Process Tomography and Choi-Jamiolkowski Isomorphism

Every piece of quantum hardware is a black box that does something to the states you feed it — some mixture of the gate you wanted and the decoherence you did not. Quantum process tomography opens that box: it takes raw measurement counts and returns a complete description of the channel $\mathcal{E}$ , as a process matrix $\chi$ , a Choi operator $J(\mathcal{E})$ , or a set of Kraus operators. The organizing principle is not an experimental trick but pure linear algebra — the channel-state duality, or Choi-Jamiołkowski isomorphism, which says that linear maps on operators are in one-to-one correspondence with operators on a doubled Hilbert space. The payoff is a genuine reduction of one hard problem to an easier one. Reconstructing a map sounds harder than reconstructing a state, but the duality collapses the distinction: feed one half of a maximally entangled pair through the channel and the resulting Choi state already contains the entire action of $\mathcal{E}$ . State tomography is solved, so channel tomography becomes state tomography in dimension $d^2$ — and this fixes how many settings are needed, which physicality constraints an estimator must respect, and how amplitude damping shows up as the spectrum of one Hermitian matrix. The isomorphism also makes the abstract structure theorems operational. A map is completely positive exactly when its Choi operator is positive semidefinite, and trace preserving exactly when a partial trace of it is the identity; the eigendecomposition of the Choi matrix hands you a canonical, minimal set of Kraus operators directly. Tomography and the theory of CPTP maps are two views of one object — which is why, even as randomized benchmarking and gate set tomography supersede naive process tomography for scalable diagnostics, the Choi picture remains the reference language for fidelity, channel distance, and entanglement-based verification.

Key takeaways

Quantum process tomography (QPT) reconstructs the complete action of an unknown channel $\mathcal{E}$ from measurement statistics on a tomographically complete set of inputs.
The Choi-Jamiołkowski isomorphism sends any linear map to one operator, $J(\mathcal{E})=(\mathcal{E}\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$ , turning channel reconstruction into state tomography on $d^2$ dimensions.
Complete positivity is equivalent to $J(\mathcal{E})\ge 0$ and trace preservation to $\mathrm{Tr}_{\mathrm{out}}J(\mathcal{E})=\mathbb{1}/d$ — channel constraints become PSD and partial-trace constraints.
The Choi rank equals the minimal Kraus count: a unitary has rank $1$ (pure Choi state), a fully depolarizing channel rank $d^2$ (maximally mixed Choi state).
Standard QPT costs $d^4-d^2$ parameters, scaling as $16^n$ ; naive linear inversion returns unphysical negative eigenvalues, cured by maximum-likelihood estimation over the CPTP set.
Key idea: a quantum channel and a quantum state are the same object seen two ways, $\mathcal{E}\longleftrightarrow J(\mathcal{E})=(\mathcal{E}\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$ , so reconstructing the channel is reconstructing one $d^2\times d^2$ state, with CP $\Leftrightarrow J\ge 0$ and TP $\Leftrightarrow\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d$ .

The core insight

A quantum channel and a quantum state are the same object viewed two ways: $\mathcal{E}\longleftrightarrow J(\mathcal{E})=(\mathcal{E}\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$ . Reconstructing the channel is therefore reconstructing one $d^2\times d^2$ state, with CP $\Leftrightarrow J\ge 0$ and TP $\Leftrightarrow\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d$ .

Where this lives today

Process tomography and its Choi language are the calibration substrate of every quantum-computing platform. On superconducting, trapped-ion, and photonic processors, one- and two-qubit gates are characterized by reconstructing $\chi$ or $J$ , from which the average gate fidelity in every hardware paper is computed via the process fidelity. Because standard QPT conflates state-preparation-and-measurement (SPAM) errors with the gate and scales as $16^n$ , high-precision work has moved to gate set tomography, which self-consistently estimates gates, preparations, and measurements together — yet still reports CPTP maps in the Choi/ $\chi$ representation. Ancilla-assisted process tomography, the direct incarnation of the duality, prepares $|\Omega\rangle$ , sends one half through the device, and tomographs the joint output to measure the Choi state itself. The spectrum of $J$ is a diagnostic in its own right: a channel is entanglement-breaking exactly when its Choi operator is separable, and non-Markovianity tests are positivity conditions on intermediate Choi maps — so access to $J(\mathcal{E})$ certifies structural properties of the noise, not just its fidelity. Compressed sensing exploits low Choi rank to cut the budget below $16^n$ ; randomized benchmarking sidesteps full tomography when only the average fidelity is wanted.

The physical picture

Think of a quantum channel as a machine you may probe but not open. Knowing its output for every input would suffice, but linearity says far less is needed: the output on a basis of input operators fixes the map. For a qubit the four states $\{|0\rangle\langle0|,|1\rangle\langle1|,|+\rangle\langle+|,|+i\rangle\langle+i|\}$ span the operator space, so measuring the output for each — state tomography four times — reconstructs the channel. That is standard process tomography, one input at a time. The Choi trick replaces "feed a basis of inputs one at a time" with "feed one half of a maximally entangled pair." Entanglement is the resource that lets a single bipartite input carry every basis operator at once: the correlations in $|\Omega\rangle$ are sculpted by the channel into the Choi state, and the full map is read off from it. The first figure makes this concrete — the channel touches only subsystem $A$ , yet the output on $A\otimes B$ encodes its entire action.

Figure

Feeding one half of a Bell pair through the channel ℰ imprints its entire action into the bipartite Choi state J(ℰ), a single d²×d² matrix in which complete positivity becomes J ≥ 0 and trace preservation becomes Tr_out J = 𝟙/d

Positivity becomes geometry. Via the duality, the set of all CPTP maps on $d$ dimensions is an affine slice of the cone of $d^2\times d^2$ positive operators — a convex body. Estimation is a constrained reconstruction inside that body, which is why maximum-likelihood and semidefinite programming are the natural tools: the constraints $J\ge 0$ and $\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d$ are the walls of the feasible set. The spectrum of the Choi matrix is meaningful in its own right. Its rank is the minimal number of Kraus operators, so a unitary channel has a rank-1 Choi state and a fully noisy one is full rank. The second figure turns this into numbers for the amplitude-damping channel, the workhorse model of energy relaxation ( $T_1$ ) in a qubit.

Figure

The Choi matrix of the amplitude-damping channel at γ = 0.2 has eigenvalues 0.90, 0.10, 0, 0 — two nonzero, so its Choi rank is 2 and exactly two Kraus operators (with weights √0.8 and √0.2) reproduce the channel

The core equations

The Choi picture rests on a short chain of exact identities. Start with the reference state, the entangled probe carrying every operator basis element at once; for $n$ qubits $d=2^n$ .

|\Omega\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle_{A}\otimes|i\rangle_{B} \qquad (1)

where: $d$ — Hilbert-space dimension; $|i\rangle$ — computational basis of input $A$ and ancilla $B$ ; $|\Omega\rangle$ — normalized maximally entangled state. Reading: one input standing in for a whole operator basis. The Choi operator is what the channel makes of that probe: since $\mathcal{E}$ acts only on $A$ , the output stores $\mathcal{E}(|i\rangle\langle j|)$ for every $i,j$ at once.

J(\mathcal{E})=(\mathcal{E}\otimes\mathrm{id}_{B})\big(|\Omega\rangle\langle\Omega|\big) \qquad (2)

where: $\mathcal{E}$ — the unknown channel on $A$ ; $\mathrm{id}_{B}$ — identity on the ancilla; $J(\mathcal{E})$ — Choi operator, a $d^2\times d^2$ Hermitian matrix. Reading: the "state" side of the duality. The duality runs both ways: given $J$ , the action on any input $\rho$ comes back by a partial trace against $\rho^T$ — the transpose being the price of packing the input into an entangled ancilla.

\mathcal{E}(\rho)=d\,\mathrm{Tr}_{B}\!\big[J(\mathcal{E})\,(\mathbb{1}_{A}\otimes\rho^{T})\big] \qquad (3)

where: $\rho$ — arbitrary input; $\rho^{T}$ — transpose in the computational basis; $\mathrm{Tr}_B$ — partial trace over the ancilla. Reading: one matrix gives the whole map. The conditions defining a physical channel become two clean statements about $J$ — one spectral, one a partial trace — the walls of every feasible set.

\mathcal{E}\ \text{is CP}\iff J(\mathcal{E})\ge 0,\qquad \mathcal{E}\ \text{is TP}\iff \mathrm{Tr}_{\mathrm{out}}J(\mathcal{E})=\tfrac{1}{d}\,\mathbb{1}_B \qquad (4)

where: $J\ge 0$ — positive semidefinite Choi operator; $\mathrm{Tr}_{\mathrm{out}}$ — partial trace over the output $A$ ; $\mathbb{1}_B$ — identity on the ancilla. Reading: physicality is positivity plus one affine constraint. Diagonalizing $J$ hands you the Kraus operators for free: reshape each eigenvector into a $d\times d$ matrix and weight by $\sqrt{d\lambda_k}$ . The count of nonzero eigenvalues is the Choi rank.

J(\mathcal{E})=\sum_{k}\lambda_k\,|v_k\rangle\langle v_k|\ \Longrightarrow\ K_k=\sqrt{d\,\lambda_k}\;\mathrm{mat}(|v_k\rangle) \qquad (5)

where: $\lambda_k,|v_k\rangle$ — eigenpairs of $J$ ; $\mathrm{mat}(\cdot)$ — reshaping a $d^2$ -vector into a $d\times d$ matrix; $K_k$ — canonical Kraus operators. Reading: the spectrum is the noise decomposition. Closer to how data is fit, one expands the channel in a fixed basis $\{E_m\}$ (the Paulis for qubits) with coefficients forming the process matrix $\chi$ .

\mathcal{E}(\rho)=\sum_{mn}\chi_{mn}\,E_m\,\rho\,E_n^{\dagger} \qquad (6)

where: $\{E_m\}$ — fixed basis, e.g. $\{I,X,Y,Z\}$ ; $\chi_{mn}$ — process matrix, $d^2\times d^2$ , Hermitian PSD, equal to $J$ up to a fixed change of basis. Reading: $\chi$ is $J$ in the Pauli frame. Finally, the number experimentalists report: the Choi/process fidelity converts into the average gate fidelity by an exact formula.

F_{\mathrm{avg}}(\mathcal{E},\mathcal{U})=\frac{d\,F_{\mathrm{ent}}+1}{d+1},\qquad F_{\mathrm{ent}}=\langle\Omega|(\mathcal{U}^{\dagger}\!\otimes\!\mathrm{id})J(\mathcal{E})|\Omega\rangle \qquad (7)

where: $F_{\mathrm{avg}}$ — average fidelity of $\mathcal{E}$ to target unitary $\mathcal{U}$ ; $F_{\mathrm{ent}}$ — entanglement (Choi/process) fidelity; $d$ — system dimension. Reading: one overlap on $J$ gives the benchmarked fidelity.

Derivation AComplete positivity is equivalent to Choi positivity

The goal is Choi's theorem, the load-bearing result of the whole subject: a linear map $\mathcal{E}$ is completely positive exactly when its Choi operator $J(\mathcal{E})$ is positive semidefinite. We prove both directions, and the proof is constructive — the eigenvectors of $J$ , suitably reshaped, are precisely the Kraus operators, so the argument that establishes physicality also hands you the channel's noise decomposition.

Step 1. For the forward direction, the projector $|\Omega\rangle\langle\Omega|$ is positive on $A\otimes B$ . Complete positivity is the statement that $\mathcal{E}\otimes\mathrm{id}_B$ carries positive operators to positive operators for an ancilla of any size, so $J(\mathcal{E})\ge 0$ follows immediately. Why: once CP is stated with an ancilla, the Choi operator is just the image of one positive input.

Step 2. For the converse, assume $J\ge 0$ and diagonalize, $J=\sum_k\lambda_k|v_k\rangle\langle v_k|$ with all $\lambda_k\ge 0$ . Hermiticity gives a real spectrum and positivity gives $\lambda_k\ge 0$ , so each $\sqrt{\lambda_k}\,|v_k\rangle$ is well defined. Why: non-negative eigenvalues are what let us take the square roots that become Kraus amplitudes.

Step 3. Reshape each eigenvector into an operator: $|v_k\rangle=\frac{1}{\sqrt d}\sum_{ij}(K_k)_{ij}|i\rangle|j\rangle$ defines a $d\times d$ matrix $K_k:=\sqrt{d\lambda_k}\,\mathrm{mat}(|v_k\rangle)$ . Why: a vector in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and a $d\times d$ matrix carry the same $d^2$ numbers — reshaping bridges "state" eigenvector to "channel" operator.

Step 4. Substitute these $K_k$ into the inverse-map formula (3): the partial trace and eigenvector sum reassemble into $d\,\mathrm{Tr}_B[J(\mathbb{1}\otimes\rho^T)]=\sum_k K_k\rho K_k^{\dagger}$ . Why: the right-hand side is manifestly a CP map — a Kraus sum can never produce a negative operator.

Step 5. Conclude: $J\ge 0$ gives the Kraus form $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ , the most general CP map, with $\mathrm{rank}\,J$ terms. Why: a positive matrix and a physical channel are the same data, and its rank is the minimal noise-operator count.

Go deeper · Derivation B · 1 minStandard process tomography as a linear inversion

Derivation BStandard process tomography as a linear inversion

The goal is to see how measured output states become the process matrix $\chi$ , and why the procedure is a plain linear inversion whose invertibility is guaranteed by tomographic completeness. Laying it out makes clear where the $d^4$ scaling comes from and why finite-statistics data lands outside the physical set, forcing the constrained estimators used in practice.

Step 1. Fix a complete input set $\{\rho_j\}_{j=1}^{d^2}$ and a fixed basis $\{E_m\}_{m=0}^{d^2-1}$ . For one qubit take $\rho_j\in\{|0\rangle\langle0|,|1\rangle\langle1|,|+\rangle\langle+|,|+i\rangle\langle+i|\}$ and $E_m\in\{I,X,Y,Z\}$ . Why: completeness means the $\rho_j$ span all operators, so their images determine the map.

Step 2. Precompute the geometry: $E_m\rho_j E_n^{\dagger}=\sum_k\beta^{jk}_{mn}\rho_k$ , where $\beta^{jk}_{mn}$ depends only on the chosen bases, not on the channel. Why: separating the known $\beta$ from the unknown $\chi$ turns a physics question into a linear system.

Step 3. Measure: state tomography of each output gives $\mathcal{E}(\rho_j)=\sum_k\lambda_{jk}\rho_k$ , the experimental data. Why: this is the only step that touches the lab — the rest is arithmetic.

Step 4. Match coefficients: $\lambda_{jk}=\sum_{mn}\beta^{jk}_{mn}\chi_{mn}$ , a system of $d^4$ equations in the $d^4$ entries of $\chi$ , invertible because the input set is complete. Why: completeness is the algebraic condition making $\beta$ nonsingular, so $\chi$ is uniquely fixed.

Step 5. Invert with caution: $\chi=\beta^{-1}\lambda$ . On finite data this generically violates $\chi\ge 0$ ; one then projects onto the PSD, trace-preserving cone by maximum likelihood. Why: the raw inverse ignores constraint (4) and wanders outside the physical body — the projection restores a real channel.

Go deeper · Derivation C · 1 minTrace preservation as a partial-trace condition on the Choi operator

Derivation CTrace preservation as a partial-trace condition on the Choi operator

The goal is to derive the second physicality constraint — that $\mathcal{E}$ preserves trace exactly when $\mathrm{Tr}_{\mathrm{out}}J(\mathcal{E})=\mathbb{1}/d$ — directly from $\mathrm{Tr}\,\mathcal{E}(\rho)=\mathrm{Tr}\,\rho$ . It complements Choi's theorem: positivity handles CP, this handles TP, and together they carve out the CPTP body every estimator respects. It is the Choi-side form of $\sum_k K_k^\dagger K_k=\mathbb{1}$ .

Step 1. Trace the inverse-map formula (3) over the output subsystem: $\mathrm{Tr}\,\mathcal{E}(\rho)=d\,\mathrm{Tr}_{AB}[J(\mathbb{1}_A\otimes\rho^T)]$ . Why: routing the output trace through $J$ moves the whole question onto the Choi operator.

Step 2. Do the output trace first: $d\,\mathrm{Tr}_{AB}[J(\mathbb{1}_A\otimes\rho^T)]=d\,\mathrm{Tr}_B[(\mathrm{Tr}_{\mathrm{out}}J)\,\rho^T]$ . Why: factoring the double trace isolates $\mathrm{Tr}_{\mathrm{out}}J$ , the object the constraint will pin down.

Step 3. Demand equality with $\mathrm{Tr}\,\rho$ for every $\rho$ . Since $\{\rho^T\}$ ranges over all operators and the Hilbert-Schmidt inner product is nondegenerate, this forces $\mathrm{Tr}_{\mathrm{out}}J=\frac{1}{d}\mathbb{1}_B$ . Why: "true for every input" upgrades a numerical equation into an operator identity — only $\mathbb{1}/d$ works.

Step 4. Read it as an affine constraint: $\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d$ is a linear condition sliced through the PSD cone. Why: this makes maximum-likelihood QPT a convex problem — both constraints bound a convex set with a unique projection.

Limiting cases

The Choi operator and the tomographic cost take clean forms in the extreme cases, which anchor what the spectrum of $J$ is telling you.

Case	Result	Meaning
Identity or unitary channel	$J(\mathcal{U})=\|(\mathbb{1}\otimes U)\Omega\rangle\langle(\mathbb{1}\otimes U)\Omega\|$	Rank-1 pure Choi state: a single Kraus operator $U$ , no noise
Fully depolarizing channel	$J(\mathcal{D})=\tfrac{1}{d^2}\,\mathbb{1}_{d}\otimes\mathbb{1}_{d}$	Maximally mixed Choi state, full rank $d^2$ : maximally noisy, all entanglement destroyed
Mixed-unitary (Pauli) channel	$\chi=\mathrm{diag}(p_0,p_x,p_y,p_z)$	Process matrix diagonal in the Pauli basis: only $d^2-1=3$ numbers per qubit, the easiest QPT case
Many qubits	$\#\text{params}=d^4-d^2,\ d=2^n$	Standard QPT scales as $16^n-4^n$ : exponential blowup, motivating compressed sensing and gate set tomography

Common trapLinear inversion returns a channel, so it must be physical

It need not be, and usually is not. Linear inversion of finite data almost always yields a Choi or process matrix with small negative eigenvalues — a valid but unphysical, non-completely-positive map. Reporting such a $\chi$ , or a fidelity from it, is a real error: a fidelity read off a non-PSD Choi matrix can even exceed one. The fix is to project onto the CPTP set with a maximum-likelihood or constrained least-squares estimator over $\{J\ge 0,\ \mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d\}$ . A subtler trap is convention: the unnormalized Choi $(\mathcal{E}\otimes\mathrm{id})(\sum_{ij}|ii\rangle\langle jj|)$ has $\mathrm{Tr}_{\mathrm{out}}=\mathbb{1}$ and trace $d$ , while the normalized version from $|\Omega\rangle=\frac{1}{\sqrt d}\sum_i|ii\rangle$ has $\mathrm{Tr}_{\mathrm{out}}=\mathbb{1}/d$ and trace $1$ . Mixing the two corrupts every factor of $d$ — fix one convention and state it.

Worked examples

1 · Choi matrix, spectrum, and Kraus operators of amplitude damping (fully worked)

The amplitude-damping channel models energy relaxation ( $T_1$ ) of a qubit with Kraus operators $K_0=\left(\begin{smallmatrix}1&0\\0&\sqrt{1-\gamma}\end{smallmatrix}\right)$ and $K_1=\left(\begin{smallmatrix}0&\sqrt{\gamma}\\0&0\end{smallmatrix}\right)$ . Take $\gamma=0.2$ . Build its normalized Choi matrix, find the spectrum, confirm it is a physical channel, and recover the Kraus operators from the eigenvectors.

Step 1. Act with $\mathcal{E}(\cdot)=K_0\cdot K_0^\dagger+K_1\cdot K_1^\dagger$ on the four matrix units: $\mathcal{E}(E_{00})=\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right)$ , $\mathcal{E}(E_{11})=\left(\begin{smallmatrix}\gamma&0\\0&1-\gamma\end{smallmatrix}\right)$ , $\mathcal{E}(E_{01})=\left(\begin{smallmatrix}0&\sqrt{1-\gamma}\\0&0\end{smallmatrix}\right)$ , and $\mathcal{E}(E_{10})=\mathcal{E}(E_{01})^\dagger$ . The excited state $|1\rangle\langle1|$ leaks probability $\gamma$ to the ground state — the physics of relaxation.

Step 2. Assemble $J=\frac12\sum_{ij}\mathcal{E}(E_{ij})\otimes|i\rangle\langle j|$ in the ordering $\{00,01,10,11\}$ :

J=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & \sqrt{1-\gamma}\\ 0 & \gamma & 0 & 0\\ 0 & 0 & 0 & 0\\ \sqrt{1-\gamma} & 0 & 0 & 1-\gamma \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 0.8944\\ 0 & 0.2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0.8944 & 0 & 0 & 0.8 \end{pmatrix}.

The trace is $\frac12(1+\gamma+1-\gamma)=1$ , as a normalized Choi state must have.

Step 3. Diagonalize. The $\{00,11\}$ block $\frac12\left(\begin{smallmatrix}1&\sqrt{1-\gamma}\\\sqrt{1-\gamma}&1-\gamma\end{smallmatrix}\right)$ has determinant zero, so its eigenvalues are $0$ and $1-\gamma/2$ ; the $\{01\}$ entry gives $\gamma/2$ and $\{10\}$ gives $0$ . The full spectrum is $\{1-\tfrac{\gamma}{2},\ \tfrac{\gamma}{2},\ 0,\ 0\}=\{\mathbf{0.90,\ 0.10,\ 0,\ 0}\}$ . All eigenvalues are $\ge 0$ , so the channel is completely positive; two are nonzero, so the Choi rank is 2.

Step 4. Check trace preservation. Tracing over the output ( $A$ ) gives $\mathrm{Tr}_{\mathrm{out}}J=\left(\begin{smallmatrix}\frac12&0\\0&\frac{\gamma}{2}+\frac{1-\gamma}{2}\end{smallmatrix}\right)=\tfrac12\mathbb{1}=\mathbb{1}/d$ . The channel is trace preserving.

Step 5. Recover the Kraus operators from (5). The eigenvector for $\lambda_1=1-\gamma/2$ is $\frac{1}{\sqrt{2-\gamma}}(|00\rangle+\sqrt{1-\gamma}\,|11\rangle)$ ; reshaping and scaling, $K_1=\sqrt{d\lambda_1}\,\mathrm{mat}=\mathrm{diag}(1,\sqrt{0.8})=K_0$ . The eigenvector for $\lambda_2=\gamma/2$ is $|01\rangle$ , reshaping to $E_{01}$ , so $K_2=\sqrt{\gamma}\,|0\rangle\langle1|=K_1$ . The eigendecomposition returns exactly the canonical Kraus pair — tomography and the Kraus structure theorem are the same computation.

2 · Depolarizing channel: process matrix and average gate fidelity

A qubit gate suffers depolarizing noise $\mathcal{E}(\rho)=(1-p)\rho+p\,\mathbb{1}/2$ with $p=0.1$ , targeting the identity. Find the process matrix $\chi$ in the Pauli basis, the Choi spectrum, and the average gate fidelity.

Step 1. Rewrite as a Pauli mixture. Using $\frac14(\rho+X\rho X+Y\rho Y+Z\rho Z)=\mathbb{1}/2$ gives $\mathcal{E}(\rho)=(1-\tfrac{3p}{4})\rho+\tfrac{p}{4}(X\rho X+Y\rho Y+Z\rho Z)$ , so in $\{I,X,Y,Z\}$ the process matrix is diagonal, $\chi=\mathrm{diag}(1-\tfrac{3p}{4},\tfrac{p}{4},\tfrac{p}{4},\tfrac{p}{4})=\mathbf{diag(0.925,\,0.025,\,0.025,\,0.025)}$ .

Step 2. Read off the Choi spectrum. Since $J=(1-p)|\Omega\rangle\langle\Omega|+p\,\mathbb{1}/4$ , the eigenvalue along $|\Omega\rangle$ is $1-\tfrac{3p}{4}=0.925$ and the three orthogonal Bell states each carry $p/4=0.025$ . The spectrum $\{0.925,0.025,0.025,0.025\}$ is full rank — depolarizing needs all four Kraus operators. For a Pauli channel the Choi eigenvalues and the $\chi$ diagonal coincide, since the Bell basis diagonalizes both.

Step 3. The entanglement fidelity to the identity is the $|\Omega\rangle$ overlap, $F_{\mathrm{ent}}=\langle\Omega|J|\Omega\rangle=\chi_{II}=1-\tfrac{3p}{4}=\mathbf{0.925}$ .

Step 4. Convert with (7) at $d=2$ : $F_{\mathrm{avg}}=\dfrac{2(0.925)+1}{3}=\dfrac{2.85}{3}=\mathbf{0.95}$ , matching the closed form $F_{\mathrm{avg}}=1-p/2$ . A depolarizing probability $p=0.1$ is a $95\%$ average-fidelity gate — and $F_{\mathrm{avg}}>F_{\mathrm{ent}}$ , the generic ordering from the $d/(d+1)$ weighting.

Physics of the portrait

The Choi-Jamiołkowski isomorphism is where a 1970s theorem in matrix analysis became the working language of every quantum-hardware laboratory — pure linear algebra dictating experimental protocol.

Origins · 1 minChoi-Jamiolkowski isomorphism and how this was discovered

Choi-Jamiolkowski isomorphism

the channel-state duality, 1972–1975 · completely positive maps and quantum operations

The isomorphism carries two names because two mathematicians reached it within three years. Andrzej Jamiołkowski, in Toruń, showed in 1972 that trace-and-positivity-preserving transformations correspond to operators on a doubled space — the "state" attached to a "map." Man-Duen Choi, an operator theorist, proved in 1975 the sharper, now more-cited result: a linear map on complex matrices is completely positive if and only if the single matrix $J(\mathcal{E})$ is positive semidefinite, and his proof builds the Kraus operators explicitly from the eigenvectors. For two decades this lived in the operator-algebra literature, disconnected from physics. Its second life began when quantum information theory needed exactly this object: a channel is a CP map, complete positivity is the non-negotiable physicality condition (a merely-positive map like the transpose becomes unphysical once an entangled ancilla is present), and the Choi matrix bridges to Kraus operators, fidelities, and entanglement-breaking tests. The experimental half arrived in 1997, when Isaac Chuang and Michael Nielsen wrote the "black box" prescription that made the duality a laboratory procedure. Today the same $J(\mathcal{E})$ Choi introduced to classify positive maps is what a calibration routine reconstructs to certify a gate, and the eigenvalues from his proof are the noise spectrum an experimentalist plots.

History

1972

— Jamiołkowski links trace-and-positivity-preserving maps to operators on a doubled space

1975

— Choi proves complete positivity is equivalent to Choi-operator positivity, with explicit Kraus form

1997

— Chuang and Nielsen give the experimental prescription for tomography of a quantum black box

2001

— D'Ariano and Lo Presti propose ancilla-assisted, entanglement-based process tomography

2013

— gate set tomography extends QPT to be robust against state-preparation and measurement errors

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, §8.4.2 and Box 8.5 · Watrous, The Theory of Quantum Information, Ch. 2 and Ch. 4 · Wilde, Quantum Information Theory, Ch. 4 · Preskill, Lecture Notes on Quantum Computation, Ch. 3 · Kitaev, Shen and Vyalyi, Classical and Quantum Computation.

Primary sources. Choi, "Completely positive linear maps on complex matrices," Lin. Alg. Appl. 10, 285 (1975) · Jamiołkowski, "Linear transformations which preserve trace and positive semidefiniteness of operators," Rep. Math. Phys. 3, 275 (1972) · Chuang and Nielsen, "Prescription for experimental determination of the dynamics of a quantum black box," J. Mod. Opt. 44, 2455 (1997). Key numerical results: Kraus operators follow from $K_k=\sqrt{d\lambda_k}\,\mathrm{mat}(|v_k\rangle)$ ; trace preservation is equivalent to $\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d$ ; a single qubit has $16-4=12$ real CPTP parameters; the amplitude-damping channel at $\gamma=0.2$ has Choi spectrum $\{0.9,0.1,0,0\}$ with Kraus recovery as worked above; and the depolarizing channel at $p=0.1$ gives $\chi=\mathrm{diag}(0.925,0.025,0.025,0.025)$ with $F_{\mathrm{avg}}=1-p/2=0.95$ .

Builds on: Quantum Operations and Completely Positive Trace-Preserving Maps · Kraus Representation of Quantum Channels · Open Quantum Systems and Reduced Dynamics Leads to: Depolarizing Channel · Amplitude Damping Channel · Phase Damping Channel · Relaxation and Dephasing: T1 and T2 Times

Ask the tutor: Ask me to walk through building the Choi matrix of any qubit channel — amplitude damping, phase damping, or depolarizing — and reading its Kraus operators, rank, and gate fidelity straight off the spectrum.

Why does one entangled input capture the entire channel when it only touches one subsystem?

Because the maximally entangled projector expands as $|\Omega\rangle\langle\Omega|=\frac1d\sum_{ij}E_{ij}\otimes E_{ij}$ , a sum over every matrix unit $E_{ij}=|i\rangle\langle j|$ . Applying $\mathcal{E}\otimes\mathrm{id}$ records $\mathcal{E}(E_{ij})$ for all $i,j$ at once, and since the $E_{ij}$ form an operator basis, linearity reconstructs $\mathcal{E}(\rho)$ for any $\rho$ . The untouched ancilla keeps the basis labels sorted, so all $d^2$ pieces ride out on one bipartite state instead of $d^2$ separate experiments.

What is the difference between the Choi matrix J and the process matrix χ?

They carry identical information in different frames. The Choi matrix $J$ is the physical bipartite state you get by sending half of $|\Omega\rangle$ through the channel. The process matrix $\chi$ is the coefficient array when you expand the channel in a fixed operator basis $\{E_m\}$ , usually the Paulis, via $\mathcal{E}(\rho)=\sum_{mn}\chi_{mn}E_m\rho E_n^\dagger$ . They are related by a fixed unitary change of basis, so both are Hermitian and PSD and complete positivity is $J\ge 0$ or equivalently $\chi\ge 0$ . Use $J$ to think of the channel as a state; use $\chi$ when you want Pauli error probabilities on the diagonal.

Why does naive linear inversion give unphysical negative eigenvalues?

Because finite statistics carry sampling noise and unconstrained inversion has no knowledge that the answer must live inside the CPTP body. Solving $\chi=\beta^{-1}\lambda$ finds the matrix best matching noisy data with no positivity requirement, so it routinely lands just outside the physical set — closer to the data than any physical channel can be. The cure is a constrained estimator: maximum likelihood or least squares over the convex set $\{J\ge 0,\ \mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d\}$ , whose optimum is unique. This projects the raw estimate to the nearest physical channel and is now standard practice.

Normalized or unnormalized Choi — which convention should I use?

Either, as long as you never mix them. The normalized convention uses $|\Omega\rangle=\frac{1}{\sqrt d}\sum_i|ii\rangle$ , giving $\mathrm{Tr}\,J=1$ and $\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}/d$ — $J$ is a genuine density matrix, handy for treating the channel as a state. The unnormalized one uses $\sum_i|ii\rangle$ , giving $\mathrm{Tr}\,J=d$ and $\mathrm{Tr}_{\mathrm{out}}J=\mathbb{1}$ , cleaner for the Kraus completeness relation. Fidelities and the factor $d$ in the inverse map shift between them, so a stray factor of $d$ is almost always a convention leak. Pick one, write it down, and check a known channel like the identity to confirm your factors.

Why has standard process tomography been superseded by randomized benchmarking and gate set tomography?

Two reasons: cost and bias. Cost — full QPT needs $d^4-d^2$ parameters, scaling as $16^n$ , so the budget explodes beyond a few qubits. Bias — standard QPT assumes perfect input states and measurements, so any SPAM error is silently blamed on the gate, capping accuracy at the SPAM level. Randomized benchmarking sidesteps both by measuring only the average error rate of random Clifford sequences, cheaply and SPAM-robustly. Gate set tomography self-consistently estimates gates, preparations, and measurements together so no element is assumed ideal. Both, however, still express their output as CPTP maps in the Choi/ $\chi$ language, so the isomorphism remains the reference frame even where the method has changed.

How do I read the Kraus operators and the Choi rank off tomographic data?

Reconstruct $J$ via a CPTP-constrained fit, diagonalize it, and apply equation (5): for each eigenpair $(\lambda_k,|v_k\rangle)$ with $\lambda_k>0$ , reshape $|v_k\rangle$ into a $d\times d$ matrix and scale by $\sqrt{d\lambda_k}$ to get $K_k$ . The count of nonzero eigenvalues is the Choi rank — rank 1 means a coherent, near-unitary process, a spread-out spectrum signals strong decoherence. In the amplitude-damping example the spectrum $\{0.9,0.1,0,0\}$ has rank $2$ , and reshaping its two eigenvectors returns exactly $K_0=\mathrm{diag}(1,\sqrt{0.8})$ and $K_1=\sqrt{0.2}\,|0\rangle\langle1|$ . The decomposition is unique up to degeneracy, giving the canonical Kraus set.