CPTP Map

A CPTP map is the correct mathematical framework for describing any operation that can be performed on a quantum system, replacing the more restrictive notion of unitary evolution.

The two conditions

Completely positive (CP): The map takes positive semidefinite operators to positive semidefinite operators, even when acting on part of a larger entangled system. This is stronger than ordinary positivity. The transpose map is positive but not completely positive: applied to half of a Bell state it produces a matrix with a negative eigenvalue, which cannot represent a physical state. Complete positivity rules out such unphysical maps.

Trace-preserving (TP): The map preserves the total probability: Tr(ε(ρ)) = Tr(ρ) = 1 for any density matrix ρ. This ensures probabilities continue to sum to one after the operation.

Why not just unitary?

Unitary maps U are the special case of CPTP maps with a single Kraus operator K₁ = U. They apply to closed systems evolving in isolation. Real quantum systems interact with environments (causing decoherence), undergo measurements (which project and renormalise), and are subject to noise. All of these processes are CPTP but not unitary.

The Kraus representation

Every CPTP map has a Kraus decomposition:

ε(ρ) = Σₖ Kₖ ρ Kₖ†

where the Kraus operators {Kₖ} satisfy Σₖ Kₖ†Kₖ = I. This completeness relation is equivalent to the trace-preserving condition. The Kraus representation is not unique: any unitary rotation of the operator set gives an equivalent channel.

The Choi matrix

Every CPTP map corresponds to a unique positive semidefinite matrix called the Choi matrix via the Choi-Jamiolkowski isomorphism. This correspondence turns questions about the physics of quantum channels into questions about the linear algebra of positive matrices, and underlies quantum process tomography.

Why it matters

The CPTP framework is foundational to quantum information theory:

• It provides the language for describing quantum noise channels (bit flip, depolarising, amplitude damping)
• Quantum error correction is the art of constructing logical operations that remain close to target CPTP maps despite physical noise
• Error mitigation techniques like probabilistic error cancellation decompose ideal CPTP maps as linear combinations of noisy ones

See also

• Kraus Operators
• Density Matrix
• Decoherence
• Quantum Error Correction