Entanglement dynamics in minimal Kitaev chains

Jorge Cayao; Vimalesh Kumar Vimal

REVIEW 3 major objections 3 minor 3 cited by

Entanglement dynamics in minimal Kitaev chains

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Small Kitaev chains can be timed to emit maximally entangled states, and parity conservation blocks only the pure W state.

desk verdict Bipartite dynamics are correct and useful, but the multipartite GME is computed against a restricted family of product states and flags product states as entangled; the central multipartite claim does not hold as written. read the letter →

arxiv 2507.17586 v2 pith:3V7PX3MR submitted 2025-07-23 quant-ph cond-mat.mes-hallcond-mat.stat-mech

Vimalesh Kumar Vimal , Jorge Cayao This is my paper

classification quant-phcond-mat.mes-hallcond-mat.stat-mech PACS 03.67.Mn03.67.-a

keywords Kitaevchainpoorman'sMajoranamodesentanglementdynamicsconcurrencegeometricmeasureofGHZstateWquantumdots

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 shows that two- and three-site Kitaev chains, tiny chains of quantum dots coupled by superconductors, are controllable sources of entanglement. Starting from a separable state in a two-site chain at the sweet spot, the dynamics swing between fully separable and maximally entangled Bell states with a period set by the pairing strength, and detuning the onsite energies reshapes that swing into tunable entanglement valleys. In a three-site chain at the genuine sweet spot, nearest-neighbour entanglement reaches half its maximum while the outer dots never entangle; a finite detuning restores the outer entanglement. Starting from an entangled three-site initial state, the chain produces maximally entangled GHZ states and imperfect W-type states, but a perfect W state cannot appear because fermion parity is conserved. These predictions matter because few-dot Kitaev chains are already built in the lab, so the exact time-evolved states give a practical schedule for generating entangled resources.

What carries the argument

The machinery is spectral decomposition of the parity-conserving many-body Hamiltonians written in the occupation-number basis; for these small chains the eigenstates are closed form, so the time-evolved states are exact. Bipartite entanglement is read from the concurrence, either directly from two-site coefficients or, in the three-site chain, from X-form reduced density matrices using $C_{ij}=2\max\{0,|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}},|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}}\}$. Multipartite entanglement is captured by the geometric measure $E_G=1-\max_\phi|\langle\phi|\psi\rangle|^2$, maximized over separable three-qubit states, with separate overlaps isolating GHZ-type and W-type components.

What would settle it

Measure time-resolved concurrence of the two quantum dots in a two-site Kitaev chain tuned to zero onsite energies with hopping equal to pairing. The paper predicts $C=\sin(2\Delta t)$: at times $t=n\pi/(2\Delta)$ the state should be separable, and at odd multiples of $\pi/(4\Delta)$ maximally entangled. A single oscillation whose minima stay above zero, or whose maxima fall clearly below one, would falsify the isolated-unitary prediction. Similarly, in a three-site chain at the genuine sweet spot the prediction $C_{13}=0$ for all times can be checked directly; a persistent nonzero outer concurrence at that tuning would rule out the claimed suppression.

Watch

Extended reading notes

Core claim

The paper's central claim is that exact unitary dynamics in minimal Kitaev chains generate entangled states in a controlled, predictable way. For the two-site chain with $\varepsilon_i=0$ and $\tau=\Delta$, the concurrence from a separable initial state is $C=\sin(2\Delta t)$, so the system reaches a Bell state at $t=(2n+1)\pi/(4\Delta)$ and is separable at multiples of $\pi/(2\Delta)$. For the three-site chain starting from $|000\rangle$, the time-evolved state is $|\psi(t)\rangle=\cos^2(\Delta t)|000\rangle-\sin^2(\Delta t)|101\rangle-\frac{i}{2}\sin(2\Delta t)(|011\rangle+|110\rangle)$, giving $C_{12}=|\sin(4\Delta t)|/2$ and $C_{13}=0$ at the genuine sweet spot. Starting from $(|000\rangle+|111\rangle)/\sqrt{2}$, the geometric measure of entanglement reaches $E_G^{GHZ}=1/2$ periodically and produces imperfect W-type states whose GME can exceed the perfect-W value $5/9$; a pure W state is excluded because parity conservation keeps the $|000\rangle$ component present throughout.

Load-bearing premise

The calculations assume each quantum dot has a single spinless level, no charging energy, no coupling to the environment, and perfect fermion-parity conservation, with the initial state prepared exactly; if real devices lose parity or dephase, the clean entanglement oscillations and the parity-based exclusion of the W state would not survive.

Editorial extensions

If this is right

At the two-site sweet spot, the chain acts as a clocked Bell-state generator: the pairing potential sets the oscillation frequency, so the duration of each maximal-entanglement pulse is set by a single externally tunable parameter.
Detuning the onsite energies changes the shape of the entanglement oscillation, converting sharp maxima into flat-topped stable plateaus or adding valleys, which gives a practical handle on how long a maximally entangled state persists.
In the three-site chain, vanishing outer concurrence for all times is a dynamical signature of the genuine sweet spot; observing finite outer concurrence at that tuning would signal that the system has moved away from localized poor-man's Majorana modes.
The three-site chain can be used to prepare GHZ states on a periodic schedule, and any W-type state it produces is necessarily imperfect, so quantum-information protocols using W states must be adapted to that constrained form.
Away from the sweet spot, the maximum outer concurrence rises toward maximal values over broad parameter regions, so detuning is not merely a source of error but a resource for generating next-nearest-neighbour entanglement.

Reading between the lines

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

The paper does not discuss readout; a natural extension is to combine these exact trajectories with single-shot parity readout, using the predicted vanishing times of the concurrence as calibration markers for initial-state preparation in real devices.
Because parity conservation is the only obstruction to a pure W state, adding a small parity-breaking perturbation such as quasiparticle poisoning could turn the imperfect W state into a near-perfect one, giving a concrete testable generalization of the paper's no-go result.
The vanishing outer concurrence at the genuine sweet spot suggests a device-level diagnostic: a gate-voltage sweep of the outer two dots' entanglement should show a sharp minimum exactly where poor-man's Majoranas localize, turning an entanglement measurement into a sweet-spot locator.
The two-site oscillation frequency being set by the pairing amplitude and not by the hopping implies an entanglement clock that could be verified experimentally by changing the two parameters independently and watching the period follow the pairing amplitude alone.

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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

Quick take: the two- and three-site bipartite entanglement calculations are done carefully and, as far as I could verify, are correct. The three-site concurrence results, particularly the vanishing C13 at the genuine sweet spot, are a legitimate and potentially useful addition. The GHZ revival dynamics from the GHZ initial state also check out. The multipartite section, however, has a load-bearing flaw. The GME is defined by maximizing the overlap over the symmetric product states of Eq. (32). That restriction is only valid for permutation-symmetric states, and the time-evolved states here are not symmetric. The decisive counterexample: at t=π/(2Δ), the even-sector state from Eq. (36) is exactly -|101>, a product state, but inserting its coefficients into Λ_W gives E_W^G = 1 - 4/27 ≈ 0.852, i.e., a product state is assigned substantial multipartite 'entanglement.' The authors themselves note that high EWG can indicate a separable configuration, which confirms the quantity is not measuring what they claim. The GHZ-type GME suffers from the same problem for non-symmetric states, so the paper's statements about generating GHZ-type and imperfect W-type multipartite entanglement are unsupported.

There is also an internal inconsistency in the three-site sweet-spot eigenvalues: Eq. (12) lists ±Δ for the even sector, but the correct eigenvalues are ±2Δ (as the paper itself uses in the dynamics, and as given for the odd sector in Appendix A). This does not invalidate the time-evolved states they write down, but it is a real error that needs fixing, along with the claim that a pure W state is excluded by parity; the actual reason is that |000> remains present in the eigenstates, not parity alone.

Bottom line: the bipartite half of the paper is a solid, correct piece of work that extends the authors' earlier two-site results to three sites. The multipartite half needs a full rework with a correctly defined entanglement measure (e.g., the full Wei-Goldbart GME or a genuine multipartite measure like the three-tangle). As written, I would not accept the multipartite claims, but the bipartite part is worth preserving. This deserves a serious referee—it is not a desk reject—but the revision demands are major.

Referee Report

3 major / 3 minor

Summary. The manuscript studies unitary entanglement dynamics in two- and three-site Kitaev chains. For the two-site chain it derives exact expressions for concurrence, geometric measure, return probability and entanglement dynamics, reporting controlled oscillations between separable and maximally entangled states, with the sweet spot playing a special role. For the three-site chain it computes nearest-neighbour and next-nearest-neighbour concurrences and introduces a multipartite geometric measure of entanglement, claiming dynamical generation of GHZ-type and imperfect W-type states and asserting that a pure W state is excluded by parity conservation. The paper is analytical and contains closed-form results in Eqs. (26), (36) and (38).

Significance. The bipartite part is a strength: the two-site formulas and the three-site concurrence expression C12=|sin(4Δt)|/2 are exact, transparent, and can be verified directly from the Hamiltonian. If the multipartite analysis were valid, the paper would offer useful predictions for quantum-dot realizations of minimal Kitaev chains. However, the multipartite GME is computed against only permutation-symmetric product states, while the time-evolved states are not symmetric, and the resulting quantity assigns large 'entanglement' to product states. The central claim of GHZ-type and W-type multipartite entanglement generation is therefore unsupported. The paper also overstates the role of parity in excluding the W state. These issues affect the main novelty of the manuscript.

major comments (3)

[Sec. V, Eq. (32)] The GME in Eq. (19) requires maximizing the overlap over all separable states, but Eq. (32) restricts |φ⟩ to the family (cosθ|0⟩+sinθ|1⟩)^{⊗3} and cites Ref. [66] for this restriction. Ref. [66] does not make this restriction: a general three-qubit product state is ⊗_i(cosθ_i|0⟩+e^{iφ_i} sinθ_i|1⟩) with independent parameters per qubit. The time-evolved states in Eqs. (36) and (39) are not permutation-symmetric; for example, in Eq. (36) β1=δ1=−(i/2)sin(2Δt) while γ1=−sin^2(Δt). Maximizing over a subset of product states can only inflate the computed value of 1−|⟨φ|ψ⟩|^2, and the resulting quantities E_GHZ^G and E_W^G in Eq. (40) are not valid entanglement measures for this dynamics.
[Sec. V.A.1, Eq. (40)] A concrete counterexample shows the failure. At t=π/(2Δ), Eq. (36) gives the exact product state |ψ⟩=−|101⟩. Substituting γ1=−1 and all other coefficients zero into Λ_W in Eq. (40) yields max_θ|sin^2θ cosθ|=2/(3√3), hence E_W^G=1−4/27≈0.852. The manuscript's own quantity therefore reports substantial multipartite 'entanglement' for a separable configuration, and the text even states that 'higher values of EWG approaching unity represent a separable configuration' in Sec. V.A.1. This invalidates the claim that the three-site chain dynamically generates GHZ-type and imperfect W-type multipartite entanglement.
[Sec. IV.A.2 and Eq. (32)] The parenthetical claim that Eq. (32) is 'a general state of three qubits up to local phase transformations' is incorrect. A general product state has independent angles and phases for each qubit, and local unitary transformations do not symmetrize the amplitudes. This incorrect identification of the separable set is the source of the invalid multipartite measure used in Sec. V.

minor comments (3)

[Sec. IV.B.2 and Fig. 5(d)] The quantity labelled EG is obtained from Eq. (25) as (1−√(1−C12^2))/2. That identity is valid only for pure two-qubit states; here C12 is the concurrence of the mixed reduced density matrix ρ12 in Eq. (37), so the plotted EG is not the geometric measure of the bipartite mixed state. The concurrence results are unaffected, but the EG curve in Fig. 5(d) should be removed or replaced.
[Abstract and Sec. V] The claim that a pure W state 'cannot be realised due to parity constraints' is imprecise. The even-parity W state |We⟩=(|011⟩+|101⟩+|110⟩)/√3 lies in the same parity sector as |000⟩, so parity conservation does not forbid it; the obstruction visible in Eq. (36) is dynamical, not parity-theoretic.
[Throughout] There are several typos and caption errors: the Introduction contains 'minimal Kitaev chains are can be utilized'; 'Kiatev' appears in Sec. II; Fig. 4's caption says 'at the sweet spot (τ=Δ)' although the section treats Δ≠τ; and Sec. V.A.1 contains 'the the return probability'.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed imperfect W-type multipartite entanglement reduces to a restricted definition: Eq. (32) maximizes over symmetric product states only, so for non-symmetric evolved states, including the exact product state -|101>, E_W^G is positive by construction. Other results are self-contained.

self definitional [Sec. V A, Eqs. (32) and (40); Sec. V A 1, Fig. 7 discussion]
"The set of separable states in a three-site system is defined as in Ref. [66], |ϕ⟩ = (cos θ |0⟩ + sin θ |1⟩)⊗3 , which is a general state of three qubits up to local phase transformations. [...] In this context, the higher values of EW G approaching unity represent a separable configuration, see yellow regions in Fig. 7(b) and green curve in Fig. 7(c)."

Eq. (32) restricts the GME maximization to the permutation-symmetric family with one shared angle θ. The evolved states of Eq. (36) are not symmetric: γ1 = -sin²(Δt) while β1 = δ1 = -(i/2)sin(2Δt). For the even-parity sector used for the W analysis, t = π/(2Δ) gives the exact product state -|101⟩; substituting into Λ_W in Eq. (40) gives maxθ|sin²θ cosθ| = 2/(3√3), hence E_W^G = 1 - 4/27 ≈ 0.852 > 0. The paper's text says 'higher values of EWG approaching unity represent a separable configuration.' So the predicted imperfect W-type multipartite entanglement is produced by the restricted definition itself, not by distance from the true separable set.

full rationale

The two-site and three-site bipartite results are exact unitary evolutions of Eqs. (1)-(2) with standard concurrence formulas and no fitted parameters; C = sin(2Δt), C12 = |sin(4Δt)|/2, and C13 = 0 at the genuine sweet spot are derived, not assumed. Self-citations, e.g. Ref. [29] for return probability and entanglement dynamics, are used only to name elementary overlaps and are not load-bearing. The single definitional circularity is the multipartite GME: Eq. (32) restricts the separable set to symmetric product states while the evolved states are non-symmetric, so E_W^G in Eq. (40) is not the geometric measure of entanglement. The explicit product-state counterexample (-|101⟩ at t = π/(2Δ) gives E_W^G ≈ 0.852) shows that the claimed W-type multipartite entanglement is an artifact of this restricted maximization. The GHZ-type claim, being a fidelity to the symmetric GHZ target, is less affected but is also labeled GME. Overall, the central bipartite dynamics is self-contained; the multipartite W-state prediction partially reduces to the definition of the measure. Correctness risk: the GME restriction is an error, but that is separate from circularity except for the W-state claim that relies on it.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The results are exact for the model but carry the burden of the idealized Hamiltonian: spinless, non-interacting, isolated, with perfectly preparable initial states. No parameters are fitted to outcomes; all scanning parameters (ε, Δ, τ) are physical inputs. No new entities are introduced.

assumptions (5)

domain assumption The spinless, non-interacting Kitaev Hamiltonians H2 and H3 (Eqs. (1)-(2)) describe the quantum-dot-superconductor chains, with only nearest-neighbor hopping and p-wave pairing.
All dynamics and entanglement claims are computed from these Hamiltonians; real devices include spin, charging energy, and additional levels, which are not modeled.
domain assumption Total fermion parity is conserved, so evolution stays in the initial parity sector.
Used to block-diagonalize the spectrum (Sec. II A) and to prove that a pure W state is inaccessible (Sec. V).
standard math The geometric measure of entanglement is the Wei-Goldbart measure, with the maximization restricted to product states of the form Eq. (32) (three copies of a single-qubit state).
Standard definition from Ref. [66], applied in Eqs. (19), (33), (40). Note: the paper mis-applies this definition by using symmetric product states.
domain assumption At the sweet spot ε_i=0, τ_i=Δ_i, the even- and odd-parity ground states are degenerate and Majorana quasiparticles are localized on the outer sites.
This defines the genuine sweet spot (from Refs. [36,56,83]) and is used to interpret the vanishing of C13 as a PMMM signature.
domain assumption The chosen initial states |000> and (|000>+|111>)/√2 can be prepared exactly, and the evolution is unitary (no decoherence or measurement).
Sec. IV B initial states; no preparation fidelity or decoherence analysis is provided.

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Cite this review

Pith. "Pith review of Entanglement dynamics in minimal Kitaev chains." pith.science (2026). https://pith.science/paper/3V7PX3MR

@misc{pith2026250717586,
  author       = {Pith},
  title        = {Pith review of: Entanglement dynamics in minimal Kitaev chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3V7PX3MR}},
  note         = {Machine review of arXiv:2507.17586}
}

read the original abstract

Minimal Kitaev chains host Majorana quasiparticles, which, although not topologically protected, exhibit spatial nonlocality and hence expected to be useful for quantum information tasks. In this work, we consider two- and three-site Kitaev chains and investigate the dynamics of bipartite and multipartite entanglement by means of concurrence and geometric measure of entanglement. In two-site Kitaev chains, we find that maximally entangled states can robustly emerge, with their stability and periodicity highly controllable by the interplay between the superconducting pair potential and the onsite energies. At the finely tuned sweet spot, where Majorana quasiparticles appear, the system exhibits oscillations between separable and entangled states, whereas detuning introduces tunable valleys in the entanglement dynamics. Extending to the three-site Kitaev chain, we uncover rich bipartite and multipartite entanglement by generalizing the concepts of concurrence and geometric measure of entanglement. At the sweet spot, the Majorana quasiparticles emerging at the edges suppress concurrence between the edges, while a finite detuning is able to restore it. Depending on the initial state, the three-site Kitaev chain can dynamically generate either a maximally entangled Greenberger-Horne-Zeilinger state or an imperfect W-type state exhibiting multipartite entanglement, although a (maximally entangled) pure W state cannot be realised due to parity constraints. Our results provide a resource for generating and characterising highly entangled states in minimal Kitaev chains, with potential relevance for quantum applications.

Figures

Figures reproduced from arXiv: 2507.17586 by the authors.

**Figure 2.** FIG. 2. Many-body energy spectra of minimal Kitaev chains [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Entanglement measures at the sweet spot ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Entanglement measures at (∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) Concurrence between the first and second QDs ( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Maximum value of [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: FIG. 7. Multipartite GME as a function of time [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

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Reference graph

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125 extracted references · 64 canonical work pages · cited by 3 Pith papers

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[1]

Here, N3 = N − e and N4 = N + e and given below Eqs

+ N 2 4 (λ3/∆)e−iλ4t([λ3/∆] − 1) , δ3(t) = N 2 3 e−iλ3t(1 − [λ4/∆]) + N 2 4 e−iλ4t(1 − [λ3/∆]) . Here, N3 = N − e and N4 = N + e and given below Eqs. (5), while λ3 = E− e , and λ4 = E+ e given by Eqs. (4). The time-evolved state given by Eq. (27) for the initially maximally entangled configu- ration |ψ3(0)⟩ takes the same form as that obtained from the se...
[2]

Dynamics of initially maximally entangled states Following the same steps as in the previous section for a separable initial state, we now consider the dynamics of the maximally entangled initial state |ψ3(0)⟩ = [|00⟩ + |11⟩]/ √ 2 in the sweet spot of a two-site Kitaev chain ∆ = τ , as given in Eq. (15). The time-evolved state for ∆ = τ and εi ̸= 0 is obt...
[3]

for C and EG, consistent with the behavior of Ed, which does not reach 0.5

Parameters: ∆ = 0 .5. for C and EG, consistent with the behavior of Ed, which does not reach 0.5. Finally, when both onsite energies are set to zero [Fig. 3(j)], the initial state becomes an eigenstate of the Hamiltonian. Consequently, the system evolves only by a global phase, and the time-evolution is simply constant in all the entanglement measures, se...
[4]

Dynamics of initially separable states We start by considering the initial separable state |00⟩ and obtain all the entanglement measures discussed in the previous subsection. In Figs. 4(a,b), we present C and EG as a function of time and onsite energy εi = ε at ∆ = 0 .5. Moreover, in Figs. 4(c-d), we present the time evolution of C, EG, Ed, and Rp for i) ...
[5]

4(f,g) shows C and EG as a func- tion of time and onsite energy εi = ε at ∆ = 0 .5

Dynamics of initially maximally entangled states In the case of the initial maximally entangled state (|00⟩ + |11⟩)/ √ 2, Figs. 4(f,g) shows C and EG as a func- tion of time and onsite energy εi = ε at ∆ = 0 .5. Also, in Figs. 4(h-j) we present the time evolution of C, EG, Ed, and Rp for distinct onsite energies εi; note that Rp is not shown because it is...
[6]

In this case, C characterises entan- glement between two qubits in a mixed state [63]

Concurrence In quantum systems with more than two sites, the state of a smaller part—such as a pair of sites—is typically not pure but mixed, as it includes the influence of the rest of the system [102]. In this case, C characterises entan- glement between two qubits in a mixed state [63]. Thus, the concurrence C between a pair of sites in the three-site ...
[7]

In particular, multipar- tite entanglement naturally arises, especially in our case, where PMMMs may overlap either with one another or with the central QD

Geometric measure of entanglement As the system size increases beyond two sites, the gen- erated entanglement is no longer purely bipartite, and concurrence alone becomes insufficient to fully charac- terise the entanglement present. In particular, multipar- tite entanglement naturally arises, especially in our case, where PMMMs may overlap either with on...

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[9]

III B 3 and given by Eq

Return probability and entanglement dynamics To find the return probability Rp and entanglement dynamics Ed, we employ the same expressions as those used in Subsec. III B 3 and given by Eq. (21) and Eq. (22), respectively. While the calculation is the same as before, we remark...
[10]

(35) The first state, |ψ5(0)⟩, is a fully separable product state and serves as a natural extension of the two-site system

Initial states in a three-site Kitaev chain To investigate both bipartite and multipartite entan- glement in the three-site Kitaev chain, we consider two distinct initial states |ψ5(0)⟩ = |000⟩ , |ψ6(0)⟩ = 1√ 2 (|000⟩ + |111⟩) . (35) The first state, |ψ5(0)⟩, is a fully separa...
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[12]

For this purpose, and following Subsec

Nearest-neighbour and next-nearest-neighbour concurrences In this section, we quantify two-site concurrences be- tween different pairs representing entanglement between nearest-neighbour and next-nearest-neighbour QDs. For this purpose, and following Subsec. IV A 1, we employ ...
[13]

7 we plot EGHZ G and EW G obtained from Eqs

At and away from the sweet spot To identify the GME with respect to the GHZ and W states, in Fig. 7 we plot EGHZ G and EW G obtained from Eqs. (40) as a function of time t and onsite energy εi = ε at the genuine sweet spot (∆ i = τi) and away from it (∆i ̸= τi). Moreover, for ...
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In this regard, EGHZ G = 1 /2 for ε = 0, a value that evolves in time following an almost periodic profile indicating the easiness to achieve a GHZ state, see Fig

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(c) Time evolution of EGHZ G , EW G , and Rp at ε = 0

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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work page 2023

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Here, N3 = N − e and N4 = N + e and given below Eqs

+ N 2 4 (λ3/∆)e−iλ4t([λ3/∆] − 1) , δ3(t) = N 2 3 e−iλ3t(1 − [λ4/∆]) + N 2 4 e−iλ4t(1 − [λ3/∆]) . Here, N3 = N − e and N4 = N + e and given below Eqs. (5), while λ3 = E− e , and λ4 = E+ e given by Eqs. (4). The time-evolved state given by Eq. (27) for the initially maximally entangled configu- ration |ψ3(0)⟩ takes the same form as that obtained from the se...

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Dynamics of initially maximally entangled states Following the same steps as in the previous section for a separable initial state, we now consider the dynamics of the maximally entangled initial state |ψ3(0)⟩ = [|00⟩ + |11⟩]/ √ 2 in the sweet spot of a two-site Kitaev chain ∆ = τ , as given in Eq. (15). The time-evolved state for ∆ = τ and εi ̸= 0 is obt...

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for C and EG, consistent with the behavior of Ed, which does not reach 0.5

Parameters: ∆ = 0 .5. for C and EG, consistent with the behavior of Ed, which does not reach 0.5. Finally, when both onsite energies are set to zero [Fig. 3(j)], the initial state becomes an eigenstate of the Hamiltonian. Consequently, the system evolves only by a global phase, and the time-evolution is simply constant in all the entanglement measures, se...

[5] [4]

Dynamics of initially separable states We start by considering the initial separable state |00⟩ and obtain all the entanglement measures discussed in the previous subsection. In Figs. 4(a,b), we present C and EG as a function of time and onsite energy εi = ε at ∆ = 0 .5. Moreover, in Figs. 4(c-d), we present the time evolution of C, EG, Ed, and Rp for i) ...

[6] [5]

4(f,g) shows C and EG as a func- tion of time and onsite energy εi = ε at ∆ = 0 .5

Dynamics of initially maximally entangled states In the case of the initial maximally entangled state (|00⟩ + |11⟩)/ √ 2, Figs. 4(f,g) shows C and EG as a func- tion of time and onsite energy εi = ε at ∆ = 0 .5. Also, in Figs. 4(h-j) we present the time evolution of C, EG, Ed, and Rp for distinct onsite energies εi; note that Rp is not shown because it is...

[7] [6]

In this case, C characterises entan- glement between two qubits in a mixed state [63]

Concurrence In quantum systems with more than two sites, the state of a smaller part—such as a pair of sites—is typically not pure but mixed, as it includes the influence of the rest of the system [102]. In this case, C characterises entan- glement between two qubits in a mixed state [63]. Thus, the concurrence C between a pair of sites in the three-site ...

[8] [7]

In particular, multipar- tite entanglement naturally arises, especially in our case, where PMMMs may overlap either with one another or with the central QD

Geometric measure of entanglement As the system size increases beyond two sites, the gen- erated entanglement is no longer purely bipartite, and concurrence alone becomes insufficient to fully charac- terise the entanglement present. In particular, multipar- tite entanglement naturally arises, especially in our case, where PMMMs may overlap either with on...