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arXiv · 2311.16349

Zero Error Correctibility and Phase Retrievability for Twirling Channels

Kai Liu·Deguang Han
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Abstract

A twirling channel is a quantum channel induced by a continuous unitary representation $π= \sum_{i}^{\oplus} m_iπ_i$, where $π_i$ are inequivalent irreducible representations. Motivated by a recent work \cite{Twirling} on minimal mixed unitary rank of $Φ_π$, we explore the connections of the independence number, zero error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel $Φ_π$ with the irreducible representation multiplicities $m_i$, the irreducible representation dimensions $\dim H_{π_i}$. In particular we show that the independence number of $Φ_π$ is the sum of the multiplicities, the orthogonal index of $Φ_π$ is exactly the sum of those representation dimensions, and the zero-error capacity is equal to $\log (\sum_{i=1}^{d}m_i)$. We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for $C^n$.

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Kai Liu, Deguang Han. 2023-11-27. Zero Error Correctibility and Phase Retrievability for Twirling Channels. https://arxiv.org/abs/2311.16349

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