| quantum channel capacity | |
|---|---|
| Name | Quantum channel capacity |
| Field | Quantum information theory |
| Related | Quantum channel, Quantum error correction, Quantum Shannon theory |
quantum channel capacity
Quantum channel capacity quantifies the maximum rate at which information can be reliably transmitted through a quantum channel under specified resources and error models. It extends the notion of Shannon entropy and classical channel capacity to settings where information carriers are quantum systems and communication may involve entanglement and quantum error correction. Quantum channel capacities are central to Quantum information theory and practical designs for quantum communication and quantum computing networks.
Quantum channel capacity is defined for a completely positive, trace-preserving map (CPTP map) representing noise on quantum systems, commonly modeled as a map between density operators on finite-dimensional Hilbert spaces associated with sender and receiver. Capacities depend on the communication task: transmitting classical information, quantum states, or entanglement. Notable capacity notions include the classical capacity of a quantum channel (the maximum classical bits per channel use), the quantum capacity (logical qubits per use), and the entanglement-assisted capacities where pre-shared entanglement between sender and receiver is allowed. Formal definitions reference coding protocols, asymptotic rates, and criteria of reliable transmission such as vanishing infidelity or trace distance constraints. Foundational contributors include Claude Shannon (classical theory), Alexander Holevo (Holevo bound), and Benjamin Schumacher (quantum noiseless coding / Schumacher compression).
Several distinct capacities are studied: - Quantum capacity (Q): rate of transmitting unknown quantum states, related to ability to transmit entanglement and correct coherent errors; mathematically connected to quantum coherent information. - Classical capacity (C): rate for reliable classical message transmission; characterized by Holevo quantities and related additivity problems. - Private capacity (P): rate for secret classical communication secure against eavesdroppers (relevant to quantum cryptography and quantum key distribution). - Entanglement-assisted capacities (C_E, Q_E): capacities when sender and receiver share unlimited entanglement, e.g., the Bennett-Shor-Smolin-Thapliyal (BSST) theorem for entanglement-assisted classical capacity. - Zero-error capacities: rates achievable with perfect fidelity, connected to combinatorial structures like noncommutative graphs. Each capacity can be considered under restrictions such as product-state encodings, adaptive strategies, or use of classical feedback. Relevant institutions producing key results include IBM Research, Microsoft Research, Perimeter Institute, and universities like MIT and Caltech.
The formalism uses operator algebra on finite-dimensional Hilbert spaces, density matrices, Kraus representations, and Stinespring dilations describing system–environment interactions. Key inequalities include the Holevo bound (constraining accessible classical information) and the Fano inequality analogues for quantum channels. The BSST theorem characterizes entanglement-assisted classical capacity in terms of quantum mutual information. The Lloyd-Shor-Devetak (LSD) theorem gives the single-letter coherent-information formula as an achievable lower bound for quantum capacity under regularization. Important proofs and techniques draw from major results by Peter Shor, Igor Devetak, John Preskill, and Charles H. Bennett. Tools such as typical subspaces, decoupling theorems, and quantum error-correcting codes underpin coding theorems.
Many capacity expressions require regularization: limits of n→∞ quantities per channel use, due to non-additivity phenomena. Additivity questions—whether single-letter formulas suffice—have driven major research: the additivity conjecture for minimum output entropy was disproved by counterexamples from Matthias B. Hastings. As a result, capacities like classical and private capacity may not be additive; expressions often involve regularized Holevo information or regularized coherent information. For entanglement-assisted capacities, additivity holds and single-letter formulas are available. Notable formulas include: - Holevo-Schumacher-Westmoreland (HSW) capacity for classical transmission via quantum channels (expressed via Holevo chi quantity, often regularized). - LSD and Devetak formulas for quantum capacity in terms of maximized coherent information and regularization. These complexities link to channel superadditivity and phenomena such as superactivation, where combining zero-capacity channels yields positive joint capacity—discovered in works by Gregory Smith and John A. Smolin.
Quantum channel capacities are operationally achieved using quantum error-correcting codes (QECCs), entanglement-assisted codes, and concatenated constructions. Stabilizer codes (stemming from work by Daniel Gottesman and others) and low-density parity-check (LDPC) quantum codes implement fault-tolerant encoding against typical noise models. Coding theorems use random coding arguments, typical subspace projectors, and decoupling approaches to demonstrate achievability of capacity rates; converse theorems bound achievable rates using entropy inequalities. Quantum capacity is directly tied to the existence of QECCs that correct errors modeled by the channel's Kraus operators, while private capacity constructions exploit privacy amplification and quantum-to-classical reduction techniques.
Physically motivated channels include: - Depolarizing channel: symmetric noise replacing state with maximally mixed state with some probability; widely analyzed for capacities. - Dephasing (phase-damping) channel: loss of quantum coherence without energy exchange; models decoherence in superconducting qubits and spin qubits. - Amplitude-damping channel: models relaxation and energy loss in two-level systems, relevant for quantum optics and NV centers in diamond. - Bosonic Gaussian channels: model optical fibers and free-space communication; include lossy channel and thermal noise channels; capacities connect to continuous-variable quantum information and results by Alexander Holevo and Alexander S. Holevo's collaborators. Analytic and numerical capacity bounds for these channels guide experimental quantum communication and error correction choices in platforms such as Google Quantum AI, Rigetti Computing, and optical quantum communication testbeds.
Estimating quantum channel capacity experimentally involves quantum process tomography, randomized benchmarking, and direct fidelity estimation to characterize channel noise. Practical bounds are often obtained from measured channel parameters: error rates, loss, and noise spectra. Techniques like channel simulation, entanglement witnesses, and semidefinite programming yield upper and lower bounds on capacities for finite resources. Experiments demonstrating quantum communication protocols, entanglement distribution, and superactivation have been reported from laboratories at Caltech, MIT, Harvard University, and industrial research groups. Ongoing challenges include finite-blocklength effects, device imperfections, and scalable estimation for high-dimensional or continuous-variable channels.
Category:Quantum information theory Category:Quantum communication