Quantum Operations and Completely Positive Trace-Preserving Maps

Closed quantum systems obey the Schrödinger equation, and their states rotate rigidly by a unitary, $\rho\mapsto U\rho U^{\dagger}$ . That picture is exact and beautiful and far too small. Every real device leaks: a qubit couples to lattice phonons, a photon scatters off a lossy mirror, a spin dephases in a fluctuating field, an experimenter measures and the state jumps. To describe any of this — noise, dissipation, measurement back-action, the deliberate discarding of a subsystem — we need the most general transformation a quantum state can physically undergo. That object is the quantum operation, formalized as a completely positive trace-preserving (CPTP) map on the space of density operators. The demand "physically allowed" turns out to have exactly three axioms. The map must be linear, because mixing ensembles and then evolving must equal evolving and then mixing. It must be trace-preserving, so probabilities keep summing to one. And it must be completely positive — not merely positive. Positivity says valid states go to valid states; complete positivity says this survives even when the map acts on one half of a state entangled with an untouched spectator of any dimension. That third axiom is the subtle one, and the whole subject turns on it: the transpose map is positive yet not completely positive, and reality forbids it. The deep payoff, the theorems of Stinespring, Kraus, and Choi, is that these three abstract axioms collapse into one physical story told three equivalent ways: couple to an environment and trace it out; write an operator sum $\sum_k E_k\rho E_k^{\dagger}$ ; or check that a single matrix is positive. This trinity is the working language of open quantum systems, error correction, and quantum Shannon theory.

Key takeaways

A quantum operation is a linear map $\mathcal{E}$ on density operators that is completely positive and trace-preserving — the most general physical transformation of a quantum state, covering unitary gates, measurement, noise, and dissipation in one structure.
Complete positivity, stronger than plain positivity, is forced by entanglement: $\mathcal{E}\otimes\mathcal{I}_R$ must stay positive for a reference $R$ of arbitrary size. The transpose is the canonical map that is positive but not CP.
Every CPTP map has a Kraus (operator-sum) form $\mathcal{E}(\rho)=\sum_k E_k\rho E_k^{\dagger}$ with $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ , and equivalently arises from unitary coupling to an environment followed by partial trace (Stinespring dilation).
The Choi–Jamiołkowski isomorphism turns dynamics into statics: $\mathcal{E}$ is CP iff its Choi matrix $J(\mathcal{E})\ge0$ , and TP iff $\mathrm{Tr}_1 J(\mathcal{E})=\mathbb{1}$ .
CPTP maps form a convex set closed under mixing, composition, and tensor product; they are the discrete-time counterpart of the Lindblad/GKSL generator.
Key idea: a transformation of quantum states is physical iff it is linear, completely positive, and trace-preserving — and by Stinespring, Kraus, and Choi this is the same as "unitary on a larger space, traced out," an operator sum $\sum_k E_k\rho E_k^{\dagger}$ with $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ , and a positive Choi operator with the correct marginal.

The core insight

A transformation of quantum states is physical if and only if it is linear, completely positive, and trace-preserving. By the theorems of Stinespring, Kraus, and Choi these axioms are equivalent to three concrete pictures: couple the system unitarily to an environment and trace it out; write an operator sum $\sum_k E_k\rho E_k^{\dagger}$ with $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ ; and produce a positive-semidefinite Choi operator with the correct marginal. Complete positivity — positivity that survives entanglement with an arbitrary spectator — is the non-obvious ingredient.

Where this lives today

CPTP maps are not a formal abstraction; they are the daily currency of experimental quantum computing. When a group at IBM, Google, or a trapped-ion lab wants to know what a gate actually does, they reconstruct its Choi matrix by quantum process tomography — preparing a complete set of input states, measuring the outputs, and fitting the most likely $J(\mathcal{E})$ subject to $J\ge0$ and the correct marginal, so the answer is guaranteed to be a physical channel. For many-qubit systems full tomography is exponentially expensive, so the field has moved to scalable summaries of the same CPTP object: randomized benchmarking extracts an average gate fidelity, gate set tomography reconstructs a self-consistent set of channels, and cycle benchmarking characterizes the effective Pauli (Kraus) error per layer. Every one of these is a measurement of a quantum operation. Downstream, the operator-sum structure is what makes fault tolerance possible at all: because a code that corrects a basis of Kraus errors automatically corrects their linear combinations, continuous physical noise "discretizes" into correctable Pauli errors, and threshold theorems then guarantee that iterated CPTP noise can be suppressed below any target. In quantum communication the same maps become the channels of quantum Shannon theory, whose classical, quantum, private, and entanglement-assisted capacities set the ultimate limits of quantum networks and QKD. And error mitigation on today's noisy devices — probabilistic error cancellation, zero-noise extrapolation — works by explicitly modeling and partially inverting the CPTP map a circuit implements. From the lab bench to the theory of capacities, the object introduced here is the interface.

The physical picture

Picture your system as a fragment of a larger closed universe. Whatever happens to it, however noisy or irreversible it looks, must be the shadow of an honest unitary acting on the system together with its environment. Prepare the environment in a fixed state, let the pair evolve by some $U$ , then look away from — trace out — the environment. The map from input state to reduced output state is what you call the channel. Stinespring's theorem says the set of maps obtainable this way is exactly the set of CPTP maps: nothing more exotic is physically realizable, and nothing physical is left out. The first figure is this one idea drawn as a circuit.

Figure

A quantum channel is the shadow of a unitary: couple the system to an environment prepared in a fixed state, evolve the pair by U, then trace out the environment. The map that survives is exactly a completely positive trace-preserving operation, and its Kraus operators are the matrix elements E_k of U between environment states.

Why "completely" positive and not just positive? Imagine your qubit is secretly one half of an entangled pair, its partner held by a distant friend you will never contact. You apply your map to your half only; the untouched partner is a spectator. The joint state must remain a legitimate state — no measurement, however cleverly correlated across the pair, may return a negative probability. A map that fails this, like the transpose, would manufacture impossible statistics on the entangled pair. Complete positivity is precisely positivity that holds up under entanglement with an arbitrary, arbitrarily large spectator. It is the axiom you would never guess from single-system reasoning, and it is the one that makes the whole framework correct. The last move in the picture is the Choi trick, which converts dynamics into statics. Feed half of a maximally entangled pair through the channel and leave the other half alone; the resulting bipartite state — the Choi matrix — is a fingerprint that stores everything about the map. Abstract properties of the map (completely positive, trace-preserving, unital) become concrete properties of a single Hermitian matrix (positive, correct partial trace, correct other marginal) that you can diagonalize on a laptop or reconstruct in a lab. The second figure shows what two representative channels do to the Bloch ball — the geometry every qubit-noise calculation comes back to, with real contraction numbers.

Figure

What single-qubit channels do to the Bloch ball, with numbers. Left: the depolarizing channel shrinks the ball isotropically toward the center by the factor 1−p, here 1−0.1 = 0.9, driving every state toward the maximally mixed state. Right: amplitude damping contracts and drifts the ball toward the ground state at the north pole, here with γ = 0.18 so the excited population multiplies by 1−γ = 0.82.

The core equations

A quantum channel is defined by what it must preserve. It takes any valid state — positive, unit trace — to another valid state, and it must do so for every input at once.

\mathcal{E}:\;\rho\ge0,\;\mathrm{Tr}\,\rho=1\;\longmapsto\;\mathcal{E}(\rho)\ge0,\;\mathrm{Tr}\,\mathcal{E}(\rho)=1 \qquad (1)

where: $\rho$ — density operator (positive semidefinite, unit trace) on the system Hilbert space; $\mathcal{E}$ — a linear map on operators, a superoperator; $\ge0$ — positive semidefinite (no negative eigenvalues). Reading: legitimate states in, legitimate states out, with linearity assumed throughout. The subtle axiom is that positivity must survive tensoring with an untouched reference of any size. This is the definition of complete positivity, and it is strictly stronger than positivity of $\mathcal{E}$ alone.

\big(\mathcal{E}\otimes\mathcal{I}_R\big)(\sigma)\ge0\quad\text{for every reference }R\text{ and every }\sigma\ge0 \qquad (2)

where: $\mathcal{I}_R$ — the identity superoperator on a reference system $R$ of arbitrary dimension; $\sigma$ — any positive operator on system $\otimes$ reference. Reading: even acting on half of an entangled pair, the map may not create a negative eigenvalue anywhere. The structure theorem gives the map a concrete form: it is a sum of one-sided products with a completeness constraint. This is the representation you compute with.

\mathcal{E}(\rho)=\sum_k E_k\,\rho\,E_k^{\dagger},\qquad \sum_k E_k^{\dagger}E_k=\mathbb{1} \qquad (3)

where: $E_k$ — Kraus operators (operation elements), not required to be Hermitian, normal, or square-summable beyond the constraint; $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ — completeness relation, which is exactly trace preservation. Reading: each $E_k$ is one "way the environment could have responded," summed coherently inside, incoherently across $k$ . Equivalently, the channel is a unitary on a dilated space with the environment traced out. This is the physical origin of the Kraus form.

\mathcal{E}(\rho)=\mathrm{Tr}_E\!\big(U\,(\rho\otimes|0\rangle_E\langle0|)\,U^{\dagger}\big) \qquad (4)

where: $U$ — a unitary on system $\otimes$ environment; $|0\rangle_E$ — a fixed initial environment state; $\mathrm{Tr}_E$ — partial trace over the environment. The Kraus operators are its matrix elements, $E_k=\langle k|_E\,U\,|0\rangle_E$ . The Choi–Jamiołkowski operator packages the entire channel into one matrix by acting on half of an (unnormalized) maximally entangled state.

J(\mathcal{E})=\big(\mathcal{E}\otimes\mathcal{I}\big)\big(|\Omega\rangle\langle\Omega|\big),\qquad |\Omega\rangle=\sum_{i=1}^{d}|i\rangle|i\rangle \qquad (5)

where: $J(\mathcal{E})$ — Choi matrix, an operator on $\mathcal{H}\otimes\mathcal{H}$ of size $d^2\times d^2$ ; $|\Omega\rangle$ — unnormalized maximally entangled state; $d$ — Hilbert-space dimension. Reading: dynamics (a map) becomes statics (a state-like operator). The CPTP axioms become two linear-algebra checks on that one matrix — the reason process tomography is tractable.

\mathcal{E}\text{ is CP}\iff J(\mathcal{E})\ge0,\qquad \mathcal{E}\text{ is TP}\iff \mathrm{Tr}_1\,J(\mathcal{E})=\mathbb{1} \qquad (6)

where: $\mathrm{Tr}_1$ — partial trace over the output factor of $J(\mathcal{E})$ . Reading: check one eigenvalue spectrum for CP and one marginal for TP. Two canonical channels anchor the theory. The depolarizing channel shrinks any state toward the maximally mixed state; amplitude damping models irreversible energy loss and is the standard non-unital example.

\mathcal{E}_{\mathrm{dep}}(\rho)=(1-p)\,\rho+p\,\frac{\mathbb{1}}{d},\qquad E_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix},\;E_1=\begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix} \qquad (7)

where: $p$ — depolarizing probability, $0\le p\le1$ ; $\mathbb{1}/d$ — maximally mixed fixed point; $\gamma$ — damping probability, related to relaxation time by $\gamma=1-e^{-t/T_1}$ ; $E_1$ — the jump operator carrying $|1\rangle\to|0\rangle$ . Reading: $\mathcal{E}_{\mathrm{dep}}$ is unital, amplitude damping is not.

Derivation AWhy positivity is not enough — the transpose counterexample

The goal of this derivation is to show that complete positivity is a genuinely stronger requirement than positivity, by exhibiting a map that sends every density operator to a legitimate density operator and yet is physically forbidden. The map is the matrix transpose, $T(\rho)=\rho^{T}$ . It preserves eigenvalues, so it preserves positivity and trace on any single system. The failure appears only when $T$ acts on half of an entangled pair, which is exactly what complete positivity tests — and this single example is why the CP axiom exists.

Step 1. Take the two-qubit maximally entangled state $|\Phi^+\rangle=\tfrac{1}{\sqrt2}(|00\rangle+|11\rangle)$ , whose density operator is $|\Phi^+\rangle\langle\Phi^+|=\tfrac12\sum_{ij}|ii\rangle\langle jj|=\tfrac12\sum_{ij}(|i\rangle\langle j|)\otimes(|i\rangle\langle j|)$ . This is a valid pure state, hence positive semidefinite. Why: we need a positive input on system-plus-reference to feed into $T\otimes\mathcal{I}$ ; an entangled one is what exposes the difference.

Step 2. Apply $T$ to the first tensor factor only. Transposing $|i\rangle\langle j|$ gives $|j\rangle\langle i|$ , so $(T\otimes\mathcal{I})|\Phi^+\rangle\langle\Phi^+|=\tfrac12\sum_{ij}|j\rangle\langle i|\otimes|i\rangle\langle j|=\tfrac12\sum_{ij}|ji\rangle\langle ij|=\tfrac12\,\mathrm{SWAP}$ . Why: the partial transpose relabels rows and columns on subsystem one, and the sum of $|ji\rangle\langle ij|$ is precisely the swap operator that exchanges the two qubits.

Step 3. The swap operator has eigenvalue $+1$ on the symmetric subspace and $-1$ on the antisymmetric one: $\mathrm{SWAP}\,|\psi^-\rangle=-|\psi^-\rangle$ with $|\psi^-\rangle=\tfrac{1}{\sqrt2}(|01\rangle-|10\rangle)$ . Therefore $(T\otimes\mathcal{I})\rho$ has an eigenvalue $-\tfrac12<0$ . Why: a negative eigenvalue means the output is not a valid state; the reference qubit, never touched, has forced $T$ to reveal itself as non-physical.

Step 4. Conclude that $T$ is positive but not completely positive, hence not a quantum operation, even though $T(\rho)=\rho^{T}$ is a perfectly good density operator for every single-system $\rho$ . Why: this is the whole point — physicality is decided on the entangled extension, and only complete positivity captures it. *Verified: $(T\otimes\mathcal{I})|\Phi^+\rangle\langle\Phi^+|=\tfrac12\,\mathrm{SWAP}$ with least eigenvalue $-\tfrac12$ ; the map is trace- and eigenvalue-preserving on any single system (Nielsen & Chuang §8.2.4, Watrous Ch. 2).*

Go deeper · Derivation B · 2 minFrom complete positivity to the operator-sum form

Derivation BFrom complete positivity to the operator-sum form

The goal here is to prove the workhorse theorem: every completely positive trace-preserving map admits a Kraus representation $\mathcal{E}(\rho)=\sum_k E_k\rho E_k^{\dagger}$ with at most $d^2$ terms and $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ . The route is Choi's: complete positivity makes the Choi matrix a positive operator, its spectral decomposition hands over the Kraus operators directly, and trace preservation becomes the completeness relation. This is the derivation that lets you go from an abstract "physical map" to explicit matrices you can multiply.

Step 1. Form the Choi operator by acting with $\mathcal{E}\otimes\mathcal{I}$ on the unnormalized entangled state: $J(\mathcal{E})=\sum_{i,j=1}^{d}\mathcal{E}(|i\rangle\langle j|)\otimes|i\rangle\langle j|$ . This single operator on the $d^2$ -dimensional space encodes $\mathcal{E}$ completely, since knowing $\mathcal{E}$ on the basis $\{|i\rangle\langle j|\}$ fixes it everywhere by linearity. Why: we trade the map for one operator so the axioms become properties of a matrix.

Step 2. Invoke complete positivity. Since $|\Omega\rangle\langle\Omega|\ge0$ and $\mathcal{E}\otimes\mathcal{I}$ preserves positivity by hypothesis, the Choi operator is positive semidefinite, $J(\mathcal{E})\ge0$ . Why: this is Choi's theorem in the forward direction, and positivity is exactly what makes the next step, a spectral decomposition with non-negative weights, legal.

Step 3. Spectrally decompose $J(\mathcal{E})=\sum_{k}\lambda_k\,|v_k\rangle\langle v_k|$ with $\lambda_k\ge0$ . A positive operator on a $d^2$ -dimensional space has at most $d^2$ nonzero eigenvalues, so the sum has at most $d^2$ terms. Why: the number of Kraus operators (the Kraus rank) is fixed here as the rank of $J$ , capped at $d^2$ .

Step 4. Reshape each eigenvector into a matrix. Writing $|v_k\rangle=\sum_{ij}(v_k)_{ij}|i\rangle|j\rangle$ , define $E_k=\sqrt{\lambda_k}\sum_{ij}(v_k)_{ij}\,|i\rangle\langle j|$ . The vectorization $|i\rangle|j\rangle\leftrightarrow|i\rangle\langle j|$ turns a $d^2$ -vector into a $d\times d$ operator, and $\sqrt{\lambda_k}$ carries the weight. Why: this is the exact bridge from the static Choi data back to operators acting on states.

Step 5. Verify the action. One checks that $\sum_k E_k|i\rangle\langle j|E_k^{\dagger}=\mathcal{E}(|i\rangle\langle j|)$ for all $i,j$ , and by linearity $\mathcal{E}(\rho)=\sum_k E_k\rho E_k^{\dagger}$ for every $\rho$ . Why: this confirms the reshaped operators actually reproduce the map, not merely something with the right Choi matrix.

Step 6. Impose trace preservation. The TP condition $\mathrm{Tr}_1 J(\mathcal{E})=\mathbb{1}$ translates term by term into $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ , the completeness relation. Why: this closes the equivalence — CP gives the operator sum, TP gives the constraint, and together they are exactly "CPTP." *Verified: $\lambda_k\ge0$ follows from $J\ge0$ ; the reshaping is the standard column-stacking isomorphism, and $\mathrm{Tr}_1 J=\mathbb{1}\Leftrightarrow\sum_k E_k^{\dagger}E_k=\mathbb{1}$ (Watrous Ch. 2, Nielsen & Chuang §8.2.3).*

Go deeper · Derivation C · 1 minStinespring dilation — every channel is a unitary in disguise

Derivation CStinespring dilation — every channel is a unitary in disguise

The goal is to run the argument the other way and show that the abstract Kraus form is not just formal bookkeeping but describes genuine unitary physics on a larger space. Given Kraus operators $\{E_k\}_{k=0}^{K-1}$ with completeness, we build an explicit environment, a fixed initial environment state, and a global unitary $U$ such that the channel is recovered by partial trace. The construction proves that open, irreversible dynamics is always closed, reversible dynamics viewed through a partial trace — the content of the first figure.

Step 1. Define the isometry $V=\sum_{k=0}^{K-1}E_k\otimes|k\rangle_E$ , mapping the system into system $\otimes$ environment with $K$ orthonormal environment states $|k\rangle_E$ . Equivalently $V|\psi\rangle=\sum_k E_k|\psi\rangle\otimes|k\rangle_E$ . Why: this bundles all the Kraus operators into one operator that adds an environment register recording "which $k$ ."

Step 2. Check that $V$ is an isometry: $V^{\dagger}V=\sum_{k,k'}E_{k'}^{\dagger}E_k\,\langle k'|k\rangle=\sum_k E_k^{\dagger}E_k=\mathbb{1}$ , using orthonormality of $|k\rangle_E$ and Kraus completeness. Why: $V^{\dagger}V=\mathbb{1}$ says $V$ preserves inner products, so it can be part of a unitary; the completeness relation is exactly the isometry condition.

Step 3. Trace out the environment: $\mathrm{Tr}_E[V\rho V^{\dagger}]=\sum_{k,k'}E_k\rho E_{k'}^{\dagger}\,\mathrm{Tr}_E(|k\rangle\langle k'|)=\sum_k E_k\rho E_k^{\dagger}=\mathcal{E}(\rho)$ , since $\mathrm{Tr}_E(|k\rangle\langle k'|)=\delta_{kk'}$ . Why: discarding the environment register collapses the coherences between different $k$ , turning one coherent unitary into the incoherent operator sum — this is where irreversibility enters.

Step 4. Extend $V$ to a unitary. Any isometry from $\mathcal{H}_S$ into $\mathcal{H}_S\otimes\mathcal{H}_E$ can be completed to a unitary $U$ on the full space by extending its columns to an orthonormal basis, with $V=U(\mathbb{1}_S\otimes|0\rangle_E)$ . This delivers the environmental form $\mathcal{E}(\rho)=\mathrm{Tr}_E[U(\rho\otimes|0\rangle\langle0|)U^{\dagger}]$ . Why: it realizes the channel as an honest closed-system evolution of system-plus-environment, which is Stinespring's theorem in finite dimensions. *Verified: $V^{\dagger}V=\mathbb{1}$ is equivalent to Kraus completeness, and the dilation reproduces $\mathcal{E}$ exactly; the minimal environment dimension equals the Kraus rank $\le d^2$ (Preskill Ch. 3, Wilde Ch. 5).*

Limiting cases

CPTP maps specialize to every familiar kind of quantum dynamics in well-defined regimes:

Case	Result	Meaning
Unitary / closed limit	$\mathcal{E}(\rho)=U\rho U^{\dagger}$	A single Kraus operator $E_0=U$ ; the environment decouples. Reversible, trace- and distance-preserving — the invertible corner of the CPTP set.
Continuous Markovian limit	$\dot\rho=-\tfrac{i}{\hbar}[H,\rho]+\sum_k\big(L_k\rho L_k^{\dagger}-\tfrac12\{L_k^{\dagger}L_k,\rho\}\big)$	A one-parameter semigroup $\mathcal{E}_t=e^{t\mathcal{L}}$ of CPTP maps; the generator $\mathcal{L}$ has Lindblad/GKSL form. Each $\mathcal{E}_t$ is itself a channel.
Classical / fully dephased	$\mathcal{E}(\rho)=\sum_i\langle i\|\rho\|i\rangle\,\|i\rangle\langle i\|$	Complete dephasing in a fixed basis reduces the channel to a stochastic matrix on the probability vector — classical Markov dynamics as a special CPTP map.
Unital sub-case	$\mathcal{E}(\mathbb{1})=\mathbb{1}$ , i.e. $\sum_k E_k E_k^{\dagger}=\mathbb{1}$	Fixes the maximally mixed state (depolarizing, Pauli, phase damping). Cannot decrease the entropy of $\mathbb{1}/d$ ; amplitude damping is the canonical non-unital exception.

Common trapConfusing positivity with complete positivity

The single most common error in this subject is to check that a map sends density operators to density operators and conclude it is a valid quantum operation. That is only positivity, and it is not enough. The transpose $T(\rho)=\rho^{T}$ preserves every eigenvalue, hence is positive, yet $(T\otimes\mathcal{I})$ applied to a Bell state yields $\tfrac12\,\mathrm{SWAP}$ with an eigenvalue of $-\tfrac12$ : physically forbidden. Physicality requires positivity even when the map touches only a subsystem entangled with an untouched reference of arbitrary dimension — that is complete positivity. A second, subtler trap: Kraus operators are not unique. Any isometric remixing $E_k'=\sum_j u_{jk}E_j$ (with $u$ the columns of an isometry, padding shorter lists with zeros) gives the identical channel. If you compare two devices by their Kraus operators you will see spurious differences; compare their Choi matrices, which are unique.

Worked examples

1 · Amplitude damping and the T₁ decay of an excited qubit (confidence builder)

A superconducting qubit has relaxation time $T_1=100\ \mu\mathrm{s}$ . It is prepared in the excited state $|1\rangle$ and left idle for $t=20\ \mu\mathrm{s}$ . Using the amplitude-damping channel, find the surviving excited-state population and confirm the map is trace-preserving.

Step 1. Fix the damping parameter from the relaxation time: $\gamma=1-e^{-t/T_1}=1-e^{-20/100}=1-e^{-0.2}=1-0.8187=\mathbf{0.1813}$ . This $\gamma$ is the probability that one quantum of energy has leaked to the environment during the idle window.

Step 2. Apply the channel to $\rho=|1\rangle\langle1|$ . With $E_0=\mathrm{diag}(1,\sqrt{1-\gamma})$ and $E_1=\begin{pmatrix}0&\sqrt\gamma\\0&0\end{pmatrix}$ , we have $E_0|1\rangle=\sqrt{1-\gamma}\,|1\rangle$ and $E_1|1\rangle=\sqrt\gamma\,|0\rangle$ . Hence $\mathcal{E}(|1\rangle\langle1|)=(1-\gamma)\,|1\rangle\langle1|+\gamma\,|0\rangle\langle0|=\mathrm{diag}(0.1813,\,0.8187)$ .

Step 3. Read off the physics: the excited-state population has fallen from $1$ to $1-\gamma=\mathbf{0.8187}$ , and a fraction $\gamma=0.1813$ has decayed to the ground state — exactly the classical exponential $e^{-t/T_1}$ for the excited population, now derived as a CPTP map. Trace preservation is manifest: $0.1813+0.8187=\mathbf{1}$ , consistent with $E_0^{\dagger}E_0+E_1^{\dagger}E_1=\mathrm{diag}(1,1-\gamma)+\mathrm{diag}(0,\gamma)=\mathbb{1}$ .

Step 4. Note non-unitality: $\mathcal{E}(\mathbb{1})=E_0E_0^{\dagger}+E_1E_1^{\dagger}=\mathrm{diag}(1+\gamma,\,1-\gamma)=\mathrm{diag}(1.18,\,0.82)\neq\mathbb{1}$ . The maximally mixed state is not fixed; the channel drags every state toward $|0\rangle$ , the signature of irreversible energy loss. Excited population $\to0.8187$ , trace $=1$ , and the channel is non-unital.

2 · Depolarizing a superposition and checking it stays a state (fully worked)

A qubit in $|+\rangle=\tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$ passes through a depolarizing channel with $p=0.1$ . Find the output state, its purity, and verify it is a legitimate (positive, unit-trace) density operator.

Step 1. Write $|+\rangle$ in Bloch form: $\rho=\tfrac12(\mathbb{1}+\vec r\cdot\vec\sigma)$ with Bloch vector $\vec r=(1,0,0)$ , a pure state on the surface of the ball. The depolarizing channel contracts the Bloch vector isotropically, $\vec r\mapsto(1-p)\vec r$ , so $\vec r\,'=(0.9,\,0,\,0)$ .

Step 2. Reconstruct the output operator: $\rho'=\tfrac12(\mathbb{1}+0.9\,X)=\begin{pmatrix}0.5&0.45\\0.45&0.5\end{pmatrix}$ . Its trace is $0.5+0.5=\mathbf{1}$ , so probability is conserved.

Step 3. Purity: $\mathrm{Tr}(\rho'^2)=\tfrac12(1+|\vec r\,'|^2)=\tfrac12(1+0.81)=\mathbf{0.905}$ . The state has been pushed off the surface toward the center — coherence in the off-diagonal has shrunk from $0.5$ to $0.45$ , and the state is now mixed.

Step 4. Positivity check: eigenvalues of $\rho'$ are $\tfrac12(1\pm|\vec r\,'|)=\tfrac12(1\pm0.9)=\{0.95,\,0.05\}$ , both non-negative. So $\rho'$ is a valid density operator — as it must be, since the depolarizing channel is CPTP for $0\le p\le1$ . $\rho'=\begin{pmatrix}0.5&0.45\\0.45&0.5\end{pmatrix}$ , purity $0.905$ , eigenvalues $\{0.95,0.05\}\ge0$ .

3 · Verifying a channel is CP by its Choi eigenvalues

Confirm directly that amplitude damping with $\gamma=0.1813$ is completely positive by computing the spectrum of its Choi matrix.

Step 1. Build the Choi matrix from $J(\mathcal{E})=\sum_{ij}\mathcal{E}(|i\rangle\langle j|)\otimes|i\rangle\langle j|$ . Using $\mathcal{E}(|0\rangle\langle0|)=|0\rangle\langle0|$ , $\mathcal{E}(|1\rangle\langle1|)=(1-\gamma)|1\rangle\langle1|+\gamma|0\rangle\langle0|$ , and $\mathcal{E}(|0\rangle\langle1|)=\sqrt{1-\gamma}\,|0\rangle\langle1|$ , assemble the $4\times4$ operator.

Step 2. Diagonalize. The eigenvalues come out to $\{0,\ 0,\ \gamma,\ 2-\gamma\}=\{0,\,0,\,\mathbf{0.181},\,\mathbf{1.819}\}$ . All four are non-negative, so $J(\mathcal{E})\ge0$ and the map is completely positive by Choi's criterion.

Step 3. Cross-check trace preservation via the marginal: $\mathrm{Tr}_1 J(\mathcal{E})=\mathbb{1}$ , equivalently $\sum_k E_k^{\dagger}E_k=\mathbb{1}$ , which we already confirmed in Example 1. The two nonzero eigenvalues, $\gamma$ and $2-\gamma$ , correspond to the two Kraus operators — the Kraus rank is $2$ . All Choi eigenvalues $\ge0$ , so amplitude damping is CPTP with Kraus rank 2.

Physics of the portrait

Behind the whole framework stands a 1955 theorem in the theory of operator algebras that had nothing to do with physics when it was proved — and everything to do with it forty years later.

Origins · 1 minW. Forrest Stinespring and how this was discovered

W. Forrest Stinespring

1929–2012 · the dilation theorem behind every quantum channel

William Forrest Stinespring was an American mathematician trained in the operator-algebra school of the University of Chicago, and his single most cited result is a 1955 paper, "Positive functions on C-algebras," published in the Proceedings of the American Mathematical Society when he was in his mid-twenties. The question he answered was purely algebraic: given a completely positive linear map from a C-algebra into the bounded operators on a Hilbert space, what does it look like? His theorem — now called Stinespring dilation — says every such map factors as a representation of the algebra sandwiched between an isometry and its adjoint. Translated into the finite-dimensional language of quantum information, this is exactly the statement that every channel is a unitary on a larger space followed by a partial trace: the environmental picture that opens this article. At the time there were no qubits, no quantum computers, and no notion of a "quantum channel"; Stinespring was extending results of Gelfand, Naimark, and Segal on how abstract algebras represent as operators. The physics arrived later. When Kraus in 1971 wrote the general state change as an operator sum, and when the quantum information community matured in the 1990s, it became clear that Kraus's operators, Choi's positive matrix, and Stinespring's dilation were three faces of one theorem — and that Stinespring had, decades early and for entirely mathematical reasons, written down the structure of physical noise. His dilation is now the first thing proved in every graduate course on open quantum systems.

History

1955

— Stinespring proves the dilation theorem for completely positive maps on C*-algebras

1961

— Sudarshan, Mathews and Rau introduce the dynamical matrix for open-system evolution

1971

— Kraus casts the most general physical state change as an operator sum

1975

— Choi characterises completely positive maps by positivity of a single matrix

1976

— Lindblad and, independently, Gorini, Kossakowski and Sudarshan derive the Markovian generator

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, Ch. 8 (Quantum noise and quantum operations) · Preskill, Quantum Computation lecture notes, Ch. 3 (Foundations II: Measurement and Evolution) · Watrous, The Theory of Quantum Information, Ch. 2 (Completely positive and trace-preserving maps) · Wilde, Quantum Information Theory, Ch. 4 (Quantum channels) · Kitaev, Shen & Vyalyi, Classical and Quantum Computation, Ch. 11 · Choi, "Completely positive linear maps on complex matrices," Linear Algebra Appl. 10 (1975) 285 · Stinespring, "Positive functions on C-algebras," Proc. Amer. Math. Soc.* 6 (1955) 211.

Provenance. The three representations follow Nielsen & Chuang Ch. 8, Preskill Ch. 3, and Watrous Ch. 2. A verifier should reproduce: the transpose counterexample $(T\otimes\mathcal{I})|\Phi^+\rangle\langle\Phi^+|=\tfrac12\,\mathrm{SWAP}$ with eigenvalue $-\tfrac12$ ; Kraus completeness $\Rightarrow$ trace preservation by cyclicity; the amplitude-damping numbers $\gamma=1-e^{-0.2}=0.1813$ , excited population $0.8187$ , non-unitality $\mathcal{E}(\mathbb{1})=\mathrm{diag}(1.18,0.82)$ , and Choi spectrum $\{0,0,0.181,1.819\}$ ; and the depolarizing action $\vec r\mapsto(1-p)\vec r$ giving $\rho'=\begin{pmatrix}0.5&0.45\\0.45&0.5\end{pmatrix}$ , purity $0.905$ , eigenvalues $\{0.95,0.05\}$ at $p=0.1$ . The relation $\gamma=1-e^{-t/T_1}$ is standard (Nielsen & Chuang §8.3.5). No nonstandard constants are introduced.

Prerequisites. Density Operators and Mixed States · Open Quantum Systems and Reduced Dynamics · Tensor Products and Partial Trace · Unitary Evolution and the Schrödinger Equation. Leads to. Kraus Representation of Quantum Channels · Process Tomography and the Choi–Jamiołkowski Isomorphism · Lindblad / GKSL Master Equation · Depolarizing Channel · Amplitude Damping Channel.

Ask the tutor: Ask me to walk through why the transpose map is positive but not completely positive, or to build the Kraus operators of any channel from its Choi matrix step by step.

What is the difference between a "quantum operation" and a "quantum channel"?

In most modern usage the two are synonyms for a completely positive, trace-preserving map, and this article uses them interchangeably. Historically "quantum operation" (following Kraus) was the broader term, allowing trace-non-increasing maps that describe a single measurement outcome occurring with probability $\mathrm{Tr}\,\mathcal{E}(\rho)<1$ ; renormalizing by that probability gives the post-measurement state. "Quantum channel" almost always means the trace-preserving case, appropriate when no information about an outcome is recorded — for instance a noise process. So every channel is an operation, but a selective (outcome-conditioned) operation need only be trace-non-increasing. When someone says "CPTP map," they mean a channel.

Why exactly d² Kraus operators — where does that bound come from?

Because the Choi matrix $J(\mathcal{E})$ lives on a Hilbert space of dimension $d^2$ (it is $d^2\times d^2$ ), and a positive operator there has at most $d^2$ nonzero eigenvalues. Each nonzero eigenvalue supplies one Kraus operator in the spectral construction, so any channel on a $d$ -dimensional system can be written with at most $d^2$ Kraus operators. The minimal number actually needed is the Kraus rank, equal to the rank of $J(\mathcal{E})$ : a unitary has rank 1, amplitude damping has rank 2, a full depolarizing channel has rank 4 for a qubit ( $d=2$ , so up to $d^2=4$ ). You can always use more operators than the rank, but never fewer, and extra ones are related to the minimal set by an isometric remixing.

If a channel is "really" a unitary on a bigger space, why is it irreversible?

The full unitary on system-plus-environment is perfectly reversible — no information is destroyed there. Irreversibility appears the instant you trace out the environment, because the partial trace throws away the correlations that were built up between system and environment. Those correlations are exactly what you would need to run the evolution backward; without access to the environment you cannot recover them. This is why the Kraus form is an incoherent sum over $k$ : the coherences between different environment outcomes $|k\rangle_E$ are precisely what the trace discards. The environment "measures" the system without telling you the result, and unread measurement is the microscopic origin of decoherence.

Are Kraus operators unique, and why does it matter?

No. Two Kraus sets $\{E_k\}$ and $\{F_j\}$ describe the same channel if and only if they are related by an isometric mixing $F_j=\sum_k u_{jk}E_k$ , where $u$ has orthonormal columns (pad the shorter list with zero operators first). For example, the bit-flip channel can be written with $\{\sqrt{1-p}\,\mathbb{1},\sqrt{p}\,X\}$ or, after a Hadamard-like remix, with a different pair — both are correct. This matters practically: you cannot compare two noise processes by staring at their Kraus operators, because the same physics wears many costumes. The Choi matrix, by contrast, is a genuine invariant of the channel, which is why tomography and channel-distance measures are all built on $J(\mathcal{E})$ rather than on any particular $\{E_k\}$ .

What makes amplitude damping "non-unital" and why does that matter physically?

A channel is unital if it leaves the maximally mixed state alone, $\mathcal{E}(\mathbb{1}/d)=\mathbb{1}/d$ . Amplitude damping fails this: it computes $\mathcal{E}(\mathbb{1})=\mathrm{diag}(1+\gamma,1-\gamma)\neq\mathbb{1}$ , because it does not treat $|0\rangle$ and $|1\rangle$ symmetrically — it drains population downhill into the ground state and never the reverse. Physically, non-unitality is the fingerprint of a channel that exchanges energy with a cold environment: it has a preferred fixed point ( $|0\rangle$ ) rather than the featureless center of the Bloch ball. Depolarizing and dephasing channels are unital because they are "unbiased" noise; relaxation ( $T_1$ decay) is non-unital because thermodynamics gives the ground state a special role.

How does a discrete CPTP map relate to the Lindblad master equation?

They are the discrete- and continuous-time faces of the same physics. A time-homogeneous Markovian evolution is a one-parameter semigroup of channels $\mathcal{E}_t=e^{t\mathcal{L}}$ , and the requirement that every $\mathcal{E}_t$ be CPTP forces the generator $\mathcal{L}$ into the Lindblad/GKSL form with its dissipators $L_k\rho L_k^{\dagger}-\tfrac12\{L_k^{\dagger}L_k,\rho\}$ . So integrating the Lindblad equation over a finite time $t$ produces a specific CPTP map, and conversely differentiating a smooth family of channels at $t=0$ recovers a Lindbladian. The Kraus operators of $\mathcal{E}_t$ for small $t$ are $E_0\approx\mathbb{1}-(iH/\hbar+\tfrac12\sum_k L_k^{\dagger}L_k)t$ and $E_k\approx\sqrt{t}\,L_k$ — the jump operators are the square-root-of-time Kraus terms.

Is measurement a quantum operation too?

Yes, and this is one of the framework's unifying wins. A general measurement with outcomes $m$ is described by a set of operators $\{M_m\}$ with $\sum_m M_m^{\dagger}M_m=\mathbb{1}$ ; outcome $m$ occurs with probability $\mathrm{Tr}(M_m\rho M_m^{\dagger})$ and leaves the state $M_m\rho M_m^{\dagger}/p_m$ . Each outcome is a trace-non-increasing operation, and if you forget which outcome occurred you sum them into a single trace-preserving channel $\mathcal{E}(\rho)=\sum_m M_m\rho M_m^{\dagger}$ — exactly the Kraus form. Projective (von Neumann) measurement is the special case $M_m=P_m$ with orthogonal projectors. So measurement, gates, and noise are not three different theories; they are three uses of one operator-sum structure.