Fault-Tolerant Logical Gate Networks for CSS Codes

Fault-tolerant logical operations for qubits encoded by CSS codes are discussed, with emphasis on methods that apply to codes of high rate, encoding k qubits per block with k>1. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into X and Z parts. Optimized methods to move logical qubits around and to achieve controlled-not and Toffoli gates are discussed. It is found that the number of time-steps required to complete a fault-tolerant quantum computation is the same when k>1 as when k=1.Comment: 13 pages, 16 figures. The material in the appendix was included in a previous quant-ph eprint, but not yet published; it has been corrected and clarified. The rest is new. Replacement version: various small corrections and clarification

Robustness of quantum Markov chains

If the conditional information of a classical probability distribution of three random variables is zero, then it obeys a Markov chain condition. If the conditional information is close to zero, then it is known that the distance (minimum relative entropy) of the distribution to the nearest Markov chain distribution is precisely the conditional information. We prove here that this simple situation does not obtain for quantum conditional information. We show that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.Comment: 14 pages, no figures; not for the feeble-minde

All Inequalities for the Relative Entropy

The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party to a smaller number $m$ of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with general access structures. A suprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.Comment: 15 pages, 3 embedded eps figure