| Information theory | |
|---|---|
| Name | Information theory |
| Field | Physics; Applied mathematics |
| Related | Quantum information science; Computer science |
| Notable institutions | Bell Labs; MIT; Microsoft Research |
| Notable people | Claude Shannon; John von Neumann |
Information theory
Information theory is the mathematical study of the representation, transmission, and processing of information. In the context of Quantum physics, it provides the formal language and measures to quantify information carried by quantum states, to characterize noise in quantum channels, and to analyze limits of quantum computation and communication. The field bridges statistical mechanics, cryptography, and computer science to yield operational results such as channel capacities, resource trade-offs, and entropic inequalities.
Classical foundations of information theory trace to Claude Shannon's 1948 paper "A Mathematical Theory of Communication", introducing the Shannon entropy and the concept of channel capacity. Fundamental notions include the entropy H, mutual information, relative entropy (Kullback–Leibler divergence), and lossless and lossy source coding theorems. These concepts underpin coding schemes like Huffman coding and the noisy-channel coding theorem. Early theoretical development occurred at Bell Labs and influenced figures such as John von Neumann who contributed to statistical formulations that later intersected with quantum theory. Classical measures remain a reference point for their quantum analogues used in quantum communication and quantum thermodynamics.
Quantum generalizations replace probability distributions with density operators on Hilbert space. The von Neumann entropy S(ρ) = −Tr(ρ log ρ) is the quantum analogue of Shannon entropy. Quantum relative entropy and trace distance quantify distinguishability of density matrices; the Umegaki relative entropy and fidelity (quantum) (Uhlmann fidelity) are central. Other measures include quantum mutual information, conditional quantum entropy (which can be negative), and entropic inequalities such as strong subadditivity proved by Lieb and Ruskai. Seminal works by Nielsen and Chuang formalized these measures in "Quantum Computation and Quantum Information". Resource-theoretic measures like quantum coherence and entanglement entropy formalize usable quantum correlations, while operational tasks link these measures to capacities and rates.
Quantum channels are completely positive, trace-preserving maps modeling noise and transmission of quantum states; mathematically they are CPTP maps and can be represented via Kraus operator decompositions or Stinespring dilation. Capacities of quantum channels generalize Shannon capacity: quantum capacity, classical capacity of a quantum channel (Holevo–Schumacher–Westmoreland theorem), and private capacity. The Holevo bound limits classical information extractable from quantum ensembles. Protocols such as quantum teleportation and superdense coding exploit entanglement to achieve communication tasks. Quantum channel discrimination and adaptive strategies relate to works by Holevo and Yuen. Experimental platforms developed at institutions like IBM Quantum, Google Quantum AI, and university groups operationalize these channels.
Quantum entanglement is a nonclassical correlation central to quantum information science; criteria such as the Peres–Horodecki criterion (PPT test) detect entanglement. Measures include entanglement of formation, concurrence, and squashed entanglement. Entanglement serves as a resource in the resource theory framework, alongside quantum discord and quantum coherence. Entropic quantities quantify correlations: quantum mutual information measures total correlations, while conditional entropies differentiate classical from quantum parts. Foundational results linking entanglement to channel capacities and to thermodynamic work extraction connect to experiments performed at facilities such as CALT (Caltech) and Harvard University quantum labs.
Quantum error correction (QEC) protects quantum information against decoherence and operational errors using redundancy and syndrome measurement. Codes include the Shor code, Steane code, and surface code; these were developed following theoretical frameworks by Peter Shor and Andrew Steane. The theory employs notions from classical coding theory (e.g., stabilizer formalism) and algebraic topology for topological codes. Fault-tolerant architectures define thresholds for reliable computation; the threshold theorem formalizes scalability under realistic error models. Implementations and demonstrations occur in laboratories at University of Waterloo's Institute for Quantum Computing, D-Wave Systems (annealing context), and industrial platforms like Rigetti and IonQ.
Connections between information and thermodynamics date back to Maxwell's demon and were formalized by Landauer's principle linking information erasure to heat dissipation. In quantum regimes, quantum fluctuation theorems and resource-theoretic thermodynamics quantify work extraction from quantum states and the role of coherence and correlations. Formal tools include the von Neumann entropy, relative entropy distances to thermal states, and completely positive maps modeling thermal operations. Research by groups at Perimeter Institute and ETH Zurich explores quantum engines, work extraction protocols, and the thermodynamic cost of quantum measurement and feedback.
Information theory underlies algorithms and security proofs in quantum computing and quantum cryptography. Quantum complexity classes like BQP relate to resource bounds; entropic uncertainty relations constrain measurement outcomes and underpin protocols in quantum key distribution (e.g., BB84 proved secure using information-theoretic arguments). Error-correcting codes enable scalable quantum processors in architectures pursued by Google Quantum AI and IBM Quantum. Quantum algorithms (e.g., Shor's algorithm, Grover's algorithm) illustrate informational advantages. Post-quantum cryptography and hybrid classical–quantum networks require integrating classical information-theoretic concepts with quantum channel capacities and privacy amplification techniques developed by researchers across academia and industry.
Category:Quantum information theory Category:Information theory