Testing non-isometry is QMA-complete

Determining the worst-case uncertainty added by a quantum circuit is shown to be computationally intractable. This is the problem of detecting when a quantum channel implemented as a circuit is close to a linear isometry, and it is shown to be complete for the complexity class QMA of verifiable quantum computation. This is done by relating the problem of detecting when a channel is close to an isometry to the problem of determining how mixed the output of the channel can be when the input is a pure state. How mixed the output of the channel is can be detected by a protocol making use of the swap test: this follows from the fact that an isometry applied twice in parallel does not affect the symmetry of the input state under the swap operation.Comment: 12 pages, 3 figures. Presentation improved, results unchange

Computational Distinguishability of Quantum Channels

The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the well-known satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixed-unitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixed-unitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.Comment: Ph.D. Thesis, 178 pages, 35 figure

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

All entangled states are useful for channel discrimination

We prove that every entangled state is useful as a resource for the problem of minimum-error channel discrimination. More specifically, given a single copy of an arbitrary bipartite entangled state, it holds that there is an instance of a quantum channel discrimination task for which this state allows for a correct discrimination with strictly higher probability than every separable state.Comment: 5 pages, more similar to the published versio

Floodplains : the forgotten and abused component of the fluvial system

River restoration is strongly focussed on in-channel initiatives driven by fisheries interests and a continued desire for river stability. This contrasts greatly with the inherently mobile nature of watercourses. What is often overlooked is the fact that many rivers have developed floodplain units that would naturally operate as integrated functional systems, moderating the effects of extreme floods by distributing flow energy and sediment transport capacity through out of bank flooding. Floodplain utilisation for farming activities and landowner intransigence when it comes to acknowledging that the floodplain is part of the river system, has resulted in floodplains being the most degraded fluvial morphologic unit, both in terms of loss of form and function and sheer levels of spatial impact. The degradation has been facilitated by the failure of regulatory mechanisms to adequately acknowledge floodplain form and function. This is testament to the ‘inward looking’ thinking behind national assessment strategies. This paper reviews the state of floodplain systems drawing on quantitative data from England and Wales to argue for greater consideration of the floodplain in relation to river management. The database is poor and must be improved, however it does reveal significant loss of watercourse-floodplain connectivity linked to direct flood alleviation measures and also to altered flood frequency as a result of river downcutting following river engineering. These latter effects have persisted along many watercourses despite the historic nature of the engineering interventions and will continue to exacerbate the risk of flooding to downstream communities. We also present several examples of the local and wider values of reinstating floodplain form and function, demonstrating major ecological gains, improvement to downstream flood reduction, elevation of water quality status and reductions in overall fine sediment loss from farmland. A re-think is required regarding our approach to managing floodplains and funding floodplain restoration, arguing for greater recognition of the natural role of the floodplain as a resource for upstream flood management and as an agent for overall biotic improvement in line with restoration objectives

Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can return to it when either taking finitely many copies of them or supplementing with a completely depolarizing channel. In these scenarios this implies that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page

The stream evolution triangle: Integrating geology, hydrology, and biology

The foundations of river restoration science rest comfortably in the fields of geology, hydrology, and engineering, and yet, the impetus for many, if not most, stream restoration projects is biological recovery. Although Lane's stream balance equation from the mid‐1950s captured the dynamic equilibrium between the amount of stream flow, the slope of the channel, and the amount and calibre of sediment, it completely ignored biology. Similarly, most of the stream classification systems used in river restoration design today do not explicitly include biology as a primary driver of stream form and process. To address this omission, we cast biology as an equal partner with geology and hydrology, forming a triumvirate that governs stream morphology and evolution. To represent this, we have created the stream evolution triangle, a conceptual model that explicitly accounts for the influences of geology, hydrology, and biology. Recognition of biology as a driver leads to improved understanding of reachscale morphology and the dynamic response mechanisms responsible for stream evolution and adjustment following natural or anthropogenic disturbance, including stream restoration. Our aim in creating the stream evolution triangle is not to exclude or supersede existing stream classifications and evolutionary models but to provide a broader “thinking space” within which they can be framed and reconsidered, thus facilitating thought outside of the alluvial box