Choi–Jamiołkowski Isomorphism

A quantum channel is, in the abstract, a completely positive trace-preserving (CPTP) linear map from operators on one Hilbert space to operators on another — a map between spaces of matrices, unwieldy to reason about or test for physicality. The Choi–Jamiołkowski isomorphism performs a small miracle: it replaces the entire map by a single finite matrix, obtained by feeding one half of a maximally entangled pair through the channel and reading out the state. Every question you might ask of the channel — is it physical, how much information does it carry, does it destroy entanglement, what is its simplest operator-sum form — becomes a question about that one matrix, the Choi matrix. The reason it works is a chain of vector-space identities. Operators on a $d$ -dimensional space are vectors in a $d^2$ -dimensional space; a linear map between operator spaces is therefore a vector in a still larger space; and the maximally entangled state makes this identification physical, probing the channel on a complete basis of inputs simultaneously. The payoff is specific: complete positivity — an infinite family of conditions, one per ancilla — collapses to the single statement $J\ge0$ ; diagonalizing the same matrix hands you a minimal set of Kraus operators; and its normalized version is a density operator whose entropies are the channel's capacities. This is why channel–state duality sits under nearly every proof in quantum Shannon theory. Two roots feed it: Jamiołkowski's 1972 bilinear form and Choi's 1975 characterization of completely positive maps, differing by a partial transpose and a normalization the careless routinely conflate.

Key takeaways

The isomorphism is a bijection between linear maps $\mathcal{N}:\mathcal{L}(\mathcal{H}_A)\to\mathcal{L}(\mathcal{H}_B)$ and Choi matrices $J(\mathcal{N})$ on $\mathcal{H}_A\otimes\mathcal{H}_B$ , built by acting with $\mathcal{N}$ on half of $|\Omega\rangle=\sum_i|ii\rangle$ .
Complete positivity of the channel is equivalent to $J(\mathcal{N})\ge 0$ ; trace preservation to the linear condition $\mathrm{Tr}_B J(\mathcal{N})=I_A$ . Two hard conditions on a map become two easy ones on a matrix.
Diagonalizing $J$ yields a Kraus (operator-sum) representation with exactly $\mathrm{rank}\,J$ operators — the minimal number, the Choi rank.
The normalized Choi state $\tau=J/d_A$ is a real density operator whose properties are the channel's: coherent information $I_c(\mathcal{N})=-H(A|B)_\tau$ , and entanglement-breaking $\Leftrightarrow$ separable $\tau$ .
The qubit depolarizing channel has Choi spectrum $\{2-\tfrac{3p}{2},\,\tfrac{p}{2},\,\tfrac{p}{2},\,\tfrac{p}{2}\}$ , so it is CP exactly for $0\le p\le \tfrac{4}{3}$ ; the transpose map is positive but its Choi matrix is SWAP, spectrum $\{+1,+1,+1,-1\}$ — the canonical proof that positive $\neq$ CP.
Key idea: acting with the channel on half of a maximally entangled state encodes the whole channel into one bipartite operator, $\mathcal{N}\leftrightarrow J(\mathcal{N})=(\mathcal{I}\otimes\mathcal{N})(|\Omega\rangle\langle\Omega|)$ ; then CP $\Leftrightarrow J\ge0$ and TP $\Leftrightarrow \mathrm{Tr}_B J=I$ .

The core insight

Acting with the channel on half of a maximally entangled state encodes the entire channel into one bipartite operator, $\mathcal{N}\leftrightarrow J(\mathcal{N})=(\mathcal{I}\otimes\mathcal{N})(|\Omega\rangle\langle\Omega|)$ . Abstract properties of the map become concrete properties of the Choi state: complete positivity is $J\ge0$ , trace preservation is $\mathrm{Tr}_B J=I$ , the minimal Kraus number is $\mathrm{rank}\,J$ .

Where this lives today

The isomorphism is the quiet engine under much modern quantum technology. In the laboratory, ancilla-assisted process tomography is the Choi construction made experimental: prepare a maximally entangled pair, send one half through the unknown device, and do full state tomography on the joint output — reconstructing $J(\mathcal{N})$ , hence the entire channel, with $J\ge0$ regularizing the fit. This is how process characterization is benchmarked on superconducting and trapped-ion processors. In certification and design, because CP is $J\ge0$ and TP is $\mathrm{Tr}_B J=I$ , questions like "is this reconstruction a legal channel?" or "what is the diamond-norm distance between two channels?" become semidefinite programs. In quantum Shannon theory, the normalized Choi state converts capacities into state entropies: the LSD quantum-capacity theorem, degradability tests, and the whole apparatus of coherent information rest on manipulating $\tau(\mathcal{N})$ . And in fault-tolerant computing, gate teleportation and magic-state injection are channel–state duality in action — a Choi resource state, consumed by Bell measurement and correction, implements the gate on demand, the route to non-transversal gates like the $T$ gate in surface-code architectures.

The physical picture

Picture the maximally entangled state $|\Omega\rangle=\sum_i|i\rangle_{A'}|i\rangle_A$ as a perfectly correlated twin pair: whatever basis you choose, measuring $A'$ tells you $A$ . Push only the $A$ half through $\mathcal{N}$ , leaving $A'$ untouched. Because $|\Omega\rangle$ is the vectorized identity — every input basis element $|i\rangle\langle j|$ present coherently at once — the joint output learns how $\mathcal{N}$ acts on the whole operator basis in one shot. That output, as an operator, is the Choi matrix $J(\mathcal{N})$ : a single bipartite object standing in for the whole map, as the first figure shows.

Figure

The Choi construction sends the A half of a maximally entangled pair Ω through the channel N while the reference A′ is left alone; the untouched A′ and output B jointly carry the Choi operator J, with Tr J = d_A = 2 for a qubit channel, so complete positivity becomes the single check J ≥ 0 and trace preservation becomes Tr over B of J = I

Why the maximally entangled state, and not some other probe? Because it is the only input whose correlations, after the channel acts, retain full information about $\mathcal{N}$ on every basis element, coherences included; a less-entangled probe blurs some inputs. This also makes the positivity test powerful: complete positivity demands $\mathcal{I}_R\otimes\mathcal{N}$ preserve positivity for every reference dimension $R$ , yet the single reference of dimension $d_A$ carried by $|\Omega\rangle$ is the worst case — if $J\ge0$ here, all larger ancillas pass automatically. The second figure turns this certificate into a quantitative dial, tracking the Choi eigenvalues of a real channel as its noise rises.

Figure

Choi spectrum of the qubit depolarizing channel N(ρ) = (1−p)ρ + p I/2 versus mixing p: the nondegenerate eigenvalue 2 − 3p/2 in blue falls to zero at p = 4/3, the exact complete-positivity boundary, while the triple eigenvalue p/2 in amber rises; the two meet at p = 1 where all four equal 1/2 and J = I/2

The core equations

Everything begins with a fixed reference state. The unnormalized maximally entangled vector is the vectorized identity — every input basis element with unit weight — and its norm is $d_A$ , not $1$ , the source of the two competing normalization conventions.

|\Omega\rangle = \sum_{i=0}^{d_A-1} |i\rangle_{A'} \otimes |i\rangle_{A} \qquad (1)

where: $|\Omega\rangle$ — unnormalized maximally entangled vector on $\mathcal{H}_{A'}\otimes\mathcal{H}_A$ with $\langle\Omega|\Omega\rangle=d_A$ ; $d_A$ — dimension of the input space; $\{|i\rangle\}$ — a fixed orthonormal computational basis. Reading: fixing this basis is what makes the isomorphism basis-dependent, so entanglement statements about $J$ must name it. The Choi matrix does all the work: act with $\mathcal{N}$ on the $A$ half of $|\Omega\rangle\langle\Omega|=\sum_{ij}|i\rangle\langle j|\otimes|i\rangle\langle j|$ . The result is a table of the channel's action on every matrix unit.

J(\mathcal{N}) = (\mathcal{I}_{A'} \otimes \mathcal{N})\big(|\Omega\rangle\langle\Omega|\big) = \sum_{i,j=0}^{d_A-1} |i\rangle\langle j|_{A'} \otimes \mathcal{N}(|i\rangle\langle j|)_B \qquad (2)

where: $J(\mathcal{N})$ — Choi matrix, a $d_Ad_B\times d_Ad_B$ operator on $\mathcal{H}_{A'}\otimes\mathcal{H}_B$ with $\mathrm{Tr}\,J=d_A$ for TP maps; $\mathcal{I}_{A'}$ — identity on the reference; $\mathcal{N}$ — the map represented; $|i\rangle\langle j|$ — matrix units, a basis of operators on $\mathcal{H}_A$ . Reading: the $A'B$ block indexed by $(i,j)$ is exactly $\mathcal{N}(|i\rangle\langle j|)$ , so $J$ stores the whole map. The construction is invertible: given $J$ , the channel is recovered by contracting against the transposed input — the inverse that makes "isomorphism" literal, no information lost.

\mathcal{N}(X) = \mathrm{Tr}_{A'}\!\big[(X^{T}\otimes I_B)\,J(\mathcal{N})\big] \qquad (3)

where: $X$ — arbitrary input operator on $\mathcal{H}_A$ ; $X^{T}$ — transpose in the fixed basis; $\mathrm{Tr}_{A'}$ — partial trace over the reference. Reading: contracting the reference index of $J$ against $X^T$ superposes the stored blocks with the coefficients of $X$ . Now the two theorems that make the isomorphism indispensable. Choi's theorem collapses complete-positivity to one semidefinite condition.

\mathcal{N}\ \text{is completely positive} \iff J(\mathcal{N}) \ge 0 \qquad (4)

where: $J(\mathcal{N})\ge 0$ — the Choi matrix is positive semidefinite on $\mathcal{H}_{A'}\otimes\mathcal{H}_B$ . Reading: one eigenvalue computation certifies physicality; a negative eigenvalue disqualifies the map. Trace preservation — probabilities summing to one — becomes an equally clean linear condition.

\mathcal{N}\ \text{is trace-preserving} \iff \mathrm{Tr}_B\,J(\mathcal{N}) = I_A \qquad (5)

where: $\mathrm{Tr}_B$ — partial trace over the output; $I_A$ — identity on the input space. Reading: (4) and (5) together say a legal CPTP channel is exactly a positive $J$ whose $B$ -marginal is the identity — the constraints of a semidefinite program. Diagonalizing $J$ produces Kraus operators directly: each de-vectorized eigenvector is a Kraus operator, and their number is $\mathrm{rank}\,J$ .

J(\mathcal{N}) = \sum_k |K_k\rangle\!\rangle\langle\!\langle K_k|, \qquad |K_k\rangle\!\rangle = \sum_{ij}(K_k)_{ji}\,|i\rangle|j\rangle \qquad (6)

where: $K_k$ — Kraus operators with $\mathcal{N}(\rho)=\sum_k K_k\rho K_k^\dagger$ ; $|K_k\rangle\!\rangle$ — vectorization of $K_k$ ; scaled eigenvectors of $J$ furnish the set. Reading: the Choi picture is the most direct route from a channel to its operator-sum form, delivering the minimal set automatically. Finally, normalizing $J$ makes a density operator whose entropies are channel information quantities — the bridge into quantum Shannon theory.

I_c(\mathcal{N}) = H(B)_{\tau} - H(AB)_{\tau} = -H(A|B)_{\tau}, \qquad \tau = \tfrac{1}{d_A}\,J(\mathcal{N}) \qquad (7)

where: $\tau$ — normalized Choi state on $A'B$ ; $H(B)_\tau$ — entropy of the output marginal; $H(A|B)_\tau$ — conditional entropy of the Choi state; $I_c(\mathcal{N})$ — coherent information for maximally mixed input. Reading: "how much quantum information the channel preserves" becomes "how negative the conditional entropy of $\tau$ is," where strong subadditivity and data processing apply.

Derivation AChoi's theorem — complete positivity equals one positivity check

The goal is to prove, in both directions, that a map $\mathcal{N}$ is completely positive if and only if its Choi matrix $J(\mathcal{N})=(\mathcal{I}\otimes\mathcal{N})(|\Omega\rangle\langle\Omega|)$ is positive semidefinite, and to see that the hard direction is constructive: positivity of $J$ literally manufactures Kraus operators, from which the operator-sum form and hence complete positivity follow. Recall $\mathcal{N}$ is completely positive precisely when $\mathcal{I}_n\otimes\mathcal{N}$ preserves positivity for every ancilla dimension $n$ — an infinite family of requirements this theorem reduces to one.

Step 1. ( $\Rightarrow$ ) Suppose $\mathcal{N}$ is completely positive. The reference $|\Omega\rangle\langle\Omega|$ is a rank-one projector, hence positive; CP with $n=d_A$ says $\mathcal{I}_{d_A}\otimes\mathcal{N}$ preserves positivity, so $J\ge 0$ . Why: the definition of CP applied to the single entangled input forces $J\ge0$ with no further work.

Step 2. ( $\Leftarrow$ ) Suppose $J\ge0$ . Diagonalize $J=\sum_k\lambda_k|v_k\rangle\langle v_k|$ with $\lambda_k\ge0$ and absorb eigenvalues into $|K_k\rangle\!\rangle=\sqrt{\lambda_k}\,|v_k\rangle$ , so $J=\sum_k|K_k\rangle\!\rangle\langle\!\langle K_k|$ . Why: a positive matrix always factorizes as a sum of $|w\rangle\langle w|$ — the only place positivity is used; the rest is bookkeeping.

Step 3. De-vectorize. Every $|K_k\rangle\!\rangle\in\mathcal{H}_{A'}\otimes\mathcal{H}_B$ corresponds to a unique operator $K_k:\mathcal{H}_A\to\mathcal{H}_B$ via $|K_k\rangle\!\rangle=(I\otimes K_k)|\Omega\rangle$ , i.e. $(K_k)_{ji}=\langle j|K_k|i\rangle$ . Why: this vector–operator dictionary is a bijection, so the eigenvectors name a genuine operator set.

Step 4. Reconstruct. Feeding this $J$ into the inverse formula (3), the algebra collapses to $\mathcal{N}(X)=\sum_k K_k X K_k^{\dagger}$ . Why: contracting the factorized $J$ reproduces the sandwich form, tying abstract $J$ to concrete operators.

Step 5. Conclude. Any operator-sum map is manifestly CP, since $\mathcal{I}_n\otimes\mathcal{N}=\sum_k(I_n\otimes K_k)(\cdot)(I_n\otimes K_k)^\dagger$ preserves positivity for every $n$ . Hence $J\ge0\Rightarrow\mathcal{N}$ CP, closing the equivalence. Why: the single positivity check is sufficient as well as necessary, exhibiting the Kraus operators explicitly. *Verified: cross-checked against Watrous Ch. 2 (Thm 2.22) and Nielsen–Chuang §8.2.3; the de-vectorization $(I\otimes K)|\Omega\rangle$ matches $(K)_{ji}=\langle j|K|i\rangle$ .*

Go deeper · Derivation B · 1 minThe inverse map and the trace-preservation condition

Derivation BThe inverse map and the trace-preservation condition

The goal is to establish two facts that together promote the Choi map to a full isomorphism with a clean physicality criterion: first, that equation (3) genuinely inverts equation (2), recovering $\mathcal{N}(X)$ from $J$ for any input $X$ ; second, that conservation of probability $\mathrm{Tr}\,\mathcal{N}(X)=\mathrm{Tr}\,X$ is equivalent to the marginal condition $\mathrm{Tr}_B J=I_A$ . Both follow from expanding in matrix units and pushing indices, using nothing beyond linearity and the partial trace.

Step 1. Expand $X=\sum_{ij}X_{ij}|i\rangle\langle j|$ ; by linearity it suffices to verify the inverse on each matrix unit and reassemble. Why: channel and trace are both linear, so tracking a basis of operators controls the general case.

Step 2. Insert the definition of $J$ . Since $(X^T)_{ij}=X_{ji}$ , the reference-block product $(X^{T}\otimes I_B)\,J=\sum_{ijkl}X_{ji}\,|i\rangle\langle j|k\rangle\langle l|\otimes\mathcal{N}(|k\rangle\langle l|)$ forces $j=k$ via $\langle j|k\rangle=\delta_{jk}$ . Why: the transpose is the twist that pairs the coefficients of $X$ with the correct stored blocks of $J$ .

Step 3. Partial-trace over $A'$ . The surviving deltas collapse the sums to $\mathrm{Tr}_{A'}[\cdots]=\sum_{jl}X_{jl}\,\mathcal{N}(|j\rangle\langle l|)=\mathcal{N}(X)$ . Why: the contraction reassembles precisely $\mathcal{N}$ acting on the original $X$ , so (3) inverts (2).

Step 4. Now impose trace preservation. On a matrix unit, $\mathrm{Tr}\,\mathcal{N}(|i\rangle\langle j|)=\delta_{ij}$ ; the reference marginal is $\langle i|(\mathrm{Tr}_B J)|j\rangle=\mathrm{Tr}_B\,\mathcal{N}(|i\rangle\langle j|)=\delta_{ij}$ , exactly $I_A$ . Why: probability conservation on every input is captured by the single equation $\mathrm{Tr}_B J=I_A$ . *Verified: direct index manipulation reproduces $\mathcal{N}(X)$ and $\mathrm{Tr}_B J=I_A$ ; consistency $\mathrm{Tr}\,J=\mathrm{Tr}\,I_A=d_A$ matches equation (2).*

Go deeper · Derivation C · 1 minFrom the Choi matrix to a minimal Kraus representation

Derivation CFrom the Choi matrix to a minimal Kraus representation

The goal is to show that the eigendecomposition of $J(\mathcal{N})$ yields a Kraus set of size $\mathrm{rank}\,J$ , that this is the smallest possible number (the Choi rank), and that the operators form a canonical orthogonal set with trace preservation doubling as the Kraus completeness relation. This sharpens the previous derivation: not only does $J\ge0$ give some Kraus operators, its spectrum gives the minimal and canonical ones.

Step 1. Spectrally decompose $J=\sum_{k=1}^{r}\lambda_k|v_k\rangle\langle v_k|$ , keeping only the $r=\mathrm{rank}\,J$ strictly positive eigenvalues. Why: zero eigenvalues contribute nothing, so the sum runs over exactly $r$ terms and fixes the operator count.

Step 2. Define $K_k=\sqrt{\lambda_k}\,\mathrm{vec}^{-1}(|v_k\rangle)$ , i.e. $(K_k)_{ji}=\sqrt{\lambda_k}\,\langle ji|v_k\rangle$ . Why: this converts the eigenbasis of $J$ into Kraus operators, one per positive eigenvalue.

Step 3. Read off completeness. Term by term, $\mathrm{Tr}_B J=I_A$ is exactly $\sum_k K_k^{\dagger}K_k=I_A$ , the Kraus completeness relation; and eigenvector orthonormality gives $\mathrm{Tr}(K_k^{\dagger}K_l)=\lambda_k\,\delta_{kl}$ , so the eigen-Kraus operators are Hilbert–Schmidt orthogonal — a canonical set. Why: the $J$ -conditions that certify physicality reappear as the standard Kraus constraints.

Step 4. Minimality. Any representation $\{L_m\}$ satisfies $J=\sum_m|L_m\rangle\!\rangle\langle\!\langle L_m|$ , so the $|L_m\rangle\!\rangle$ span the range of $J$ , of dimension $r$ ; no representation uses fewer than $r$ operators, and the eigen-construction attains the bound. Why: the range of $J$ is a representation-independent invariant pinning the minimal Kraus number at $\mathrm{rank}\,J$ . *Verified: Choi rank $=$ minimal Kraus number is standard (Wilde Ch. 4); orthogonality via $\langle\!\langle K_k|K_l\rangle\!\rangle=\mathrm{Tr}(K_k^\dagger K_l)=\lambda_k\delta_{kl}$ ; for the depolarizing channel below, $r=4$ recovers the four Pauli Kraus operators.*

Limiting cases

Special channels wear especially transparent Choi matrices — each a check the formalism must reproduce.

Case	Result	Meaning
Identity channel	$J(\mathcal{I}) = \vert\Omega\rangle\langle\Omega\vert$	The Choi matrix is the unnormalized maximally entangled state — rank one; $\tfrac12 J$ is the Bell state $\Phi^+$ , entanglement entropy $1$ bit.
Completely depolarizing	$J(\mathcal{N}_{\mathrm{dep}}) = \tfrac{1}{d}\,I_A\otimes I_B$	Outputting $I/d$ gives a product (uncorrelated) Choi operator; for a qubit $\tfrac12 I_4$ , the $p=1$ point of the figure.
Unitary channel	$J(\mathcal{U}) = (I\otimes U)\vert\Omega\rangle\langle\Omega\vert(I\otimes U^{\dagger})$	Rank one and maximally entangled for any $U$ ; Choi entanglement entropy $\log d$ , a unitary preserving all information.
Entanglement-breaking	$J(\mathcal{N}) = \sum_k p_k\,\sigma_k\otimes\rho_k$	Separable Choi state — the defining characterization: a channel breaks entanglement iff its Choi state is separable.

Common trapThe two "Choi matrices" and the hidden partial transpose

Two distinct objects are both called "the Choi matrix," and confusing them is a common error. One convention uses the unnormalized $|\Omega\rangle=\sum_i|ii\rangle$ , giving $\mathrm{Tr}\,J=d$ and the marginal $\mathrm{Tr}_B J=I_A$ ; the other normalizes by $1/d$ to a density operator $\tau=J/d$ with $\mathrm{Tr}\,\tau=1$ . Quoting a marginal condition without the normalization is meaningless. Separately, Jamiołkowski's bilinear form differs from Choi's matrix by a partial transpose on the reference. For positivity this is invisible — both conventions have a clean CP criterion — but decisive for entanglement, since a partial transpose interchanges entanglement structure with PPT structure. Fix both the normalization and the transpose convention before discussing separability, PPT, or entanglement of $J$ .

Worked examples

1 · The qubit depolarizing channel — reading CP and rank off the spectrum (fully worked)

The qubit depolarizing channel is $\mathcal{N}(\rho)=(1-p)\rho+p\,\mathrm{Tr}(\rho)\,\tfrac{I}{2}$ . Build its Choi matrix, find the spectrum, and determine for which $p$ it is a legal channel and how many Kraus operators it needs.

Step 1. Compute $\mathcal{N}$ on the operator basis (using $\mathrm{Tr}(|i\rangle\langle j|)=\delta_{ij}$ ): $\mathcal{N}(|0\rangle\langle0|)=\mathrm{diag}(1-\tfrac{p}{2},\,\tfrac{p}{2})$ , $\mathcal{N}(|1\rangle\langle1|)=\mathrm{diag}(\tfrac{p}{2},\,1-\tfrac{p}{2})$ , off-diagonals shrinking: $\mathcal{N}(|0\rangle\langle1|)=(1-p)|0\rangle\langle1|$ , $\mathcal{N}(|1\rangle\langle0|)=(1-p)|1\rangle\langle0|$ . These four blocks are the $A'B$ blocks of $J$ .

Step 2. Assemble $J=\sum_{ij}|i\rangle\langle j|\otimes\mathcal{N}(|i\rangle\langle j|)$ in the basis $\{00,01,10,11\}$ :

J = \begin{pmatrix} 1-\tfrac{p}{2} & 0 & 0 & 1-p \\ 0 & \tfrac{p}{2} & 0 & 0 \\ 0 & 0 & \tfrac{p}{2} & 0 \\ 1-p & 0 & 0 & 1-\tfrac{p}{2} \end{pmatrix}

The $\{01,10\}$ sector is diagonal with entries $\tfrac{p}{2}$ ; the $\{00,11\}$ sector is $\begin{pmatrix}1-p/2 & 1-p\\ 1-p & 1-p/2\end{pmatrix}$ .

Step 3. Diagonalize. The $2\times2$ block has eigenvalues $(1-\tfrac{p}{2})\pm(1-p)=\{\,2-\tfrac{3p}{2},\ \tfrac{p}{2}\,\}$ , so the full spectrum is $\boxed{\{\,2-\tfrac{3p}{2},\ \tfrac{p}{2},\ \tfrac{p}{2},\ \tfrac{p}{2}\,\}}$ with $\mathrm{Tr}\,J=2=d_A$ . This is the spectrum plotted in the second figure.

Step 4. Read off complete positivity. All eigenvalues $\ge0$ : the triple needs $p\ge0$ , the singleton $2-\tfrac{3p}{2}\ge0$ , i.e. $p\le\tfrac{4}{3}$ . Hence $\mathcal{N}$ is CP exactly for $0\le p\le\tfrac{4}{3}$ — a nontrivial physical bound from Choi eigenvalues alone. At $p=1.4>\tfrac43$ the singleton is $-0.1<0$ and the "channel" is unphysical.

Step 5. Count Kraus operators. For generic $p\in(0,\tfrac43)$ the rank is $4$ , so the minimal set is four operators — the familiar $\{\sqrt{1-\tfrac{3p}{4}}\,I,\ \sqrt{\tfrac{p}{4}}\,X,\ \sqrt{\tfrac{p}{4}}\,Y,\ \sqrt{\tfrac{p}{4}}\,Z\}$ . At the endpoints the rank drops: $p=0$ gives rank $1$ , $p=\tfrac43$ gives rank $3$ . CP range $p\in[0,\tfrac43]$ , Choi rank $4$ generically.

2 · The transpose map — positive but not completely positive

The transpose $\mathcal{T}(\rho)=\rho^{T}$ maps density operators to density operators, so it is positive. Is it a channel?

Step 1. Apply the definition: $J(\mathcal{T})=\sum_{ij}|i\rangle\langle j|\otimes(|i\rangle\langle j|)^{T}=\sum_{ij}|i\rangle\langle j|\otimes|j\rangle\langle i|$ — the SWAP operator on two qubits.

Step 2. SWAP acts as $+1$ on the symmetric subspace (dimension 3) and $-1$ on the antisymmetric singlet, so its spectrum is $\{+1,+1,+1,-1\}$ .

Step 3. The eigenvalue $-1$ means $J(\mathcal{T})\not\ge0$ , so by Choi's theorem the transpose is not completely positive, though it is positive. Concretely, $(\mathcal{I}\otimes\mathcal{T})$ on a Bell state gives an operator with a negative eigenvalue — the partial transpose behind the PPT entanglement criterion. Positive $\neq$ CP; the witness is the eigenvalue $-1$ of the SWAP.

3 · Coherent information from the Choi state

Use the normalized Choi state to compute the coherent information of the qubit depolarizing channel and find where it vanishes.

Step 1. Normalize: $\tau=J/2$ has spectrum $\{1-\tfrac{3p}{4},\,\tfrac{p}{4},\,\tfrac{p}{4},\,\tfrac{p}{4}\}$ and $B$ -marginal $I/2$ , so $H(B)_\tau=1$ bit for all $p$ . Then $I_c=H(B)_\tau-H(AB)_\tau=1-H(\tau)$ .

Step 2. At $p=0$ , $H(\tau)=0$ and $I_c=1$ bit. Increasing $p$ raises $H(\tau)$ : at $p=0.25$ , $H(\tau)\approx0.993$ so $I_c\approx0.007$ bit, barely positive; by $p=\tfrac13$ , $I_c\approx-0.21$ bit. The single-letter coherent information crosses zero near $p\approx0.25$ — a capacity statement from a state entropy alone.

Physics of the portrait

The isomorphism is where operator-algebra theorems from the 1950s–1970s became the daily tools of quantum information; two names for two conventions differing by a partial transpose.

Origins · 1 minChoi–Jamiołkowski isomorphism and how this was discovered

Choi–Jamiołkowski isomorphism

channel–state duality J(N) = (I⊗N)( · Ω⟩⟨Ω

The result grew from operator-algebra theory, decades before "quantum channel" was a phrase anyone used. In 1955 W. Forrest Stinespring proved his dilation theorem, characterizing completely positive maps as compressions of *-homomorphisms — the ancestor of every channel representation. In 1972 the Polish mathematical physicist Andrzej Jamiołkowski associated to each linear map a bipartite operator (a bilinear form), the first version of the correspondence now bearing his name; his convention is basis-independent and sits one partial transpose from the modern matrix. In 1975 Man-Duen Choi, a Hong Kong-born operator theorist then at Toronto, proved the sharp statement: a linear map on complex matrices is completely positive iff the single Choi matrix $\sum_{ij}E_{ij}\otimes\mathcal{N}(E_{ij})$ is positive semidefinite, its eigen-decomposition yielding the operator-sum form. For two decades this lived in mathematics; then quantum information found its central objects — channels, capacities, degradability — most naturally studied as properties of the Choi state. The 2003 characterization of entanglement-breaking channels by Horodecki, Shor and Ruskai (separable Choi state) is emblematic, and reading $\tau=\tfrac1d J$ as a state whose entropies are capacities completes the arc from abstract theorem to workhorse.

History

1955

— Stinespring proves the dilation theorem for completely positive maps

1972

— Jamiołkowski associates a bipartite operator to each linear map

1975

— Choi characterizes complete positivity by positivity of the Choi matrix

2003

— Horodecki, Shor and Ruskai tie entanglement-breaking to separable Choi states

Going further

References. Nielsen & Chuang, Quantum Computation and Quantum Information, §8.2 · Watrous, The Theory of Quantum Information, Ch. 2 and 4 · Wilde, Quantum Information Theory (2nd ed.), Ch. 4–5, §13 · Preskill, Lecture Notes for Physics 219, Ch. 3 · Kitaev, Shen, Vyalyi, Classical and Quantum Computation, Ch. 11 · M.-D. Choi, "Completely positive linear maps on complex matrices," Lin. Alg. Appl. 10 (1975) 285 · A. Jamiołkowski, Rep. Math. Phys. 3 (1972) 275 · Horodecki, Shor, Ruskai, "Entanglement breaking channels," Rev. Math. Phys. 15 (2003) 629.

Provenance. Definitions and theorems follow Watrous (Ch. 2), Wilde (Ch. 4–5), and Nielsen & Chuang (§8.2). A verifier should: (1) confirm the inverse-map and TP derivations by direct index manipulation; (2) check the transpose map's Choi matrix is SWAP with spectrum $\{+1,+1,+1,-1\}$ ; (3) verify the depolarizing Choi spectrum $\{2-\tfrac{3p}{2},\tfrac{p}{2},\tfrac{p}{2},\tfrac{p}{2}\}$ and CP window $p\in[0,\tfrac43]$ (numerically confirmed); (4) verify $I_c=-H(A|B)_\tau$ . Anchor numbers (Bell-state entropy $1$ bit, $\mathrm{Tr}\,J=d_A$ , Choi rank $=$ minimal Kraus number, depolarizing bound $\tfrac43$ ) are standard. Attributions: Stinespring (1955), Jamiołkowski (1972), Choi (1975), Horodecki–Shor–Ruskai (2003).

Prerequisites. Quantum Channels and CPTP Maps · Kraus Representation of Quantum Channels · Von Neumann Entropy. Leads to. Stinespring Dilation Theorem · Complementary Channels and Degradability · Data-Processing Inequality · Holevo Bound.

Ask the tutor: Want to see the Choi matrix, its eigenvalues, and a minimal Kraus set worked out for any channel you name — amplitude damping, dephasing, or your own noisy gate?

Why does one entangled input certify complete positivity, when the definition quantifies over all ancilla dimensions?

Because the maximally entangled reference of dimension $d_A$ is the worst case. CP asks that $\mathcal{I}_n\otimes\mathcal{N}$ preserve positivity for every $n$ , but if it passes on the maximally entangled state of the input dimension, it passes for every larger ancilla automatically: $|\Omega\rangle$ already carries maximal correlations across the full input space, so no bigger ancilla builds a harder input. This is the whole miracle — an infinite family of checks reduces to one eigenvalue computation on $J$ .

What is the difference between the Choi matrix and the Choi state?

Only normalization, but the distinction matters. The Choi matrix $J=(\mathcal{I}\otimes\mathcal{N})(|\Omega\rangle\langle\Omega|)$ uses the unnormalized $|\Omega\rangle$ , has $\mathrm{Tr}\,J=d_A$ , and obeys the clean marginal law $\mathrm{Tr}_B J=I_A$ . The Choi state $\tau=J/d_A$ is a genuine density operator with $\mathrm{Tr}\,\tau=1$ . Use $J$ for CP/TP certification and Kraus extraction; use $\tau$ for information theory, where you need a normalized state for von Neumann entropies. Mixing the two normalizations is a frequent slip.

How is the Choi–Jamiołkowski isomorphism related to Kraus and Stinespring?

They are three faces of one channel. Diagonalizing the Choi matrix and de-vectorizing its eigenvectors gives Kraus operators directly, the count equalling the minimal (Choi-rank) number. Purifying the Choi state and tracing out the environment recovers the Stinespring dilation, in which the channel is an isometry followed by a partial trace. So Choi $\to$ Kraus is an eigen-decomposition and Choi $\to$ Stinespring is a purification; the isomorphism lets you move freely between representations, choosing whichever suits the proof at hand.

Why does the transpose map show that positive is weaker than completely positive?

The transpose $\rho\mapsto\rho^T$ sends every density operator to a valid density operator, so it is a positive map. But its Choi matrix is the SWAP operator, whose spectrum $\{+1,+1,+1,-1\}$ has a negative eigenvalue, so it fails $J\ge0$ and is not completely positive. Applied to half of an entangled pair, the transpose produces a joint operator with a negative eigenvalue — impossible for a physical process, and exactly the signal exploited by the PPT entanglement test. No accident that the textbook "positive but not CP" map also underlies a famous entanglement criterion.

Does the Choi matrix depend on the choice of basis, and does that ever cause trouble?

Yes — it is defined relative to the fixed basis named by $|\Omega\rangle=\sum_i|ii\rangle$ . For the CP and TP criteria this is harmless: whether $J\ge0$ or $\mathrm{Tr}_B J=I$ is unchanged under a coordinated basis rotation. Where it bites is entanglement and separability statements about $J$ , and the relation to Jamiołkowski's convention, which differs by a partial transpose. Since a partial transpose swaps entanglement structure with PPT structure, pin down both the basis and the transpose convention before claiming anything about $J$ 's entanglement — the practical content of the "two Choi matrices" warning.

Can the isomorphism be used experimentally, or is it only a proof technique?

Very much experimentally. Ancilla-assisted process tomography is the Choi construction run in a lab: prepare a maximally entangled pair, send one half through the device, and do full state tomography on the joint output. The reconstructed bipartite state is a normalization of the Choi matrix, from which you obtain the complete channel — every Kraus operator, the process fidelity, the diamond-norm error — with $J\ge0$ imposed during the fit. This is often more robust and sample-efficient than preparing a tomographically complete set of separate inputs, and it is standard in benchmarking real processors.