1/2+1/2>1 for quantum error correction

Abstract: Quantum error correction assisted by entanglement helps to transmit the encoded qudits through quantum channels with some of them being noiseless. Here we consider a more realistic scheme for experiments what we called as partial-noisy quantum channels in which, instead of completely free of noise, only part of the qudit suffers no noise. In this scenario we show by an explicit example that two half-noisy channels work better than one noiseless channel, a queer phenomenon showing 1/2+1/2>1. Our example also saturates a unified quantum Singleton bound, valid for the standard and entanglement-assisted codes as well. Furthermore, as illustrated by a mixed-alphabet code with half-noisy channels, a higher dimensional physical qudit can so improve the performance of a partial-noisy channel that it even outperforms a noiseless channel.