Zero-Error Capacity of a Quantum Channel

The zero-error capacity of a discrete classical channel was first defined by Shannon as the least upper bound of rates for which one transmits information with zero probability of error [C. Shannon, IRE Trans. Inform. Theory, IT-2(3):8–19, 1956]. Here, we define the quantum zero-error capacity, a new kind of classical capacity of a noisy quantum channel C represented by a trace-preserving map (TPM) E(·). Moreover, the necessary requirement for which a quantum channel has zero-error capacity greater than zero is also given. All proofs will appear in the full paper. Indeed, we will show the connection between quantum zero-error capacity and capacity of a graph. Let X be the set of possible input states to the quantum channel E(·), belonging to a d-dimensional Hilbert space H, and let ρ ∈ X . We denote σk = E(ρ) the received quantum state when ρ is transmitted through the quantum channel. Because knowledge of post-measurement states is not important, measurements are performed by means of a POVM (Positive OperatorValued Measurements) {Ej}, where ∑ j Ej = I . Assume that p(j|i) denotes the probability of Bob measures j given that Alice sent the state ρi. Then, p(j|i) = tr [σiEj ]. We define the zero-error capacity of a quantum channel for product states. A product of any n input states will be called an input quantum codeword, ρi = ρi1 ⊗ · · · ⊗ ρin , belonging to a d -dimensional Hilbert space H. A mapping of K classical messages (which we may take to be the integers 1, . . . ,K) into a subset of input quantum codewords will be called a quantum block code of length n. Thus, 1 n logK will be the rate for this code. A piece of n output indices obtained from measurements performed by means of a POVM {Ej} will be called an output word, w = {1, . . . , N}. A decoding scheme for a quantum block code of length n is a function that univocally associates each output word with integers 1 to K representing classical messages. The probability of error for this code is greater than zero if the system identifies a different message from the message sent.