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arXiv:2305.15741 · Physical Review A

Coherence filtration under strictly incoherent operations

C. L. Liu, Chuan Sun

Published May 31, 2023• 4 citations
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Abstract

We study the task of coherence filtration under strictly incoherent operations. The aim of this task is to transform a given state $\ensuremath{\rho}$ into another one ${\ensuremath{\rho}}^{\ensuremath{'}}$ whose fidelity with the maximally coherent state is maximal by using stochastic strictly incoherent operations. We find that the maximal fidelity between ${\ensuremath{\rho}}^{\ensuremath{'}}$ and the maximally coherent state is given by a multiple of the $\mathrm{\ensuremath{\Delta}}$ robustness of coherence, $R(\ensuremath{\rho}\ensuremath{\parallel}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}):=min{\ensuremath{\lambda}|\ensuremath{\rho}\ensuremath{\le}\ensuremath{\lambda}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}}$, which provides $R(\ensuremath{\rho}\ensuremath{\parallel}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho})$ an operational interpretation. Finally, we provide a coherence measure based on the task of coherence filtration.

DOI
10.1103/physreva.107.052424

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We study the task of coherence filtration under strictly incoherent operations. The aim of this task is to transform a given state \ensuremathρ\ensuremath{\rho}\ensuremathρ into another one \ensuremathρ\ensuremath′{\ensuremath{\rho}}^{\ensuremath{'}}\ensuremathρ\ensuremath′ whose fidelity with the maximally coherent state is maximal by using stochastic strictly incoherent operations. We find that the maximal fidelity between \ensuremathρ\ensuremath′{\ensuremath{\rho}}^{\ensuremath{'}}\ensuremathρ\ensuremath′ and the maximally coherent state is given by a multiple of the \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ robustness of coherence, R(\ensuremathρ\ensuremath∥\ensuremathΔ\ensuremathρ):=min\ensuremathλ∣\ensuremathρ\ensuremath≤\ensuremathλ\ensuremathΔ\ensuremathρR(\ensuremath{\rho}\ensuremath{\parallel}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}):=min{\ensuremath{\lambda}|\ensuremath{\rho}\ensuremath{\le}\ensuremath{\lambda}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}}R(\ensuremathρ\ensuremath∥\ensuremathΔ\ensuremathρ):=min\ensuremathλ∣\ensuremathρ\ensuremath≤\ensuremathλ\ensuremathΔ\ensuremathρ, which provides R(\ensuremathρ\ensuremath∥\ensuremathΔ\ensuremathρ)R(\ensuremath{\rho}\ensuremath{\parallel}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho})R(\ensuremathρ\ensuremath∥\ensuremathΔ\ensuremathρ) an operational interpretation. Finally, we provide a coherence measure based on the task of coherence filtration.
In this paper, we address the issue of enhancing coherence of a state under stochastic strictly incoherent operations. Based on the l1{l}_{1}l1​ norm of coherence, we obtain the maximal value of coherence that can be achieved for a state undergoing a stochastic strictly incoherent operation and the maximal probability of obtaining the maximal coherence. Our findings indicate that a pure state can be transformed into a maximally coherent state under a stochastic strictly incoherent operation if and only if all the components of the pure state are nonzero while a mixed state can never be transformed into a maximally coherent state under a stochastic strictly incoherent operation.
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Manipulation and quantification of quantum resources are fundamental problems in quantum physics. In the asymptotic limit, coherence distillation and dilution have been proposed by manipulating infinite identical copies of states. In the nonasymptotic setting, finite data-size effects emerge, and the practically relevant problem of coherence manipulation using finite resources has been left open. This Letter establishes the one-shot theory of coherence dilution, which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations, dephasing-covariant incoherent operations, incoherent operations, or strictly incoherent operations. We introduce several coherence monotones with concrete operational interpretations that estimate the one-shot coherence cost-the minimum amount of maximally coherent states needed for faithful coherence dilution. Furthermore, we derive the asymptotic coherence dilution results with maximally incoherent operations, incoherent operations, and strictly incoherent operations as special cases. Our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources, such as quantum key distribution and random number generation.
Quantum state transformation under free operations plays a central role in the resource theory of coherence. In this paper, we investigate the transformation from a mixed coherent state into a pure one by using both incoherent operations and stochastic incoherent operations. We show that contrary to the strictly incoherent operations and the stochastic strictly incoherent operations, both the incoherent operations and the stochastic incoherent operations can increase the dimension of the maximal pure coherent subspace of a state. This means that the incoherent operations are generally stronger than the strictly incoherent operations when we want to transform a mixed coherent state into a pure coherent one. Our findings can also be interpreted as confirming the ability of incoherent operations to enhance the coherence of mixed states relative to certain coherence monotones under strictly incoherent operations.
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An important problem in the coherent resource theory is the convertibility of coherent states by free operations. In this paper we consider such a problem from a mixed coherent state into a pure one by using both stochastic incoherent operations and incoherent operations. We prove that stochastic incoherent operations can transform a mixed coherent state into a pure coherent state if and only if the density matrix of the initial state contains a singular principal submatrix. Then we provide two sufficient conditions with explicit expressions for the probabilistic transformation and the deterministic transformation. These two sufficient conditions facilitate the construction of concrete operators of stochastic incoherent operations and incoherent operations. These results demonstrate that incoherent operations are strictly more powerful than strictly incoherent operations within the probabilistic transformation as well as the deterministic transformation.