ChoiJamiolkowski

Choi–Jamiołkowski isomorphism, named after Man-Duen Choi and József Jamiołkowski, is a fundamental result in quantum information theory that establishes a one-to-one correspondence between linear maps on matrix algebras and bipartite operators. For a linear map Φ: M_d → M_d, the Choi matrix J(Φ) is defined by J(Φ) = ∑_{i,j=1}^d Φ(E_{ij}) ⊗ E_{ij}, where E_{ij} are the standard matrix units. Equivalently, J(Φ) = (Φ ⊗ I)(|Ω⟩⟨Ω|) with |Ω⟩ = ∑_{i=1}^d |i⟩ ⊗ |i⟩ / √d, a maximally entangled state on C^d ⊗ C^d.

Under this isomorphism, a linear map Φ is completely positive if and only if its Choi matrix J(Φ) is positive semidefinite. The map is trace-preserving if a corresponding partial-trace condition holds on J(Φ), depending on the chosen convention; completely positive trace-preserving (CPTP) maps thus correspond to Choi matrices that are PSD and satisfy the trace constraint. The action of Φ on a matrix X can be recovered from J(Φ) by Φ(X) = Tr_1[(I ⊗ X^T) J(Φ)], where Tr_1 denotes the partial trace over the first subsystem.

The isomorphism provides a practical framework for analyzing quantum channels, including their positivity properties, entanglement structure of the associated Choi states, and optimization problems in quantum information, such as channel discrimination and capacity bounds. It remains a central tool in studying the structure and behavior of quantum operations.