| completely positive maps | |
|---|---|
| Name | Completely positive map |
| Caption | Schematic of a quantum operation acting on a subsystem |
| Field | Quantum information theory |
| Introduced | 1970s |
| Related | Kraus representation, Choi–Jamiołkowski isomorphism, Quantum channel |
completely positive maps
A completely positive map is a linear map between C*-algebras (or between spaces of bounded operators) that preserves positivity even when tensored with the identity map on an auxiliary system. In Quantum mechanics and Quantum information theory, completely positive maps formalize allowed physical transformations of density operators and are fundamental for describing open quantum system dynamics, measurement, and noise. Their stronger condition relative to mere positivity ensures physical consistency under composition with entangled ancillary systems such as those encountered in experiments at Bell test scales.
A linear map Φ: B(H) → B(K) between algebras of bounded operators on Hilbert spaces H and K is positive if Φ(X) is positive semidefinite for every positive semidefinite X ∈ B(H). Φ is called completely positive (CP) if for all integers n ≥ 1 the ampliation Φ ⊗ id_n: B(H) ⊗ M_n(ℂ) → B(K) ⊗ M_n(ℂ) is positive, where id_n is the identity map on n×n matrices M_n(ℂ). Equivalently, CP requires positivity of Φ ⊗ id_L for any auxiliary Hilbert space L. In the operator-algebraic approach of GNS theory and C*-algebra theory, CP maps play a central role in dilation theorems such as the Stinespring dilation theorem, which characterizes CP maps via an isometry into a larger Hilbert space and a **-representation.
In quantum theory a physical state is represented by a density operator ρ (a positive trace-class operator) on H. A physically realizable transformation acting only on the system must map density operators to density operators even when the system is correlated with an external ancilla. This requirement motivates complete positivity: if Φ were merely positive but not CP, applying Φ locally to part of an entangled state could yield a non-physical (non-positive) global state, violating the operational demands of LOCC scenarios and experimental procedures at places such as the Quantum Information Science divisions of institutions like IBM Research or QuTech.
Every completely positive, trace-non-increasing map on B(H) admits an operator-sum (Kraus) decomposition Φ(ρ) = Σ_i K_i ρ K_i^† with Kraus operators K_i: H → K satisfying Σ_i K_i^†K_i ≤ I_H; equality yields trace preservation. This representation originates from work by Karl Kraus and is widely used in descriptions of quantum noise models such as the depolarizing channel, amplitude damping channel, and phase damping. The Kraus form is directly connected to the physical picture of system–environment interaction followed by partial tracing, as formalized by the Stinespring dilation theorem and modeled in experiments in laboratories like Max Planck Institute for Quantum Optics and MIT quantum optics groups.
Positivity alone does not guarantee physicality in the presence of entanglement. A canonical example is the transpose map T on matrices: T is positive but not completely positive because T ⊗ id acting on a maximally entangled state produces an operator with negative eigenvalues. The partial transpose criterion used in entanglement detection (Peres–Horodecki criterion) exploits this property. On the other hand, unitary conjugations U(·)U^† and convex mixtures thereof (random unitary channels) are completely positive. Other examples include the depolarizing channel and maps described by Lindblad generators in the Markovian limit.
A quantum channel is a completely positive, trace-preserving (CPTP) map and represents the most general deterministic evolution of a quantum state between input and output Hilbert spaces. Time-continuous Markovian dynamics of open systems are typically generated by Lindblad superoperators described by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form, whose dynamical maps are CPTP semigroups under suitable conditions. Non-Markovian dynamics and memory effects can yield families of CP maps parameterized by time; characterizing divisibility and CP-divisibility connects to experimental control in platforms like superconducting qubits and trapped ions.
The Choi–Jamiołkowski isomorphism relates linear maps Φ to a bipartite operator (the Choi matrix) C_Φ = (Φ ⊗ id)(|Φ^+⟩⟨Φ^+|) constructed from a maximally entangled state |Φ^+⟩. Choi's theorem states that Φ is completely positive iff C_Φ is positive semidefinite. The Choi matrix provides a concrete numerical criterion and enables semidefinite programming tests for CP-ness and complete positivity of processes reconstructed by quantum process tomography in experimental groups such as those at NIST and Google Quantum AI.
Completely positive maps underpin quantum information protocols including quantum error correction, quantum cryptography (security proofs often model adversarial actions as CPTP maps), and channel capacities (e.g., Holevo capacity). In open systems, CP maps capture decoherence and dissipation in models ranging from quantum optics to condensed matter. The failure of complete positivity of certain reduced dynamics motivates careful derivation of master equations; researchers from E. B. Davies to modern groups have addressed conditions ensuring CP. CP and its violation also inform entanglement criteria: the action of non-CP positive maps, like the reduction map or partial transpose, produces operational separability tests such as the Peres criterion and subsequent refinements by the Horodeckis.
Category:Quantum information theory Category:Operator theory