The Choi-Jamiolkowski Isomorphism

Every quantum channel corresponds to a single bipartite state, turning questions about maps into questions about states.

Channels as states

The Choi-Jamiolkowski isomorphism encodes a channel E in one operator: apply E to half of a maximally entangled pair. The resulting Choi state J(E) = (E tensor I)(|Omega> = sum_i |ii>, contains complete information about the channel. Knowing J(E) is equivalent to knowing E; the map and the state are two faces of the same object.

The dictionary

• E is completely positive if and only if J(E) is positive semidefinite
• E is trace-preserving if and only if the B-marginal of J(E) is the identity (up to normalization)
• The Kraus operators of E are read off from the eigenvectors of J(E)
• Applying E to any input can be computed from J(E) by a partial-trace formula

This dictionary converts abstract properties of maps into familiar linear-algebra checks on a matrix, which is why the isomorphism is so useful in proofs and computation.

Recovering the action

The channel output on an arbitrary state rho is recovered by E(rho) = Tr_A[(rho^T tensor I) J(E)], where the transpose acts on the input register. So a single fixed measurement of the Choi state, obtained by sending in one half of a Bell pair, characterizes the channel's response to every possible input, the principle behind ancilla-assisted process tomography.

import numpy as np
def choi(kraus, d=2):
    Om = np.zeros((d*d,)); 
    for i in range(d): Om[i*d+i]=1
    Om = Om/np.sqrt(d)
    rho = np.outer(Om,Om).reshape(d,d,d,d)
    # apply channel on first factor
    out = sum(np.einsum('ij,jbkd->ibkd', K, rho.reshape(d,d,d,d)) for K in kraus)
    return out
# used to test complete positivity via eigenvalues of Choi matrix

Why it matters

Complete positivity, the defining constraint that makes a map physical, reduces to a positivity check on the Choi matrix, giving a practical test. Optimization over channels becomes optimization over states with linear constraints, amenable to semidefinite programming. The isomorphism connects channel capacities, entanglement of the Choi state, and error rates, making it a central tool for both theory and the numerical certification of quantum devices.