Quantum Error Correction for Communication.

We show how procedures which can correct phase and amplitude errors can be directly applied to correct errors due to quantum entanglement. We specify general criteria for quantum error correction, introduce quantum versions of the Hamming and the Gilbert-Varshamov bounds and comment on the practical implementations of quantum codes. Suppose we want to transmit a block of l qubits (i.e. two-state quantum systems) in some unknown quantum state (pure or mixed) over a noisy quantum channel. Here ‘noisy’ means that each transmitted qubit may, with some small probability p, become entangled with the channel. In order to increase the probability of the error-free transmission we can encode the state of l qubits into a set of n qubits and try to disentangle a certain number of qubits from the channel at the receiving end. This paper specifies conditions under which such encoding and disentanglement are possible. Let us start with introducing convenient definitions and notation. Amplitude errors in a block of n qubits are defined as a sequence of σx transformations performed on qubits at locations specified by a binary n-tuple α (non-zero entries of α mark the locations of the affected qubits). In a selected basis {v} the amplitude errors can be written as Aα | v〉 = | v + α〉 , (1) where the addition is performed modulo 2. Analogously, phase errors are defined as a sequence of σz transformations performed on qubits at locations