by, F l,φ ( t ) = | α | (1 − x l ( t )) + | β | x l ( t ) + 2 Re ( α ∗ βy l ( t )) . (16)The quantity F l,φ ( t ) depends on the parameters α and β of the initial state. Theaverages over the parameters in the above equations are given as | α | = | β | = , | α | | β | = and | β | = . Taking the average over all possible pure states on the Blochsphere characterised by α and β the average state transfer fidelity[1] becomes, F l ( t ) = 12 + 16 | G l ( t ) | + 13 Re ( e i(cid:15) t G l ( t )) . (17)The state transfer fidelity F l ( t ) with unitary dynamics is shown in Fig. 2(a) as a functionof time for different site indices. Since the function G l ( t ) is a Bessel function with timeas argument it falls inversely with time and the fidelity of state transfer decreases withtime for a particular site and over a large time saturates to the value 0.5. It alsodeceases with the increase in the distance between the first site and the decoding sitel. The maximum fidelity for the site l is obtained around time l/ J = 1 / J between the spins.
3. Local quantum decohering process
We now consider the effect of a local QDP intervening the smooth unitary dynamics,thus effect the state transfer protocol. The time evolution of a general state involves,as we have seen in the previous section, the coherent movement of the down spins, dueto the unitary dynamics generated by the Heisenberg Hamiltonian we are considering.A local incoherent process or QDP can be any single qubit quantum gate operation, forexample a local projective measurement[20, 29]. The QDP we consider below occursinstantaneously at a time t = t , operating on a certain spin, thus changing the whole nterference of the signal of a local process with the quantum state propagation (a) (b) (c)(d) (e) (f) Figure 2. (a) State transfer fidelity F l ( t ) as function of site index( l ) and time( t ) for Heisenberg dynamics with with open boundary condition without anydecohering process. (b) Differences of state transfer fidelities with and withoutQDP (∆ F l , m = 1( t ; t = 0 . l ) and time ( t ); wherethe QDP occurs at the site ( m = 1) and time ( t = 0 . F l , m = 20( t ; t =10 .
0) as function of site index( l ) and time ( t ); where the QDP occurs at thesite ( m = 20) and time ( t = 10 . F l , m = 1( t ; t = 10 .
0) as functionof site index( l ) and time ( t ); where the QDP occurs at the site ( m = 1) andtime ( t = 10 . F l,m ( t ; t ) andwithout QDP F l ( t ) for the site l = 100, site of QDP m = 100 and time of QDP t = 50 .
0. (f) Maximum fidelities with and without QDP as a function of siteindex l ; the site of QDP m = l and time of QDP m = l/ N = 100. many-qubit state instantaneously. That is is the QDP occurs between the time t = t − and the time t = t +0 . The dynamics is unitary for t < t − , and again for t > t +0 ,and it is given by the Heisenberg Hamiltonian. The QDP can be non-unitary, whichcauses explicit decoherence of the many-qubit state, considered in this section, or theQDP can be another coherent quantum operation that will be discussed in the nextsection. First, we consider a non-unitary QDP that is a local decohering process, asan example, a projective measurement of σ zm which has two outcomes, correspondingprojectors P = (1+ σ zm ) / P = (1 − σ zm ) /
2. A more general measurement operators,for example measuring an arbitrary component of (cid:126)σ m , have been considered and seen tobe qualitatively similar[21]. Thus, now we have marked three qubits, at three differentlocations, namely, the first qubit where the initial information is coded, the l ’th qubit nterference of the signal of a local process with the quantum state propagation m ’th qubit where a local QDPoccurs. We will see below the case of l > m , in which both the desired state fromthe first spin and the effect of QDP will have to travel in the same direction to reachthe recovery site, and the case of m > l in which the two effects travel to the recoverysite from opposite directions. There will be interesting quantum interferences effects,particularly if the QDP is a coherent operation that will be discussed in the next section.The initial state ρ (0) = | ψ (0) (cid:105)(cid:104) ψ (0) | undergoes three different evolutions here inthree steps, as depicted in Fig.1 In the first step, the state is evolved through theunitary operation using the Heisenberg Hamiltonian. The evolution of the state up to t = t − results in the state, ρ ( t − ) = U t , ρ (0) U † t , , where the unitary operator is givenby U t ,t = exp − iH ( t − t ), where we have absorbed the Planck constant in the timeitself or set ¯ h = 1. The state is given by, ρ ( t − ) = e − iHt | ψ (0) (cid:105)(cid:104) ψ (0) | e iHt = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | . (18)We have discussed the computation of | ψ ( t ) (cid:105) in the last section, along with the statetransfer fidelity. In the second step, an instantaneous operation of the local QDP at the l ’th site, the state transforms from ρ ( t − ) to ˜ ρ ( t +0 ). Using the Kraus operator formalismwith the two operators P and P , the state is now given by˜ ρ ( t +0 ) = P ρ ( t − ) P † + P ρ ( t − ) P † . (19)As we can see from the above, the state here is a mixed state with two different pure-state components, suffering decoherence due to the occurrence of the QDP at m ’th site.Just after the QDP, the state can be written in terms of two pure states as,˜ ρ ( t + ) = | ˜ ψ + ( t ) (cid:105)(cid:104) ˜ ψ + ( t ) | + | ˜ ψ − ( t ) (cid:105)(cid:104) ˜ ψ − ( t ) | , (20)where, | ˜ ψ ± ( t ) (cid:105) ≡ ± σ zm | ψ ( t ) (cid:105) . In the third step, the state undergoes a unitary timeevolution from t = t +0 to time t using the Heisenberg Hamiltonian, we have˜ ρ ( t ) = U t,t ˜ ρ ( t +0 ) U † t,t . (21)In this step, the unitary Hamiltonian evolution occurs for an interval t − t . Finally, wecan rewrite the state using the two pure states | ˜ ψ ± (cid:105) , as˜ ρ ( t ) = | ˜ ψ + ( t ) (cid:105)(cid:104) ˜ ψ + ( t ) | + | ˜ ψ − ( t ) (cid:105)(cid:104) ˜ ψ − ( t ) | , (22)where, | ˜ ψ + ( t ) (cid:105) = αe − i(cid:15) t | ... (cid:105) + β (cid:88) y (cid:48) H y (cid:48) ( m, t, t ) | y (cid:48) (cid:105) , | ˜ ψ − ( t ) (cid:105) = β (cid:88) y (cid:48) K y (cid:48) ( m, t, t ) | y (cid:48) (cid:105) , (23)Here, the time-dependent wave functions[21] are given by, H y (cid:48) y ( m, t, t ) = (cid:88) y (cid:48)(cid:48) (cid:54) = m G y (cid:48)(cid:48) y ( t ) G y (cid:48) y (cid:48)(cid:48) ( t − t ) ,K y (cid:48) y ( m, t, t ) = G my ( t ) G y (cid:48) m ( t − t ) . (24) nterference of the signal of a local process with the quantum state propagation l th site aregiven by, ˜ x l ( t ) = (cid:104) | ˜ ρ l ( t ) | (cid:105) = | β | ( | H l ( m, t, t ) | + | K l ( m, t, t ) | ) , ˜ y l ( t ) = (cid:104) | ˜ ρ l ( t ) | (cid:105) = αβ ∗ e − i(cid:15) t H ∗ l ( m, t, t ) . (25)The average state transfer fidelity averaged over all pure states on the Bloch spherefor l th site is given by,˜ F l,m ( t ; t ) = 12 + 16 ( | H l ( m, t, t ) | + | K l ( m, t, t ) | ) + 13 Re ( e i(cid:15) t H l ( m, t, t ))= 12 + 16 | G l ( t ) | + 13 Re ( e i(cid:15) t ( G l ( t ) − K l ( m, t, t )) − G ∗ l ( t ) K l ( m, t, t ))+ 56 | K l ( m, t, t ) | . (26)The maximum fidelity saturates to the value 1 / t because the function G l ( t ) is basically a Bessel function which falls as 1 /t . In case of a local decoheringprocess the dynamics is not very different or in other words the QDP does not ”disturb”the system significantly. Hence, the difference in fidelity with and without QDP for l th site is given by ∆ F l,m ( t ; t ) = ˜ F l,m ( t ; t ) − F l ( t ), we have∆ F l,m ( t ; t ) = 56 | K l ( m, t, t ) | − Re (( e i(cid:15) t + G ∗ l ( t )) K l ( m, t, t )) . (27)However, the quantum state transfer process (which is a coherent or unitary process)and the effect of the QDP (which is a decohering process) can interfere with each otherand change the fidelity slightly. If the QDP is performed at a particular site at atime when the target state is at site while moving the fidelity changes slightly. FromEq. 27 maximum difference of fidelity with and without QDP can be estimated to be | K l ( m, t, t ) | at time t = l/
2, because the second term is merely a oscillating term.The Fig. 2(b) depicts the change in fidelity for one extreme case where the QDPoccurs at the first site just after the evolution starts. In that case the maximum differencein fidelity is around 0.4. However, when the QDP occurs at some other site at a latertime this difference is much less. For example in Fig. 2(c) maximum change in fidelityis around 0.15 when the QDP is occurred at the site m = 20 at the time t = 10. Ifthe QDP occurs at a site where where the basic fidelity without the QDP itself is verysmall the interference effect is negligible and the fidelity does not change significantly.In Fig.2(d) depicts the case where the QDP occurs at the first site but at much latertime t = 10 .
0, the difference is seen to be negligible. In Fig. 2(e) the dependence of thefidelity for the site l = 100 is plotted with and without the QDP, where the location m of the QDP is same, and the time of QDP is t is 50 .
0. Both the fidelities show peaksat a time slightly after t = 50 .
0, but the maximum fidelity with the QDP is slightlygreater than that without the QDP. The time at which the fidelity is maximum for isaround t = l/
2, for the QDP occurring at the location l . In Fig. 2(f) the maximum nterference of the signal of a local process with the quantum state propagation l obtained with and without QDP, with thelocation and the time of QDP being m = l and t = l/
2. The difference of fidelities issmall for sites close to the first site, from where the state transfer has been started. Thedifference of fidelities is almost constant for further sites, as seen in the figure, and thedifference is around 0 .
4. Local quantum coherent process
In this section, we will turn our attention to the case of an instantaneous local unitaryQDP intervening the unitary dynamics of the Heisenberg Hamiltonian. Similar to thelast section, the QDP operates on the m ’th spin at time t = t . But unlike the previoussection, the unitary QDP does not cause decoherence. The three steps of evolution thatare involved here, as shown in Fig.1, are all unitary evolutions, as a result the initialpure state evolves into another pure state. The intervening QDP being unitary, it can begenerated using a Hamiltonian. The instantaneous local unitary QDP intervening theunitary Hamiltonian evolution can be viewed as an evolution with a kicked HeisenbergHamiltonian. The new Hamiltonian ˜ H can be written as a sum of two terms, theHeisenberg Hamiltonian H , given in Eq. 2 that generates the background unitaryevolution, and a magnetic field term H (cid:48) = (cid:126)σ m . ˆ n with a delta-function kick, where ˆ n is the direction of the magnetic field at the m ’th site. We can consider ˆ n to be a unitvector in the x-y plane. Thus, the total Hamiltonian covering all the three steps can bewritten as, ˜ H = H + H (cid:48) δ ( t/t − . (28)The second term in the Hamiltonian represents the instantaneous QDP operating onthe given spin at t = t . The unitary evolution operator for t > t , is a product ofthree unitary operators corresponding to the three time steps, i.e., U t , = e − iHt for theevolution up to t = t − , followed by V m = e − t (cid:126)σ m . ˆ n for the instantaneous unitary QDP,and U t,t = e − i ( t − t ) H for the evolution from t = t +0 up to time t . Thus, an initial state | ψ (0) (cid:105) prepared at time t = 0 evolves to the state | ˜ ψ ( t ) (cid:105) = ˜ U t, | ψ (0) (cid:105) , the evolutionoperator is given by,˜ U t, = U t,t V m U t , . (29)Now, similar to the non-unitary QDP, we start with an initial state | ψ (0) (cid:105) = α | .. (cid:105) + β | . (cid:105) . In the first step, the state evolves unitarily upto a time t − withthe Hamiltonian H , yielding | ψ ( t − ) (cid:105) . This part is the same as discussed in the lastsection, as shown in Eq. 10. In the second step, the state is changed by an operationof V m , the m th qubit undergoes a local unitary operation. This coherent or unitaryprocess, which is a local quantum gate operation, is represented by an instantaneousunitary operation that acts on the state of the given qubit between time t = t − andtime t +0 . The operation of the unitary operator V m on the basis states of m ’th spin isgiven by , V m | (cid:105) = γ | (cid:105) + δ | (cid:105) , V m | (cid:105) = − δ ∗ | (cid:105) + γ | (cid:105) , (30) nterference of the signal of a local process with the quantum state propagation t , the components n x , n y of the unit vectorˆ n as, γ = cos t and ( n y + in x ) sin t = δ . The state | ˜ ψ ( t +0 ) (cid:105) , immediately after theoperation at t = t is given by, | ˜ ψ ( t + ) (cid:105) = V m [ αe − i(cid:15) t | F (cid:105) + β (cid:88) y G y ( t ) | y (cid:105) ]= ( αγe − i(cid:15) t − βδ ∗ G m ( t )) | F (cid:105) + αδe − i(cid:15) t | m (cid:105) + βγ (cid:88) y (cid:48) G y (cid:48) ( t ) | y (cid:48) (cid:105) + βδ (cid:88) y (cid:48) (cid:54) = m G y (cid:48) ( t ) | m, y (cid:48) (cid:105) . (31)Here, after the coherent operation, we can see that two-magnon states are also generatingfrom one magnon states, that were absent in the last section. Now, the state is amixture of zero, one and two magnon states.This is the simplest possible operationwhere three different magnon sectors are obtained. Each of these sectors will have theirown dynamics, during the further evolution using the Heisenberg Hamiltonian, in thethird step. The state at a later time t becomes, | ˜ ψ ( t ) (cid:105) = U t,t | ˜ ψ ( t + ) (cid:105) = ( αγe − i(cid:15) t − βδ ∗ G m ( t ) e − i(cid:15) ( t − t ) ) | F (cid:105) + (cid:88) y [ αδe − i(cid:15) t G ym ( t − t ) + βγX y ( m, t, t )] | y (cid:105) + βδ (cid:88) y ,y ; y 0. without taking two magnon scatteringstates into account for anisotropy constant ∆ = 1 . (cid:80) y (cid:48) (cid:54) = l | L l,y (cid:48) ,m ( m, t, t ) | as function of site index( l ) and time ( t ) for Heisenberg dynamics with closedboundary condition with local unitary operation on the tenth site at t = 5 . . F l,m =15 ( t ; t = 7 . 5) as function of site index( l ) and time ( t )for Heisenberg dynamics with closed boundary condition with local unitaryoperation on the tenth site at t = 7 . . F l,m = l ( t = l/ t = l/ 2) as function of site index l for different valuesof anisotropy constants ∆. ˜ F l,m =1 ( t ; t = 0 + ) = 12 + 16 ( | γ | − | δ | ) | G l ( t ) | + | γ | Re (( e i(cid:15) t G l ( t )) , ˜ F l,m =1 ( t ; t = 0 + ) = 12 + 16 Re (( e i(cid:15) t G l ( t )) . (38)The last terms of the above equations are merely oscillating terms. The secondequation suggests that state transfer will not take place. Setting | γ | = | δ | (Hadamardgate operation)in the first one also shows the same. nterference of the signal of a local process with the quantum state propagation (a) (b)(c) (d) Figure 4. The state transfer fidelity F l ( t ) as function of site index( l ) and time ( t ) forkicked Harper dynamics with open boundary condition with parameters (a) g = 1 . τ = 0 . 1, (b) g = 3 . τ = 0 . g = 1 . τ = 0 . 4, (d) g = 1 . τ = 0 . η = √ | ψ (0) (cid:105) = √ . | ... (cid:105) + √ . | .. (cid:105) . The speed of state transfer is if the order of J (here J is taken to be 1 / t = J l = l/ 2, which means that the evolution isinterrupted at a particular site when the target state is passing through. In context ofquantum state transfer where the target state is unknown at the receiver end one canvary the parameters γ and δ to get the maximum fidelity. The values of m , t and t isset l , l/ l/ γ = 0 and δ = 1 (bit flip or X gate)for large values of l and the difference infidelity with and without the local operation is given by,∆ F l,m = l ( t = l t = l F l,m = l ( t = l t = l − F l ( t = l (cid:88) y (cid:54) = l | L l,y ,l ( m = l, t = l , t = l | − | G lm ( t = l | − Re ( e i(cid:15) l/ G l ( t = l − . (39) nterference of the signal of a local process with the quantum state propagation 5. Local decohering process in non-integrable quantum systems In the previous sections, we have considered the effect of an intervening QDP, eithernon-unitary or unitary, on the background unitary Hamiltonian evolution, governed bythe Heisenberg Hamiltonian, that belongs to the integrable class of dynamical systems.It will be interesting to see the effect of a QDP intervention on a unitary evolution,corresponding to non-integrable Hamiltonian systems. The signal propagation froma local QDP and its detection with a background non-integrable dynamics has beeninvestigated in XY model with transverse and longitudinal magnetic fields[21].Moreover,there are sharp differences in the eigenvalue spacing distribution and the structure ofthe eigenfunctions in the two cases, that have been widely investigated.To see the effect of QDP in non integrable systems we consider a simple modelHamiltonian with a tuneable parameter, to go continuously from completely integrableto completely non-integrable regimes. We use a one-dimensional periodically-kickedHarper model, a simple model of fermions hopping on a chain with an inhomogeneoussite potential, appearing as a kick at regular intervals. The Hamiltonian is given by H ( t ) = N (cid:88) j =1 [ˆ c † j ˆ c j +1 + H.C. ] + g cos( 2 πjηN )ˆ c † j ˆ c j ∞ (cid:88) n = −∞ δ ( t/τ − n )] , (40)where c † j is a creation operator at site j , g is a potential strength parameter, η isa parameter to make the potential either commensurate or incommensurate with thelattice, and τ is the kicking interval. The first term represents the kinetic energy of thefermion or hopping term, and the second term is the kicked potential energy operator.The Hilbert space for a site is two-dimensional, either occupied site or unoccupied site,thus it can be mapped to the spin language. The first term will turnout to be XYterm, shown in In the Eq.2, and the second term becomes a transverse field that isinhomogeneous. The fermion occupation can be mapped to the down spin occupationin the spin states. Unlike the Heisenberg Hamiltonian that incorporates an interactionof down spins on neighbouring sites, there are no many-body interaction effects here.The kicked Harper model has been investigated for the entanglement distribution anddynamics[25]. The effect of the magnetic field or the potential is through a train ofkicking pulses with an interval τ . For τ → τ the dynamics is completely chaotic (see Figure 4in the work of Lakshminarayan and Subrahmanyam[25] for further details)[26].Similar to the previous sections, we will consider an initial state | ψ (cid:105) = α | .. (cid:105) + β | .. (cid:105) , a linear combination of zero-particle state and one-particle state localised atfirst site. Through the time evolution, the particle can hop around to other sites. Wewill consider evolution at discrete times, viz. t = τ + , τ + etc, that is at instant justafter a kick. The evolution operator U n at a time just after n kicks (corresponding to t = nτ ), is given by U ( n ) = (cid:18) e − iτ (cid:80) j c † j c j +1 + H.C. e − iτg (cid:80) j cos πjηN c † j c j (cid:19) n , (41) nterference of the signal of a local process with the quantum state propagation (a) (b)(c) (d)(e) (f) Figure 5. The detector function f l ( t ) as function of site index( l ) and time ( t − t ) forkicked Harper dynamics with open boundary condition with local unitary operationon the first site after 5 kicks ( t = 5 τ ) with parameters (a) g = 1 . τ = 0 . g = 3 . τ = 0 . g = 10 . τ = 0 . 1, (d) g = 1 . τ = 0 . g = 2 . τ = 0 . 4, (f) g = 1 . τ = 0 . η = √ | ψ (0) (cid:105) = √ . | ... (cid:105) + √ . | .. (cid:105) . where the two operator factors appearing above do not commute. This evolutionoperator evolves the initial state to give | ψ n (cid:105) = U ( n ) | ψ (cid:105) . nterference of the signal of a local process with the quantum state propagation t = n τ + . Thus, the evolution proceeds inthree steps as shown in Fig. 1. In the first step, the initial state will become the state | ψ ( n ) (cid:105) at time t = n τ . In the second step, the state ρ ( n ) , in analogy with Eq. 19,will result in the state ˜ ρ ( n ) = P ρ ( n ) P † + P ρ ( n ) P † . The third step, the state evolvesunitarily to become,˜ ρ ( n ) = U ( n − n )( P ρ ( n ) P † + U ( n − n ) P ρ ( n ) P † ) U † ( n − n ) . (42)The quantum state transfer fidelity can also be calculated numerically, by calculatingthe matrix elements of the reduced density matrix ρ l ( n ) for the l ’th site, noting that ˜ x l is just the expectation value of the fermion number operator. We have, analogous toEq. 25, ˜ x l ( n ) = T r c † l c l ˜ ρ ( n ) , ˜ y l ( n ) = T r c † l ˜ ρ ( n ) , (43)where we have written the matrix elements as expectation value of operators. The statetransfer fidelity can be calculated numerically using the above. The average fidelityfor an initial state | ψ (cid:105) = √ . | (cid:105) + √ . | (cid:105) has been shown in Fig. 4(a)-4(d) for adifferent values of τ , g as a density plot, for various target sites and times. For a smallervalues of τ , the dynamics is still closer to an integrable one which can be seen in Fig.4(a). For small times, we can see larger fidelity only for small l . As the time increases,for larger values of l get larger fidelity. This is as expected, the state is spreading at agiven speed. For larger value of potential strength parameter g the spreading of the statedoes not take place. This can be seen in Fig. 4(b) where the density plot for fidelity hasbeen plotted for τ = 0 . g = 3 . 0. However, for larger value of τ as shown in 4(d)even for small times too there are larger fidelities for far away sites. This implies, wecannot define a speed of propagation for the non-integrable case. This transition occursat some intermediate value near τ = 0 . f l ( t = nτ ) = ˜ x l ( t ) − x l ( t ) . (44)Here, the two quantities are the diagonal matrix elements of the RDM from the twostates ˜ ρ ( n ) and ρ ( n ) evolved with and without the QDP occurrence. This is a usefuldetector function to see the effect of the QDP, similar to the Loschmidt echo, but a lotsimpler to calculate. This function has been studied in detail for various Hamiltonians inconjunction with a non-unitary QDP[21]. One can also study similar detector functionsconstructed from the off-diagonal matrix elements, and the von-Neumann entropy ofthe RDM. It has been see that they all show similar structure. The results for the abovedetector function are shown in Fig. 5(a) to Fig. 5(f) for various combination of valuesfor g and τ .For smaller values of kicking interval τ and potential strength g the detector function f l ( t ) shows behaviour similar to that of integrable models like Heisenberg model, XYmodel with transverse magnetic field [21]. The signal of the local QDP propagates nterference of the signal of a local process with the quantum state propagation τ and g . If the valueof g is increased the signal propagation does not take place or in other words f l ( t ) iszero for farther sites. This is depicted in Fig. 5(b), Fig. 5(c) and Fig. 5(e). For largervalues of τ the speed of propagation abruptly becomes very high, which can taken as asignature of transition from integrability to non integrability. Fig. 5(d) shows this for τ = 0 . g = 1. For τ = 0 . g > τ , the signal does not propagate to larger distances,and we cannot define a speed, as shown in Fig.5(b), (c) and (e). On the other hand,for g = 1, and larger values of τ , as seen in Fig. 5(d) and 5(f), the signal spreads outalmost instantaneously to larger distances, and we cannot get a finite speed. 6. Conclusions In this paper, we have investigated the effect of a local quantum dynamical process(QDP), both unitary and non-unitary processes, intervening the process of quantumstate transfer through a unitary evolution for both integrating and non integratingmodels. For Heisenberg model, that represents an integrable dynamics, we haveanalytically calculated the state transfer fidelity in terms of time-dependent Greenfunctions. The information coded in the state of the first spin, spreads out to otherlocation through the unitary dynamics, We have seen that the signal from the localQDP can interfere with the dynamics of the state transfer, and it changes the fidelitydepending on the time and the location of the QDP. In the case of a non-unitary QDP,there is a small change in the state transfer fidelity, even if the QDP occurs at thelocation of the particular qubit where the coded state arrives at the time of QDP.However, for a coherent or unitary QDP, the fidelity can increase or decrease dependingon the constructive or destructive interference of the propagation of the coded state,and the signal from the QDP. For appropriate location and the time, the state transferfidelity can increase substantially. In the case of unitary QDP, an initial state with acombination of zero and one-magnon states, becomes a superposition of zero, one andtwo magnon eigenfunctions. The state transfer has contributions from both two-magnonbound states and scattering states, which are quantitatively calculated separately usingthe two-magnon Bethe ansatz wave functions.Finally, we have investigated the dynamics of Kicked Harper model in the contextof quantum state transfer. The dynamics of this model changes from integrable tonon integrable by increasing the kicking interval time. We show that the spreading ofthe coded state is dependent on the kicking interval time. For smaller values of τ thedynamics is similar to the integrable model and larger values of τ the spreading takesplace quickly which is a possible signature of non integrability.The signal propagationalso depends on the potential strength parameter g and τ . The signal propagates with nterference of the signal of a local process with the quantum state propagation 20a finite speed below a certain value of τ and above that value the propagation takesplace too quickly to define a speed. Whereas, for larger value of g the signal does notreach the farther sites. So, the signal of the QDP gets localised. Acknowledgement: SS acknowledges the financial support from CSIR, India. Wethank Professor A. Lakshminarayan for useful discussion of non-integrable systems. Appendix A The time-dependent two-particle Green’s function G x (cid:48) ,x (cid:48) x ,x ( t ) has contributions fromboth scattering and bound states. We can write the function as two parts, as G x (cid:48) ,x (cid:48) x ,x ( t ) = G x (cid:48) ,x (cid:48) ( B ); x ,x ( t ) + G x (cid:48) ,x (cid:48) ( S ); x ,x ( t ) . (45)In Bethe Ansatz solution for the two-magnon eigenfunctions for a chain with periodicboundary conditions, the momenta can be parametrised by the relation, p i = 2 cot − (2 λ i ) . (46)The S-matrix in the Bethe Ansatz wave function is given in the form, S ( p , p ) = e iθ ( p ,p ) , (47)Where, the phase factor θ is given by, θ ( p , p ) = 2 tan − (cid:104) ∆ sin[( p − p ) / p + p ) / − ∆ cos[( p − p ) / (cid:105) . (48)Bound state solutions are complex and are given by the form, λ = q + i/ λ = q − i/ . (49)Here, we consider the case of infinitely long isotropic ferromagnetic chain where∆ = 1 to calculate the bound state contribution to state transfer fidelity. In this casethe the number q varies continuously within the limits −∞ < q < + ∞ . The S-matrixfor bound state wave function can be calculated using the solutions given above usingEq. 49, e iθ ( λ ,λ ) = e i tan − ( λ − λ ) = e log i − λ λ i + λ − λ = 0 . (50)Two magnon bound state wave function apart from a normalisation factor is given by, ψ x ,x p ,p = e i ( p x + p x ) (51)Since the momenta p and p are related as shown in Eq. 49, the wave function for thebound state can be written in terms of the quantity q as, ψ x ,x q = A ( q ) e i [ x cot − (2 q + i )+ x cot − (2 q − i )] = A ( q ) e x log( iq − iq )+ x log( iqiq +1 ) = A ( q )( q q ) ( x − x ) / e i ( x + x ) tan − ( q ) . (52) nterference of the signal of a local process with the quantum state propagation A ( q ) is the normalisation factor.The normalisation factor for the wave functioncan be calculated as, | A ( q ) | − = N − (cid:88) r =1 r ( q q ) N − r (53)The function G x (cid:48) ,x (cid:48) ( B ); x ,x ( t ) for an infinitely long ferromagnetic isotropic chain thenbecomes (using x = x + x (cid:48) − x − x (cid:48) and ˜ x = x − x (cid:48) + x − x (cid:48) ), G x (cid:48) ,x (cid:48) ( B ); x ,x ( t ) = lim N →∞ N π (cid:90) + ∞−∞ ψ x ,x q ψ ∗ x (cid:48) ,x (cid:48) q e − it(cid:15) ( q ) ∂p∂q dq = 12 π (cid:90) + ∞−∞ dq ∂p∂q q (cid:16) q q (cid:17) x / e i ˜ x tan − (1 /q ) − it(cid:15) ( q ) , (54)Where, (cid:15) ( q ) is the energy associated with the state and given in the form, (cid:15) ( q ) = (cid:15) + 21 + q (55)Two magnon scattering state wave function is given by, ψ x ,x p ,p = A ( p , p ) e iθ ( p ,p ) / ( e i [ p x + p x − θ ( p ,p ) / − e i [ p x + p x + θ ( p ,p ) / ) . (56)Where, the normalisation factor A ( p , p ) for the wave function is given by, | A ( p , p ) | − = 4 N − (cid:88) r =1 r cos [ ( N − r )( p − p ) − θ ( p , p )2 ] (57)The function G x (cid:48) ,x (cid:48) ( S ); x ,x ( t ) becomes, G x (cid:48) ,x (cid:48) ( S ); x ,x ( t ) = (cid:88) p ,p | A ( p , p ) | ψ x ,x p ,p ψ ∗ x (cid:48) ,x (cid:48) p ,p e − i(cid:15) ( p ,p ) t . (58)Where, the energy eigenvalues for Heisenberg chain with anisotropy constant ∆ aregiven by, (cid:15) ( p , p ) = (cid:15) + 2(∆ − cos p ) + 2(∆ − cos p ) . (59)For an infinitely long chain it can be shown that the magnon momenta p and p become independent and takes continuous values in an interval depending on the valueof anisotropy constant ∆. For an infinitely long chain with anisotropy constant ∆ = 1and ∆ ( − < ∆ < 1) Eq. 58 becomes, G x (cid:48) ,x (cid:48) ( S ); x ,x ( t ) = 14 π (cid:90) π (cid:90) π dp dp ψ x ,x p ,p ψ ∗ x (cid:48) ,x (cid:48) p ,p e − i(cid:15) ( p ,p ) t . 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