Quantum f -divergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f -divergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f -divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f -divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f -divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.
We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called "operator algebra quantum error correction"). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of information flows in quantum interactions.Error correction methods are of crucial importance for quantum computing and the so far most general framework, called operator quantum error correction (OQEC) [1,2], encompasses active error correction [4,5,6,7,8] (QEC), together with the concepts of decoherence-free and noiseless subspaces and subsystems [9,10,11,12,13,14,15]. The OQEC approach has enabled more efficient correction procedures in active error correction [16,17,18,19], has led to improved threshold results in fault tolerant quantum computing [20], and has motivated the development of a structure theory for passive error correction [21,22] which has recently been used in quantum gravity [23,24,25,26].In this paper, we introduce a natural generalization of this theory. To this end, we change the focus from that of states to that of observables: conservation of a state by a given noise model implies the conservation of all of its observables, and is therefore a rather strong requirement. This can be alleviated by specifically selecting only some observables to be conserved. In this context it is natural to consider algebras of observables [27]. Hence our codes take the form of operator algebras that are closed under Hermitian conjugation; that is, finite dimensional C * -algebras [28]. As a convenience, we shall simply refer to such operator algebras as "algebras". Correspondingly we refer to the new theory as "operator algebra quantum error correction" (OAQEC). We present results that establish testable conditions for correctability in OAQEC. We also discuss illustrative examples and consider applications to quantum teleportation and information flow in quantum interactions. We shall present the proofs and more examples in [29].Noise models in quantum information are described by channels, which are (in the Schrödinger picture) tracepreserving (TP) and completely positive (CP) linear maps E on mixed states, ρ, which are operators acting on a Hilbert space H. If ρ is a density matrix we can always write ρ → E(ρ) = a E a ρE † a where {E a } is a non-unique family of channel elements. The QEC framework addresses the question of whether a given subspace of states P H, called the code, can be corrected in the sense that there exists a correction channel R such that R(E(ρ)) = ρ for all states ρ in the subspace; that is, all ρ which satisfy ρ = P ρP . This amounts to asking for a subspace on which E has a left inverse that is a ph...
We derive necessary and sufficient conditions for the approximate correctability of a quantum code, generalizing the Knill-Laflamme conditions for exact error correction. Our measure of success of the recovery operation is the worst-case entanglement fidelity. We show that the optimal recovery fidelity can be predicted exactly from a dual optimization problem on the environment causing the noise. We use this result to obtain an estimate of the optimal recovery fidelity as well as a way of constructing a class of near-optimal recovery channels that work within twice the minimal error. In addition to standard subspace codes, our results hold for subsystem codes and hybrid quantum-classical codes.
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