Asymptotic entanglement in 2D quantum walks

For a localized initial state at the origin with initial coin |L,L>, the asymptotic entropy of entanglement is E ≃ 1.744.

“E = -trace (ρ c log 2 ρc ) ≃ 1.744”

For localized separable initial conditions, the asymptotic CPE is E(θ1,φ1;θ2,φ2)=E1(θ1,φ1)+E1(θ2,φ2).

“Using additivity, one can obtain the CPE for localized, separable initial conditions (eq.32) from this expression as E(θ 1 , φ 1 ; θ 2 , φ 2 ) = E 1 (θ 1 , φ 1 ) + E 1 (θ 2 , φ 2 ).”

The method is applied to a Hadamard walk, yielding the dependence of CPE on initial conditions.

“The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained.”

Initial states leading to maximum or minimum CPE are identified.

“Initial states leading to maximum or minimum CPE are identified”

The CPE obtained from separable initial states and separable coin operations satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases.

“The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding onedimensional cases.”

• Also stated in: § A. Separable initial states · plain

  “In other words, for a separable initial state and separable coin operation, the CPE for a two-dimensional QW can be obtained from the CPE of the underlying onedimensional quantum walks.”

For a two-dimensional Hadamard walk with coin operation H⊗H, the eigenproblem for U_k can be solved exactly.

“The eigenproblem for this operator can be solved exactly.”

For the initial separable coin state |χ(I)(θ,φ) = |L ⊗ (cosθ|L + e^{iφ} sinθ|R), the asymptotic entanglement as a function of initial parameters is E(θ,φ)=E0+E1(θ,φ), with E0≃0.872.

“E(θ, φ) = E 0 + E 1 (θ, φ)(38) with E 1 given by eq. ( 34) and E 0 = E 1 (0, φ) ≃ 0.872.”

For separable initial coin states and localized positions, the CPE satisfies an additivity property which allows it to be expressed as the sum of the CPE for the corresponding one-dimensional motions.

“In this case, the CPE satisfies an additivity property which allows it to be expressed as the sum of the CPE for the corresponding one-dimensional motions.”

For extended non-locality with initial coin χ(II), the elements of the long-time reduced density operator result from evaluating the expressions for P(k) obtained in the localized case at kx=ky=0.

“In this case, the elements of the long-time reduced density operator, ρc , result from evaluating the expressions for P(k) obtained in the localized case, for k x = k y = 0.”

Extended non-locality increases significantly the range of variation for asymptotic CPE compared with the local case.

“Comparison with the local case in Fig. 3 shows that extended non-locality increases significantly the range of variation for asymptotic CPE.”

For extended non-locality, full asymptotic entanglement E_max = 2 results for the initial coin |Ψ+>.

“Two aspects of this surface are remarkable: (i) full asymptotic entanglement E max = 2 results for the initial coin |Ψ + .”

For extended non-locality, the minimum entanglement is exactly E_min = 1 for the initial coin |Ψ->.

“And (ii) the minimum entanglement is now (exactly) E min = 1 for the initial coin |Ψ -, as shown in the lower panel of Fig. 7.”

The relative phase of the initial coin state can cause significant changes in CPE.

“Notice that the relative phase of the initial coin state can cause significant changes in CPE.”

If both coins are tunable, the maximum CPE, E = 2, can be obtained.

“If both coins are tunable, the maximum CPE, E = 2, can be obtained.”

Values between E = 1 and E = 2 can be obtained depending on the initial coin state.

“Values between E = 1 and E = 2 can be obtained depending on the initial coin state.”

The relation E(χ(III); θ, φ) = E(χ(II); -θ, φ) is satisfied.

“A comparison between both panels in Fig. 3 shows that the relation E(χ (III) ; θ, φ) = E(χ (II) ; -θ, φ) is satisfied.”

For extended non-locality with initial coin χ(II), the diagonal elements of ρc are given by ρc(1,1) = (sin(2θ) cos φ + 3)/16, ρc(2,2) = (1 - sin(2θ) cos φ + 8 cos^2 θ)/16, ρc(3,3) = (9 - sin(2θ) cos φ - 8 cos^2 θ)/16, ρc(4,4) = (3 + sin(2θ) cos φ)/16.

“The diagonal elements are ρc (1, 1) = (sin(2θ) cos φ + 3)/16 ρc (2, 2) = (1 -sin(2θ) cos φ + 8 cos 2 θ)/16 ρc (3, 3) = (9 -sin(2θ) cos φ -8 cos 2 θ)/16 ρc (4, 4) = (3 + sin(2θ) cos φ)/16”

The maximum CPE depends on whether one can tune both initial coins or not.

“The maximum CPE depends on wether one can tune both initial coins or not.”

For the first family χ(II), |F|^2 equals |α2|^2 cos^2 θ + |α3|^2 sin^2 θ + sin(2θ) R α*2 α3 e^{-iφ}.

“For the first family, |χ (II) , straightforward evaluation of eq. ( 20) leads to |F | 2 = |α 2 | 2 cos 2 θ + |α 3 | 2 sin 2 θ + sin(2θ) ℜ α * 2 α 3 e -iφ .”

When extended non-local states are considered, large variations of CPE with respect to the local case may be obtained.

“However, when extended non-local states are considered, large variations of CPE with respect to the local case may be obtained.”

For the initial coin families χ(II) and χ(III), the initial coin-coin entanglement S depends only on the parameter θ.

“As shown in Fig. 4 (b), it depends on the parameter θ alone.”

For family χ(III), a similar expression for |F|^2 holds with (α2, α3) replaced by (α1, α4).

“A similar expression holds for |χ (III) with (α 2 , α 3 ) replaced by (α 1 , α 4 ).”

For extended non-locality, the initial product state |L, R results in a low CPE level of E ≈ 1.20.

“Finally, the initial product state |L, R results in a low CPE level of E ≃ 1.20.”

Initial superpositions of two close sites can produce small changes in CPE (smaller than 1%) with respect to the localized case.

“Initial superpositions of two close sites can produce small changes in CPE (smaller than 1%) with respect to the localized case.”

For separable initial states, the CPE obtained is above 73% of the maximum value in all cases considered, except for the extended nonlocality discussed below.

“The CPE obtained is rather high, above 73% of the maximum value in all cases considered, except for the extended nonlocality discussed below.”

Extreme values for CPE (maxima or minima) appear associated with maximum values of CCE.

“Extreme values for CPE (maxima or minima) appear associated with maximum values of CCE.”

There are maximally entangled initial coins which do not lead to maximum CPE.

“However, there are maximally entangled initial coins which do not lead to maximum CPE.”

The overall effect of restricted non-locality (small delocalizations about the origin) on CPE is small.

“The overall effect of this restricted non-locality on CPE is small, but this was to be expected since we are considering only small delocalizations about the origin.”

• Also stated in: § C. Extended non-locality · plain

  “We have considered initial position states with a small degree of non-locality, i.e. the initial amplitudes that are non-zero only in a few sites around the origin and found that the changes in asymptotic CPE due to initial non-zero amplitudes at sites x = ±1 or y = ±1 as compared with the local case x = y = 0, are small.”

The largest variations in CPE due to non-locality are associated with balanced superpositions, α = ±π/4.

“The largest variations due to non-locality are associated to balanced superpositions, α = ±π/4.”

The exact diagonalization of the reduced density operator leads to the asymptotic entanglement as a function of the initial parameters.

“The exact diagonalization of this operator leads to the asymptotic entanglement as a function of the initial parameters, E(θ, φ).”

In the uniform limit, the variation for CPE is large.

“In the uniform limit, the variation for CPE is large.”

Equations (17) and (20) imply P'(k) = |ã(k)|^2 P(k).

“The calculation of the reduced density operator is simplified by noting that eqs.( 17) and (20) imply P ′ (k) = |ã(k)| 2 P(k).(50)”

Diagonalization of the non-local reduced density operator leads to the asymptotic CPE E(χ; α, β), which depends on the initial coin state and the two parameters α, β that specify the initial position.

“Diagonalization of this operator leads to the corresponding asymptotic entropy of entanglement, E(χ; α, β), which depends on the initial coin state and the two parameters that specify the initial position.”

For both entangled families, the asymptotic CPE E depends on the initial coin state parameters (θ, φ) as shown in Fig. 3.

“For both cases, the dependence of this quantity on the initial coin state parameters (θ, φ) is shown in Fig. 3.”

For extended non-locality with initial coin χ(II), the off-diagonal elements of ρc are given by the expressions in the text.

“The expressions for the off-diagonal elements are, ρc (1, 2) = (1 -4 cos 2 θ + sin(2θ) cos φ)/16 ρc (1, 3) = (-3 + sin(2θ) cos φ + 4 cos 2 θ)/16 ρc (1, 4) = (-1 + sin(2θ) cos φ)/16 ρc (2, 3) = (-1 + sin(2θ) cos φ)/16 ρc (2, 4) = (-1 -sin(2θ) cos φ + 4 cos 2 θ)/16 ρc (3, 4) = (3 -sin(2θ) cos φ -4 cos 2 θ)/16.”

The maximum E=2 for |Ψ+> is basically flat in the φ direction, so it is robust against small variations in the relative phase of the initial coin state.

“Note that this maximum is basically flat in the φ direction, so it is robust against small variations in the relative phase of the initial coin state.”

Comparison of the resulting CPE surfaces E(α, β) with the corresponding localized cases shows, in most cases, variations due to position non-locality.

“A comparison of the resulting CPE surfaces E(α, β) with the corresponding localized cases values of E show, in most cases, variations due to position non-locality.”

The calculation of the asymptotic density operator for non-local initial conditions can be reduced to that of the local case with the same initial coin state, provided an additional integration factor is included in the final step.

“We have shown that the calculation of the asymptotic density operator for this case can be reduced to that of the local case with the same initial coin state, provided an additional integration factor is included in the final step.”

The paper provides an analytical method for exact calculation of the asymptotic reduced density operator and corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice, with arbitrary initial conditions and coin operations.

“In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice.”

• Also stated in: § I. INTRODUCTION · plain

  “Our approach provides exact results for the two-dimensional QW with arbitrary initial conditions and coin operations.”

• AssumptionRests on an assumption: Most algorithmic applications require higher dimensions.
• AssumptionRests on an assumption: Higher dimensional quantum walks open new possibilities, such as the preparation of initially entangled states within the coin or position subspaces.

After several steps, coin-position entanglement converges to a well-defined value determined by the initial state for a given evolution operator.

“After several steps, the coin-position entanglement (CPE) converges to a well defined value which depends on the initial state.”

• Also stated in: § I. INTRODUCTION · plain

  “After several steps, it converges to a well defined value which, for a given evolution operator, is determined by the initial state [21].”

• cites [21] Carneiro I, 2005, New J. Phys. sourceCheck pending

The CPE for the case where both initial coin states are superposition states (φ1=φ2=0) is consistent with the additivity property, eq. (35).

“The result is consistent with the additivity property, eq. ( 35).”

For the |L,L initial state, the asymptotic CPE is exactly twice that of the one-dimensional case.

“Thus, for |L, L we obtain exactly twice CPE as in the onedimensional case.”

The subadditivity property of von Neumann entropy for a separable density operator implies E(ρ_C1⊗ρ_C2)=E(ρ_C1)+E(ρ_C2).

“The subadditivity property of the von Neumann entropy for a separable density operator [35] implies, E(ρ C1 ⊗ ρ C2 ) = E(ρ C1 ) + E(ρ C2 ).”

• cites [35] Nielsen M, 2000, Quantum Computation and Quantum InformationNot checked yet

The non-local case can be easily obtained from the local case by adding a weight factor to the final integration.

“In particular, we show that the non-local case can be easily obtained from the local case by adding a weight factor to the final integration.”

For a one-dimensional Hadamard walk starting at the origin with |L or |R as the initial coin, the asymptotic CPE is E0 ≃ 0.872.

“In this cases, the asymptotic CPE is E 0 ≃ 0.872, [21,31].”

• cites [21] Carneiro I, 2005, New J. Phys. sourceCheck pending
• cites [31] Abal G, 2006, Phys. Rev. sourceNot checked — no full text of the source

These are new results, as previous analytical work based on one-dimensional walks did not allow for this possibility.

“These are new results, as previous analytical work based on one-dimensional walks [31] did not allow for this possibility.”

• cites [31] Abal G, 2006, Phys. Rev. sourceNot checked — no full text of the source

• AssumptionRests on an assumption: The one-dimensional case does not allow for initially entangled (coin or position) states as the 2D case does.

The relation E(χ(III); θ, φ) = E(χ(II); -θ, φ) can be traced to the reduced density operators ρc(χ(III); θ, φ) and ρc(χ(II); -θ, φ) having the same eigenvalues.

“From a mathematical point of view this may be traced to the fact that the reduced density operators ρ c (χ (III) ; θ, φ) and ρ c (χ (II) ; -θ, φ) have the same eigenvalues.”

The Fourier transform is extremely useful when single-step displacements are involved because the evolution operator is diagonal in k-space.

“The Fourier transform, as first noted in this context by Nayak and Vishwanath [11], is extremely useful when single-step displacements are involved because the evolution operator is diagonal in k-space.”

• cites [11] Nayak A Vishwanath A, 2000Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The work describes a method for the exact characterization of the long-time (asymptotic) coin-position entanglement (CPE) of a discrete-time quantum walk on a two-dimensional lattice or, alternatively, of two independent walkers on a line.

“This work describes a method for the exact characterization of the long-time (asymptotic) coin-position entanglement (CPE) of a discrete-time quantum walk on a two-dimensional (2D) lattice or, alternatively, of two independent walkers on a line.”

The small variations in CPE due to initial non-local positions in the neighborhood of the origin are contextualized because variations may be larger when more extended initial states in position space are considered.

“This result also contextualizes the small variations in CPE due to initial non-local positions in the neighborhood of the origin, as the variations may be larger when more extended initial states in position space are considered.”

The relation between asymptotic CPE and the initial state allows one to choose the initial conditions that will lead to the desired entanglement level.

“It also allows one to choose the initial conditions that will lead to the desired entanglement level.”

For the extended position distribution, the asymptotic CPE depends only on the initial coin state, which determines |F|^2.

“Thus for the extended position distribution, the asymptotic CPE depends only on the initial coin state which determines |F | 2 .”

For non-local initial conditions, the matrix P'(k) is expressed in terms of the corresponding matrix P(k) for the local case with the same initial coin state.

“Thus, the matrix P ′ (k) for the non-local case is expressed in terms of the corresponding matrix P(k) for the local case with the same initial coin state.”

The maximum entanglement E=N can be obtained if all initial states are properly prepared.

“The maximum entanglement of E = N can be obtained of all initial states are properly prepared.”

The additivity property can be generalized to the case of N independent walkers with separable initial condition and coin operation.

“This additivity property can be generalized to the case of N independent walkers with separable initial condition and coin operation.”

For the Hadamard coin operation U_C = H ⊗ H, the initial coin-coin entanglement is preserved by the evolution.

“Note that, since the coin operation U C = H ⊗ H is separable, the initial CCE is preserved by the evolution.”

In the asymptotic limit n≫1, according to the stationary phase theorem, only terms with ω_k=ω'_k contribute in eq. (14).

“In the asymptotic limit n ≫ 1, according to the stationary phase theorem, only terms with ω k = ω ′ k contribute in eq. ( 14) as discussed in detail in Ref. [11].”

• cites [11] Nayak A Vishwanath A, 2000Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.