Probabilistic Monads, Domains and Classical Information

Shannon's classical information theory uses probability theory to analyze channels as mechanisms for information flow. In this paper, we generalize results of Martin, Allwein and Moskowitz for binary channels to show how some more modern tools - probabilistic monads and domain theory in particular - can be used to model classical channels. As initiated Martin, et al., the point of departure is to consider the family of channels with fixed inputs and outputs, rather than trying to analyze channels one at a time. The results show that domain theory has a role to play in the capacity of channels; in particular, the (n x n)-stochastic matrices, which are the classical channels having the same sized input as output, admit a quotient compact ordered space which is a domain, and the capacity map factors through this quotient via a Scott-continuous map that measures the quotient domain. We also comment on how some of our results relate to recent discoveries about quantum channels and free affine monoids.Comment: In Proceedings DCM 2011, arXiv:1207.682

Agents, subsystems, and the conservation of information

Dividing the world into subsystems is an important component of the scientific method. The choice of subsystems, however, is not defined a priori. Typically, it is dictated by experimental capabilities, which may be different for different agents. Here we propose a way to define subsystems in general physical theories, including theories beyond quantum and classical mechanics. Our construction associates every agent A with a subsystem SA, equipped with its set of states and its set of transformations. In quantum theory, this construction accommodates the notion of subsystems as factors of a tensor product Hilbert space, as well as the notion of subsystems associated to a subalgebra of operators. Classical systems can be interpreted as subsystems of quantum systems in different ways, by applying our construction to agents who have access to different sets of operations, including multiphase covariant channels and certain sets of free operations arising in the resource theory of quantum coherence. After illustrating the basic definitions, we restrict our attention to closed systems, that is, systems where all physical transformations act invertibly and where all states can be generated from a fixed initial state. For closed systems, we propose a dynamical definition of pure states, and show that all the states of all subsystems admit a canonical purification. This result extends the purification principle to a broader setting, in which coherent superpositions can be interpreted as purifications of incoherent mixtures.Comment: 31+26 pages, updated version with new results, contribution to Special Issue on Quantum Information and Foundations, Entropy, GM D'Ariano and P Perinotti, ed

Factor Graphs for Quantum Information Processing

[...] In this thesis, we are interested in generalizing factor graphs and the relevant methods toward describing quantum systems. Two generalizations of classical graphical models are investigated, namely double-edge factor graphs (DeFGs) and quantum factor graphs (QFGs). Conventionally, a factor in a factor graph represents a nonnegative real-valued local functions. Two different approaches to generalize factors in classical factor graphs yield DeFGs and QFGs, respectively. We proposed/re-proposed and analyzed generalized versions of belief-propagation algorithms for DeFGs/QFGs. As a particular application of the DeFGs, we investigate the information rate and their upper/lower bounds of classical communications over quantum channels with memory. In this study, we also propose a data-driven method for optimizing the upper/lower bounds on information rate.Comment: This is the finial version of the thesis of Michael X. Cao submitted in April 2021 in partial fulfillment of the requirements for the degree of doctor of philosophy in information engineering at the Chinese university of Hong Kon

Belief propagation decoding of quantum channels by passing quantum messages

Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the blocklength of the code. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a polar decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.Comment: v3: final version for publication; v2: improved discussion of the algorithm; 7 pages & 2 figures. v1: 6 pages, 1 figur

Gaussian Entanglement Distribution via Satellite

In this work we analyse three quantum communication schemes for the generation of Gaussian entanglement between two ground stations. Communication occurs via a satellite over two independent atmospheric fading channels dominated by turbulence-induced beam wander. In our first scheme the engineering complexity remains largely on the ground transceivers, with the satellite acting simply as a reflector. Although the channel state information of the two atmospheric channels remains unknown in this scheme, the Gaussian entanglement generation between the ground stations can still be determined. On the ground, distillation and Gaussification procedures can be applied, leading to a refined Gaussian entanglement generation rate between the ground stations. We compare the rates produced by this first scheme with two competing schemes in which quantum complexity is added to the satellite, thereby illustrating the trade-off between space-based engineering complexity and the rate of ground-station entanglement generation.Comment: Closer to published version (to appear in Phys. Rev. A) 13 pages, 6 figure