Discrete-time quantum walks: Continuous limit and symmetries
Open accessAbstract
The continuous limit of one-dimensional discrete-time quantum walks with time-and space-dependent coefficients is investigated. A given quantum walk does not generally admit a continuous limit but some families (1-jets) of quantum walks do. All families (1-jets) admitting a continuous limit are identified. The continuous limit is described by a Dirac-like equation or, alternately, a couple of Klein-Gordon equations. Variational principles leading to these equations are also discussed, together with local invariance properties.
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TL;DR
The paper identifies which 1-jets of one-dimensional discrete-time quantum walks with time- and space-dependent SU(2) coins admit a formal continuous limit. When it exists, the limit is described by a Dirac-like equation or, alternately, coupled Klein-Gordon equations, with variational and local symmetry properties.
Key findings
- Generically a given discrete-time quantum walk does not admit a continuous limit; only certain families (1-jets) do, and only a few 1-jets admit one. The Hadamard walk cannot have a formal continuous limit.
- The richest scaling is α = β = δ = 1, where all contributions balance. In this scaling the continuous limit is a first-order Dirac-like equation DΨ = 0; each component also obeys an uncoupled Klein-Gordon equation at points where θ does not vanish, but these contain time-irreversibility terms.
- The Dirac-like dynamics derives from a Lagrangian density. It is not invariant under SU(2), but it belongs to a family of operators D(B) that exhibits a local invariance under the transformation given in the paper; for j = 3 this coincides with a standard gauge transformation along σ₃.
- Autonomous variational principles for the individual Klein-Gordon equations exist only in special cases where the irreversible terms vanish; otherwise there is only a single variational principle delivering the coupled Dirac-like equations for Ψ = (ψ−, ψ+).
Methods
- Theoretical analysis of one-dimensional discrete-time quantum walks on N × Z, driven by time- and space-dependent SU(2) coins parameterized by Euler angles θ, ξ, ζ, acting on a two-component wave function.
- Continuous limit investigated by introducing time step Δt, space step Δx, scales τ, λ, infinitesimal ε, and δ; Taylor expanding the discrete equations and identifying 1-jets whose angles tend to 0 or π as ε → 0.
- Main analysis focuses on the scaling α = β = δ = 1, derives equations in null coordinates u− and u+, and studies variational principles and local symmetries of the resulting operators.
Limitations
- The paper states that only the α = β = δ = 1 scaling is studied in detail; other scalings should be studied separately, numerical simulations are needed, and extensions to higher-dimensional space-times, higher-dimensional coins, space-time geometry, and decoherent/relativistic stochastic processes are left for future work.
- The paper states that autonomous variational principles for the separate Klein-Gordon equations exist only when dissipative/irreversible terms vanish; otherwise the two components cannot be decoupled without losing symmetry and simple variational principles.
- Evident from the provided text: many displayed equations are omitted or not decoded, so explicit derivations and coefficients cannot be verified from this copy.
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Claims the paper makes
0 of 28 cited sources checked · 0 supported · 0 flagged
- Findingreasoning§ B. Second order form of the equations · p. 10
The mass M- appearing in (36) must be real for the Hamilton equations obeyed by ψ- and ψ-† to be equivalent.
“Note also that the mass M -appearing in (36) must be real for the Hamilton equations obeyed by ψ -and ψ -† to be equivalent, as they should.”
Quote in the paperSubstantial claim - Findingreasoning§ A. First order form of the equations · p. 8
By (29) and (17), the total probability preservation property is preserved under a transformation (26) if and only if ∂-α = ∂+α, i.e. ∂x α = 0.
“By (29) and (17), this property is preserved under a transformation (26) if and only if ∂ -α = ∂ + α i.e. ∂ x α = 0.”
Quote in the paperSubstantial claim - Findingreasoning§ A. First order form of the equations · p. 8
The Lagrangian density L_D can be brought to a form which closely resembles the Dirac Lagrangian density.
“The Lagrangian density L D can also be brought to a form which closely resembles the Dirac Lagrangian density.”
Quote in the paperNot a substantial claim - Findingnone given§ A. First order form of the equations · p. 7
The Lagrangian density (24) leads to the first-order equations of motion (15, 16).
“leads to the first-order equations of motion (15, 16).”
Quote in the paperSubstantial claim - Findingreasoning§ III. EQUATIONS OF MOTION · p. 5
The expansion of the discrete equations around ǫ = 0 leads to coupled first-order equations of motion for ψ^- and ψ^+.
“The expansion of the discrete equations around ǫ = 0 then leads to the following equations of motion for ψ -and ψ + :”
Quote in the paperSubstantial claim
- Findingreasoning§ A. First order form of the equations · p. 8
The operator D given by equation (17) is the most general D(B) operator which preserves the total probability as an identity of the dynamics.
“The operator D given by equation (17) is thus the most general D ( B ) operator which preserves the total probability as an identity of the dynamics.”
Quote in the paperSubstantial claim - Findingreasoning§ A. First order form of the equations · p. 8
The form (29) coincides with the Dirac form of the continuous quantum walk, except that ξ- is not necessarily identical to ξ+.
“This coincides with the Dirac form of the continuous quantum walk, except for the fact that ξ -is not necessarily identical to ξ + .”
Quote in the paperSubstantial claim - Findingreasoning§ A. First order form of the equations · p. 7
For j = 3, the transformation (26) coincides with a standard gauge transformation along the σ3 generator of SU(2), because σ3 commutes with all D(B).
“Note however that, for j = 3, the above transformation coincides with a standard gauge transformation along the σ 3 generator of SU (2), because σ 3 commutes with all D ( B ).”
Quote in the paperNot a substantial claim - Findingreasoning§ B. Second order form of the equations · p. 9
The second-order equations (22) and (23) are not time-reversible.
“As already noted, the second order equations (22) and (23) are not time-reversible.”
Quote in the paperSubstantial claim - Findingnone given§ A. First order form of the equations · p. 8
A generic D(B) operator does not conserve the total probability ∫ dx [|ψ-|^2 + |ψ+|^2] for all ψ- and ψ+.
“Let us remark that a generic D ( B ) operator does not conserve the total probability ∫ x dx [ | ψ -( t, x ) | 2 + | ψ -( t, x ) | 2 ] for all ψ -and ψ + .”
Quote in the paperNot a substantial claim - Findingreasoning§ B. Second order form of the equations · p. 11
There are situations for which one or the other autonomous Klein-Gordon equation follows from a variational principle.
“We thus conclude that there are indeed situations for which one or the other autonomous Klein-Gordon equation follows from a variational principle, but that these situations are not particularly interesting, at least as far as the dynamics of the retained component of Ψ is concerned.”
Quote in the paperNot a substantial claim - Findingnone given§ A. Summary · p. 11
The 1-jets of 1D discrete-time quantum walks which admit a continuous limit have been identified.
“We have identified the 1-jets of 1 D discrete-time quantum walks which admit a continuous limit.”
Quote in the paperSubstantial claim - Findingnone given§ B. Discussion · p. 12
The strict formal continuous limit of discrete time-and space-dependent walks, when it exists, is given by differential equations obeyed by Ψ.
“presenting, when it exists, the strict formal continuous limit of discrete time-and space-dependent walks in the form of differential equations obeyed by Ψ.”
Quote in the paperSubstantial claim - Findingreasoning§ B. Discussion · p. 12
The Hadamard walk cannot have a formal continuous limit.
“we have proved by a rather trivial argument that the Hadamard walk cannot have a formal continuous limit”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. Second order form of the equations · p. 11
The two components of Ψ are really coupled by the quantum walk and cannot be decoupled without losing symmetry and the possibility of writing simple variational principles.
“It only tells us that the two components of Ψ are really coupled by the quantum walk and cannot be decoupled without paying the price i.e. loosing symmetry and the possibility of writing simple variational principles.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. First order form of the equations · p. 7
The invariance considered here is a formal invariance of an operator family, not an invariance of the equations of motion.
“The invariance considered here is a formal invariance of an operator family, not an invariance of the equations of motions.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ B. Scaling for the continuous limit · p. 4
For the continuous limit to exist, at least formally, the operator B(θ,ξ,ζ) defined by the functions θ, ξ and ζ must tend to unity as ǫ tends to zero.
“For the continuous limit to exist, at least formally, the operator B ( θ, ξ, ζ ) defined by the functions θ , ξ and ζ must also tend to unity as ǫ tends to zero.”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. Scaling for the continuous limit · p. 4
The third Euler angle ζ does not have to tend to zero with ǫ because of the sin θ factor in front of e^{iζ} in (2).
“Note that the third Euler angle ζ does not have to tend to zero with ǫ because of the sin θ factor in front of e iζ in (2).”
Quote in the paperSubstantial claim - Interpretationreasoning§ III. EQUATIONS OF MOTION · p. 6
The operator D defines the formal continuous limit of the quantum walk.
“The operator D defines the formal continuous limit of the quantum walk.”
Quote in the paperSubstantial claim - Premisecitation§ I. INTRODUCTION · p. 2
One-dimensional discrete-time quantum walks are entirely determined by three possibly time- and space-dependent Euler angles.
“one-dimensional discrete-time quantum walks, which are entirely determined by three possibly time- and space-dependent Euler angles 10 .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
- Premisecitation§ I. INTRODUCTION · p. 2
The continuous limit of discrete time quantum walks with constant coefficients has already been addressed by several authors with very different and apparently not reconcilable approaches.
“The continuous limit of discrete time quantum walks with constant coefficients has already been addressed by several authors 6,10,17,20,32 , with very different and apparently not reconcilable approaches.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [6] 2004, Quantum random walks and piecewise deterministic evolutions, Phys. Rev. Lett. source
Not checked in the free run.
- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
- cites [17] 1998, Quantum computation and decision trees, Phys. Rev. A source
Not checked in the free run.
- Interpretationnone given§ I. INTRODUCTION · p. 2
The Dirac-like equation derives from a variational principle that exhibits a symmetry.
“the Dirac-like equation derives from a variational principle which exhibits an interesting symmetry.”
Quote in the paperNot a substantial claim - Interpretationnone given§ B. Scaling for the continuous limit · p. 4
For α = β = δ = 1, all contributions to (10) are of equal importance.
“The richest and most interesting case corresponds to α = β = δ = 1 because all contributions to (10) are then of equal importance.”
Quote in the paperSubstantial claim - Interpretationreasoning§ III. EQUATIONS OF MOTION · p. 6
The first-order equations imply that each component of Ψ obeys, at points where θ does not vanish, an uncoupled Klein-Gordon equation.
“These first-order equations imply that each component of Ψ obeys, at points where θ does not vanish, an uncoupled Klein-Gordon equation.”
Quote in the paperSubstantial claim - Interpretationnone given§ I. INTRODUCTION · p. 2
Only a few 1-jets admit a continuous limit.
“only a few 1-jets admit a continuous limit”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. Scaling for the continuous limit · p. 4
The Hadamard walk cannot have a continuous limit.
“Note also that the Hadamard walk cannot have a continuous limit.”
Quote in the paperNot a substantial claim - Interpretationnone given§ III. EQUATIONS OF MOTION · p. 6
The natural initial conditions for the first order equations (15) and (16) are the values of ψ^- and ψ^+ at time t = 0, i.e. on the submanifold u^- = -u^+.
“The natural initial conditions for the first order equations (15) and (16) are simply the values of ψ -and ψ + at time t = 0 i.e. on the submanifold u -= -u + .”
Quote in the paperNot a substantial claim - Interpretationreasoning§ III. EQUATIONS OF MOTION · p. 6
By equation (15), giving oneself these initial conditions is equivalent to giving oneself both ψ^- and ψ^+ at time t = 0.
“By equation (15), giving oneself these initial conditions is equivalent to giving oneself both ψ -and ψ + at time t = 0.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 2
Generically, the object which can admit a continuous limit is not a given discrete-time walk, but rather a family (1-jet) of discrete-time walks whose defining angles depend on ǫ and tend to zero with ǫ.
“generically, the object which can admit a continuous limit is not a given discrete-time walk, but rather a 1-jet of discrete-time walks, which regroups all walks which share a common expansion in the time-and length steps as these tend to zero”
- Also stated in: § B. Scaling for the continuous limit · p. 4 · plain
“In other words, it generally does not make sense to consider the continuous limit of a given walk, defined by ǫ -independent angles, but rather the limit of a family of walks indexed by ǫ , whose defining angles depend on ǫ and tend to zero with this infinitesimal.”
Quote in the paperSubstantial claim - Also stated in: § B. Scaling for the continuous limit · p. 4 · plain
- Premisecitation§ B. Discussion · p. 12
The differential equations proposed in 6,20 admit discretizations which start with terms identical to those found in the discrete quantum walk dynamics.
“the differential equations proposed in 6,20 rather admit discretizations which start with terms identical to those found in the discrete quantum walk dynamics.”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ B. Discussion · p. 12
The crucial condition that θ must tend to zero together with the time and length step is overlooked in reference 10.
“but the crucial condition that θ must tend to zero together with the time and length step is overlooked.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
- Interpretationreasoning§ B. Discussion · p. 12
Once the correct scaling law is applied, the equations in reference 10 turn out to be identical to the power laws and logarithm appearing in this work.
“Once the correct scaling law is applied, these equations turn out to be identical to the power laws and logarithm appearing in our work.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
- Premisenone given§ B. Discussion · p. 12
The differential equations proposed in 6,20 are not obeyed by the original field Ψ, but rather by two composite fields which mix ψ- and ψ+ in a time dependent manner.
“the differential equations proposed in 6,20 are not obeyed by the original field Ψ, but rather by two composite fields which mix ψ -and ψ + in a time dependent manner.”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ B. Discussion · p. 12
General situations where none of the Euler angles vanishes and where these angles depend on time and/or space are not discussed in previous literature.
“The following four points are not discussed in the previous literature and have thus been addressed here for the first time: (i) general situations where none of the the Euler angle vanishes and where these angles depend on time and/or space (ii) the existence of several scaling laws (iii) the existence of variational principles for the continuous limit (iv) the symmetry properties of the contin”
Quote in the paperSubstantial claim - Premisenone given§ B. Discussion · p. 12
The existence of several scaling laws is not discussed in previous literature.
“The following four points are not discussed in the previous literature and have thus been addressed here for the first time: (i) general situations where none of the the Euler angle vanishes and where these angles depend on time and/or space (ii) the existence of several scaling laws (iii) the existence of variational principles for the continuous limit (iv) the symmetry properties of the contin”
Quote in the paperSubstantial claim - Premisenone given§ B. Discussion · p. 12
The symmetry properties of the continuous limit are not discussed in previous literature.
“The following four points are not discussed in the previous literature and have thus been addressed here for the first time: (i) general situations where none of the the Euler angle vanishes and where these angles depend on time and/or space (ii) the existence of several scaling laws (iii) the existence of variational principles for the continuous limit (iv) the symmetry properties of the contin”
Quote in the paperSubstantial claim - Premisereasoning§ B. Discussion · p. 12
The differential equations proposed in 6,20 are not the strict formal equivalent of the random walk equations when both time- and length steps tend to zero.
“the differential equations proposed in 6,20 are not the strict formal equivalent of the random walk equations when both time- and length steps tend to zero”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisenone given§ B. Discussion · p. 12
The existence of variational principles for the continuous limit is not discussed in previous literature.
“The following four points are not discussed in the previous literature and have thus been addressed here for the first time: (i) general situations where none of the the Euler angle vanishes and where these angles depend on time and/or space (ii) the existence of several scaling laws (iii) the existence of variational principles for the continuous limit (iv) the symmetry properties of the contin”
Quote in the paperNot a substantial claim - Premisereasoning§ B. Discussion · p. 12
The authors of references 6,20 do not use a strict concept of continuous limit.
“the authors of references 6,20 do not use a strict concept of continuous limit.”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ B. Discussion · p. 12
Reference 10 only addresses situations where ξ = ζ = 0 and θ is a constant.
“Reference 10 only addresses situations where ξ = ζ = 0 and θ is a constant.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
- Premisenone given§ B. Discussion · p. 12
References 17,32 address only the special case where θ and ζ identically vanish and ξ is a constant.
“References 17,32 address only the special case where θ and ζ identically vanish and ξ is a constant.”
Quote in the paperSubstantial claim0 of 2 sources checked
- May combine two claims. 17
- The quote only partly states this claim. 2
- Classified as probably not a substantial claim. 11
May combine two claims.
Findingreasoning§ B. Second order form of the equations · p. 9For the continuous quantum walks dynamics considered here, there are no autonomous variational principles which deliver separately the Klein-Gordon equations obeyed by ψ- and ψ+ when dissipative terms do not vanish, but there is always a single variational principle which delivers the coupled Dirac-like equations obeyed by Ψ = (ψ-, ψ+).
“For the continuous quantum walks dynamics considered here, there are indeed no autonomous variational principles which deliver separately the Klein-Gordon equations obeyed by ψ -and ψ + when disspative terms do not vanish, but there is always a single variational principle which delivers the coupled Dirac-like equations obeyd by the Ψ = ( ψ -, ψ + ).”
Quote in the paperSubstantial claimThe quote only partly states this claim.
Findingreasoning§ A. First order form of the equations · p. 8If total probability conservation is imposed as an identity valid for all values of ψ- and ψ+, then D(B) necessarily takes the form (29), where p is an arbitrary integer and ξ-, ξ+, ̄θ and ζ are arbitrary possibly time- and position-dependent real quantities.
“If one imposes total probability conservation as an identity valid for all values of ψ -and ψ + , one obtains that D ( B ) necessarily takes the form:”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ A. First order form of the equations · p. 7The operators D(B) do not commute with σ1 and σ2, and the invariance considered here is not an invariance of the continuous quantum walk under the action of SU(2).
“Let us stress that the operators D ( B ) do not commute with σ 1 and σ 2 ? and that the invariance considered here is not an invariance of the continuous quantum walk under the action of SU (2).”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ A. First order form of the equations · p. 7The generator σ3 commutes with D, but σ1 and σ2 do not.
“The generator σ 3 does commute with D , but σ 1 and σ 2 do not.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ B. Second order form of the equations · p. 9The irreversibility is made manifest in (22) by the (∂+ ln ̄θ)∂-ψ- term and in (23) by the (∂- ln ̄θ)∂+ψ+ term.
“This irreversibility is made manifest in (22) by the ( ∂ + ln ¯ θ ) ∂ -ψ -term and in (23) by the ( ∂ -ln ¯ θ ) ∂ + ψ + term.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ A. First order form of the equations · p. 7The operator D characterizing the continuous dynamics of the quantum walks considered in this article belongs to the family D(B).
“It is obvious that the operator D characterizing the continuous dynamics of the quantum walks considered in this article belongs to this family.”
Quote in the paperNot a substantial claimMay combine two claims. The quote only partly states this claim.
Findingnone given§ A. Summary · p. 11The Lagrangian density and hence the dynamics is not invariant under the global or local action of SU(2).
“Neither the dynamics of the discrete-time walk nor the dynamics of its continuous limit are invariant under SU (2) , but the continuous dynamics nevertheless belongs to a family of Dirac-like equations which exhibit an elegant symmetry.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ A. First order form of the equations · p. 7The family of operators D(B) displays local invariance with respect to transformation (26), and the operator fixing the continuous quantum walk dynamics is one member of this locally invariant family.
“The family of operators D ( B ) thus displays local invariance with respect to the transformation (26) and the operator fixing the continuous quantum walk dynamics is one member of this locally invariant family.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ B. Second order form of the equations · p. 10The only non-possibly-vanishing components of the electromagnetic tensor F are F-+ = F+-; these components vanish if the conditions from identifying (36) with (22) are satisfied.
“The only non possibly vanishing components of the electromagnetic tensor F ( i.e. the curvature of the connection A ) are F -+ = F + -. A straightforward calculation shows that these components actually vanish if the above conditions are satisfied.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ B. Second order form of the equations · p. 11In the other situations, each Klein-Gordon equation contains terms which induce time-irreversibility and the equations do not follow from autonomous variational principles.
“In the other situations, each Klein-Gordon equation contains terms which induce time-irreversibility and the equations do not follow from autonomous variational principles.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ B. Discussion · p. 12The scaling studied in this article is not the only one which allows for a continuous limit of the discrete walk.
“the scaling studied in this article is not the only one which allows for a continuous limit of the discrete walk.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingnone given§ A. Summary · p. 11The continuous limit is described by a simple Dirac-like equation which derives from a variational principle.
“The continuous limit is described by a simple Dirac-like equation which derives from a variational principle.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ B. Second order form of the equations · p. 10The conditions for ψ+ make the electromagnetic tensor F+ vanish; thus A+ does not represent a real, physical electromagnetic field.
“As above, these conditions makes the electromagnetic tensor F + vanish. Thus, A + does not represent a real, physical electromagnetic field.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Interpretationreasoning§ III. EQUATIONS OF MOTION · p. 6The natural initial conditions for (22) are the values of ψ^- and ∂_t ψ^- on the sub-manifold t = 0.
“The natural initial conditions for (22) are the values of ψ -and ∂ t ψ -on the sub-manifold t = 0.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ B. Scaling for the continuous limit · p. 4The two functions θ and ξ must depend on ǫ and tend to 0 or π as ǫ tends to zero.
“This trivial remark implies that the two functions θ and ξ must actually depend on ǫ and tend to 0 or π as ǫ tends to zero.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ B. Second order form of the equations · p. 10A mass with imaginary part would generate an imaginary part in the frequency (energy) or wave-vector (impulse) of the particle, and this would signal irreversibility.
“The physical reason for this restriction on the mass M -is that a mass with imaginary part would generate an imaginary part in the frequency (energy) or wave-vector (impulse) of the particle, and this would signal irreversibility.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationnone given§ I. INTRODUCTION · p. 2When it exists, the continuous limit can be described by a 2D Dirac-like equation or, alternately, two independent Klein-Gordon equations.
“when it exists, the continuous limit can be described by a 2 D Dirac -like equation or, alternately, two independent Klein-Gordon equations”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ A. First order form of the equations · p. 7σ1 and σ2 interchange ψ- and ψ+ while these two components of Ψ play fundamentally different roles, one propagating towards the left while the other propagates towards the right.
“This reflects the fact that σ 1 and σ 2 interchange ψ -and ψ + while these two components of Ψ play fundametally different roles, one propagating towards the left while the other propagates towards the right.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ B. Second order form of the equations · p. 10The potential A- can be eliminated by the zero of phase for ψ- and does not represent a real, physical electromagnetic field.
“Thus, the potential A -can be eliminated by the zero of phase for ψ -and it does not represent a real, physical electromagnetic field.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone given§ A. First order form of the equations · p. 9Ψ belongs to a two-dimensional Hilbert space and is not a Dirac spinor.
“Of course, Ψ here belongs to a two -dimensional Hilbert space, and is not a Dirac spinor.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ B. Discussion · p. 11At least formally, discrete time quantum walks become continuous time quantum walks when the time step tends to zero.
“It is shown in 32 that, at least formally, discrete time quantum walks become continuous time quantum walks when the time step tends to zero.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [32] 1986, Feynman and the visualization of space-time processes, Rev. Mod. Phys. source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ B. Discussion · p. 12References 17,32 mention that a Dirac-like equation obeyed by Ψ can be formally obtained from a discrete quantum walk when the angle ξ tends to zero with the time and length steps.
“These references mention that a Dirac-like equation obeyed by Ψ can be formally obtained from a discrete quantum walk when the angle ξ tends to zero with the time and length steps”
Quote in the paperNot a substantial claim0 of 2 sources checkedClassified as probably not a substantial claim.
Premisecitation§ B. Discussion · p. 12References 6,20 propose continuous limits for several discrete walks with constant coefficients, including the Hadamard walk.
“References 6,20 propose continuous limits for several discrete walks with constant coefficients, including the Hadamard walk.”
Quote in the paperSubstantial claim0 of 2 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ B. Discussion · p. 12The equations presented in 10 contain various trigonometric functions of θ.
“the equations presented in 10 contain various trigonometric functions of θ .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
Classified as probably not a substantial claim.
Premisenone given§ B. Discussion · p. 12References 17,32 offer little detailed computations.
“but they offer little detailed computations”
Quote in the paperSubstantial claim0 of 2 sources checkedClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ B. Discussion · p. 12Reference 10 presents a detailed computation and both Dirac and Klein-Gordon equations are obtained.
“This reference does present a detailed computation and both Dirac and Klein-Gordon equations are obtained”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10] D. Burgarth, Quantum State Transfer with Spin Chains, Ph.D. dissertation (University College, London, 2006).
Not checked in the free run.
Classified as probably not a substantial claim.
Premisenone given§ B. Discussion · p. 12References 17,32 do not mention the existence of Klein-Gordon equations.
“and do not mention the existence of Klein-Gordon equations.”
Quote in the paperSubstantial claim0 of 2 sources checked
The authors state this as an assumption.
All functions can be chosen at least C^2 in both space and time variables for all sufficiently small values of ∆t and ∆x.
“We now suppose, that all functions can be chosen at least C 2 in both space and time variables for all sufficiently small values of ∆ t and ∆ x .”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Dirac Hamiltonian for a non-zero mass particle has been recovered from QW by taking the evolution into the continuum limit and changing to null coordinates, or by taking each step evolution operator to continuous form and introducing a rotation.
“In order to recover the DE for a non-zero mass particle from the QW evolution operator, the evolution was taken into continuum limit and the coordinate was changed to null coordinate48. DH for a non-zero mass particle was also recovered by taking each step evolution operators to continuous form and by introducing a rotation 47.”
From Dirac Cellular Automaton from Split-step Quantum Walk (2016) · cites this paper as [48]
Section III (p. 5) derives the continuous-limit equations using null coordinates and calls them the Dirac form of the walk dynamics. The First order form section (pp. 8–9) identifies a mass-like term, but describes the result as Dirac-like and explicitly says the two-component wave function is not a Dirac spinor. The separate claim about a rotation is attached to citation 47, not this source. (LLM, conf=0.84)
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Total-probability expression repeats the minus component
MinorThe later expression counts |ψ−|2 twice and omits |ψ+|2, contradicting the paper's earlier definition of total probability.
“∫ x dx [ | ψ -( t, x ) | 2 + | ψ -( t, x ) | 2 ]”
“The total probability π j = ∑ m ( | ψ -j,m | 2 + | ψ + j,m | 2 ) is independent of j i.e. conserved by the walk.”
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“τ τ θ ≈ − − + ( ) ( ) H a k 1 0 0 1 0 1 1 0 (48) DA 2 ”
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- Authors
- di Molfetta, G., & Debbasch, F.
- Affiliations
- Université Paris 6 , ERGA-LERMA, UMR 8112, 3, rue Galilée, F-94200 Ivry, France
- Laboratoire d’Etudes du Rayonnement et de la Matière en Astrophysique et Atmosphères (FR)
- ORCID
- Debbasch, F.
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 2012-11-08
- Volume
- 53
- Issue
- 12
- Pages
- 123302
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2012636081
- Cited by
- 60
- References
- 45
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 1 · 2023: 3 · 2022: 3 · 2021: 4 · 2020: 3 · 2019: 6 · 2018: 5 · 2017: 6 · 2016: 9 · 2015: 7 · 2014: 5 · 2013: 3 · 2012: 2
- FWCI
- 5.31
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Keywords
- discrete-time quantum walks, continuum limit, time-dependent coefficients, space-dependent coefficients, Dirac equation, Klein-Gordon equation, variational principle, local invariance, symmetry, one-dimensional quantum walk
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