| completely positive map | |
|---|---|
| Name | Completely positive map |
| Caption | Schematic of a quantum operation acting on a density operator |
| Field | Quantum information theory |
| Introduced | 1970s |
| Related | Kraus representation, Stinespring dilation theorem, Choi matrix |
completely positive map
A completely positive map is a linear map between algebras of operators that preserves positivity even when tensored with the identity on an auxiliary system. In Quantum Physics and quantum information theory it formalizes physical transformations of density operators, guaranteeing that composite systems remain in valid states. Completely positive maps underpin descriptions of quantum channels, open system dynamics, and error models relevant to experimental platforms such as IBM Quantum, Google Quantum AI, and Rigetti Computing.
A completely positive (CP) map Φ: A → B is a linear map between C*-algebras (commonly matrix algebras M_n(ℂ)) for which Φ ⊗ id_k is positive for all k ≥ 1, where id_k is the identity map on M_k(ℂ). The requirement differs from mere positivity (Φ positive) by demanding stability under arbitrary addition of ancilla systems; this is essential for consistent composition with entangled states used in protocols like quantum teleportation and superdense coding. Basic properties include convexity of the set of CP maps, closure under composition and tensor product, and characterization via operator inequalities. Important named contributors include Göran Lindblad, Alexander Kraus (Kraus), Walter Stinespring, and Man-Duen Choi.
By the Kraus representation theorem, any completely positive map Φ: M_n(ℂ) → M_m(ℂ) that is also trace-nonincreasing can be written in operator-sum form Φ(ρ) = Σ_i K_i ρ K_i† with Kraus operators K_i ∈ M_{m×n}(ℂ). For trace-preserving CP maps (quantum channels) the Kraus operators satisfy Σ_i K_i† K_i = I. The operator-sum formalism is central in modeling noise in laboratory platforms such as superconducting qubit devices and trapped ion systems. The formalism connects to experimental tomography techniques like quantum process tomography and to numerical simulation in packages developed by Qiskit and Cirq.
Complete positivity is stronger than positivity: a map can be positive but fail to be CP, violating physicality when acting on entangled inputs. The Choi–Jamiołkowski isomorphism associates Φ with its Choi matrix C_Φ = (Φ ⊗ id)(|Φ^+⟩⟨Φ^+|) built from a maximally entangled state |Φ^+⟩. Man-Duen Choi proved that Φ is CP iff C_Φ is positive semidefinite. The Choi criterion enables algorithmic checks of complete positivity in software used at institutions like Los Alamos National Laboratory and in academic research; it also relates to semidefinite programming and resource-theoretic characterizations used by groups at Perimeter Institute and MIT.
Trace-preserving CP maps are called quantum channels and model physical evolutions including unitary dynamics, decoherence, and dissipative processes described by master equations such as the Lindblad equation. The Gorini–Kossakowski–Sudarshan–Lindblad framework provides generators for continuous CP trace-preserving (CPTP) semigroups relevant to quantum optics experiments at institutions like Max Planck Institute for Quantum Optics. Quantum channels are the mathematical backbone of quantum communication protocols, quantum cryptography standards considered by organizations like National Institute of Standards and Technology (NIST), and analyses of noise in near-term devices.
Common CP maps used as noise models include: - Dephasing channel: models phase randomization, with Kraus operators derived from projectors; relevant to nuclear magnetic resonance and superconducting qubits. - Depolarizing channel: replaces state with maximally mixed state with some probability; used in benchmarking and randomized benchmarking protocols developed at IBM Research. - Amplitude damping channel: models energy relaxation (T1 processes) in qubits; derived from a physical system coupled to a bath and captured by Kraus operators K_0, K_1. These channels are employed in theoretical studies (e.g., capacity theorems) and experimental error characterization at labs including Harvard Quantum Initiative and University of Oxford.
CP maps are naturally framed within operator algebra theory. The Stinespring dilation theorem states that any CP map Φ: A → B(H) admits a representation Φ(a) = V† π(a) V, where π is a *-representation on an enlarged Hilbert space and V is a bounded operator; this provides a unitary dilation and links CP maps to system–environment models. Stinespring's result underlies constructive approaches to simulate open dynamics and implement channels via unitary couplings in laboratory setups and quantum simulators developed by groups at ETH Zurich and Caltech.
Completely positive maps are foundational for quantum error correction (QEC) theory, where noise channels are modeled as CP maps and recovery operations are designed to be CPTP maps; landmark works by Peter Shor and Andrew Steane rely on this framework. Understanding CP maps informs resource theories, channel capacities, and fault-tolerance thresholds critical to equitable deployment of quantum technologies. Policy and equity implications arise when noise modeling affects access: ensuring open benchmarks, transparent noise characterization (e.g., by Quantum Open Source Foundation initiatives), and capacity-building in underserved regions helps prevent concentration of capability. Democratising tools and sharing CP map characterizations can support more equitable participation in research and industry around quantum computing.
Category:Quantum information theory Category:Operator theory