Coherenceremaining

Coherenceremaining is a term used in quantum information science to denote the amount of quantum coherence that persists in a system after it interacts with an environment that causes decoherence. It is a way to quantify how much of the system's ability to exhibit interference, superposition, and phase relations remains as time evolves or after noise acts on the system.

In formal terms, if the system is described by a density matrix ρ(t), coherence is associated with the off-diagonal elements ρij(t) for i ≠ j. Several measures—such as the l1-norm of coherence C_l1(ρ) = sum_{i≠j} |ρij| and the relative entropy of coherence—are used to define coherenceremaining as either the instantaneous value C(ρ(t)) or its ratio to the initial coherence C(ρ(0)).

In a simple two-level system subject to pure dephasing at rate γ, ρ01(t) = ρ01(0) e^{-γ t}, so coherenceremaining decays exponentially with time. In Markovian environments, coherence may vanish asymptotically, while in non-Markovian settings partial revivals of coherenceremaining can occur due to information back-flow.

Studying coherenceremaining is important for evaluating and improving quantum devices, including sensors, computers, and communication systems. Techniques such as dynamical decoupling, quantum error correction, and decoherence-free subspaces aim to maximize the coherenceremaining of an operational qubit or register.

Related concepts include quantum coherence, decoherence, and measures of coherence. Coherenceremaining provides a practical language for describing how environmental interactions impact interference capabilities and resource availability in quantum technologies.