ENTROPIC BOUNDS ON CODING FOR NOISY QUANTUM CHANNELS

analogy with its classical counterpart, a noisy quantum channel is cha racterized by a loss, a quantity that depends on the channel input and the quantum operation performed by the channel. The loss reflects the transmission quality: if the loss is zero, quantum information can be perfectly transmitted at a rate measured by the quantum source entrop y. By using block coding based on sequences of II entangled symbols, t he average loss (defined as the overall loss of the joint n-symbol cha nnel divided by n, when n-->infinity) can be made lower than the loss for a single use of the channel. In this context, we examine several u pper bounds on the rate at which quantum information can be transmitte d reliably via a noisy channel, that is, with an asymptotically vanish ing average loss while the one-symbol loss of the channel is nonzero. These bounds on the channel capacity rely on the entropic Singleton bo und on quantum error-correcting codes [Phys. Rev. A 56, 1721 (1997]. F inally, we analyze the Singleton bounds when the noisy quantum channel is supplemented with a classical auxiliary channel.