Spatial entanglement using a quantum walk on a many-body system
aa r X i v : . [ qu a n t - ph ] M a y Spatial entanglement using a quantum walk on a many-body system
Sandeep K. Goyal ∗ The Institute of Mathematical Sciences, CIT campus, Chennai 600 113, India
C. M. Chandrashekar † Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada andPerimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
The evolution of a many-particle system on a one-dimensional lattice, subjected to a quantumwalk can cause spatial entanglement in the lattice position, which can be exploited for quantuminformation/communication purposes. We demonstrate the evolution of spatial entanglement andits dependence on the quantum coin operation parameters, the number of particles present in thelattice and the number of steps of the quantum walk on the system. Thus, spatial entanglementcan be controlled and optimized using a many-particle discrete-time quantum walk.
I. INTRODUCTION
Entanglement in a quantum state has been the fundamental resource in many quantum information and com-putation protocols, such as cryptography, communication, teleportation and algorithms [1, 2]. To implement theseprotocols, generating an entangled state is very important. Similarly, studies on the interface between condensedmatter systems and quantum information have shown entanglement as a signature of quantum phase transition [3–5].To understand the phases and dynamics in many-body systems an analysis of entanglement in many-body systems isvery important. Hence, various schemes have been proposed for entanglement generation in quantum systems [6–9]and for understanding entanglement in many-body systems [10]. Quantum walk (QW) is one such process in whichan uncorrelated state can evolve to an entangled state and be used to analyze the evolution of entanglement [11, 12].The QW, which was developed as a quantum analog of the classical random walk (CRW), evolves a particle into anentanglement between its internal and position degrees of freedom. It has played a significant role in the development ofquantum algorithms [13]. Furthermore, the QW has been used to demonstrate coherent quantum control over atoms,quantum phase transition [14], to explain the phenomena such as breakdown of an electric field-driven system [15]and direct experimental evidence for wavelike energy transfer within photosynthetic systems [16, 17]. Experimentalimplementation of the QW has also been reported [18–21], and various other schemes have been proposed for itsphysical realization [22–26]. Therefore, studying entanglement during the QW process will be useful from a quantuminformation theory perspective and also contribute to further investigation of the practical applications of the QW.In this direction, evolution of entanglement between single particle and position with time (number of steps of thediscrete-time QW) has been reported [12].In this paper, we consider a multipartite quantum walk on a one-dimensional lattice and study the evolution of spatial entanglement , entanglement between different lattice points. All the particles considered in the system areidentical and indistinguishable with two internal states (sides of the quantum coin). Spatial entanglement generatedusing a QW can be controlled by tuning different parameters, such as parameters in the quantum coin operation,number of particles in the system and evolution time (number of steps). To quantify entanglement in the system weare using Meyer-Wallach multipartite entanglement measure.In Sec. II, we describe single-particle and many-particle discrete-time QWs. In Sec. III, entanglement between aparticle and position space and spatial entanglement using single- and many-particle QWs are discussed. In Sec. IV,we present the measure for spatial entanglement of the system using the Meyer-Wallach global entanglement measurescheme for particles in a one-dimensional lattice and in a closed chain ( n − cycle). We also demonstrate control overspatial entanglement by exploiting the dynamical properties of the QW. We conclude with the summary in Sec. V. ∗ Electronic address: [email protected] † Electronic address: [email protected]
II. QUANTUM WALK
Classical random walk (CRW) describes the dynamics of a particle in position space with a certain probability. TheQW is the quantum analog of CRW-developed exploiting features of quantum mechanics such as superposition andinterference of quantum amplitudes [27–29]. The QW, which involves superposition of states, moves simultaneouslyexploring multiple possible paths with the amplitudes corresponding to the different paths interfering. This makesthe variance of the QW on a line to grow quadratically with the number of steps which is in sharp contrast to thelinear growth for the CRW.The study of QWs has been largely divided into two standard variants: discrete-time QW (DTQW) [29–31] anda continuous-time QW (CTQW) [32]. In the CTQW, the walk is defined directly on the position
Hilbert space H p ,whereas for the DTQW it is necessary to introduce an additional coin Hilbert space H c , a quantum coin operationto define the direction in which the particle amplitude has to evolve. The connection between these two variants andthe generic version of the QW has been studied [33, 34]. However, the coin degree of freedom in the DTQW is anadvantage over the CTQW as it allows control of dynamics of the QW [35, 36]. Therefore, we take full advantage ofthe coin degree of freedom in this work and study the DTQW on a many-particle system. A. Single-particle quantum walk
The DTQW is defined on the Hilbert space H = H c ⊗ H p . In one dimension, the coin Hilbert space H c , spannedby the basis state | i and | i , represents two sides of the quantum coin, and the position Hilbert space H p , spannedby the basis states | ψ j i , j ∈ Z , represent the positions in the lattice. To implement the DTQW, we will consider athree-parameter U(2) operator C ξ,θ,ζ of the form C ξ,θ,ζ ≡ (cid:18) e iξ cos( θ ) e iζ sin( θ ) e − iζ sin( θ ) − e − iξ cos( θ ) (cid:19) (1)as the quantum coin operation [36]. The quantum coin operation is applied on the particle state ( C ξ,θ,ζ ⊗ ) whenthe initial state of the complete system is | Ψ in i = (cid:2) cos( δ ) | i + e iη sin( δ ) | i (cid:3) ⊗ | ψ i . (2)The state cos( δ ) | i + e iη sin( δ ) | i is the state of the particle and | ψ i is the state of the position at the lattice position j = 0.The quantum coin operation on the particle is followed by the conditional unitary shift operation S which acts onthe complete Hilbert space of the system: S = exp( − iσ z ⊗ P l ) , (3)where P is the momentum operator, σ z is the Pauli spin operator in the z direction and l is the length of each step.The eigenstates of σ z are denoted by | i and | i . Therefore, S , which delocalizes the wave packet over the positions( j −
1) and ( j + 1), can also be written as S = | ih | ⊗ X j ∈ Z | ψ j − ih ψ j | + | ih | ⊗ X j ∈ Z | ψ j +1 ih ψ j | . (4)The process of W ξ,θ,ζ = S ( C ξ,θ,ζ ⊗ ) (5)is iterated without resorting to intermediate measurement to help realize a large number of steps of the QW. Theparameters δ and η in Eq. (2) can be varied to obtain different initial states of the particle. The three parameters ξ , θ and ζ of C ξ,θ,ζ can be varied to choose the quantum coin operation. By varying parameter θ the variance canbe increased or decreased according to the functional form, σ ≈ (1 − sin( θ )) t , where t is the number of steps of theQW, as shown in Fig. 1. Biased coin operation and biased QW:
The most widely studied form of the DTQW is the walk using the Hadamardoperation H = 1 √ (cid:18) − (cid:19) , (6) −100 −80 −60 −40 −20 0 20 40 60 80 10000.020.040.060.080.10.120.140.16 P r obab ili t y Particle position(a) = (0 ° , 15 ° , 0 ° )(b) = (0 ° , 45 ° , 0 ° )(c) = (0 ° , 75 ° , 0 ° ) FIG. 1: (color online) Spread of the probability distribution for different values of θ using the quantum coin operator C ,θ, .The distribution is wider for (a) (0 , θ,
0) = (0 , π ,
0) than for (b) (0 , θ,
0) = (0 , π ,
0) and (c) (0 , θ,
0) = (0 , π , θ . The initial state of the particle is | Ψ ins i = √ ( | i + i | i ) ⊗ | ψ i and the distribution isfor 100 steps. corresponding to the quantum coin operation with ξ = ζ = 0 and θ = π/ | Ψ ins i = 1 √ | i + i | i ] ⊗ | ψ i , (7)obtained by choosing δ = π/ η = π/ W ξ,θ,ζ | Ψ ins i = 1 √ (cid:2)(cid:0) e iξ cos( θ ) + ie iζ sin( θ ) (cid:1) | i| ψ − i + (cid:0) e − iζ sin( θ ) − ie − iξ cos( θ ) (cid:1) | i| ψ +1 i (cid:3) . (8)If ξ = ζ , Eq. (8) has left-right symmetry in the position probability distribution, but not otherwise. That is, theparameters ξ and ζ introduce asymmetry in the position space probability distribution. Therefore, a coin operationwith ξ = ζ in Eq. (1) can be called as a biased quantum coin operation which will bias the QW probability distributionof the particle initially in a symmetric superposition state (Fig. 2) [36]. However, we should note that irrespective ofthe quantum coin operation used, QW can also be biased by choosing an asymmetric initial state of the particle (forexample, the Hadamard walk of a particle initially in the state | i or the state | i ). B. Many-particle quantum walk
To define a many-particle QW in one dimension, we will consider an M -particle system with one non-interactingparticle at each position (Fig. 3). The M identical particles in M lattice points with each particle having its owncoin and position Hilbert space will have a total Hilbert space H = ( H c ⊗ H p ) M . We assume the particles to bedistinguishable. −100 −80 −60 −40 −20 0 20 40 60 80 10000.020.040.060.080.10.120.14 P r obab ili t y Particle position(a) = (30 ° , 30 ° , 0 ° )(b) = (0 ° , 30 ° , 30 ° )(c) = (75 ° , 60 ° , 0 ° )(d) = (0 ° , 60 ° , 75 ° ) FIG. 2: Spread of probability distribution for different values of ξ , θ , ζ using the quantum coin operator C ξ,θ,ζ . The parameter ξ shifts the distribution to the left: (a)( ξ, θ, ζ ) = ( π , π ,
0) and (c) ( ξ, θ, ζ ) = ( π , π , ζ shifts it to the right:(b) ( ξ, θ, ζ ) = (0 , π , π ) and (d) ( ξ, θ, ζ ) = (0 , π , π ). The initial state of the particle | Ψ ins i = √ ( | i + i | i ) ⊗ | ψ i and thedistribution is for 100 steps. The evolution of each step of the QW on the M -particle system is given by the application of the operator W ⊗ M ,θ, .The initial state that we will consider for the many-particle system in one dimension will be | Ψ Mins i = j = M − O j = − M − (cid:18) | i + i | i√ (cid:19) ⊗ | ψ j i . (9) M particles and 2t+M+1 lattice positions|0> shifts to left|1> shifts to right
FIG. 3: Many-particle state with one non-interacting particle at each position space.
For an M -particle system after t steps of the QW, the Hilbert space consists of the tensor product of single latticeposition Hilbert space which is (2 t + M + 1) in number. That is, after t steps of the QW, the M particles are spreadbetween ( j − t ) to ( j + t ). In principle, each lattice point is associated with a Hilbert space spanned by two subspaces,a zero-particle subspace and one-particle subspace spanned by two possible states of the coin, | i and | i . Therefore,the dimension of each lattice point will be 3 M and the dimension of total Hilbert space is (3 M ) ⊗ M . Fig. (4) showsthe probability distribution of the many-particle system with an increase in number of steps of the QW. III. ENTANGLEMENT
To efficiently make use of entanglement as a physical resource, the amount of entanglement in a given system hasto be quantified. Therefore, entanglement in a pure bipartite system or a system with two Hilbert spaces is quantifiedusing standard measures known as entropy of entanglement or Schmidt number [1]. The entropy of entanglement −80 −60 −40 −20 0 20 40 60 8000.20.40.60.81 Lattice position P r obab ili t y initial dist.10 steps25 steps40 steps60 steps FIG. 4: (color online) Probability distribution of 40 particles initially with one particle in each position space when subjectedto the QW of different number of steps. The initial state of all the particles is √ ( | i + i | i ) and is evolved in position spaceusing the Hadamard operator, C ,π/ , as the quantum coin. The distribution spreads in the position space with an increasein number of steps. corresponds to the von Neumann entropy, a functional of the eigenvalues of the reduced density matrix, and a Schmidtnumber is the number of non-zero Schmidt coefficients in its Schmidt decomposition. For a multipartite state, there arequite a few good entanglement measures that have been proposed [37–43]. However, as the number of particles in thesystem increases, the complexity of finding an appropriate entanglement measure also increases, making scalabilityimpractical. Among the proposed measures, to address this scalability problem, Mayer and Wallach proposed a scalable global entanglement measure (polynomial measure) to quantify entanglement in many-particle systems [40].In this section, we will first discuss the entanglement of a particle with position space quantified using entropy ofentanglement. Later we will discuss spatial entanglement quantified using the Mayer-Wallach (M-W) measure. Spatialentanglement has been explored earlier using different methods. For example, in an ideal bosonic gas it has beenstudied using off-diagonal long-range order [44]. For our investigations, we consider a distinguishable many-particlesystem, implement QW and use the M-W measure to quantify spatial entanglement. In this system the dynamics ofparticles can be controlled by varying the quantum coin parameters, the initial state of the particles, the number ofparticles in the system and the number of steps of the QW. In particular, we choose the particles in one-dimensionalopen and closed chains. The spatial entanglement thus created can be used for example to create entanglementbetween distant atoms in an optical lattice [45] or as a channel for state transfer in spin chain systems [46–48]. A. Single-particle - position entanglement
QW entangles the particle (coin) and the position degrees of freedom. To quantify it, let us consider a DTQW ona particle initially in a state given by Eq. (7) with a simple form of a coin operation C ,θ, ≡ (cid:18) cos( θ ) sin( θ )sin( θ ) − cos( θ ) (cid:19) . (10)After the first step, W ,θ, = S ( C ,θ, ⊗ ), the state takes the form | Ψ i = W ,θ, | Ψ ins i = γ ( | i ⊗ | ψ j − i ) + δ ( | i ⊗ | ψ j +1 i ) (11)where γ = (cid:16) cos( θ )+ i sin( θ ) √ (cid:17) and δ = (cid:16) sin( θ ) − i cos( θ ) √ (cid:17) . The Schmidt rank of | Ψ i is 2 which implies entanglement in thesystem. The value of entanglement with an increase in the number of steps can be further quantified by computingthe von Neumann entropy of the reduced density matrix of the position subspace. Number of steps E n t ang l e m en t θ = 15 ° θ = 30 ° θ = 45 ° θ = 60 ° θ = 75 ° FIG. 5: (color online) Entanglement of a single particle with position space when subjected to the QW. The initial state ofa particle is √ ( | i + i | i ) and is evolved in position space using different values for θ in the quantum coin operation C ,θ, .The entanglement initially oscillates and approaches an asymptotic value with an increase in the number of steps. For smallervalues of θ the entanglement is higher and decreases with an increase in θ . Initial oscillation is also larger for higher θ . E n t ang l e m en t θ = 15 ° θ = 30 ° θ = 45 ° θ = 60 ° θ = 75 ° FIG. 6: (color online) Entanglement of single particle with position space when subjected to the QW. The initial state of theparticle is given by Eq. (2) with δ = π and η = π and is evolved in position space using different values for θ in the quantumcoin operation C ,θ, . The entanglement initially oscillates and approaches an asymptotic value with an increase in the numberof steps. For smaller values of θ the entanglement is higher and decreases with an increase in θ . Initial oscillation is also largerfor higher θ . Fig. 5 shows a plot of the entanglement against the number of steps of the QW on a particle initially in a symmetricsuperposition state using different values for θ in the operation W θ . The von Neumann entropy of the reduced densitymatrix of the coin is used to quantify entanglement between the coin and the position in Fig. 5. That is, E c ( t ) = − X j λ j log ( λ j ) (12)where λ j are eigenvalues of the reduced density matrix of the coin after t steps (time). The entanglement initiallyoscillates and reaches an asymptotic value with increasing number of steps. In the asymptotic limit, the entanglementvalue decreases with an increase in θ and this dependence can be attributed to the spread of the amplitude distributionin position space. That is, with an increase in θ , constructive interference of quantum amplitudes toward the originbecomes prominent narrowing the distribution in the position space. In Fig. 6, the process is repeated for a particleinitially in an asymmetric superposition state | Ψ in i = (cid:2) cos( π ) | i + e i π sin( π ) | i (cid:3) ⊗ | ψ i . Comparing Fig. 6 withFig. 5, we can note the increase in entanglement and decrease in the oscillation. This observation can be explained bygoing back to our earlier note on biased QW in Sec. II A. In Fig. 2 we note that biasing of the coin operation leadsto an asymmetry in the probability distribution, with an increase in peak height on one side and a decrease on theother side (increase and decrease are in reference to the symmetric distribution). A similar biasing effect can also bereproduced by choosing an asymmetric initial state of the particle. The biased distribution with an increased value ofprobability at one side in the distribution contributes to a reduced oscillation in the distribution. This in turn resultsin the increase of the von Neumann entropy: entanglement. B. Spatial entanglement
Spatial entanglement is the entanglement between the lattice points. This entanglement takes the form of non-localparticle number correlations between spatial modes. To observe spatial entanglement we first need to associate thelattice with the state of a particle. Then we need to consider the evolution of a single-particle QW followed by theevolution of a many-particle QW, in order to understand spatial entanglement.
1. Using a single-particle quantum walk
In a single-particle QW, each lattice point is associated with a Hilbert space spanned by two subspaces. The firstis the zero-particle subspace which does not involve any coin (particle) states. The other is the one-particle subspacespanned by the two possible states of the coin, | i and | i . To obtain the spatial entanglement we will write the stateof the particle in the form of the state of a lattice. Following from Eq. (11), the state of the particles after first twosteps of QW takes the form | Ψ i = W ,θ, | Ψ i = γ [cos( θ ) | i| ψ j − i + sin( θ ) | i| ψ j i ] + δ [sin( θ ) | i| ψ j i − cos( θ ) | i| ψ j +2 i ] . (13)In order to obtain the state of the lattice we can redefine the position state in the following way: the occupied positionstate | ψ j i as | j i , which means that the j th position is occupied and the rest of the lattice is empty. Therefore, wecan rewrite Eq. (13) as | Ψ i = γ [cos( θ ) | i| j − i + sin( θ ) | i| j i ] + δ [sin( θ ) | i| j i − cos( θ ) | i| j +2 i ] . (14)Since we are interested in the spatial entanglement, we project this state into one of the coin state so that we canignore the entanglement between the coin and the position state and consider only the lattice states. Here we willchoose the coin state to be | i and take projection to obtain the state of the lattice in the form | Ψ lat i = | i ( γ cos( θ ) | j − i + δ sin( θ ) | j i ) . (15)Each lattice site j can be considered as a Hilbert space with basis states | j i (occupied state) and | j i (unoccupiedstate). Then, the above Eq. (15) in the extended Hilbert space of each lattice can be rewritten in terms of occupiedand unoccupied lattice states as | Ψ ′ lat i = γ cos( θ ) | j − j i + δ sin( θ ) | j − j i . (16)We can see that after first two steps of the QW the lattice points j and ( j −
2) are entangled. One can check that thelattice points j and ( j + 2) are entangled if we choose the coin state to be | i . With an increase in the number of steps,the state of the particle spreads in position space and the projection over one of the coin state reduces that state toa pure state, for which one may compute spatial entanglement, according to the above prescription. Therefore, withan increase in the number of steps, the spatial entanglement from a single-particle QW decreases.
2. Using many-particle quantum walk
We will extend the study of evolution of spatial entanglement as the QW progresses on a many-particle system.Let us first consider the analysis of first two steps of the Hadamard walk ( θ = π/ | Ψ pins i = +1 O j = − (cid:18) | i + i | i√ (cid:19) ⊗ | ψ j i . (17)We will label the three particles at positions −
1, 0 and 1 as A, B and C, respectively. Since evolution of these particlesis independent, we write down the state after the first step as a tensor product of each of the three particles: | Ψ p i = W ⊗ ,θ, | Ψ pins i = [ γ | i| − i + δ | i| i ] A ⊗ [ γ | i| − i + δ | i| + 1 i ] B ⊗ [ γ | i| i + δ | i| + 2 i ] C , (18)where γ = (1 + i ) / δ = (1 − i ) /
2. After two steps the tensor product of each of the three particles is given by | Ψ p i = (cid:20) γ (cid:18) | i| − i + | i| − i√ (cid:19) + δ (cid:18) | i| − i − | i| + 1 i√ (cid:19)(cid:21) A ⊗ (cid:20) γ (cid:18) | i| − i + | i| i√ (cid:19) + δ (cid:18) | i| i − | i| + 2 i√ (cid:19)(cid:21) B ⊗ (cid:20) γ (cid:18) | i| − i + | i| + 1 i√ (cid:19) + δ (cid:18) | i| + 1 i − | i| + 3 i√ (cid:19)(cid:21) C . (19)By projecting this state into one of the coin states (we choose state | i ⊗ | i ⊗ | i ) we can obtain a state of the latticefor which spatial entanglement may be computed. Then the state of the lattice after projection and normalization is | Ψ lat i = γ | A i − | B i − | C i − + γ δ ( | A i − | B i − | C i + | A i − | B i | C i − + | AC i − | B i − )+ γδ ( | A i − | B i | C i + | AC i − | B i + | A i − | B i − | C i )+ δ | A i − | B i | C i , (20)where A, B and C represent the particle labels and the subscripts represent the position labels. In a similar mannerwe can obtain | Ψ lat i for a system with a large number of particles. Then the next task is to calculate the spatialentanglement. IV. CALCULATING SPATIAL ENTANGLEMENT IN A MULTIPARTITE SYSTEM
In a system with two particles, the state is separable if we can write it as a tensor product of individual particlestates, and entangled if not. For a system with
M > | ψ i = | φ i ⊗ | φ i ⊗ · · · | φ k i , (21)when k = M . | φ i i will then denote the state of the i th particle. When k < M a state is said to be partially entangledand when k = 1 the state will be fully entangled.Rather than using the von Neumann entropy to quantify multipartite entanglement of a given state ρ , one sometimesoften prefers to consider purity, which corresponds (up to a constant) to linear entropy, that is the first-order term inthe expansion of the von Neumann entropy around its maxima, given by E = dd − (cid:2) − Tr ρ (cid:3) (22)for a d -dimensional particle Hilbert space [49]. To quantify the entanglement of multipartite pure states, one measurecommonly, used is the Meyer- Wallach (M-W) measure [40]. It is the entanglement measure of a single particle to therest of the system, averaged over the whole of the system and is given by E MW = dd − " − L L X i =1 Tr ρ i (23)where L is the system size and ρ i is the reduced density matrix of the i th subsystem. The M-W measure does notdiverge with increasing system size and is relatively easy to calculate.In a multipartite QW the dimension at each lattice point, after projection over one particular state of coin, is 2 M where M is the number of particles. Hence, the expression for entanglement will be E MW ( | ψ lat i ) = 2 M M − − t + M + 1 t + M X j = − ( t + M ) tr ρ j (24)where t is the number of steps and ρ j is the reduced density matrix of j th lattice point. The reduced density matrix ρ j can be written as ρ j = X k p jk | k ih k | (25)where | k i is one of the 2 M possible states available for a lattice point and p jk can be calculated once we have theprobability distribution of an individual particle on the lattice.Since we have M distinguishable particles, we have 2 M configurations depending upon whether a given particle ispresent in the lattice point or not after freezing the state of the particle. This set of configurations forms the basis fora single-lattice point Hilbert space. Now we can calculate p jk , the probability of k th configuration of a particle in the j th lattice point as follows. Let us say a ( l i ) j is the probability of the i th particle to be or not to be in the j th latticepoint depending on l i . If l i is 1, then it gives us the probability of the particle to be in the lattice point. If l i is 0,then a ( l i ) j is the probability of a particle not to be in the lattice point, that is, a (0) j = 1 − a (1) j . Hence, we can write p jk = Y i a l i j . (26)Once we have the probability of each particle at a given lattice position, the spatial entanglement can be convenientlycalculated. Since the QW is a controlled evolution, one can obtain a probability distribution of each particle overall lattice positions. In fact, one can easily control the probability distribution by varying quantum coin parametersduring the QW process and hence the entanglement. Number of particles N u m be r o f s t ep s FIG. 7: (color online) Evolution of spatial entanglement with an increase in the number of steps of the QW for different numberof particles in an open one-dimensional lattice chain. The entanglement first increases and with further increase in the numberof steps, the number of lattice positions exceeds the number of particles in the system resulting in the decrease of the spatialentanglement. The distribution is obtained by implementing the QW on particles in the initial state √ ( | i + i | i ) and theHadamard operation C ,π/ , as quantum coin operation. Fig. 7 shows the phase diagram of the spatial entanglement using a many-particle QW. Data for the phase diagramwere obtained numerically by subjecting the many-particle system with different number of particles to the QW withincreasing number of steps. The quantity of spatial entanglement was computed using Eq. (24).0
0 30 60 90 120 150 18000.050.10.150.20.25
Quantum coin parameter θ E n t ang l e m en t (a) 10 particles 10 steps (sym.)(b) 20 particles 20 steps (sym.) (c) 20 particles 20 steps (|0>) FIG. 8: (color online) Quantity of spatial entanglement for 10 particles after 10 steps and 20 particles after 20 steps of the QWon a one-dimensional lattice using different values of θ in the quantum coin operation C ,θ, . For (a) and (b), the distributionis for particles initially in the symmetric superposition state, √ ( | i + i | i ), and for (c) the particle’s initial state is | i (will bethe same for state | i ). Quantity of entanglement is higher for θ closer to 0 and π/ θ = π/
2, for every even number of steps of the QW, the system returns to theinitial state where entanglement is 0. Entanglement is 0 for θ = 0. Number of steps E n t ang l e m en t
10 particles14 particles20 particles
FIG. 9: (color online) Evolution of spatial entanglement for a system with different number of particles in a closed chain. Withan increase in the number of steps, the entanglement value remains close to asymptotic value with some peaks in between. Thepeaks can be accounted for the crossover of leftward and rightward propagating amplitudes of the internal state of the particleduring the QW. The peaks are more for a chain with a smaller number of particles. An increase in the number of particles inthe system results in the decrease of the entanglement value. The distribution is obtained by using √ ( | i + i | i ) as the initialstates of all particles and the Hadamard operation C ,π/ , as quantum coin operation. Here, we have chosen the Hadamard operation C ,π/ , and √ ( | i + i | i ) as the quantum coin operation and initialstate of the particles, respectively, for the evolution of the many-particle QW. To see the variation of entanglementfor a fixed number of particles with an increase in steps, we can pick a line parallel to the y axis. That is, fix thenumber of particles and see the variation of entanglement with the number of steps.In Fig. 7, we see that for a fixed number of particles, the entanglement at first increases to some value beforegradually falling. For M = 12 we can note that the peak value is about 0 . θ E n t ang l e m en t
20 particles on a closed chain
FIG. 10: (color online) Quantity of spatial entanglement for 20 particles on a closed chain after 20 steps of the QW usingdifferent values of θ in the quantum coin operation C ,θ, . The distribution is for particles initially in the state √ ( | i + i | i ).Since the system is a closed chain, the QW does not expand the position Hilbert space, and therefore for all values of θ from0 to π/ θ . For θ = 0 when thenumber of steps equal to the number of particles, the amplitudes goes round the chain and returns to its initial state makingthe entanglement 0 and for θ = π/
2, for every even number of steps of the QW, the system returns to the initial state whereentanglement is again 0. equal to the number of particles should be noted. This is because for the Hadamard walk the spread of a probabilitydistribution after t steps is between − t √ and t √ [36].If we fix the number of steps and measure the entanglement by increasing the number of particles in the system,the quantity of spatial entanglement first decreases and then it starts increasing with an increase in the number ofparticles.To show the variation of spatial entanglement with the quantum coin parameter θ , we plot the spatial entanglementby varying the parameter θ for a system with 10 particles after 10 steps of the QW and for a system with 20 particlesafter 20 steps of the QW in Fig. 8. In this figure, (a) and (b) are plots that use the symmetric superposition state √ ( | i + i | i ) (unbiased QW) as an initial state of all the particles, and (c) is the plot with all the particles in oneof the basis states | i or | i (biased QW) as the initial state. We note that the quantity of entanglement is higherfor θ values closer to 0 and π/ ξ, ζ in the coin operation C ξ,θ,ζ on particles initially in a symmetricsuperposition state. Closed chain:
Since most physical systems considered for implementation will be of a definite dimension, we extendour calculations to one of the simplest examples of closed geometry, an n − cycle. For a QW on an n − cycle, the shiftoperation, Eq. (4), takes the form S = | ih | ⊗ n − X j =0 | ψ j − mod n ih ψ j | + | ih | ⊗ n − X j =0 | ψ j +1 mod n ih ψ j | . (27)When we consider a many-particle system in a closed chain, with the number of lattice positions equal to the numberof particles M , the QW process does not expand the position Hilbert space like it does on an open chain (line).Therefore the spatial entanglement does not decrease at later times as it does for a walk on an open chain, butremains close to the asymptotic value. Fig. 9 shows the evolution of entanglement for a system with different numberof particles in a closed chain. The peaks seen in the plot can be accounted for by the crossover of the leftward andrightward propagating amplitudes of the internal state of the particle during the QW process. The frequency of thepeaks is more for a smaller number of particles (smaller closed chain). Also, note that the increase in the number ofparticles and the number of lattice points in the closed cycle results in the decrease in spatial entanglement of thesystem.2In Fig. 10, the value of spatial entanglement for 20 particles on a closed chain after 20 steps of the QW using differentvalues of θ in the quantum coin operation C ,θ, is presented. For all values of θ from 0 to π/ θ . For θ = 0, the amplitude goes round the ring andreturns to its initial state making the spatial entanglement value = 0. For θ = π/
2, for every even number of steps ofthe QW, the system returns to the initial state where spatial entanglement is again 0.Therefore, spatial entanglement on a large lattice space can be created, controlled and optimized for a maximumentanglement value by varying the quantum coin parameters and number of particles in the multi-particle QW.
V. CONCLUSION
We have presented the evolution of spatial entanglement in a many-particles system subjected to a QW process.By considering many particle in the one-dimensional open and closed chain we have shown that spatial entanglementcan be generated and controlled by varying the quantum coin parameters, the initial state and the number of steps inthe dynamics of the QW process. The spatial entanglement generated can have a potential application in quantuminformation theory and other physical processes.
Acknowledgement
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