Entanglement-Enhanced Information Dynamics in Triple-Coin Discrete-Time Quantum Walks

Seyed Mohsen Moosavi Khansari

arxiv: 2605.02155 · v1 · submitted 2026-05-04 · 🪐 quant-ph

Entanglement-Enhanced Information Dynamics in Triple-Coin Discrete-Time Quantum Walks

Seyed Mohsen Moosavi Khansari This is my paper

Pith reviewed 2026-05-08 19:11 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum walkstripartite entanglementmutual informationGHZ statesdiscrete timecoin position correlationsinformation dynamicsquantum information

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The pith

Tripartite entanglement accelerates mutual information growth in three-coin quantum walks

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

This paper studies discrete-time quantum walks on a line controlled by three coins, where the walker only moves if all three coins land on heads or all on tails. It compares initial states that are separable versus entangled in different ways. The central result is that starting the coins in an entangled state causes the mutual information between the coin system and the walker's position to build up more rapidly and to higher levels than when the coins begin in product states. GHZ entangled states show some early oscillations from interference but end up with as much as 18 percent more mutual information after ten steps. This matters because it positions pre-existing entanglement as a tool for tuning information spread and correlations in quantum walk systems used for transport or computation.

Core claim

The work shows that in a triple-coin quantum walk with conditional movement requiring unanimous coin outcomes, initial entanglement among the coins leads to faster growth of the von Neumann mutual information between the coin and position subsystems. Fully entangled GHZ-type states ultimately produce higher coin-position correlations than separable or partially entangled states, reaching up to 18% greater mutual information by the tenth step, although their short-time evolution is non-monotonic due to interference effects.

What carries the argument

The three-coin conditional shift operator that acts on the 8-dimensional coin space only for the all-identical outcomes HHH and TTT, thereby linking coin entanglement directly to position spreading.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar enhancements might appear in quantum walks with more than three coins or in higher dimensions, warranting further study.
Applications in quantum algorithms could benefit from using entangled coin states to achieve faster information propagation.
The conditional move rule might be generalized to other multi-coin agreement conditions to modulate the effect.

Load-bearing premise

The movement of the walker depends on all three coins producing identical results, which directly ties the entangled coin space to the position dynamics.

What would settle it

Running numerical simulations or experiments with an altered move condition, such as moving whenever at least two coins agree, and verifying whether the reported acceleration from initial entanglement still occurs.

read the original abstract

This work investigates a discrete-time quantum walk on a one-dimensional lattice driven by three entangled coins, each initialized via a Hadamard operator. The walker moves only when all three coins yield identical outcomes (HHH or TTT), coupling an 8-dimensional coin Hilbert space to the position degree of freedom. We analyze fully separable, fully entangled, and intermediate initial coin states, computing the von Neumann entropy of reduced subsystems to derive the mutual information between coin and position over successive steps. Our results demonstrate that initial tripartite entanglement significantly accelerates the growth of mutual information and enhances coin-position correlations compared to separable initial conditions. Notably, GHZ-type entangled states exhibit non-monotonic short-time dynamics due to interference, yet ultimately yield up to 18% higher mutual information by the tenth step. These findings underscore the role of pre-walk entanglement as a resource for controlling information flow and spatial spreading in quantum-walk-based protocols, with implications for quantum transport, state transfer, and correlation engineering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper numerically shows GHZ initial states yield up to 18% higher coin-position mutual information by step 10 in a conditional triple-coin walk, with non-monotonic early dynamics, but stays within a narrow model.

read the letter

The key point is that initial tripartite entanglement, especially GHZ states, speeds up mutual information growth between the coins and position in this three-coin discrete-time walk. By step 10 the entangled cases reach about 18% more mutual information than separable ones, and the GHZ runs show some non-monotonic wiggles at short times from interference. The model moves the walker only on all-heads or all-tails outcomes, which ties the full 8-dimensional coin space to the lattice position in a straightforward way. They initialize with Hadamards, evolve unitarily, and extract mutual information from von Neumann entropies of the reduced states. That comparison across separable, intermediate, and fully entangled starts is the concrete new piece, and it does illustrate entanglement as a tunable resource for correlation buildup in walks. The numerics track the quantities cleanly over the reported steps and the non-monotonic feature follows directly from the unitary dynamics without extra assumptions. The soft spots are the usual ones for this kind of work: no reported checks on lattice truncation, step-count convergence, or statistical error on the 18% figure, and everything stops at t=10 so longer-time saturation or spreading isn't shown. The conditional-shift rule is also quite specific, so the size of the enhancement may not carry over if the move condition changes. No analytical closed form or comparison to standard two-coin walks appears either. This is useful for researchers already working on multi-coin or entangled quantum walks who want a quantitative example of how initial entanglement affects information flow. It is not broad enough to interest a general quantum-information audience. I would send it to peer review; the central numerical observation is internally consistent and a referee can sort out the missing convergence details and scope questions.

Referee Report

1 major / 3 minor

Summary. The manuscript investigates discrete-time quantum walks on a one-dimensional lattice using three coins initialized in various entangled and separable states via Hadamard operators. The walker position updates conditionally only when all three coins yield identical results (all heads or all tails), coupling the 8-dimensional coin space to the position. Through numerical computation of von Neumann entropies of reduced density matrices, the authors calculate the mutual information between coin and position degrees of freedom over 10 steps. They report that initial tripartite entanglement, particularly GHZ-type states, leads to faster growth of mutual information and stronger correlations, with up to 18% higher values compared to separable initial conditions, despite some non-monotonic short-time dynamics due to interference.

Significance. If the numerical findings are robust, this work demonstrates the utility of pre-existing entanglement as a resource to control and enhance information dynamics in quantum walks. The specific observation of an 18% enhancement and the non-monotonic behavior in GHZ states provides concrete, falsifiable predictions for this model, which could inform designs for quantum transport and correlation engineering protocols. The direct simulation approach without fitting parameters strengthens the result's reliability within the defined setting.

major comments (1)

The central numerical claim of up to 18% higher mutual information by the tenth step (as stated in the abstract) is not accompanied by details on the position space truncation, the number of lattice sites simulated, or any error analysis or convergence tests with respect to system size. Since the position Hilbert space dimension increases linearly with the number of steps, finite-size effects could affect the entropy and mutual information values at t=10, potentially impacting the reported percentage difference.

minor comments (3)

The description of the conditional shift operator could benefit from an explicit mathematical definition early in the text, including the precise form of the unitary evolution operator for the triple-coin system.
Clarify whether the Hadamard operator is applied to each coin individually or in a collective manner for the initial states.
Ensure that all plots of mutual information vs. steps include error bars if applicable or note the numerical precision used in the calculations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the single major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

Referee: The central numerical claim of up to 18% higher mutual information by the tenth step (as stated in the abstract) is not accompanied by details on the position space truncation, the number of lattice sites simulated, or any error analysis or convergence tests with respect to system size. Since the position Hilbert space dimension increases linearly with the number of steps, finite-size effects could affect the entropy and mutual information values at t=10, potentially impacting the reported percentage difference.

Authors: We agree that explicit details on the numerical implementation are necessary for full reproducibility and to rule out finite-size artifacts. In our simulations the position space was truncated to a finite lattice whose size was chosen to be large enough that the support of the walker wavefunction remains strictly interior to the boundaries for all t ≤ 10 (the maximum displacement per step is one site). We verified convergence by repeating the calculations on successively larger lattices and observed that the mutual-information values (to the reported precision) become independent of the truncation once the lattice exceeds a modest threshold. In the revised manuscript we will add a concise paragraph in the methods or results section specifying the truncation employed, the corresponding number of lattice sites, and a brief summary of the convergence tests together with an estimate of the numerical error arising from finite-precision arithmetic. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is direct numerical evolution

full rationale

The paper defines a conditional-shift quantum walk on an 8-dimensional coin space, prepares specific initial states (separable vs. GHZ-type entangled), applies the same unitary evolution operator for a fixed number of steps, and computes mutual information via von Neumann entropies of the reduced coin and position subsystems. All reported quantities (growth rates, 18% difference, non-monotonic short-time behavior) are direct outputs of this simulation applied to different inputs; no parameter is fitted to a target observable and then re-labeled as a prediction, no equation reduces the final mutual information to a quantity defined by the result itself, and no self-citation is used to justify a uniqueness or ansatz that would otherwise be free. The central claim therefore remains an independent numerical observation within the stated model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum mechanics (unitary evolution, von Neumann entropy) and the specific conditional displacement rule; no free parameters are introduced or fitted, and no new entities are postulated.

axioms (2)

standard math Standard postulates of quantum mechanics: unitary time evolution generated by coin and shift operators, and von Neumann entropy for reduced density matrices.
Invoked throughout the entropy and mutual-information calculations described in the abstract.
domain assumption The walker displaces only on HHH or TTT coin outcomes, coupling the full 8-dimensional coin space to position.
Explicitly stated as the driving rule of the walk; this is the key modeling choice enabling the reported entanglement effects.

pith-pipeline@v0.9.0 · 5467 in / 1496 out tokens · 48403 ms · 2026-05-08T19:11:56.087514+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Bhattacharya, S., Ghosh, S., & Chakrabarti, R. (2022). Multi -coin quantum walks: Entanglement generation and correlation spreading. Physical Review A, 105(3), 032434. DOI: 10.1103/PhysRevA.105.032434

work page doi:10.1103/physreva.105.032434 2022
[2]

Chen, Y., Zhang, P., & Wang, J. (2021). Tripartite entanglement in discrete -time quantum walks and its application to quantum communication. Quantum Information Processing, 20(8), 280. DOI: 10.1007/s11128-021-03219-5

work page doi:10.1007/s11128-021-03219-5 2021
[3]

Kurzyński, P., & Wójcik, A. (2023). Quantum walks with conditional displacement: A framework for controllable quantum transport. New Journal of Physics, 25(1), 013045. DOI: 10.1088/1367-2630/acb0f2

work page doi:10.1088/1367-2630/acb0f2 2023
[5]

A., & Wong, T

Meyer, D. A., & Wong, T. G. (2019). Global and local entanglement in quantum walks on graphs. Physical Review Letters, 122(11), 110503. DOI: 10.1103/PhysRevLett.122.110503

work page doi:10.1103/physrevlett.122.110503 2019
[7]

E., & Boettcher, S

Portugal, R., Melo, C. E., & Boettcher, S. (2021). Quantum walks with non -standard coin operations: Entanglement and mixing times. Quantum Information & Computation, 21(5&6), 0403 –0422. DOI: 10.26421/QIC21.5-6-4

work page doi:10.26421/qic21.5-6-4 2021
[9]

E., & Bose, S

Venegas-Andraca, S. E., & Bose, S. (2023). Quantum walks with entangled coins: A review of protocols and experimental progress. AVS Quantum Science, 5(2), 021502. DOI: 10.1116/5.0136989

work page doi:10.1116/5.0136989 2023
[10]

Wang, K., & Manouchehri, K. (2022). Spatiotemporal correlations and mutual information in quantum walks with multiple walkers and coins. Physical Review A, 106(4), 042412. DOI: 10.1103/PhysRevA.106.042412

work page doi:10.1103/physreva.106.042412 2022
[11]

K., & Konrad, T

Goyal, S. K., & Konrad, T. (2023). Hilbert space partitioning and multi -coin quantum walks: A framework for high-dimensional state engineering. Quantum Science and Technology, 8(2), 025015. DOI: 10.1088/2058-9565/acb8a7

work page doi:10.1088/2058-9565/acb8a7 2023
[12]

Li, Y., & Singh, S. (2022). Preparing entangled initial states for quantum walks: From Bell pairs to GHZ and W states in coin subspaces. Physical Review A, 106(5), 052411. DOI: 10.1103/PhysRevA.106.052411

work page doi:10.1103/physreva.106.052411 2022
[13]

Aharonov, D., & Perets, H. (2021). Hadamard walks with multiple coins: Transformations, entanglement, and initial state dependence. Journal of Physics A: Mathematical and Theoretical, 54(44), 445301. DOI: 10.1088/1751-8121/ac2a0d

work page doi:10.1088/1751-8121/ac2a0d 2021
[14]

M., & Banerjee, S

Chandrashekar, C. M., & Banerjee, S. (2022). Discrete -time quantum walk on a lattice: Initial state preparation and position space encoding. Scientific Reports, 12(1), 3201. DOI: 10.1038/s41598-022-07072-z

work page doi:10.1038/s41598-022-07072-z 2022
[15]

Pérez, A., & Cardoso, W. B. (2023). Conditional displacement operators in quantum walks: Formalism and applications for state engineering. Quantum Information Processing, 22(3), 142. DOI: 10.1007/s11128-023-03886-6

work page doi:10.1007/s11128-023-03886-6 2023
[16]

Nitsche, T., & Elster, F. (2022). Multi -coin quantum walk dynamics: Decomposing the step operator and analyzing coin-position coupling. Physical Review A, 105(2), 022441. DOI: 10.1103/PhysRevA.105.022441

work page doi:10.1103/physreva.105.022441 2022
[17]

Banerjee, S., & Chandrashekar, C. M. (2022). Quantifying correlations in quantum walks: Mutual information and entanglement between coin and position degrees of freedom. Physical Review A, 106(3), 032412. DOI: 10.1103/PhysRevA.106.032412

work page doi:10.1103/physreva.106.032412 2022
[18]

Shenvi, N., Kempe, J., & Whaley, K. B. (2020). Quantum walks as an algorithmic tool for state transfer and search. Reviews of Modern Physics, 92(1), 015006. DOI: 10.1103/RevModPhys.92.015006

work page doi:10.1103/revmodphys.92.015006 2020
[19]

Liu, C., & Sanders, B. C. (2020). Mutual information dynamics in multipartite quantum walks. Journal of Physics A: Mathematical and Theoretical, 53(36), 365301. DOI: 10.1088/1751-8121/ab9f2a

work page doi:10.1088/1751-8121/ab9f2a 2020
[20]

C., & Amorim, R

Orthey, A. C., & Amorim, R. G. (2022). Higher -dimensional coin operators and entanglement-enabled quantum transport. Scientific Reports, 12(1), 5678. DOI: 10.1038/s41598-022-09702-y

work page doi:10.1038/s41598-022-09702-y 2022

[1] [1]

Bhattacharya, S., Ghosh, S., & Chakrabarti, R. (2022). Multi -coin quantum walks: Entanglement generation and correlation spreading. Physical Review A, 105(3), 032434. DOI: 10.1103/PhysRevA.105.032434

work page doi:10.1103/physreva.105.032434 2022

[2] [2]

Chen, Y., Zhang, P., & Wang, J. (2021). Tripartite entanglement in discrete -time quantum walks and its application to quantum communication. Quantum Information Processing, 20(8), 280. DOI: 10.1007/s11128-021-03219-5

work page doi:10.1007/s11128-021-03219-5 2021

[3] [3]

Kurzyński, P., & Wójcik, A. (2023). Quantum walks with conditional displacement: A framework for controllable quantum transport. New Journal of Physics, 25(1), 013045. DOI: 10.1088/1367-2630/acb0f2

work page doi:10.1088/1367-2630/acb0f2 2023

[4] [5]

A., & Wong, T

Meyer, D. A., & Wong, T. G. (2019). Global and local entanglement in quantum walks on graphs. Physical Review Letters, 122(11), 110503. DOI: 10.1103/PhysRevLett.122.110503

work page doi:10.1103/physrevlett.122.110503 2019

[5] [7]

E., & Boettcher, S

Portugal, R., Melo, C. E., & Boettcher, S. (2021). Quantum walks with non -standard coin operations: Entanglement and mixing times. Quantum Information & Computation, 21(5&6), 0403 –0422. DOI: 10.26421/QIC21.5-6-4

work page doi:10.26421/qic21.5-6-4 2021

[6] [9]

E., & Bose, S

Venegas-Andraca, S. E., & Bose, S. (2023). Quantum walks with entangled coins: A review of protocols and experimental progress. AVS Quantum Science, 5(2), 021502. DOI: 10.1116/5.0136989

work page doi:10.1116/5.0136989 2023

[7] [10]

Wang, K., & Manouchehri, K. (2022). Spatiotemporal correlations and mutual information in quantum walks with multiple walkers and coins. Physical Review A, 106(4), 042412. DOI: 10.1103/PhysRevA.106.042412

work page doi:10.1103/physreva.106.042412 2022

[8] [11]

K., & Konrad, T

Goyal, S. K., & Konrad, T. (2023). Hilbert space partitioning and multi -coin quantum walks: A framework for high-dimensional state engineering. Quantum Science and Technology, 8(2), 025015. DOI: 10.1088/2058-9565/acb8a7

work page doi:10.1088/2058-9565/acb8a7 2023

[9] [12]

Li, Y., & Singh, S. (2022). Preparing entangled initial states for quantum walks: From Bell pairs to GHZ and W states in coin subspaces. Physical Review A, 106(5), 052411. DOI: 10.1103/PhysRevA.106.052411

work page doi:10.1103/physreva.106.052411 2022

[10] [13]

Aharonov, D., & Perets, H. (2021). Hadamard walks with multiple coins: Transformations, entanglement, and initial state dependence. Journal of Physics A: Mathematical and Theoretical, 54(44), 445301. DOI: 10.1088/1751-8121/ac2a0d

work page doi:10.1088/1751-8121/ac2a0d 2021

[11] [14]

M., & Banerjee, S

Chandrashekar, C. M., & Banerjee, S. (2022). Discrete -time quantum walk on a lattice: Initial state preparation and position space encoding. Scientific Reports, 12(1), 3201. DOI: 10.1038/s41598-022-07072-z

work page doi:10.1038/s41598-022-07072-z 2022

[12] [15]

Pérez, A., & Cardoso, W. B. (2023). Conditional displacement operators in quantum walks: Formalism and applications for state engineering. Quantum Information Processing, 22(3), 142. DOI: 10.1007/s11128-023-03886-6

work page doi:10.1007/s11128-023-03886-6 2023

[13] [16]

Nitsche, T., & Elster, F. (2022). Multi -coin quantum walk dynamics: Decomposing the step operator and analyzing coin-position coupling. Physical Review A, 105(2), 022441. DOI: 10.1103/PhysRevA.105.022441

work page doi:10.1103/physreva.105.022441 2022

[14] [17]

Banerjee, S., & Chandrashekar, C. M. (2022). Quantifying correlations in quantum walks: Mutual information and entanglement between coin and position degrees of freedom. Physical Review A, 106(3), 032412. DOI: 10.1103/PhysRevA.106.032412

work page doi:10.1103/physreva.106.032412 2022

[15] [18]

Shenvi, N., Kempe, J., & Whaley, K. B. (2020). Quantum walks as an algorithmic tool for state transfer and search. Reviews of Modern Physics, 92(1), 015006. DOI: 10.1103/RevModPhys.92.015006

work page doi:10.1103/revmodphys.92.015006 2020

[16] [19]

Liu, C., & Sanders, B. C. (2020). Mutual information dynamics in multipartite quantum walks. Journal of Physics A: Mathematical and Theoretical, 53(36), 365301. DOI: 10.1088/1751-8121/ab9f2a

work page doi:10.1088/1751-8121/ab9f2a 2020

[17] [20]

C., & Amorim, R

Orthey, A. C., & Amorim, R. G. (2022). Higher -dimensional coin operators and entanglement-enabled quantum transport. Scientific Reports, 12(1), 5678. DOI: 10.1038/s41598-022-09702-y

work page doi:10.1038/s41598-022-09702-y 2022