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From Trace Distance to Coherence in a Dephasing Qubit: A Single-Qubit View of

Humberto Arrabal

Published Apr 25, 2026• 0 citations
Publisher

Abstract

We consider a single qubit undergoing pure dephasing with fixed populations and show that, fora natural pair of phase-opposite states, the trace distance coincides exactly with the ℓ1-norm ofcoherence. This allows the Breuer–Laine–Piilo (BLP) non-Markovianity indicator to be expresseddirectly in terms of single-qubit coherence dynamics within this setting, avoiding the usual optimizationover initial states. We also derive a simple relation between the entropy rate and coherence,showing that coherence revivals necessarily correspond to intervals of decreasing von Neumann entropy.Together, these results provide a transparent way to connect coherence, distinguishability,and information flow in dephasing qubits.

DOI
10.6084/m9.figshare.32095843

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We consider a single qubit undergoing pure dephasing with fixed populations and show that, fora natural pair of phase-opposite states, the trace distance coincides exactly with the ℓ1-norm ofcoherence. This allows the Breuer–Laine–Piilo (BLP) non-Markovianity indicator to be expresseddirectly in terms of single-qubit coherence dynamics within this setting, avoiding the usual optimizationover initial states. We also derive a simple relation between the entropy rate and coherence,showing that coherence revivals necessarily correspond to intervals of decreasing von Neumann entropy.Together, these results provide a transparent way to connect coherence, distinguishability,and information flow in dephasing qubits.
We derive an analytical relation between the von Neumann entropy rate and the coherence dynamicsof a qubit undergoing pure dephasing without assuming Markovianity. The result expresses˙S(t) solely in terms of the ℓ1-coherence C(t) and its time derivative ˙C (t), for fixed populations. TheMarkovian exponential decay is recovered as a special case. In non-Markovian regimes, the formulapredicts intervals of negative entropy rate associated with coherence revivals, providing a compactlink between information backflow and entropy dynamics.
We derive a closed-form relation between the von Neumann entropy rate and the coherence dynamicsof a qubit undergoing pure dephasing without assuming Markovianity. For fixed populations,the entropy rate is expressed solely in terms of the ℓ1-coherence and its time derivative. The Markovianexponential decay is recovered as a limiting case. In non-Markovian regimes, the relationpredicts intervals of negative entropy production associated with coherence revivals. In realisticquantum devices, deviations from the ideal relation provide a direct diagnostic of non-ideal dephasing,enabling detection and mitigation of noise and relaxation effects from coherence measurementsalone.
For a qubit undergoing pure dephasing, the trace distance between a natural pair of phase-oppositestates equals the ℓ1 coherence of the qubit. This identity provides a simple way to interpret distinguishabilitybackflow as single-qubit coherence recovery. Using this observation, we motivate aninformation-mirror interpretation of coherence revivals in structured environments: environmentaldegrees of freedom may temporarily store phase information and later make part of it locally visibleagain in the qubit. We define a cycle-resolved reflection coefficient, R = B/L, where L is the coherencelost during a decay segment and B is the coherence recovered during the following revival. Fora recovery cycle referenced to its preceding coherence maximum, ordinary partial recovery satisfies0 ≤ R ≤ 1, with R = 1 corresponding to complete return to the previous maximum. This coefficientdoes not replace standard non-Markovianity measures, but provides a compact diagnostic of howstrongly a structured environment returns previously lost coherence.
How quantum coherence influences thermodynamic behavior remains an open question in quantum thermodynamics. Here we investigate this relation within the pure dephasing framework, where a central qubit interacts with a finite Ising-like spin environment. Although the system's internal energy remains constant, the interaction induces decoherence and gives rise to nontrivial thermodynamic features. Within the two-point measurement approach, we show that the heat dissipated into the environment matches the coherent energy contribution appearing in a reformulated first law of quantum thermodynamics. Numerical calculations reveal oscillatory coherence dynamics, with revivals associated with information backflow and non-Markovian effects, as quantified by the Breuer-Laine-Piilo measure. We find that heat and coherence exhibit intertwined temporal behavior, with enhanced heat dissipation during coherence decay and reduced heat during revivals. These results suggest a connection between coherence dynamics and thermodynamic quantities in finite, closed composite systems undergoing pure dephasing.
We derive an exact relation between quantum coherence decay and von Neumann entropy productionfor a qubit undergoing Markovian pure dephasing. For a general initial state with fixedpopulations, the entropy production rate is expressed in closed form in terms of the instantaneouscoherence and Bloch radius. In the equal-population case, the expression reduces to a coherenceonlylaw. The result provides a direct analytical connection between coherence loss and entropygeneration in open two-level systems.
We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, XXX states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between l1{l}_{1}l1​-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both lp{l}_{p}lp​-norm- and Schatten-ppp-norm--based measures violate the (strong) monotonicity for all $p\ensuremath{\in}(1,\ensuremath{\infty})$.