The Choi–Jamiołkowski Isomorphism

There is a perfect dictionary between channels and states: every channel corresponds to a single bipartite state, and vice versa. This Choi–Jamiołkowski isomorphism (channel–state duality) turns hard questions about maps into ordinary questions about states, and it is the engine behind Choi's positivity test from earlier.

The construction

Let $\mathcal{E}: \mathcal{L}(\mathcal{H}_A)\to\mathcal{L}(\mathcal{H}_B)$ with $\dim\mathcal{H}_A = d$ . Take a second copy $A'$ of the input, form the (unnormalized) maximally entangled vector

|\Omega\rangle = \sum_{i=0}^{d-1} |i\rangle_{A'} |i\rangle_A,

and run the channel on the $A$ half only. The result is the Choi operator

J(\mathcal{E}) \;=\; (\mathbb{1}_{A'}\otimes\mathcal{E})\big(|\Omega\rangle\langle\Omega|\big) \;=\; \sum_{i,j=0}^{d-1} |i\rangle\langle j|_{A'} \otimes \mathcal{E}\big(|i\rangle\langle j|_A\big).

This is a single operator on $\mathcal{H}_{A'}\otimes\mathcal{H}_B$ . The map $\mathcal{E}\mapsto J(\mathcal{E})$ is linear and invertible: knowing how $\mathcal{E}$ acts on the $d^2$ basis operators $|i\rangle\langle j|$ is knowing $\mathcal{E}$ completely, and those actions are exactly the blocks of $J$ .

Recovering the channel

The dictionary runs both ways. Given the Choi operator you reconstruct the action of the channel by

\mathcal{E}(\rho) = \operatorname{tr}_{A'}\!\Big[\big(\rho^{\mathsf{T}}\otimes\mathbb{1}_B\big)\,J(\mathcal{E})\Big],

where the transpose is taken in the chosen basis. So $J(\mathcal{E})$ is a complete fingerprint of the channel — no information is lost.

The dictionary of properties

The power of the isomorphism is that structural properties of $\mathcal{E}$ become familiar properties of the operator $J$ :

Complete positivity $\;\Longleftrightarrow\; J(\mathcal{E}) \succeq 0$ . (Choi's theorem.) This is why CP is finitely checkable: positivity of one operator.
Trace preservation $\;\Longleftrightarrow\; \operatorname{tr}_B\,J(\mathcal{E}) = \mathbb{1}_{A'}$ .
A channel (CPTP map) $\;\Longleftrightarrow\;$ a bipartite operator $J\succeq 0$ with $\operatorname{tr}_B J = \mathbb{1}_{A'}$ . Normalizing by $d$ makes $\tfrac1d J$ a bona-fide density operator with maximally mixed marginal on $A'$ .
The Kraus rank of $\mathcal{E}$ $\;\Longleftrightarrow\;$ the rank of $J$ . A spectral decomposition $J = \sum_k|v_k\rangle\langle v_k|$ "unvectorizes" into a minimal Kraus set.

Example: the Choi state of the identity

For the qubit identity channel, $J(\mathbb{1}) = \sum_{ij}|i\rangle\langle j|\otimes|i\rangle\langle j| = |\Omega\rangle\langle\Omega|$ , i.e. (up to normalization) the projector onto the Bell state $|\Phi^+\rangle$ . The noisier the channel, the more mixed its Choi state: a depolarizing channel's Choi state is the Werner state $(1-p)|\Phi^+\rangle\langle\Phi^+| + p\,\mathbb{1}/4$ , and the channel is entanglement-breaking exactly when its Choi state is separable. With the unnormalized $|\Omega\rangle$ used here, the transpose map's "Choi operator" is $\mathrm{SWAP}$ (the normalized Choi state would be $\tfrac{1}{d}\,\mathrm{SWAP}$ ), whose negative eigenvalue is the precise reason transpose is not a channel.

Why it matters

Channel–state duality lets you import the entire toolkit of state analysis — eigenvalues, partial traces, entanglement measures, semidefinite programming — to study maps. Channel distinguishability, capacity bounds, and tests for special channel classes are routinely phrased and solved on the Choi side. It is also the theoretical basis of process tomography: measuring the Choi state of an unknown device reconstructs the device's channel.

The takeaway

The Choi–Jamiołkowski isomorphism is an invertible linear bijection between channels on $A\to B$ and bipartite operators on $A'\otimes B$ , built by sending half of a maximally entangled state through the channel. Under it, complete positivity becomes positivity of the Choi operator, trace preservation becomes a partial-trace condition, and Kraus rank becomes operator rank — converting questions about maps into questions about states.