Channel covariance, twirling, contraction, and some upper bounds on the quantum capacity

Abstract

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 $\mathcal{N}$ and a finite projective group of unitaries $\mathcal{V}$, we show that the $\mathcal{V}$-twirl of $\mathcal{N}$ has a quantum capacity at most the coherent information of $\mathcal{N}$ maximized over a $\mathcal{V}$-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 -- $d$-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.

33 pages, 3 figures. Title changed with slight revisions and to appear in QIC

References in corpus (5)

• Degenerate Quantum Codes for Pauli Channels
• Quantum Capacities of Channels with small Environment
• Trade-off capacities of the quantum Hadamard channels
• New lower bounds on the non-zero capacity of Pauli Channels
• Qubit channels with small correlations

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• Estimating Quantum and Private capacities of Gaussian channels via degradable extensions
• Resonant Multilevel Amplitude Damping Channels
• Capacities of the covariant Pauli channel
• Estimate distillable entanglement and quantum capacity by squeezing useless entanglement
• The noisy Werner-Holevo channel and its properties
• Allowing leakage can increase quantum transmission
• Explicit construction of optimal witnesses for input-output correlations attainable by quantum channels
• The superadditivity effects of quantum capacity decrease with the dimension for qudit depolarizing channels