| completely positive map | |
|---|---|
| Name | Completely positive map |
| Field | Quantum mechanics; Operator theory |
| Introduced | Early 1970s |
| Notable examples | Kraus operators, Stinespring dilation, Choi matrix |
completely positive map
A completely positive map is a linear map between algebras of operators (typically between matrix algebras) that sends positive operators to positive operators even when tensored with the identity on an arbitrary ancillary space. In Quantum Physics and Quantum information theory such maps are the mathematical models of physically admissible transformations of quantum states and of noise acting on quantum systems. They matter because physical evolution of density operators, including open system dynamics and quantum channels, must be completely positive to preserve positivity under entanglement with external systems.
A linear map Φ: A → B between C*-algebras or matrix algebras is positive if Φ(ρ) is positive for every positive ρ ∈ A. Φ is completely positive (CP) if, for every integer n ≥ 1, the amplified map Φ ⊗ id_n acting on A ⊗ M_n(ℂ) is positive. The requirement is stronger than positivity and is essential to guarantee that Φ preserves positivity on subsystems entangled with arbitrary ancillae such as those arising in experiments at IBM, Google Quantum AI, or Rigetti Computing. Basic properties include closure under convex combinations, composition, and tensor products with other CP maps. The set of CP maps between fixed finite-dimensional algebras is a convex cone; when combined with the trace-preserving constraint it forms the convex set of quantum channels often studied in quantum entropy contexts.
The operator-sum or Kraus representation theorem states that any CP map Φ on finite-dimensional matrix algebras admits a decomposition Φ(ρ) = Σ_k E_k ρ E_k† where the are called Kraus operators or operation elements. The representation is nonunique; different sets {E_k} related by an isometry yield the same map. First established in work by Karl Kraus and formalized in operator algebra literature, the theorem underpins practical descriptions of decoherence channels used in experimental studies at institutions such as Harvard University and California Institute of Technology. The operator-sum form makes explicit how noise and measurements act on density matrices and facilitates computation of output states, fidelities, and entropic quantities.
A CP map that also preserves the trace is a CPTP map and is commonly termed a quantum channel or quantum operation. Quantum channels model physically realizable state transformations including unitary evolution, projective and generalized measurements, and noisy processes implemented in quantum computing hardware. CPTP maps satisfy Σ_k E_k† E_k = I in the Kraus form. Channels are central to the study of capacities (classical and quantum), error correction such as Shor code and Steane code, and communication protocols investigated at conferences like the IEEE International Symposium on Information Theory.
Common CP maps include: - Unitary channels: Φ_U(ρ) = UρU† with U ∈ U(n), representing closed-system evolution under Schrödinger equation. - Depolarizing channel: Φ_p(ρ) = (1−p)ρ + p I/d, modeling isotropic noise and used in threshold estimates for architectures like IonQ. - Amplitude damping: models energy relaxation (spontaneous emission) with Kraus elements derived from coupling to a bath; relevant to experiments at NIST and superconducting qubit platforms. Other classes include dephasing, phase-damping, and entanglement-breaking channels; the latter break entanglement with any ancillary system and are characterized by specific structural criteria.
The Choi matrix (or Choi–Jamiołkowski isomorphism) associates a linear map Φ with an operator C_Φ = (Φ ⊗ id)(|Ω⟩⟨Ω|) built from a maximally entangled vector |Ω⟩; Φ is CP iff C_Φ is positive semidefinite. The Choi test is an algorithmic criterion widely used in theoretical work and numerical verification. The Stinespring dilation theorem provides a structure theorem: any CP map admits an isometric embedding into a larger Hilbert space followed by a partial trace over an environment, Φ(ρ) = Tr_E(V ρ V†). Stinespring’s result clarifies the physical picture of CP maps arising from unitary evolution on a system-plus-environment and underlies derivations of Markovian master equations such as the Lindblad equation.
In open quantum systems theory, CP maps describe discrete-time steps of reduced dynamics and, in the continuous-time limit under certain assumptions, generate semigroups whose generators have the Lindblad form. CP maps ensure that reduced dynamics do not produce unphysical negative eigenvalues in density matrices when systems are entangled with external degrees of freedom, a concern in modeling experiments by groups at Los Alamos National Laboratory and Max Planck Institute for Quantum Optics. In quantum information processing, CP and CPTP formalism provides the language for error models, quantum process tomography, and design of fault-tolerant protocols such as those used in surface code research.
CP maps enjoy stability properties: compositions of CP maps are CP, convex mixtures are CP, and tensor products of CP maps are CP. Continuity properties in operator norm and completely bounded norm are relevant for approximation and numerical analysis; the theory of completely bounded maps refines stability analysis in infinite dimensions and in C*-algebraic approaches exemplified by work in operator algebras and at institutions like Institute for Advanced Study. The convex geometry of CPTP maps is exploited in optimization tasks (semidefinite programming) for channel discrimination, capacity bounds, and quantum control problems.
Category:Quantum information theory Category:Operator theory