fractional-quantum-information-memory

name	fractional-quantum-information-memory
description	Fractional quantum information methodology using Riemann-Liouville derivative formalism with memory effects. Covers quantum information measures in fractional quantum mechanics, generalized entropy measures, and non-Markovian dynamics.
category	quantum-physics
metadata	{"arxiv_id":"2606.13525","published_date":"2026-06-12"}

Context

Fractional quantum mechanics extends standard quantum mechanics by replacing the standard time/space derivatives with fractional derivatives (typically Riemann-Liouville or Caputo types). This introduces memory effects and non-local dynamics that are relevant for open quantum systems and anomalous diffusion. This paper analyzes quantum information measures within this fractional framework.

Core Methodology

1. Riemann-Liouville Fractional Derivative: Replace the standard Schrödinger equation time derivative with the Riemann-Liouville fractional derivative: D^α_t ψ(t) = (1/Γ(1-α)) d/dt ∫_0^t (t-τ)^{-α} ψ(τ) dτ where 0 < α < 1.

2. Generalized Quantum Information Measures: Redefine von Neumann entropy, mutual information, and other information-theoretic quantities in the fractional framework. The fractional dynamics modify the density matrix evolution.

3. Memory Effect Characterization: The fractional derivative introduces a memory kernel that weights past states. Characterize the memory depth and its effect on information preservation/degradation.

4. Fractional Entropy Production: Analyze how entropy production rates differ between standard and fractional quantum evolution. The non-Markovian nature of fractional dynamics can lead to entropy decrease (information backflow).

Implementation Steps

1. Define the fractional Schrödinger equation with Riemann-Liouville derivative
2. Solve for the fractional propagator (Green's function) using Laplace transforms
3. Compute the fractional density matrix evolution: ρ(t) = U_α(t) ρ(0) U_α†(t)
4. Calculate fractional von Neumann entropy: S_α(ρ) = -Tr[ρ log_α ρ] using appropriate fractional logarithm
5. Analyze mutual information between subsystems under fractional evolution
6. Characterize memory effects via the autocorrelation function decay

Key Results

• Fractional quantum mechanics naturally incorporates memory effects without explicit environment modeling
• Quantum information measures differ quantitatively from standard QM, with α controlling the degree of deviation
• Memory effects can protect quantum information from decoherence for certain α regimes
• The fractional framework bridges Markovian and fully non-Markovian dynamics

Pitfalls

• Riemann-Liouville vs Caputo: The choice of fractional derivative definition affects initial condition handling. Riemann-Liouville requires fractional initial conditions, while Caputo uses standard initial conditions.
• Normalization: Fractional evolution may not preserve trace; ensure proper normalization of the density matrix.
• Physical Interpretation: Fractional parameters (α) lack direct physical interpretation — they are phenomenological parameters that must be fitted to experimental data.
• Numerical Stability: Fractional derivative computations involve singular integrals; use specialized quadrature methods (Grünwald-Letnikov, L1 scheme).

Verification

1. Verify that α → 1 recovers standard quantum mechanics results
2. Check trace preservation of the fractional density matrix
3. Verify that fractional entropy reduces to von Neumann entropy in the α → 1 limit
4. Test memory effect predictions against known non-Markovian dynamics models

Activation

fractional quantum mechanics, riemann-liouville derivative, memory effects, quantum information, non-markovian dynamics, fractional entropy, anomalous diffusion, quantum decoherence