arxiv: 2605.07033 · v1 · submitted 2026-05-07 · 🪐 quant-ph

Analytic C_{ell₁} norm of Coherence Evolution for Bell States under a Two-Qubit Superconducting Hamiltonian

Seyed Mohsen Moosavi Khansari This is my paper

Pith reviewed 2026-05-11 01:26 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Bell statescoherence dynamicssuperconducting qubitsunitary evolutionC_l1 normanalytic expressionstwo-qubit Hamiltonianquantum information

0 comments p. Extension

The pith

Exact analytic formulas show that two Bell states maintain constant coherence while the others exhibit tunable oscillations in a two-qubit superconducting system.

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

The paper computes the exact time evolution of coherence in Bell states for a two-qubit superconducting system using the full unitary operator. Closed-form density matrices are obtained for each initial Bell state, leading to an analytic formula for the l1-norm of coherence as a function of time. Two Bell states prove invariant, keeping constant coherence, while the remaining two display oscillations whose frequencies and amplitudes depend on the Hamiltonian's coupling and tunneling strengths. These results enable precise prediction and control of coherence through parameter choice. They also establish a clean starting point for models that include dissipation.

Core claim

Deriving the complete time evolution operator for the two-qubit superconducting Hamiltonian and applying it to Bell state initial conditions yields closed-form time-dependent pure-state density matrices. An explicit analytic expression follows for the C_{l1} norm of coherence. Two Bell states remain invariant with constant coherence under the dynamics. The other two exhibit oscillations governed by two distinct frequency scales that correspond directly to the circuit coupling and tunnelling parameters, permitting predictable tuning of amplitude and periodicity.

What carries the argument

The analytically derived time evolution operator of the two-qubit superconducting Hamiltonian applied to Bell states to produce the C_l1 coherence norm.

If this is right

Two Bell states maintain constant C_l1 coherence over time.
The other two Bell states display coherence oscillations with two frequency components.
Oscillation amplitude and period can be tuned by adjusting the Hamiltonian parameters.
Specific operating regimes allow transient enhancement or suppression of coherence.
The expressions provide compact tools for parameter optimization in coherence control.

Where Pith is reading between the lines

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

The unitary results can be used as a reference to quantify the impact of adding dissipation terms to the model.
Similar analytic approaches might apply to other entangled states or multi-qubit systems in superconducting architectures.
Experimental realization in superconducting qubit devices could directly measure the predicted oscillation frequencies.
The parameter mapping offers a route to design circuits that stabilize or modulate coherence on demand.

Load-bearing premise

The evolution is purely unitary under the two-qubit superconducting Hamiltonian with no dissipation or decoherence present.

What would settle it

Numerical or experimental measurement of the time-dependent coherence for the oscillating Bell states in a real superconducting two-qubit device, compared against the closed-form predictions.

Figures

Figures reproduced from arXiv: 2605.07033 by Seyed Mohsen Moosavi Khansari.

**Figure 1.** Figure 1: A comprehensive depiction of a two-qubit superconducting system (TQS). 2.2 Matrix Form of the Hamiltonian The Hamiltonian matrix in the computational basis {∣ 00⟩, ∣ 01⟩, ∣ 10⟩, ∣ 11⟩} is: ℋ𝑇𝑄𝑆 = ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Graph of Cℓ1 (ρ Φ+ (t)) according to changes in time and for values ℏ = 1, EJ = 1/2, Em = 3/2 This figure illustrates the temporal evolution of 𝐶ℓ1 (𝜌 Φ+ (𝑡)). The horizontal axis represents time 𝑡 ranging from 0 to 10, while the vertical axis spans values from 0 to 5. The red curve shows a periodic, non-sinusoidal oscillation. Starting at approximately 1.0 at 𝑡 = 0, the value rises to a peak of about 3.0 … view at source ↗

**Figure 3.** Figure 3: Three-dimensional graph of Cℓ1 (ρ Φ+ (t)) in terms of changes in EJ and time, for ℏ = 1, Em = 3/2 This 3D surface plot visualizes 𝐶ℓ1 (𝜌 Φ+ (𝑡)) as it varies with two parameters: 𝑡 (0 to 10) and 𝐸𝐽 (0 to 0.5). The vertical axis represents the function's value, spanning 0 to 3. The colour bar quantifies the magnitude, with darker blues indicating lower values (around 1.0) and brighter yellows signifying hig… view at source ↗

**Figure 4.** Figure 4: Three-dimensional graph of Cℓ1 (ρ Φ+ (t)) in terms of changes in Em and time, for ℏ = 1, EJ = 1/2 This 3D surface plot depicts the behavior of 𝐶ℓ1 (𝜌 Φ+ (𝑡)) as a function of 𝑡 (0 to 10) and 𝐸𝑚 (0 to 1.5). The most striking feature is the presence of distinct periodic oscillations along the 𝐸𝑚 axis, creating a series of parallel ridges and valleys. This suggests a strong, perhaps sinusoidal or wave-like, d… view at source ↗

read the original abstract

We present an exact analytic study of unitary coherence dynamics in a minimal two qubit superconducting system. By deriving the full time evolution operator and propagating Bell state initial conditions, we obtain closed form time dependent pure state density matrices and an explicit analytic expression for the $C_{l_1}$ norm of coherence. Two of the Bell states are shown to be invariant under the model dynamics with constant coherence, while the other two exhibit controlled, parameter dependent coherence oscillations. The oscillatory behaviour is governed by two distinct frequency scales that map directly onto the circuit coupling and tunnelling parameters, allowing predictable tuning of amplitude and periodicity. Numerical visualizations clarify operating regimes for transient enhancement or suppression of coherence. These results deliver compact, analytically tractable tools for parameter optimisation and provide a clear foundation for incorporating dissipation and for experimental validation.

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

The paper supplies closed-form expressions for the l1-coherence of Bell states under one specific two-qubit superconducting Hamiltonian, with direct dependence on the two circuit parameters.

read the letter

The central result is an explicit time-dependent formula for the C_l1 norm starting from each Bell state. Two of the states keep constant coherence while the other two oscillate at frequencies set by the coupling and tunneling terms. The authors obtain this by writing down the Hamiltonian matrix, computing its time-evolution operator through standard diagonalization, propagating the initial pure-state density matrices, and extracting the off-diagonal sum that defines the coherence measure. The resulting piecewise expressions are algebraic and contain no hidden approximations or fitting parameters. The numerical plots simply illustrate the analytic behavior across different parameter regimes. That part of the work is clean and reproducible from the stated Hamiltonian. The main limitation is the strict unitary assumption. The model contains no dissipation, no decoherence channels, and no environment terms, so the expressions apply only to the ideal closed system. The paper also stays inside two qubits and these four initial states, with no extension to larger registers or error-corrected encodings. These restrictions are explicit and consistent with the scope, but they cap the immediate practical reach. The work is aimed at people who design or simulate small superconducting circuits and want analytic tuning rules for coherence rather than repeated numerical runs. It is not a broad conceptual advance, yet the closed forms are concrete and falsifiable once the Hamiltonian is accepted. I would bring it to a reading group only if the group already works on superconducting qubit models. I would not cite it in my own papers unless I needed exactly this Hamiltonian. It deserves peer review because the derivations are algebraic, the claims are narrow and precise, and the results can be checked directly from the given matrix.

Referee Report

0 major / 3 minor

Summary. The manuscript derives the full time-evolution operator for a two-qubit superconducting Hamiltonian and applies it to the four Bell states to obtain closed-form time-dependent pure-state density matrices. It then extracts an explicit analytic expression for the l1-norm of coherence C_l1(t), demonstrating that two Bell states remain invariant with constant coherence while the other two exhibit oscillations controlled by two frequency scales that map directly to the circuit coupling strength and tunneling parameter.

Significance. If the algebraic derivations are correct, the work supplies compact, parameter-explicit analytic expressions for coherence dynamics in a minimal superconducting qubit model. These results enable direct tuning of oscillation amplitude and period via the two circuit parameters and provide a clean starting point for extensions that include dissipation, making them useful for both theoretical analysis and experimental parameter optimization in quantum information processing.

minor comments (3)

[Abstract] The abstract states that the oscillatory behaviour is governed by two distinct frequency scales but does not introduce the symbols for the coupling and tunneling parameters at that point; adding them would improve immediate readability.
[Numerical visualizations] The numerical visualizations are described as clarifying operating regimes, yet the figure captions do not list the specific parameter values or ranges plotted; explicit labels would aid reproducibility.
[Bell-state definitions] Standard Bell-state notation (e.g., |Φ⁺⟩, |Ψ⁺⟩) is used but not defined in the main text; a brief reminder of the computational-basis expansions would help readers unfamiliar with the convention.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as the recommendation for minor revision. The referee's assessment correctly identifies the key analytic results on coherence dynamics for Bell states under the two-qubit superconducting Hamiltonian. No specific major comments were provided in the report, so we have no individual points to address in detail. We will incorporate minor improvements to the presentation and clarity in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct algebraic propagation from given Hamiltonian

full rationale

The manuscript begins with an explicitly stated two-qubit superconducting Hamiltonian, derives the unitary time-evolution operator via standard matrix exponentiation or diagonalization, applies it to the four Bell-state initial conditions to produce closed-form time-dependent density matrices, and then computes the l1-norm of coherence as an explicit function of the two circuit parameters. All steps are purely algebraic identities with no fitted parameters, no self-referential definitions, and no load-bearing self-citations. The oscillatory frequencies map directly onto the Hamiltonian matrix elements by construction of the Schrödinger equation, but this is the expected non-circular outcome of solving the model rather than a reduction of the claimed result to its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results rest on the assumption of a specific time-independent two-qubit Hamiltonian and purely unitary evolution; these are standard domain assumptions rather than new postulates, but they are not derived within the paper.

free parameters (2)

circuit coupling strength
Input parameter that sets one of the two oscillation frequencies in the coherence expression; treated as a tunable constant rather than fitted to data.
tunnelling parameter
Input parameter that sets the second oscillation frequency; appears directly in the closed-form expressions.

axioms (2)

domain assumption The two-qubit system evolves unitarily according to the given superconducting Hamiltonian with no dissipation.
Invoked to justify the existence of a closed-form time-evolution operator and the absence of decoherence terms.
domain assumption The initial states are exactly the four Bell states.
Used to propagate the density matrices and obtain the coherence expressions.

pith-pipeline@v0.9.0 · 5441 in / 1454 out tokens · 50017 ms · 2026-05-11T01:26:32.528457+00:00 · methodology

discussion (0)

Reference graph

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