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Bath-state (In)dependence of Non-Markovianity Measures in Quadratic Open Systems

This paper demonstrates that while certain non-Markovianity measures (BLP and LPP) are independent of the bath state in fermionic quadratic open systems, other measures (RHP and Megier et al.) remain sensitive to reservoir parameters like temperature and chemical potential, with numerical results showing that the former vanish only at the bound-state threshold while the latter remain finite.

Original authors: Mirko Consiglio, Ilia Khomchenko, Tony J. G. Apollaro

Published 2026-10-30

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CC BY 4.0

Original authors: Mirko Consiglio, Ilia Khomchenko, Tony J. G. Apollaro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the smallest particles rarely exist in isolation. They are constantly bumping into, exchanging energy with, and being influenced by their surroundings. Physicists call this surrounding environment a "bath." When a quantum system interacts with this bath, information can flow out, but it can also flow back in. If information returns, the system remembers its past interactions, a behavior known as non-Markovian dynamics. This is distinct from the simpler, more predictable behavior where information flows out and is lost forever. Understanding when and how this memory effect happens is crucial for building reliable quantum computers and sensors, as these devices must protect their delicate states from the noisy environment. A central question in this field has been whether the specific conditions of the environment—such as how hot it is or how many particles it contains—change the way we measure this memory.

A team of researchers at the University of Malta has investigated this question using a specific type of quantum system: a single particle that behaves like a fermion, coupled to a large chain of similar particles acting as a reservoir. They examined five different mathematical tools used to quantify non-Markovianity, or the degree of memory in the system. Their work reveals a surprising split in how these tools behave. Two of the most common measures, which rely on tracking the distance between two evolving states or the volume of space those states can occupy, turn out to be completely blind to the state of the environment. Whether the reservoir is hot or cold, or whether it is nearly empty or nearly full of particles, these two measures give the exact same result. The researchers proved this mathematically for any system where the interactions follow a specific quadratic pattern, showing that the environment's temperature and chemical potential simply do not matter for these particular calculations.

However, the story changes for the other three measures. These tools, which look at entanglement with a helper particle, the divisibility of the process over time, or the entropy of the system, are highly sensitive to the environment. The researchers found that these measures change significantly as the temperature of the reservoir shifts or as the chemical potential is adjusted. In fact, they identified a specific scenario where the two environment-blind measures drop to zero, suggesting the system has no memory, while the three environment-sensitive measures remain active and finite. This means that depending on which tool you use, you could look at the exact same physical situation and conclude either that the system is behaving in a simple, memoryless way or that it is deeply entangled with its surroundings.

To reach these conclusions, the team modeled the environment as a semi-infinite chain of non-interacting particles, a setup that allows for precise mathematical analysis. They derived an exact equation to describe how the single particle evolves over time, accounting for the flow of information back and forth. They then ran extensive numerical simulations across a wide range of temperatures and chemical potentials. In one set of simulations, they looked at a situation where the system's energy was high enough to create a stable, localized state that does not decay. Here, the two blind measures remained constant across a temperature range spanning three orders of magnitude, while the sensitive measures varied by up to sixty percent. In another scenario, they tuned the system to a critical point where the localized state just barely exists. At this threshold, the blind measures vanished completely, yet the sensitive measures still detected non-zero memory effects, proving that the environment's state was still influencing the system in a way those other tools missed.

The researchers also explored what happens when the system's energy is too low to support a stable state, causing the particle's influence to fade away through the chain. In this regime, the environment-sensitive measures showed a much weaker dependence on the bath's state compared to the high-energy case, but they still did not vanish. The study confirms that the choice of measurement tool is not just a matter of preference; it fundamentally alters the picture of the system's behavior. The work establishes that for a broad class of fermionic systems, the memory of the environment is not a universal property detectable by all methods. Instead, some measures capture only the structural flow of information, while others are deeply entangled with the thermal and chemical details of the surroundings. This distinction is vital for anyone trying to characterize or control quantum systems, as it implies that a system might appear perfectly memoryless to one observer while remaining rich with memory to another.

Technical Summary: Bath-state (In)dependence of Non-Markovianity Measures in Quadratic Open Systems

Problem Statement
In open quantum systems, non-Markovian dynamics arise when the environment's relaxation timescale is comparable to the system's, leading to information backflow. While various measures exist to quantify this non-Markovianity (e.g., Breuer–Laine–Piilo (BLP), Lorenzo–Plastina–Paternostro (LPP), Rivas–Huelga–Plenio (RHP), and Megier–Smirne–Vacchini (MSV)), it is generally expected that these measures depend on environmental parameters such as temperature ( $T$ ) and chemical potential ( $\mu$ ). Previous studies on bosonic systems and specific fermionic models have confirmed such dependencies. However, the general mechanism governing whether non-Markovianity measures are sensitive to the specific state of a fermionic reservoir, particularly for arbitrary quadratic Hamiltonians, had not been established. This work addresses the gap in understanding how different non-Markovianity measures respond to the temperature and chemical potential of a fermionic quadratic reservoir.

Methodology
The authors analyze a single fermionic site coupled to a reservoir of $K$ fermionic sites via an arbitrary quadratic Hamiltonian, which includes both normal and pairing terms. The system and bath are initially uncorrelated, and the bath state is assumed to be parity-even (commuting with the parity operator), a condition satisfied by thermal Gibbs states.

Analytical Derivation: The authors derive the exact time-local master equation and the dynamical map for the reduced system. They demonstrate that for quadratic fermionic systems, the dynamical map takes an affine form where the bath state enters only through a single scalar term, $Z_\beta(t)$ , which represents a state-independent shift along the $\hat{\sigma}_z$ axis.
Measure Analysis: They apply this dynamical structure to five distinct non-Markovianity measures:
- BLP: Based on the trace distance between evolving states.
- LPP: Based on the volume of accessible states (determinant of the Bloch matrix).
- MSV: Based on the quantum Jensen–Shannon divergence (QJSD).
- RHPE: Based on entanglement with an ancilla.
- RHP: Based on the divisibility of the dynamical map.
Specific Model: To validate analytical findings, the authors model the reservoir as a semi-infinite free-fermionic chain (mapped from an XX spin chain via Jordan–Wigner transformation). They derive exact propagators and noise kernels for this model, considering regimes above, on, and below the bound-state threshold.
Numerical Simulation: Extensive numerical simulations are performed across a wide range of temperatures and chemical potentials to evaluate the five measures.

Key Contributions and Results

Bath-State Independence of BLP and LPP: The primary theoretical result (Theorem 1) proves that the BLP and LPP measures are strictly independent of the reservoir state. Because these measures rely on the trace distance or the volume of the Bloch ball, they depend only on the linear part of the affine dynamical map. The bath-dependent shift ( $Z_\beta(t)$ ) cancels out in the calculation of the trace distance between state pairs and does not affect the determinant of the Bloch matrix. Consequently, these measures are invariant under changes in temperature and chemical potential for any quadratic fermionic Hamiltonian.
Bath-State Dependence of MSV, RHPE, and RHP: In contrast, the authors prove that the MSV, RHPE, and RHP measures remain sensitive to the reservoir state.
- The MSV measure depends on the mean occupation of the state pair, which is shifted by $Z_\beta(t)$ . Theoretical analysis shows that the QJSD grows as the pair drifts away from half-filling, a condition directly influenced by $T$ and $\mu$ .
- The RHP measure (divisibility) depends on individual decay rates $\Lambda_\pm(t)$ , which retain explicit dependence on the bath distribution function $f(\omega)$ . The authors derive a bath-independent lower bound for this measure, showing that the excess non-Markovianity arises specifically from the variation of the Fermi edge within the spectral band.
- The RHPE measure (entanglement) similarly depends on the bath state through the concurrence, which is sensitive to the thermal population of the bath.
Threshold Phenomena: Numerical results reveal a critical regime at the bound-state threshold ( $|E| = \Omega - \Gamma/2$ ). In this regime, the BLP and LPP measures vanish identically (indicating Markovian behavior regarding trace distance), yet the bath-sensitive measures (MSV, RHP, RHPE) remain finite. This demonstrates that the "Markovianity" of a system is measure-dependent; a system can appear Markovian to trace-distance-based measures while exhibiting non-Markovian features detectable by entropic or divisibility-based measures due to the bath-dependent translation of the Bloch ball.
Chemical Potential Dependence: Scans of the chemical potential confirm that while BLP and LPP remain constant, the other three measures vary significantly, particularly when the Fermi edge lies within the spectral band of the reservoir.

Significance
The paper establishes a fundamental distinction in how non-Markovianity is perceived depending on the chosen quantifier in quadratic fermionic systems. The authors demonstrate that the independence of BLP and LPP from bath parameters is a structural property of quadratic fermionic dynamics, relying on the affine nature of the map and the parity-even condition of the bath. This finding clarifies previous numerical observations in specific models and extends them to arbitrary quadratic Hamiltonians.

The work highlights that the "non-Markovianity" of a system is not an absolute property but is relative to the measure used. Specifically, it identifies a parameter regime where standard trace-distance measures vanish while other measures detect non-Markovianity driven entirely by the bath's thermal state. The authors note that this independence is specific to fermionic quadratic systems and does not extend to bosonic quadratic models (where temperature dependence is known to exist), suggesting a fundamental difference in the information flow mechanisms between fermionic and bosonic environments. The results imply that for quadratic fermionic systems, any observed dependence of BLP or LPP on bath parameters would necessitate either non-quadratic interactions or initial system-bath correlations.

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