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Register a new userGeometry of Information: classical and quantum aspects
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Added by
Matilde Marcolli
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this article, we describe various aspects of categorification of the structures appearing in information theory. These aspects include probabilistic models both of classical and quantum physics, emergence of F-manifolds, and motivic enrichments.
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Technology of data collection and information transmission is based on various mathematical models of encoding. The words Geometry of information refer to such models, whereas the words Moufang patterns refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.
In 2018, Renes [IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 577-592 (2018)] (arXiv:1701.05583) developed a general theory of channel duality for classical-input quantum-output (CQ) channels. That result showed that a number of well-known duality results for linear codes on the binary erasure channel could be extended to general classical channels at the expense of using dual problems which are intrinsically quantum mechanical. One special case of this duality is a connection between coding for error correction (resp. wire-tap secrecy) on the quantum pure-state channel (PSC) and coding for wire-tap secrecy (resp. error correction) on the classical binary symmetric channel (BSC). While this result has important implications for classical coding, the machinery behind the general duality result is rather challenging for researchers without a strong background in quantum information theory. In this work, we leverage prior results for linear codes on PSCs to give an alternate derivation of the aforementioned special case by computing closed-form expressions for the performance metrics. The noted prior results include optimality of the square-root measurement (SRM) for linear codes on the PSC and the Fourier duality of linear codes. We also show that the SRM forms a suboptimal measurement for channel coding on the BSC (when interpreted as a CQ problem) and secret communications on the PSC. Our proofs only require linear algebra and basic group theory, though we use the quantum Dirac notation for convenience.
We examine the role of information geometry in the context of classical Cramer-Rao (CR) type inequalities. In particular, we focus on Eguchis theory of obtaining dualistic geometric structures from a divergence function and then applying Amari-Nagoakas theory to obtain a CR type inequality. The classical deterministic CR inequality is derived from Kullback-Leibler (KL)-divergence. We show that this framework could be generalized to other CR type inequalities through four examples: $alpha$-version of CR inequality, generalized CR inequality, Bayesian CR inequality, and Bayesian $alpha$-CR inequality. These are obtained from, respectively, $I_alpha$-divergence (or relative $alpha$-entropy), generalized Csiszar divergence, Bayesian KL divergence, and Bayesian $I_alpha$-divergence.
A renowned information-theoretic formula by Shannon expresses the mutual information rate of a white Gaussian channel with a stationary Gaussian input as an integral of a simple function of the power spectral density of the channel input. We give in this paper a rigorous yet elementary proof of this classical formula. As opposed to all the conventional approaches, which either rely on heavy mathematical machineries or have to resort to some external results, our proof, which hinges on a recently proven sampling theorem, is elementary and self-contained, only using some well-known facts from basic calculus and matrix theory.
Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in $mathbb{C}^{N times N}$ as a partial $2m times 2m$ binary symplectic matrix, where $N = 2^m$. We state and prove two theorems that use symplectic transvections to efficiently enumerate all symplectic matrices that satisfy a system of linear equations. As an important corollary of these results, we prove that for an $[![ m,m-k ]!]$ stabilizer code every logical Clifford operator has $2^{k(k+1)/2}$ symplectic solutions. The desired physical circuits are then obtained by decomposing each solution as a product of elementary symplectic matrices. Our assembly of the possible physical realizations enables optimization over them with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the $[![ 6,4,2 ]!]$ CSS code. We conclude with a classical coding-theoretic perspective for constructing logical Pauli operators for CSS codes. Since our circuit synthesis algorithm builds on the logical Pauli operators for the code, this paper provides a complete framework for constructing all logical Clifford operators for CSS codes. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a
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