Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD

Statement

Setup. Let $\Phi:\mathcal L(\mathcal H_A)\to\mathcal L(\mathcal H_B)$ be a linear map on operators, with $\dim\mathcal H_A=d$. Fix the unnormalized maximally entangled vector on $\mathcal H_A\otimes\mathcal H_{A'}$ (with $\mathcal H_{A'}\cong\mathcal H_A$) $$ |\Omega\rangle=\sum_{i=1}^{d}|i\rangle\otimes|i\rangle,\qquad |\Omega\rangle\langle\Omega|=\sum_{i,j=1}^d |i\rangle\langle j|\otimes|i\rangle\langle j|. $$

Definition (Choi matrix / Choi–Jamiołkowski isomorphism). The Choi matrix of $\Phi$ is the operator on $\mathcal H_B\otimes\mathcal H_{A'}$ $$ J(\Phi)=(\Phi\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)=\sum_{i,j=1}^{d}\Phi(|i\rangle\langle j|)\otimes|i\rangle\langle j|. $$ The assignment $\Phi\mapsto J(\Phi)$ is a linear bijection (isomorphism of vector spaces) between linear maps $\mathcal L(\mathcal H_A)\to\mathcal L(\mathcal H_B)$ and operators on $\mathcal H_B\otimes\mathcal H_{A'}$: given $J(\Phi)$ one recovers $\Phi(X)=\operatorname{Tr}_{A'}\!\big[(\mathbb 1_B\otimes X^{\mathsf T})\,J(\Phi)\big]$. This is state–channel duality: quantum channels (dynamics) are put in one-to-one correspondence with bipartite operators (states), up to normalization. (Source: en.wikipedia.org/wiki/Choi–Jamiołkowski_isomorphism, direct fetch; Choi 1975; Jamiołkowski 1972.)

Complete-positivity criterion (the load-bearing theorem). $$ \Phi \text{ is completely positive (CP)} \iff J(\Phi)\ \text{is positive semidefinite (PSD)}. $$ Equivalently: the smallest $k$ such that $\Phi\otimes\mathrm{id}_k$ is positive is already $k=d$, so testing positivity of one $d^2\times d^2$ matrix decides complete positivity for *all* ancilla dimensions at once. (Source: en.wikipedia.org/wiki/Choi–Jamiołkowski_isomorphism — "since $\mathcal E$ is completely positive, $(I_A\otimes\mathcal E)(|\Phi^+\rangle\langle\Phi^+|)$ is a nonnegative operator," and conversely any nonnegative operator arises this way.)

Trace-preservation criterion. $$ \Phi \text{ is trace-preserving} \iff \operatorname{Tr}_{B}\,J(\Phi)=\mathbb 1_{A'}. $$ Hence a quantum channel (CPTP map) corresponds exactly to a Choi matrix that is PSD with reduced state $\propto$ identity on the input side. (Source: same Wikipedia page, partial-trace condition.)

Normalization conventions. Two conventions coexist; keep them straight. The Choi convention above uses the *unnormalized* $|\Omega\rangle=\sum_i|ii\rangle$, so $\operatorname{Tr} J(\Phi)=d$ for a channel. The Jamiołkowski convention divides by $d$, using the normalized maximally entangled state $|\Phi^+\rangle=\tfrac1{\sqrt d}\sum_i|ii\rangle$; then $\rho_\Phi=(\Phi\otimes\mathrm{id})(|\Phi^+\rangle\langle\Phi^+|)$ is a genuine density matrix (the Jamiołkowski state / Choi state). The two differ only by the scalar $d$ and by whether a fixed maximally entangled basis is used, so PSD-ness, separability, and PPT-ness of the Choi object are convention-independent. (Source: Wikipedia normalization note + Watrous Ch. 2, WebSearch-corroborated.)

Facts

- **Why it is the workhorse for PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4.** Every structural property of a channel becomes a *matrix* property of its Choi operator, checkable in closed form or by SDP on a single $d^2\times d^2$ matrix: CP ⇔ Choi PSD; entanglement-breaking ⇔ Choi separable (see Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement); PPT map ⇔ Choi PSD and Choi has positive partial transpose (see PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose). The PPT² conjecture — that $\Phi_2\circ\Phi_1$ is entanglement-breaking whenever $\Phi_1,\Phi_2$ are PPT — is therefore restated entirely on Choi matrices, which is why every attack (Singh–Nechita factor-width, the $n=3$ Schmidt-number-≤2 proofs, Jin's SDP search) lives in Choi space. - Complete positivity is decided in one shot. The isomorphism's power is that positivity of the *single* matrix $J(\Phi)$ certifies $\Phi\otimes\mathrm{id}_n\ge0$ for every $n$ — you never have to check infinitely many ancilla sizes. This is the exact reason the "positive but not completely positive" gap (the source of all entanglement witnesses and of the transpose map's failure) is detectable. - Composition is not a simple product on Choi matrices. $J(\Phi_2\circ\Phi_1)$ is *not* $J(\Phi_2)J(\Phi_1)$ or a tensor product; it is a "link product" (a partial trace of a suitably matched tensor of the two Choi matrices). This non-multiplicativity is precisely why PPT² is hard: PPT-ness (a positivity condition on each factor's Choi matrix) does not transfer to the composite's Choi matrix by any elementary algebra. - Rank of Choi = Kraus number. The rank of $J(\Phi)$ equals the minimal number of Kraus operators of $\Phi$; a rank-1 Choi matrix corresponds to a single-Kraus (e.g. unitary or isometric) channel. Eigenvectors of $J(\Phi)$, reshaped $d\times d$, give a canonical Kraus set. This links the operational (Kraus/Stinespring) and matricial (Choi) pictures. - Transpose map is the prototype non-CP positive map. The transpose $T$ has Choi matrix $J(T)=\sum_{ij}|j\rangle\langle i|\otimes|i\rangle\langle j|$ = the SWAP operator, which is Hermitian but not PSD (eigenvalues $\pm1$). So $T$ is positive but not CP — this single fact powers the Peres–Horodecki PPT criterion and the whole partial-transpose story (Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3).

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the PPT-squared conjecture is stated and attacked entirely in Choi space; this isomorphism is the translation layer that turns "channel" statements into checkable matrix statements. - Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement — EB ⇔ Choi matrix separable; the isomorphism is what makes that equivalence meaningful. - PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose — PPT ⇔ Choi PSD and partial-transpose PSD; a direct decoration of the CP-⇔-PSD criterion with an extra partial-transpose condition. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ — witnesses are the dual objects to positive-but-not-CP maps; the isomorphism (specifically the transpose map ↔ SWAP example) is the bridge between the map picture and the witness picture. - Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — the PPT separability test is exactly "apply $T\otimes\mathrm{id}$ to a state and check positivity," i.e. the isomorphism applied to the prototypical non-CP positive map.