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arXiv:2311.16349

Zero Error Correctibility and Phase Retrievability for Twirling Channels

Kai Liu, Deguang Han

Published Nov 27, 2023• 0 citations
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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 $C^n$.

DOI
10.48550/arxiv.2311.16349

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This survey introduces the role of frame phase retrieval in pure-state identification and information preservation by quantum channels. For unit vectors, the lift x↦xx∗x\mapsto xx^*x↦xx∗ identifies vectors that differ only by a global phase with the same rank-one quantum state, converts frame intensities into linear functionals of the lifted state, and makes the adjoint pullback of output observables an operator-valued measurement on the input system. From this viewpoint, a channel is phase retrievable exactly when it is injective on pure states. We develop this correspondence through Choi-rank-two linear combinations of Kraus operators, higher-rank relative spectra, structural obstructions, and constructions with prescribed Choi rank. We distinguish injective identification from full tomography, perfect one-shot discrimination, zero-error classical communication, and exact quantum correction. Twirling channels then provide a structured setting in which commutants, irreducible dimensions, multiplicities, orbit-frame orthogonality, coding indices, and phase-retrievable subspaces can be read from group representations.
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Entanglement distillation from noisy bipartite states and reliable quantum communication over noisy channels are fundamental tasks in quantum information theory. Yet existing coding schemes typically require prior knowledge of the underlying state or channel, or assume independent or weakly correlated quantum information sources. Here, using Schur-Weyl duality, we establish a universal quantum error-correction principle that overcomes these limitations. We show that the irreducible permutation spaces arising from the local Schur-Weyl decomposition of multiple copies of an unknown bipartite mixed state form maximally entangled quantum code spaces whose error structure is governed entirely by representation theory. Correcting these symmetry-resolved errors yields protocols for entanglement distillation and quantum communication that achieve coherent-information rates with exponentially vanishing error and optimal second-order corrections, even in the case of almost i.i.d. resources. Our approach also provides sharp finite-copy guarantees that are worst-case optimal at every finite block length, including in the sample-efficient regime, where the number of copies scales only polynomially with the number of qubits. Together, these results establish Schur-Weyl duality as a general mechanism for universal, robust, and optimal quantum coding, unifying entanglement distillation, quantum error correction, and quantum communication within a common representation-theoretic framework.
Evaluating the quantum capacity of quantum channels is an important but difficult problem, even for channels of low input and output dimension. Smith and Smolin showed that the quantum capacity of the Clifford-twirl of a qubit amplitude damping channel (a qubit depolarizing channel) has a quantum capacity that is at most the coherent information of the qubit amplitude damping channel evaluated on the maximally mixed input state. We restrict our attention to obtaining upper bounds on the quantum capacity using a generalization of Smith and Smolin's degradable extension technique. Given a degradable channel $\cN$ and a finite projective group of unitaries $\cV$, we show that the $\cV$-twirl of $\cN$ has a quantum capacity at most the coherent information of $\cN$ maximized over a $\cV$-contracted space of input states. As a consequence, degradable channels that are covariant with respect to diagonal Pauli matrices have quantum capacities that are their coherent information maximized over just the diagonal input states. As an application of our main result, we supply new upper bounds on the quantum capacity of some unital and non-unital channels -- ddd-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.