| Quantum channel capacity | |
|---|---|
| Name | Quantum channel capacity |
| Caption | Schematic of information transmission through a noisy quantum channel |
| Field | Quantum information theory |
| Introduced | 1990s |
| Notable contributors | Peter Shor, Charles H. Bennett, Gilles Brassard, Bennet–Shor Smolin Thapliyal, Alexander Holevo, Benjamin Schumacher |
| Applications | Quantum communication, Quantum cryptography, Quantum computing |
Quantum channel capacity
Quantum channel capacity quantifies the maximum rate at which information can be reliably transmitted through a noisy quantum channel given particular resources and error criteria. It generalizes the classical Shannon entropy notion of channel capacity to the quantum domain, distinguishing between transmission of classical information, quantum information, and private communication. Quantum channel capacities underpin protocols in quantum key distribution and fault-tolerant quantum computing and guide design of physical links such as optical fibers and satellite links developed by organizations like IBM, Google Quantum AI, and research labs such as Los Alamos National Laboratory and Perimeter Institute.
A quantum channel is mathematically a completely positive trace-preserving (CPTP) map that models noise and decoherence acting on quantum states prepared by a sender. Quantum channel capacity denotes the asymptotic number of information units per channel use that can be transmitted with vanishing error in the limit of many uses. Multiple capacity notions exist depending on whether the transmitted information is classical bits, qubits, or confidential information, and on available auxiliary resources such as prior entanglement or classical communication. Foundational work by Alexander Holevo, Benjamin Schumacher, and Peter Shor established the basic limits and stimulated development of coding theorems and capacity formulas.
The principal capacities are: - Classical capacity (C): maximum reliable transmission rate of classical bits over a quantum channel, related to the Holevo bound and Holevo–Schumacher–Westmoreland (HSW) theorem. - Quantum capacity (Q): rate of reliable transmission of quantum states (qubits); connected to the Lloyd-Shor-Devetak (LSD) theorem and coherent information. - Private capacity (P): optimal rate for sending classical information with information-theoretic secrecy against an eavesdropper, relevant to quantum cryptography and linked to the Devetak result. - Entanglement-assisted capacity (C_E, Q_E): capacities when sender and receiver share unlimited prior entanglement; the Bennett-Shor-Smolin-Thapliyal (BSST) theorem gives a single-letter formula using quantum mutual information. Each capacity can differ markedly for the same physical channel; examples and counterexamples were studied by researchers including Gilles Brassard, Igor Devetak, and Peter Shor.
Capacity formulas employ quantum information measures: von Neumann entropy S(ρ)=−Tr(ρ log ρ), quantum mutual information I(A;B)=S(A)+S(B)−S(AB), and coherent information I_c(ρ, N)=S(N(ρ))−S((I⊗N)(|ψ⟩⟨ψ|)). The Holevo quantity χ quantifies accessible classical information in ensembles and enters the HSW classical capacity expression. Capacities often require optimization over input ensembles or density operators and may involve regularization: limits of n→∞ of 1/n times an optimized quantity for N^⊗n. Mathematical tools from operator theory, von Neumann algebra, and quantum Shannon theory are routinely used, as in works by Holevo (Alexander S.) and Mark Wilde.
Coding theorems establish direct (achievability) and converse (upper bound) results for capacities. The HSW theorem provides achievability for classical capacity via typical subspace and measurement designs; the LSD theorem provides quantum rate achievability via quantum error-correcting codes and decoupling arguments. Techniques include random coding, typicality, entanglement distillation, and privacy amplification used in proofs by Devetak (Igor), Winter (Andreas), and Hayden (Patrick). Converse bounds use monotonicity of quantum relative entropy and data-processing inequalities. Finite-blocklength analyses and strong converse statements have been developed by groups at institutions like Caltech and MIT.
A central technical challenge is additivity: whether single-letter formulas suffice or regularization across many channel uses is necessary. Early conjectures that capacities were additive were disproved by counterexamples (notably entanglement-breaking and superadditive behaviors). Superadditivity of coherent information and nonadditivity of private capacities were demonstrated in constructions by researchers including Hastings (Matthew B.), producing channels where joint coding outperforms independent uses. These phenomena force regularized capacity formulas and complicate numerical evaluation and coding design, motivating study of degradable channels and conditions guaranteeing additivity.
Common channel models include the depolarizing channel, amplitude damping channel, phase damping channel, erasure channel, and thermal bosonic channels modeling optical loss. The quantum and classical capacities of the erasure channel are exactly solvable; the depolarizing channel displays regions of partial additivity known from numerical studies. Bosonic Gaussian channels, relevant to free-space and fiber optics and studied by Serafini (Alessio) and Weedbrook (Christian), have capacity formulas under energy constraints, and entanglement-assisted Gaussian capacities admit closed forms via quantum mutual information.
Quantum channel capacities inform design and performance limits of quantum repeaters, quantum networks, and long-distance quantum key distribution systems such as those implemented by the China Micius satellite program and commercial testbeds by AT&T and European Space Agency. Capacities determine required entanglement distribution rates for distributed quantum computation and fault tolerance thresholds in architectures by IBM and Google. Understanding capacity trade-offs guides resource allocation (classical vs entanglement assistance) and informs standardization efforts for quantum communication protocols at bodies like the ITU.