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.

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.

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.Keywords: relative entropy, inequalities, cone, secret sharing. I. ENTROPY AND RELATIVE ENTROPYEntropy inequalities play a central role in information theory [5], classical or quantum. This is so because practically all capacity theorems are formulated in terms of entropy, and the same, albeit to a lesser degree, holds for many monotones, of, for example, entanglement: e.g., the entanglement of formation [2] or squashed entanglement [4]. It may thus come as a surprise that until recently [11] essentially the only inequality known for the von Neumann entropies in a composite system is strong subadditivityproved by Lieb and Ruskai [8]. We use the notation ρ ABC for the density operator representing the state of the system ABC, with the notation ρ BC = Tr A ρ ABC etc. for the reduced states. The relative entropy of two states ρ, σ (density operators of trace 1) is defined aswhere supp ρ is the supporting subspace of the density operator ρ. Note that in this paper, log always denotes the logarithm to base 2. Like von Neumann entropy, the relative entropy is used extensively in quantum information and entanglement theory to obtain capacity-like quantities and monotones. The most prominent example may be the relative entropy of entanglement [19,20]. Many other applications of the relative entropy are illustrated in the review [18]. In this paper we study the universal relations between the relative entropies in a composite system and for general pairs of states. For the most part we shall restrict ourselves to finite dimensional spaces. * Electronic address: ben.ibinson@bris.ac.uk † Electronic address: n.linden@bristol.ac.uk ‡ Electronic address: a.j.winter@bris.ac.uk