Maneuvering measurement-coherence into measurement-entanglement

TL;DR

The paper develops a comprehensive resource-theoretic framework for measurement-based quantum resources, showing that measurement-cohering power of a channel can be converted into measurement-entangling power using only free operations and a dephasing step, thereby completing the dynamical coherence-entanglement correspondence for measurements. It introduces a canonical structure for incoherent measurements and proves dualities between state- and measurement-based dynamical powers for unital channels via adjoint maps, notably $C(\mathcal{E}_A)=E(\boldsymbol{\triangle}_{AB}\circ\mathcal{E}_A\circ\mathcal{U}_{\mathrm{CNOT}}^{\dag})$. The work provides explicit definitions and monotones for measurement-coherence and measurement-entanglement powers, derives conversion bounds, and demonstrates that the maximal conversion is achieved with a CNOT-adjoint construction, paralleling known state-based results. These findings unify static and dynamic resource theories across states and measurements, offering a foundation for resource-efficient quantum technologies and future exploration of multipartite entanglement dynamics.

Abstract

Quantum dynamics governs the transformation of static quantum resources, such as coherence and entanglement, in both quantum states and measurements. Prior studies have established that a quantum channel's state-cohering power can be converted into the state-entangling power without additional coherence. Here, we complete this coherence-to-entanglement paradigm by demonstrating that a channel's measurement-cohering power can likewise be converted into the measurement-entangling power. This result reinforces, on the dynamical level, the intuition that entanglement emerges as a manifestation of coherence. To formalize this picture, we develop resource theories for measurement-cohering and measurement-entangling powers and characterize the structure of incoherent measurements to analyze measurement-coherence generation. Furthermore, we show that the state-cohering power of a quantum channel is equivalent to the measurement-cohering power of its adjoint map, and a corresponding equivalence also exists between the state-entangling power and the measurement-entangling power.

Figures (4)

• Figure 1: Quantum dynamics can modify static quantum resources in both quantum states and quantum measurements.
• Figure 2: The measurement-cohering power of a quantum channel $\mathcal{E}_{A}$ can be converted to the measurement-entangling power of a quantum channel $\widetilde{\mathcal{E}}_{AB}$ by composing it with a $\mathbf{DIO}$ channel $\mathcal{K}_{AB}$, a unital $\mathbf{DIO}$ channel $\mathcal{L}_{AB}$, and the dephasing channel $\Delta_{AB}$.
• Figure 3: A quantum measurement $\mathcal{M}_{A}$ transforms to another quantum measurement $\widetilde{\mathcal{M}}_{A}$ under a pre-processing channel $\mathcal{E}_{A}$ and a classical post-processing channel $\mathcal{S}_{A'}$. The pre-processing channel can generate measurement-resource.
• Figure 4: (a) Measurement-cohering power of a quantum channel $\mathcal{E}_{A}$ can convert to measurement-entangling power of a quantum channel $\widetilde{\mathcal{E}}_{AB}$ under a $\mathbf{DIO}$ channel $\mathcal{K}_{AB}$, a unital $\mathbf{DIO}$ channel $\mathcal{L}_{AB}$, and the dephasing channel $\Delta_{AB}$. (b) State-cohering power of a quantum channel $\mathcal{E}_{A}$ can convert to state-entangling power of a quantum channel $\widetilde{\mathcal{E}}_{AB}$ under $\mathbf{MIO}$ channels $\mathcal{K}_{AB}$ and $\mathcal{L}_{AB}$, and the dephasing channel $\Delta_{AB}$theurer2020QuantifyingDynamicalCoherence.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 1
• Lemma 2: Structure of an incoherent measurement