The role of coherence on two-particle quantum walks

Li-Hua Lu; Shan Zhu; You-Quan Li

REVIEW 2 major objections 5 minor 33 references

The role of coherence on two-particle quantum walks

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper derives a general density-matrix formula for the two-particle correlation function of a quantum walk and shows that, for a concrete two-photon scheme, the degree of coherence and the relative phase of the initial state…

desk verdict A useful but incremental density-matrix extension of two-boson quantum walks; the central formula is sound for constant tunneling but the derivation has a sign slip that undermines the advertised generality for site-dependent potentials. read the letter →

arxiv 1908.04987 v1 pith:CFVXGVU6 submitted 2019-08-14 quant-ph

Li-Hua Lu , Shan Zhu , You-Quan Li This is my paper · ORCID

classification quant-ph PACS 03.67.Ac03.67.Lx05.40.Fb05.90.+m

keywords two-particlequantumwalkdegreeofcoherencetwo-photoncorrelationfunctionmixedstatepurewaveguidearraystight-bindingmodelinterference

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how partial coherence—how much the input photons can still interfere with each other—changes the dynamics of two indistinguishable photons performing a continuous-time quantum walk on a one-dimensional lattice. It derives a general analytical formula for the two-photon correlation function that works for any pure or mixed two-boson input state, expressed purely in terms of the initial density matrix and the single-particle transition amplitudes. For a concrete setup in which two mutually incoherent beams are split by gratings and sent into a uniform waveguide array, the formula reduces to a compact expression whose interference term is proportional to $\gamma\cos\varphi$, with $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$; here $\eta\in[0,1]$ is the degree of coherence, $\varphi$ is the relative phase, and $\alpha$ is the initial photon distribution. The central claim is that these three quantities jointly control the two-photon correlation and the average distance between the two walkers, so coherence and phase are tunable parameters of the walk rather than mere sources of error. If the paper is right, two-photon quantum-walk experiments can engineer bunching or antibunching by choosing the initial degree of coherence and phase, not just the input sites.

What carries the argument

The load-bearing object is the input density matrix $\rho_{qr,q'r'}$ on the two-particle occupation basis together with the single-particle propagator $U_{q,r}(t)$ obtained from the equation of motion $\mathrm{i}\partial_t\hat a_q^\dagger=\beta_q\hat a_q^\dagger+T_{q,q+1}\hat a_{q+1}^\dagger+T_{q,q-1}\hat a_{q-1}^\dagger$. Eq. (8) combines them into the exact two-particle correlation function, separating the cases where the two particles start on the same site from distinct sites. In the concrete scheme, all coherence effects are concentrated in one term, $2\gamma\,\mathrm{Re}(e^{i\varphi}U_{q0}U_{r0}U_{r1}^*U_{q1}^*)$, with $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$; this is the term that produces two-photon interference, that grows as $\eta$ increases, and that flips sign when $\varphi$ changes from $0$ to $\pi$. Its sign, through $\cos\varphi$, decides whether the two photons are pushed apart or pulled together in the numerical correlation maps and in the average-distance expression (13).

What would settle it

Perform the proposed two-photon walk in a uniform waveguide array with $\alpha=0.5$ and a fixed partial coherence (for instance $\eta=0.5$), and measure the average distance $d$ between the two photons at time $t=4/C$ while the relative phase $\varphi$ is scanned from $0$ to $\pi$. Equation (13) predicts that $d$ varies sinusoidally with $\varphi$ with amplitude proportional to $\sqrt{\eta}$, and that at $\eta=0$ the curve is exactly flat. Observing no phase dependence at $\eta=0.5$, or observing a phase dependence at $\eta=0$, would falsify the paper's central claim.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the two-particle correlation function $\Gamma_{k,l}(t)=\langle\hat a_k^\dagger(t)\hat a_l^\dagger(t)\hat a_l(t)\hat a_k(t)\rangle$ for a two-boson quantum walk is determined by the input density matrix through Eq. (8), with no assumption that the input is pure. For the proposed mixed two-photon state, $\rho=\cos^2\delta|\psi_1\rangle\langle\psi_1|+\sin^2\delta|\psi_2\rangle\langle\psi_2|$ with $\psi_1=\cos(\theta/2)|2\rangle_1+\sin(\theta/2)e^{i\varphi}|2\rangle_0$ and $\psi_2=|2\rangle_1$, the correlation becomes $\Gamma_{q,r}=2\gamma\,\mathrm{Re}(e^{i\varphi}U_{q0}U_{r0}U_{r1}^*U_{q1}^*)+2\alpha|U_{r0}U_{q0}|^2+2(1-\alpha)|U_{r1}U_{q1}|^2$. In a uniform periodic lattice this is (12), $\Gamma_{q,r}(\tau)=-2\gamma\cos\varphi\, J_q(\tau)J_r(\tau)J_{r-1}(\tau)J_{q-1}(\tau)+2\alpha[J_q(\tau)J_r(\tau)]^2+2(1-\alpha)[J_{r-1}(\tau)J_{q-1}(\tau)]^2$. The coherence of the input enters through $\eta=2\,\mathrm{Tr}(\rho^2)-1$ inside $\gamma$; when $\eta=0$ the interference term vanishes, when $\eta=1$ the pure-state two-photon interference is recovered. The average distance between the two photons inherits the same $\gamma\cos\varphi$ factor, which is why the distance grows with $\eta$ for $\varphi=0$, shrinks with $\eta$ for $\varphi=\pi$, and is insensitive to $\eta$ at $\varphi=\pi/2$.

Load-bearing premise

The quantitative predictions of the proposed scheme rest on the assumption that the two mutually incoherent beams, after passing through gratings and entering the waveguide array, really produce the density matrix in Eq. (9) with exactly the stated $\psi_1$ and $\psi_2$, so that the off-diagonal element is $\rho_{00,11}=\frac12 e^{i\varphi}\sqrt{\eta-1+4\alpha(1-\alpha)}$; any residual coherence, spectral distinguishability, or multi-pair emission that changed this element would alter the interference term in Eq. (11) and invalidate the predicted photon separation.

Editorial extensions

If this is right

Experimentalists can compute the full two-photon coincidence map for any pure or mixed input by substituting the initial density matrix into Eq. (8), without re-solving the two-particle problem.
The degree of coherence $\eta$ becomes a tunable control: at $\varphi=0$ raising $\eta$ increases the average photon distance, at $\varphi=\pi$ it decreases it, and at $\varphi=\pi/2$ it leaves the distance unchanged.
The relative phase $\varphi$ acts as a switch between bunching and antibunching, so the sign of two-photon interference can be chosen in advance by setting the initial phase.
In the fully incoherent limit $\eta=0$, the correlation is a simple incoherent sum of single-particle probability products, and the nonclassical two-photon interference signature disappears entirely.
Because the general formula depends on the lattice only through the single-particle amplitude $U_{q,r}(t)$, it also applies to lattices with defects, non-uniform tunnel couplings, or other single-particle geometries.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same coherence-parameter route could be applied to fermionic or interacting walkers: the sign and magnitude of the interference term would then encode exchange statistics or interaction phases, making a partially coherent two-particle walk a possible probe of statistical effects.
Because the amplitude of the phase-dependent part of the average distance $d$ is proportional to $\sqrt{\eta}$ at fixed $\alpha$, a measurement of $d(\varphi)$ in this setup would give a direct experimental estimate of the degree of coherence $\eta$; the framework thereby doubles as a coherence metrology tool.
The simple formula $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$ works because the mixed state has only one off-diagonal pair; for inputs with several mutually coherent sectors, a single scalar $\eta$ would no longer suffice and the full off-diagonal density matrix would be needed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The genuinely new thing here is Eq. (8): a closed expression for the two-boson correlation function from an arbitrary density matrix. That is not in the earlier pure-state papers, and it is a natural and potentially usable tool for people planning waveguide experiments with partially coherent inputs. The pure-state limit reproduces Ref. [16], and the Bessel form Eq. (12) is consistent with the known single-particle propagator. I would also credit the authors for clearly parameterizing the mixed state via the coherence measure eta and for showing, in Figs. 1–3, that the HBT interference term and the average photon distance respond to eta and phi in a straightforward way.

Now the soft spots, in proportion. The sign in Eq. (6) is wrong: commuting a†_q with the Hamiltonian in Eq. (1) gives i∂_t a†_q = -β_q a†_q + T_{q,q+1}a†_{q+1} + T_{q,q-1}a†_{q-1}. The paper has a plus sign on the beta term. For constant beta, that only changes a global phase in U and cancels in the correlation function, so the main demonstration survives. But the paper explicitly allows site-dependent beta_q, including the boundary-defect case. For non-uniform beta, the U obtained from the wrong equation is not the propagator of Eq. (1), so Eq. (8) as derived does not have the advertised generality. This is a real flaw in the presentation, though not one that destroys the constant-tunneling result.

The derivation of Eq. (8) is omitted. In a short paper that is forgivable if the final formula is checkable, and I did check the pure-state limit. Still, a referee should ask for the intermediate steps, especially the diagonal/half-filling terms with the √2 factors.

A smaller point: the statement that the numerical results are 'explained' by the analytical formula is tautological, since the numerics are evaluations of the same formula. The qualitative discussion of HBT bunching versus anti-bunching is fine, but the phrasing should be softened.

The mixed-state preparation via two incoherent beams is reasonable but not deeply analyzed. Residual coherence between the beams or spectral distinguishability would change the off-diagonal rho_{00,11}, and the paper does not discuss tolerances. That is a minor concern for a theory paper.

Overall: the central physical claim—coherence and relative phase control two-photon correlations in a continuous-time quantum walk—holds for the constant-tunneling case. The paper is incremental and has a load-bearing sign issue in the general formalism, but the specific demonstration is solid and the tool could be useful.

Recommendation: I would send this to peer review with a request to fix the sign in Eq. (6), provide the derivation of Eq. (8), and narrow the generality claim to match what is actually proven. A competent referee can handle that without much trouble.

Referee Report

2 major / 5 minor

Summary. This paper studies a continuous-time quantum walk of two noninteracting bosons on a one-dimensional lattice, described by the tight-binding Hamiltonian (1). The authors develop a density-matrix formalism that allows the initial two-particle state to be pure or mixed, and they derive a general expression, Eq. (8), for the two-boson correlation function Γ_{k,l}(t) in terms of single-particle propagators U_{q,r}(t) and the initial density-matrix elements. For a concrete mixed initial state parameterized by an occupation probability α, a coherence parameter η = 2Tr(ρ^2) − 1, and a relative phase φ, they obtain the closed form Eq. (11) and its Bessel-function version Eq. (12) for a uniform lattice. They also compute the average inter-photon distance, Eq. (13), and present numerical plots showing that the correlation function and the distance depend on α, η, and φ, with the coherent first term in Eq. (11) responsible for Hanbury Brown–Twiss-type interference. The pure-state limit is shown to reproduce a known result from Ref. [16].

Significance. If correct, the paper provides a useful analytic tool for two-particle bosonic quantum walks with partially coherent initial states. The general formula (8) goes beyond earlier pure-state treatments and could serve as a reference for experiments in waveguide lattices. The derivation is standard second-quantization, and the pure-state limit matches the known result of Bromberg et al. The concrete prediction that the coherence parameter η and the relative phase φ control the interference contribution to the two-photon correlation and to the average distance is the main value of the work. The paper does not present machine-checked proofs or reproducible code; the numerical results are evaluations of the analytic formulas. Overall, this is a reasonable contribution to the quantum-walk literature, though not a conceptual breakthrough.

major comments (2)

[§II, Eq. (6)] Equation (6) has a sign error in the β term. For the Hamiltonian in Eq. (1), the Heisenberg equation of motion is i∂_t a†_q = −β_q a†_q + T_{q,q+1} a†_{q+1} + T_{q,q−1} a†_{q−1}, because [a†_q, a†_j a_j] = −a†_j δ_{q,j}. The published Eq. (6) has +β_q, which corresponds to a Hamiltonian with −β_q in the potential term, not the one written in Eq. (1). For the constant-β case used in Sec. III, this only changes a global phase in U_{q,r}(t) and that phase cancels in Γ, so Eqs. (11)–(13) and Figs. 1–3 remain valid. However, Sec. II explicitly allows site-dependent β_q, including the boundary-defect case, and for such β_q the U obtained from Eq. (6) is not the propagator of Eq. (1). Since Eq. (8) is advertised as a general result for arbitrary initial states, this sign inconsistency must be corrected and the derivation re-examined.
[§II, Eq. (8)] The derivation of Eq. (8) from Eqs. (6)–(7) is omitted. The formula contains four sums with explicit factors of √2 in the double-occupancy contributions, and without an explicit derivation or a precise reference the reader cannot verify that all combinatoric factors are correct. Because Eq. (8) is the central analytical claim of the paper, the authors should include the derivation (for example in an appendix) or give a reference that contains the general two-boson correlation formula for arbitrary density matrices.

minor comments (5)

[§III, Eq. (9)] The text states the intensity relation as "cos 2δ : sin2δ", which appears to mean cos^2 δ : sin^2 δ; please write this unambiguously to match Eq. (9).
[§III, after Eq. (10)] The reparameterization ρ00,00 = α and ρ11,11 = 1−α should be stated explicitly in terms of the original parameters δ and θ, i.e., α = cos^2δ sin^2(θ/2) and 1−α = cos^2δ cos^2(θ/2)+sin^2δ, so that the range constraints and the expression for γ are transparent.
[§III, Eq. (11)] The expression for ρ00,11 contains "η − 1 − 4α^2 + 4α"; writing it as η − 1 + 4α(1 − α) would make the nonnegativity of the radicand clearer.
[Figures 2 and 4] The captions should specify the number of lattice sites L for all figures; currently only Fig. 4 states L = 15, while Figs. 2 and 3 do not, which makes the numerical plots hard to reproduce.
[§III, proposed scheme] The preparation scheme leading to Eq. (9) assumes that two mutually incoherent beams, after grating splitting, produce exactly the mixture cos^2δ|ψ1⟩⟨ψ1| + sin^2δ|ψ2⟩⟨ψ2|. The paper should state this as an idealization and briefly comment on how residual coherence between the beams, spectral distinguishability, or multi-pair emission would affect the off-diagonal element ρ00,11 and hence the interference term in Eq. (11).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the correlation formulas are derived from the stated density matrix and propagator, with no fitted target or self-citation chain.

full rationale

The derivation chain is self-contained. The input is the two-boson density matrix (10); the coherence parameter eta is explicitly defined in the text as eta = 2Tr(rho^2) - 1, and the off-diagonal element rho_{00,11} is parameterized through it. Equation (11) follows from the general trace formula (8) by direct evaluation with this density matrix, and Eq. (12) substitutes the known Bessel-function propagator U_{q,r}(t) = e^{2iCt} i^{q-r} J_{q-r}(2Ct). No parameter is fitted to the correlation or distance data; the numerical figures merely evaluate the derived analytical expressions. The pure-state limit reproduces the cited result of Bromberg et al. (Ref. [16]). The only self-citation, Ref. [29], supplies a standard definition of the coherence measure that is restated explicitly in the present paper, so it is not load-bearing. The sign discrepancy between Eq. (6) and the Hamiltonian in Eq. (1) is a correctness concern about the claimed generality of the derivation, not a circularity: Eq. (8) is not assumed as an input but is presented as a derived consequence. Therefore no step reduces the paper's predictions to its inputs by construction.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a standard non-interacting tight-binding model, the Bessel-function propagator, and an idealized mapping from incoherent beams to the density matrix (9). No additional new entities are introduced; the listed alpha, eta, phi are state-preparation parameters, not fitted constants.

free parameters (3)

alpha
Diagonal weight of the two-photon state on site 0; chosen by hand to scan initial distributions.
eta
Purity measure 2Tr(rho^2)-1; controls the magnitude of the coherent term in Eq. (11).
phi
Relative phase between the two components; controls the sign of the interference term.

assumptions (3)

domain assumption The two photons are non-interacting bosons evolving under the tight-binding Hamiltonian (1).
The entire correlation formula (8) assumes free (non-interacting) propagation; interactions would add extra terms to H and change the correlation.
standard math The single-particle propagator on a uniform lattice is U_{q,r}(tau)=e^{i2Ct} i^{q-r}J_{q-r}(2Ct).
Standard Bessel-function solution of the tight-binding model for uniform tunneling and constant on-site term, used in Eqs. (12) and (13).
ad hoc to paper The two incoherent beams produce the density matrix rho = cos^2(delta)|psi_1><psi_1| + sin^2(delta)|psi_2><psi_2| with psi_1 and psi_2 as in Eq. (9).
This is an idealized modeling assumption specific to the proposed scheme; no experimental validation is given.

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Cite this review

Pith. "Pith review of The role of coherence on two-particle quantum walks." pith.science (2026). https://pith.science/paper/CFVXGVU6

@misc{pith2026190804987,
  author       = {Pith},
  title        = {Pith review of: The role of coherence on two-particle quantum walks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFVXGVU6}},
  note         = {Machine review of arXiv:1908.04987}
}

read the original abstract

We investigate the dynamical properties of the two-bosons quantum walk in system with different degrees of coherence, where the effect of the coherence on the two-bosons quantum walk can be naturally introduced. A general analytical expression of the two-bosons correlation function for both pure states and mixed states is given. We propose a possible two-photon quantum-walk scheme with a mixed initial state and find that the two-photon correlation function and the average distance between two photons can be influenced by either the initial photon distribution, or the relative phase, or the degree of coherence. The propagation features of our numerical results can be explained by our analytical two-photon correlation function.

Figures

Figures reproduced from arXiv: 1908.04987 by the authors.

**Figure 2.** FIG. 2: (Color online) The time evolution of the distance bet [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: (Color online) The dependence of the distance betwee [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: (Color online) The time evolution of the von Neumann e [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [16]

Karski,L

M. Karski,L. F¨ orster, J. M. Choi, A. Steﬀen, W. Alt, D. M eschede, A. Widera: ”Quantum Walk in Position Space with Single Optically Trapped Atoms ”, Science, Vol. 325, (2009) , PP. 174

work page 2009

[2] [1]

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work page

[3] [2]

Collaborative Innovation Center of Advanced Microstruc tures, Nanjing, P. R. China (Received August 15, 2019) We investigate the dynamical properties of the two-bosons q uantum walk in system with diﬀerent degrees of coherence, where the eﬀect of the coherence on the two-bosons quantum walk can be naturally introduced. A general analytical expression of t...

work page 2019

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Y. Aharonov, L. Davidovich, and N. Zagury: ”Quantum rand om walks” Phys. Rev. A, Vol. 48, (1993), PP. 1687

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M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik : ”Environment-assisted quantum walks in photosynthetic energy transfer”, J. Chem. Phys., Vol. 129, (2008), PP. 1741 06

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