Channel kets, entangled states, and the location of quantum information

The well-known duality relating entangled states and noisy quantum channels is expressed in terms of a channel ket, a pure state on a suitable tripartite system, which functions as a pre-probability allowing the calculation of statistical correlations between, for example, the entrance and exit of a channel, once a framework has been chosen so as to allow a consistent set of probabilities. In each framework the standard notions of ordinary (classical) information theory apply, and it makes sense to ask whether information of a particular sort about one system is or is not present in another system. Quantum effects arise when a single pre-probability is used to compute statistical correlations in different incompatible frameworks, and various constraints on the presence and absence of different kinds of information are expressed in a set of all-or-nothing theorems which generalize or give a precise meaning to the concept of ``no-cloning.'' These theorems are used to discuss: the location of information in quantum channels modeled using a mixed-state environment; the $CQ$ (classical-quantum) channels introduced by Holevo; and the location of information in the physical carriers of a quantum code. It is proposed that both channel and entanglement problems be classified in terms of pure states (functioning as pre-probabilities) on systems of $p\geq 2$ parts, with mixed bipartite entanglement and simple noisy channels belonging to the category $p=3$, a five-qubit code to the category $p=6$, etc.; then by the dimensions of the Hilbert spaces of the component parts, along with other criteria yet to be determined.Comment: Latex 32 pages, 4 figures in text using PSTricks. Version 3: Minor typographical errors correcte

Code properties from holographic geometries

Almheiri, Dong, and Harlow [arXiv:1411.7041] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes which admit a holographic interpretation. We introduce a new quantity called `price', which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit `uberholography', meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales small compared to the AdS curvature radius.Comment: 17 pages, 5 figure

Tema Con Variazioni: Quantum Channel Capacity

Channel capacity describes the size of the nearly ideal channels, which can be obtained from many uses of a given channel, using an optimal error correcting code. In this paper we collect and compare minor and major variations in the mathematically precise statements of this idea which have been put forward in the literature. We show that all the variations considered lead to equivalent capacity definitions. In particular, it makes no difference whether one requires mean or maximal errors to go to zero, and it makes no difference whether errors are required to vanish for any sequence of block sizes compatible with the rate, or only for one infinite sequence.Comment: 32 pages, uses iopart.cl

Finite correlation length implies efficient preparation of quantum thermal states

Preparing quantum thermal states on a quantum computer is in general a difficult task. We provide a procedure to prepare a thermal state on a quantum computer with a logarithmic depth circuit of local quantum channels assuming that the thermal state correlations satisfy the following two properties: (i) the correlations between two regions are exponentially decaying in the distance between the regions, and (ii) the thermal state is an approximate Markov state for shielded regions. We require both properties to hold for the thermal state of the Hamiltonian on any induced subgraph of the original lattice. Assumption (ii) is satisfied for all commuting Gibbs states, while assumption (i) is satisfied for every model above a critical temperature. Both assumptions are satisfied in one spatial dimension. Moreover, both assumptions are expected to hold above the thermal phase transition for models without any topological order at finite temperature. As a building block, we show that exponential decay of correlation (for thermal states of Hamiltonians on all induced subgraph) is sufficient to efficiently estimate the expectation value of a local observable. Our proof uses quantum belief propagation, a recent strengthening of strong sub-additivity, and naturally breaks down for states with topological order.Comment: 16 pages, 4 figure