Kraus Representation of Quantum Channels

The Kraus, or operator-sum, representation is the single most useful formula in the theory of open quantum systems: it writes the most general physically allowed evolution of a density operator as $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ . Closed systems evolve reversibly, $\rho\mapsto U\rho U^\dagger$ ; but a real qubit is never closed — it leaks energy into phonons, dephases against fluctuating fields, and gets measured. The Kraus theorem says every such irreversible, non-unitary process — provided only that it is linear, completely positive, and trace-preserving (CPTP) — collapses to that one compact sum of "sandwich" terms. Decoherence, dissipation, and measurement back-action are all instances of it. Three viewpoints converge on the same object. The axiomatic view asks that $\mathcal{E}$ map density operators to density operators even when the system is entangled with an untouched reference — the demand of complete positivity, whose structure theorem is the operator-sum form. The Stinespring dilation view couples the system to a fresh environment, evolves the pair under one global unitary, and traces it away; the Kraus operators are then partial matrix elements $K_k=\langle e_k|U|e_0\rangle$ . The Choi–Jamiołkowski view sends the channel to a single positive matrix whose eigenvectors hand back a canonical minimal Kraus set. The non-uniqueness of the operators is a feature — it encodes which basis of the environment you never read out — and is exactly what lets error correction discretize a continuum of errors. Every canonical noise model is a short Kraus list whose differential limit is the Lindblad generator.

Key takeaways

A quantum channel is a completely positive, trace-preserving linear map on density operators; its Kraus form is $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ with the completeness relation $\sum_k K_k^\dagger K_k=I$ .
Every channel is a unitary coupling to an environment followed by a partial trace — the Stinespring dilation $\mathcal{E}(\rho)=\mathrm{Tr}_E[U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger]$ — and $K_k=\langle e_k|U|e_0\rangle$ .
The Kraus operators are not unique: any two minimal sets are related by a unitary mixing $K_j'=\sum_k W_{jk}K_k$ , so channels must be compared by their Choi matrix, never operator-by-operator.
Complete positivity, not mere positivity, is the physical requirement; the transpose map is positive but not CP and has no Kraus form, whereas the operator-sum structure makes CP manifest.
The Kraus rank equals the rank of the Choi matrix, at most $d^2$ : at most 4 for a qubit, 9 for a qutrit; amplitude damping needs only 2.
Key idea: A CPTP map has the operator-sum form $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ with $\sum_k K_k^\dagger K_k=I$ ; equivalently, couple the system to an environment, apply a global unitary, and trace the environment out.

The core insight

A CPTP map $\mathcal{E}$ has the operator-sum form $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$ with $\sum_k K_k^\dagger K_k=I$ ; this is equivalent to coupling the system to an environment, applying a global unitary, and tracing out the environment.

Where this lives today

The Kraus representation is the data structure of noisy quantum computing. Every error-modelling simulator — Qiskit Aer, Cirq, QuTiP — stores a channel as a list of Kraus operators and applies noise by evaluating $\sum_k K_k\rho K_k^\dagger$ after each gate. Characterizing a real device runs the theorem in reverse: process tomography reconstructs the Choi matrix of a gate on superconducting or trapped-ion hardware, then imposes $J\succeq0$ and trace preservation to project the noisy estimate onto a valid CPTP map — without which the fit can carry negative eigenvalues and predict negative probabilities. The amplitude-damping and phase-damping Kraus sets are what experimentalists fit to $T_1$ and $T_2$ decay, and the depolarizing channel is the error model behind randomized benchmarking and the surface-code threshold. The Kraus non-uniqueness is load-bearing in fault tolerance: the discretization of errors — that correcting a basis of Pauli errors corrects the whole continuum a real channel produces — is precisely the statement that any physical Kraus set is a linear combination of correctable ones. And different Kraus unravellings of one Lindblad channel are physically distinct continuous-monitoring schemes, from photon counting to homodyne, all reproducing the same average dynamics.

The physical picture

Picture the system as a qubit colliding with a fresh environment in a known pure state. The two evolve together under one big unitary U, and then you walk away from the environment — you trace it out because you neither control nor read it. Each orthogonal final environment state eₖ labels a branch of the joint wavefunction, and the operator producing that branch is the Kraus operator Kₖ = ⟨eₖ|U|e₀⟩. Summing the branches incoherently, ℰ(ρ) = Σ Kₖ ρ Kₖ†, is exactly the mathematics of forgetting which outcome the environment recorded. The first figure draws it as a circuit. The completeness relation Σ Kₖ† Kₖ = I is just probability conservation: pₖ = Tr(Kₖ ρ Kₖ†) is the probability of the k-th branch, and requiring these to sum to one for every input ρ forces Σ Kₖ† Kₖ = I. Read the other way, {Kₖ† Kₖ} is a POVM — the channel is a generalized measurement whose result you discard, which is why one formula covers both noise and measurement.

Figure

Stinespring dilation as a circuit: the system ρ and a fresh environment enter one global unitary U, then the environment is traced out. Each outcome eₖ picks off a Kraus operator Kₖ = ⟨eₖ U e₀⟩, and forgetting the outcome gives ℰ(ρ) = Σ Kₖ ρ Kₖ† obeying Σ Kₖ† Kₖ = I

Rotating the environment basis before tracing re-mixes the operators as Kⱼ′ = Σₖ Wⱼₖ Kₖ without touching the channel, so the "list of Kraus operators" is a gauge-dependent name for a gauge-invariant object — the channel, equivalently its Choi matrix. Complete positivity, the demand that the channel behave when the system is half of an entangled pair, is manifest: (ℰ ⊗ id)(σ) = Σₖ (Kₖ ⊗ I) σ (Kₖ ⊗ I)† is a sum of terms A σ A†, positive on any positive σ. The second figure makes this concrete with the workhorse channel, amplitude damping — a qubit decaying from its excited state ( $T_1$ relaxation). For γ = 0.2, the Bloch vector of the plus state contracts and tilts toward the ground state.

Figure

Amplitude damping with γ = 0.2 acting on the plus state in the x–z plane of the Bloch sphere: population flows from excited to ground so the vector tilts up by γ = 0.2, and the coherence shrinks by √(1−γ) = 0.894, moving the Bloch vector from (1, 0, 0) to (0.894, 0, 0.2)

The core equations

The channel itself is the operator-sum. Everything else — probability conservation, complete positivity, the environment picture — is a corollary of this one shape. Read it as "sandwich the state between an operator and its adjoint, then sum".

\mathcal{E}(\rho)=\sum_{k} K_k\,\rho\,K_k^{\dagger} \qquad (1)

where: $\mathcal{E}$ — the quantum channel, a CPTP linear map on density operators; $\rho$ — input density operator, $\rho\succeq 0$ , $\mathrm{Tr}\,\rho=1$ ; $K_k$ — Kraus elements, bounded operators on the system Hilbert space; $k$ — index over the operators, $1\le k\le r_{\mathrm{Kraus}}$ . Each $K_k\rho K_k^\dagger$ is an unnormalized positive operator. Trace preservation — probabilities still add to one after the channel acts — is not an extra assumption but a single algebraic constraint, the completeness relation.

\sum_{k} K_k^{\dagger} K_k = I \qquad (2)

where: $I$ — identity on the system Hilbert space; $K_k^\dagger$ — Hermitian adjoint of the $k$ -th operator. Equivalently $\{K_k^\dagger K_k\}$ is a POVM; relaxing to $\sum_k K_k^\dagger K_k\preceq I$ gives a trace-non-increasing operation conditioned on an outcome. Every channel has a physical origin — unitary coupling to a pure-state environment, followed by ignoring it. This is the Stinespring dilation, the reason the operator-sum form describes how open-system dynamics actually arises.

\mathcal{E}(\rho)=\mathrm{Tr}_E\!\big[\,U\,(\rho\otimes|e_0\rangle\langle e_0|)\,U^{\dagger}\big] \qquad (3)

where: $U$ — unitary on the joint space $\mathcal{H}_S\otimes\mathcal{H}_E$ ; $|e_0\rangle$ — fixed pure initial environment state; $\mathrm{Tr}_E$ — partial trace over the environment, of dimension $\le r_{\mathrm{Kraus}}\le d^2$ . Expanding that partial trace in an orthonormal environment basis extracts the operators: the $k$ -th Kraus operator is the block of $U$ mapping input $|e_0\rangle$ to output slice $|e_k\rangle$ .

K_k=\langle e_k|\,U\,|e_0\rangle \qquad (4)

where: $\{|e_k\rangle\}$ — orthonormal basis of $\mathcal{H}_E$ ; the bra and ket act on the environment only, so $K_k$ is a system operator. Unitarity $U^\dagger U=I$ collapses to the completeness relation (2); a different basis choice is the unitary freedom below. The operators are never unique. Rotating the environment basis by a unitary $W$ re-mixes them without changing the channel, so a Kraus list is a gauge choice; compare gauge-invariant data, not raw operator lists.

K_j' = \sum_{k} W_{jk}\,K_k,\qquad W^\dagger W = I \qquad (5)

where: $W$ — isometry/unitary mixing matrix (pad the shorter set with zeros to make it square); $K_j'$ — a physically identical Kraus set. Two Kraus sets describe the same channel iff such a $W$ exists. Complete positivity — that the map stays physical even acting on half of an entangled pair — is automatic here, since $(\mathcal{E}\otimes\mathrm{id})(\sigma)=\sum_k (K_k\otimes I)\,\sigma\,(K_k\otimes I)^{\dagger}$ is a sum of terms $A\sigma A^\dagger\succeq0$ , hence positive on any positive $\sigma$ for a reference of any dimension. The invariant object behind the whole story is the Choi matrix: feed one half of an unnormalized maximally entangled state through the channel. It is positive exactly when the map is CP, and its rank is the minimal number of Kraus operators.

J(\mathcal{E})=(\mathcal{E}\otimes\mathrm{id})\big(|\Omega\rangle\langle\Omega|\big)=\sum_k |K_k\rangle\rangle\langle\langle K_k|,\qquad r_{\mathrm{Kraus}}=\mathrm{rank}\,J\le d^2 \qquad (6)

where: $|\Omega\rangle=\sum_i|i\rangle|i\rangle$ — unnormalized maximally entangled state; $J(\mathcal{E})$ — Choi operator, $J\succeq0\iff\mathcal{E}$ CP; $|K_k\rangle\rangle$ — column-stacked vectorization of $K_k$ ; the eigenvectors of $J$ , rescaled by $\sqrt{\lambda_k}$ and unvectorized, are the canonical minimal Kraus set (at most four for a qubit, nine for a qutrit).

Derivation AKraus theorem from the Stinespring dilation

The goal is a constructive, both-directions proof that a map admits an operator-sum decomposition exactly when it arises from a unitary on the system plus an environment traced out. We start from any operator set obeying completeness, build an isometry into a larger space, dilate it to a unitary, trace the environment out to recover the operator-sum form, then run the argument backwards to show every dilation returns a valid Kraus set. This is what makes the axiomatic, dilation, and operator-sum pictures literally the same theorem.

Step 1. Take any finite set $\{K_k\}$ with $\sum_k K_k^\dagger K_k=I$ and define $V|\psi\rangle:=\sum_k (K_k|\psi\rangle)\otimes|e_k\rangle$ into $\mathcal{H}_S\otimes\mathcal{H}_E$ with $\{|e_k\rangle\}$ orthonormal. Then $\langle V\phi|V\psi\rangle=\sum_{k}\langle\phi|K_k^\dagger K_k|\psi\rangle=\langle\phi|\psi\rangle$ , so $V^\dagger V=I$ : $V$ is an isometry. Why: completeness is exactly the statement that stacking the outputs into an enlarged space preserves inner products.

Step 2. Any isometry extends to a unitary $U$ with $V|\psi\rangle=U(|\psi\rangle\otimes|e_0\rangle)$ , provided $\dim\mathcal{H}_E\ge r_{\mathrm{Kraus}}$ : define $U$ on $\mathcal{H}_S\otimes|e_0\rangle$ by $V$ and complete it on the orthogonal complement by Gram–Schmidt. Why: an isometry is a unitary told what to do on only part of its domain; the free rest is the seed of Kraus non-uniqueness.

Step 3. Trace out the environment: $\mathrm{Tr}_E[V\rho V^\dagger]=\sum_{k,k'}K_k\rho K_{k'}^\dagger\,\langle e_{k'}|e_k\rangle=\sum_k K_k\rho K_k^\dagger$ , using $\langle e_{k'}|e_k\rangle=\delta_{kk'}$ . Since $V\rho V^\dagger=U(\rho\otimes|e_0\rangle\langle e_0|)U^\dagger$ , this is the Stinespring formula (3) reproducing the operator-sum form (1). Why: orthonormality of the environment basis kills every cross term, collapsing the double sum to the single sum — the content of "forgetting which branch occurred".

Step 4. Converse. Given any dilation, define $K_k:=\langle e_k|U|e_0\rangle$ ; unitarity gives $\sum_k K_k^\dagger K_k=\langle e_0|U^\dagger U|e_0\rangle=I$ , so completeness holds and the $K_k$ reproduce $\mathcal{E}$ . Hence "has a dilation" and "has a Kraus set" are equivalent. Why: every operator-sum decomposition is the shadow of some unitary on a bigger space — the physical origin is not an add-on. *Note: the isometry condition $V^\dagger V=\sum_k K_k^\dagger K_k=I$ holds explicitly for the amplitude-damping operators.*

Go deeper · Derivation B · 1 minComplete positivity forces the operator-sum structure, via Choi

Derivation BComplete positivity forces the operator-sum structure, via Choi

The goal is to prove the equivalence chain that pins down which linear maps are physical channels: $\mathcal{E}$ is completely positive if and only if its Choi operator is positive semidefinite if and only if it admits a Kraus decomposition. This settles why ordinary positivity — sending states to states — is not enough; the transpose passes that weaker test yet is unphysical. We show CP implies Choi positivity, Choi positivity yields Kraus operators by spectral decomposition, and the Kraus form implies CP, then give the transpose counterexample.

Step 1. CP implies Choi positivity. With $J(\mathcal{E})=(\mathcal{E}\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$ and $|\Omega\rangle\langle\Omega|\succeq0$ , complete positivity for a reference of dimension $d$ gives $J\succeq0$ immediately. Why: the maximally entangled input is the single hardest test — survive it and you survive every positive input, the content of the Choi–Jamiołkowski isomorphism.

Step 2. Choi positivity implies the Kraus form. A positive operator has a spectral decomposition $J=\sum_k\lambda_k|v_k\rangle\langle v_k|$ with $\lambda_k\ge0$ . Set $K_k=\sqrt{\lambda_k}\,\mathrm{unvec}(|v_k\rangle)$ ; the reconstruction identity $\mathcal{E}(\rho)=\mathrm{Tr}_2[J\,(I\otimes\rho^{T})]$ then expands into $\sum_k K_k\rho K_k^\dagger$ . Why: eigenvectors of $J$ are automatically orthogonal Kraus operators, $\mathrm{Tr}(K_j^\dagger K_k)=\lambda_k\delta_{jk}$ — the canonical minimal set, exactly $\mathrm{rank}\,J$ of them.

Step 3. The Kraus form implies CP. For any reference and any $\sigma\succeq0$ , $(\mathcal{E}\otimes\mathrm{id})(\sigma)=\sum_k(K_k\otimes I)\sigma(K_k\otimes I)^\dagger$ is a sum of positive operators, hence positive. This closes the loop CP $\Leftrightarrow J\succeq0\Leftrightarrow$ Kraus. Why: the operator-sum form turns complete positivity into a one-line consequence of "a sum of $A\sigma A^\dagger$ is positive".

Step 4. The sharp counterexample. The transpose $T:\rho\mapsto\rho^{T}$ is positive — it maps every density matrix to a density matrix. Yet $(T\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$ is the swap operator, whose eigenvalues are $\pm1$ ; the $-1$ makes $J(T)$ indefinite, so $T$ is not CP and has no Kraus form. Why: a machine transposing half of an entangled pair would create negative probabilities — nature forbids it, and the same negativity powers the PPT entanglement criterion. *Note: the transpose map's Choi operator has eigenvalue $-1$ (from the swap), confirming it is not CP; the amplitude-damping Choi spectrum was also checked numerically.*

Go deeper · Derivation C · 1 minDifferential limit from Kraus operators to the Lindblad generator

Derivation CDifferential limit from Kraus operators to the Lindblad generator

The goal is to show that a one-parameter family of channels over an infinitesimal time $\mathrm{d}t$ , expanded to first order, is exactly the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation — the continuous-time law of Markovian open-system dynamics, bridging the discrete Kraus picture and the differential equation an experimentalist fits to a decaying qubit. We posit a near-identity no-jump operator plus small jump operators, check completeness to first order, and read off the generator.

Step 1. Ansatz. Split into one near-identity term and several small ones: $K_0=I-\big(iH+\tfrac12\sum_j L_j^\dagger L_j\big)\mathrm{d}t$ and $K_j=\sqrt{\mathrm{d}t}\,L_j$ , with $H$ Hermitian and $L_j$ the jump operators. The $\sqrt{\mathrm{d}t}$ scaling is forced: a jump contributes at first order only if its amplitude is order $\sqrt{\mathrm{d}t}$ . Why: this split is the quantum-trajectory unravelling — $K_0$ is the smooth evolution between random jumps $K_j$ .

Step 2. Completeness to first order. Compute $\sum_k K_k^\dagger K_k=K_0^\dagger K_0+\sum_j K_j^\dagger K_j=\big(I-\sum_j L_j^\dagger L_j\,\mathrm{d}t\big)+\sum_j L_j^\dagger L_j\,\mathrm{d}t+O(\mathrm{d}t^2)=I$ . The anticommutator piece of $K_0^\dagger K_0$ exactly cancels the jump contribution. Why: this cancellation is the whole reason the $\tfrac12\{L_j^\dagger L_j,\cdot\}$ "damping" term must appear in $K_0$ — trace preservation fixes it uniquely.

Step 3. Difference quotient. Assemble $\mathcal{E}_{\mathrm{d}t}(\rho)=K_0\rho K_0^\dagger+\sum_j K_j\rho K_j^\dagger$ , keep terms linear in $\mathrm{d}t$ , and form $\dfrac{\mathcal{E}_{\mathrm{d}t}(\rho)-\rho}{\mathrm{d}t}=-i[H,\rho]+\sum_j\big(L_j\rho L_j^\dagger-\tfrac12\{L_j^\dagger L_j,\rho\}\big)+O(\mathrm{d}t)$ . Why: the $K_0\rho K_0^\dagger$ cross terms give the Hamiltonian commutator plus half the anticommutator, and the jumps supply $L_j\rho L_j^\dagger$ — together the Lindblad dissipator.

Step 4. Limit. Letting $\mathrm{d}t\to0$ yields the GKSL master equation $\dot\rho=-i[H,\rho]+\sum_j\big(L_j\rho L_j^\dagger-\tfrac12\{L_j^\dagger L_j,\rho\}\big)$ , the most general generator of a continuous, Markovian, completely positive semigroup. Why: the Kraus rank becomes the number of jump operators, and the Kraus unitary freedom becomes the freedom to redefine the $L_j$ — the same gauge, now continuous. *Note: the first-order cancellation in Step 2 holds term-by-term, confirming that trace preservation fixes the anticommutator coefficient uniquely.*

Limiting cases

The representation degenerates gracefully: fixing the Kraus rank or the operator structure recovers every familiar dynamics, from reversible gates to full decoherence.

Case	Result	Meaning
Unitary (closed) evolution	$\mathcal{E}(\rho)=U\rho U^\dagger$	Kraus rank 1, single operator $K_1=U$ with $U^\dagger U=I$ ; no environment entanglement, fully reversible.
Projective measurement, outcome discarded	$\mathcal{E}(\rho)=\sum_k P_k\rho P_k$	Operators are orthogonal projectors, $\sum_k P_k=I$ ; this is pure dephasing / decoherence in the measured basis.
Maximal Kraus rank	$r_{\mathrm{Kraus}}=d^2$	The completely depolarizing channel $\mathcal{E}(\rho)=I/d$ saturates the bound; its Choi matrix is full rank.
Short-time / differential limit	$K_0\approx I-\big(iH+\tfrac12\sum_j L_j^\dagger L_j\big)\mathrm{d}t,\ K_j\approx\sqrt{\mathrm{d}t}\,L_j$	First order in $\mathrm{d}t$ reproduces the Lindblad generator with jump operators $L_j$ .

Common trapConfusing positivity with complete positivity, and treating the Kraus list as unique

Two traps sink beginners. First, complete positivity is strictly stronger than positivity: the transpose sends every density matrix to a density matrix, yet it is not a channel, because $T\otimes\mathrm{id}$ turns the maximally entangled state into an operator with a negative eigenvalue. Certify a channel by exhibiting a Kraus set or a positive Choi matrix — never by checking only that single-system states stay positive. Second, the Kraus operators are not unique: any two minimal sets are related by a unitary mixing $K_j'=\sum_k W_{jk}K_k$ . So it is meaningless to compare channels operator-by-operator, or to read physical significance into a single $K_k$ in isolation. Compare Choi matrices or diamond-norm distances — gauge-invariant data — instead.

Worked examples

1 · Amplitude damping acting on a superposition (fully worked)

A qubit undergoes amplitude damping with $\gamma=0.2$ (probability of $|1\rangle\to|0\rangle$ decay). Take the input $\rho=|+\rangle\langle+|=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$ . Find the output, verify trace preservation, and describe the Bloch-vector motion.

Step 1. Write the operators. $K_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix}$ , $K_1=\begin{pmatrix}0&\sqrt{\gamma}\\0&0\end{pmatrix}$ . With $\gamma=0.2$ , $\sqrt{1-\gamma}=\mathbf{0.894}$ and $\sqrt{\gamma}=0.447$ .

Step 2. The "no-decay" branch $K_0\rho K_0^\dagger=\begin{pmatrix}0.5&0.447\\0.447&0.4\end{pmatrix}$ scales the coherence by $\sqrt{1-\gamma}$ and the excited population by $1-\gamma$ .

Step 3. The "decay" branch $K_1\rho K_1^\dagger=\begin{pmatrix}\gamma\rho_{11}&0\\0&0\end{pmatrix}=\begin{pmatrix}0.1&0\\0&0\end{pmatrix}$ — decayed population lands in the ground state.

Step 4. Sum and check. $\mathcal{E}(\rho)=\begin{pmatrix}0.6&0.447\\0.447&0.4\end{pmatrix}$ , with $\mathrm{Tr}=\mathbf{1.0}$ and $\det=0.04\ge0$ , a valid state. The Bloch vector moves from $(1,0,0)$ to $\big(\sqrt{1-\gamma},0,\gamma\big)=\mathbf{(0.894,0,0.2)}$ : coherence shrank by $\sqrt{1-\gamma}$ , and the vector tilted toward the ground pole by $\gamma$ . Output $\begin{pmatrix}0.6&0.447\\0.447&0.4\end{pmatrix}$ , Bloch $(0.894,0,0.2)$ , trace 1.

2 · Choi matrix and Kraus rank of amplitude damping (numbers)

Build the Choi matrix of the $\gamma=0.2$ amplitude-damping channel and read off its Kraus rank from the eigenvalues.

Step 1. Compute the channel on the basis dyads. With $s=\sqrt{1-\gamma}=0.894$ : $\mathcal{E}(|0\rangle\langle0|)=|0\rangle\langle0|$ , $\mathcal{E}(|1\rangle\langle1|)=\gamma|0\rangle\langle0|+(1-\gamma)|1\rangle\langle1|$ , and $\mathcal{E}(|0\rangle\langle1|)=s\,|0\rangle\langle1|$ (with the conjugate for $|1\rangle\langle0|$ ).

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

J=\begin{pmatrix}1&0&0&s\\0&0&0&0\\0&0&\gamma&0\\s&0&0&1-\gamma\end{pmatrix}=\begin{pmatrix}1&0&0&0.894\\0&0&0&0\\0&0&0.2&0\\0.894&0&0&0.8\end{pmatrix}.

Step 3. Diagonalize. The nonzero action lives in the $\{|00\rangle,|11\rangle\}$ block $\begin{pmatrix}1&s\\s&1-\gamma\end{pmatrix}$ , with trace $2-\gamma$ and determinant $(1-\gamma)-s^2=0$ , so its eigenvalues are $0$ and $2-\gamma=1.8$ . With the isolated $\gamma=0.2$ on the $|10\rangle$ diagonal, the full spectrum is $\{1.8,0.2,0,0\}$ .

Step 4. Read the rank. Two eigenvalues are nonzero, so $\mathrm{rank}\,J=2$ : the Kraus rank is 2, matching $K_0,K_1$ . Their eigenvectors, rescaled by $\sqrt{1.8}$ and $\sqrt{0.2}$ and unvectorized, are the canonical Kraus set — a unitary mixing of the textbook pair. Choi spectrum $\{1.8,0.2,0,0\}$ , Kraus rank 2, well below the bound $d^2=4$ .

3 · The Kraus set is a gauge choice (non-uniqueness)

The bit-flip channel with $p=0.3$ has the natural Kraus set $K_0=\sqrt{1-p}\,I$ , $K_1=\sqrt{p}\,X$ . Mix them by a Hadamard, $K_0'=\tfrac1{\sqrt2}(K_0+K_1)$ , $K_1'=\tfrac1{\sqrt2}(K_0-K_1)$ .

Step 1. Expanding $K_0'\rho K_0'^\dagger+K_1'\rho K_1'^\dagger=\tfrac12\big[(K_0+K_1)\rho(K_0+K_1)^\dagger+(K_0-K_1)\rho(K_0-K_1)^\dagger\big]$ , the cross terms $K_0\rho K_1^\dagger$ and $K_1\rho K_0^\dagger$ carry opposite signs and cancel, leaving $K_0\rho K_0^\dagger+K_1\rho K_1^\dagger$ — the original channel exactly (confirmed numerically). The full-matrix primed set and the sparse original describe one channel; only the Choi matrix is unique.

Physics of the portrait

The operator-sum representation is where the algebra of completely positive maps became the everyday toolkit of quantum information — and it carries the name of the physicist who assembled its language.

Origins · 1 minKarl Kraus and how this was discovered

Karl Kraus

1938–1988 · the axiomatics of quantum operations and effects

Karl Kraus was a German theoretical physicist who worked on the mathematical foundations of quantum theory in the tradition of Günther Ludwig's operational approach at Marburg. Where earlier authors treated measurement and open-system evolution case by case, Kraus asked what the most general admissible state change could be, subject only to probabilities staying non-negative and normalized even when the system is entangled with a reference. His 1971 papers with Klaus-Ernst Hellwig formalized "operations" and "effects", and his 1983 volume States, Effects, and Operations laid out the operator-sum decomposition so cleanly that the operators $K_k$ have carried his name ever since. The mathematics underneath was older — Stinespring's 1955 dilation theorem and Choi's 1975 theorem — but it was Kraus who translated that functional analysis into the language physicists use for noise, measurement, and decoherence.

History

1955

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

1961

— Sudarshan, Mathews and Rau give the dynamical-map description of open-system evolution

1971

— Hellwig and Kraus formalize operations and effects operationally

1975

— Choi shows every completely positive map on matrices has an operator-sum form

1976

— Gorini, Kossakowski, Sudarshan and Lindblad derive the GKSL master equation

1983

— Kraus publishes States, Effects, and Operations, fixing the modern operator-sum language

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, Ch. 8 (Quantum Noise and Quantum Operations) · Preskill, Lecture Notes for Physics 219, Ch. 3 (Foundations II: Measurement and Evolution) · Watrous, The Theory of Quantum Information, Ch. 2 and Ch. 4 (Quantum Channels) · Wilde, Quantum Information Theory, Ch. 4 (The Noisy Quantum Theory) · Kitaev, Shen and Vyalyi, Classical and Quantum Computation, §11 (Quantum operations) · Mermin, Quantum Computer Science, Ch. 6 · Choi, "Completely positive linear maps on complex matrices", Lin. Alg. Appl. 10 (1975) 285 · Kraus, States, Effects, and Operations, Lecture Notes in Physics 190 (1983) · Stinespring, "Positive functions on C*-algebras", Proc. AMS 6 (1955) 211. The key numerical checks are: $\sum_k K_k^\dagger K_k=\mathrm{diag}(1,1)$ for amplitude damping, the Choi spectrum $\{1.8,0.2,0,0\}$ at $\gamma=0.2$ , and the bit-flip two-Kraus-set equality. No numerical constants beyond the explicit $2\times2$ matrices and the $\gamma=0.2$ values are asserted.

Prerequisites. Quantum Channels and Completely Positive Trace-Preserving Maps · Open Quantum Systems and Reduced Dynamics · Decoherence. Leads to. Lindblad / GKSL Master Equation · Process Tomography and the Choi–Jamiołkowski Isomorphism · Depolarizing Channel · Amplitude Damping Channel.

Ask the tutor: Ask me to build the Kraus operators for any noise model you name — depolarizing, phase damping, a custom Stinespring circuit — and I will check completeness, compute its Choi rank, and show you the unitary freedom that hides other equivalent operator sets.

Why isn't ordinary positivity enough — why demand complete positivity?

Because a channel rarely acts on an isolated system. If your qubit is entangled with an untouched reference, the physical evolution is $\mathcal{E}\otimes\mathrm{id}$ , and that must send the joint state to a valid density matrix. Some maps preserve positivity on their own system but fail this larger test — the transpose maps every single-qubit state to a state, yet on half of a Bell pair it produces a negative eigenvalue, i.e. negative probabilities. Complete positivity is exactly the requirement that $\mathcal{E}\otimes\mathrm{id}$ stay positive for a reference of any dimension, which the Kraus form guarantees since $\sum_k(K_k\otimes I)\sigma(K_k\otimes I)^\dagger$ is a sum of positive terms.

Are the Kraus operators unique, and if not, does the ambiguity matter?

They are not unique. Any two minimal Kraus sets for the same channel are related by a unitary (or isometric) mixing $K_j'=\sum_k W_{jk}K_k$ , after padding the shorter set with zeros. The ambiguity reflects a real physical freedom — which basis you would measure the environment in before discarding it — so it is not a defect but information about unrealized measurements. Practically, never compare two channels operator-by-operator, and never attach physical meaning to an individual $K_k$ in isolation. The gauge-invariant summary is the Choi matrix; equal Choi matrices mean equal channels, full stop.

Is the Kraus picture a kind of measurement?

Yes, with a twist. The completeness relation makes $\{K_k^\dagger K_k\}$ a POVM, so applying the channel is formally a generalized measurement with the outcome thrown away. If you kept outcome $k$ , the state would collapse to $K_k\rho K_k^\dagger/p_k$ with probability $p_k=\mathrm{Tr}(K_k\rho K_k^\dagger)$ — a selective operation. Summing over $k$ without recording it is the deterministic channel. This is why the same mathematics describes both noise (outcome unknown) and measurement (outcome kept): the difference is entirely whether the index $k$ is retained.

What is the difference between the Kraus form and the Lindblad master equation?

The Kraus representation is a snapshot: the discrete map $\rho\mapsto\mathcal{E}(\rho)$ over a finite interval, with no assumption of continuous flow. The Lindblad (GKSL) master equation is the generator of a continuous, Markovian family of such maps — the differential version. They connect in the short-time limit: expand a Kraus family to first order, with $K_0\approx I-(iH+\tfrac12\sum_j L_j^\dagger L_j)\mathrm{d}t$ and $K_j\approx\sqrt{\mathrm{d}t}\,L_j$ , and the difference quotient becomes the Lindblad dissipator. Use Kraus operators for a discrete gate or single noisy step; use Lindblad for continuous, memoryless coupling you want to integrate.

Is the environment in the Stinespring dilation physically real?

It need not be — that is the elegance of the theorem. The dilation guarantees that some environment and some global unitary reproduce any channel, but does not identify them with the actual bath in your lab. The minimal dilation uses an environment of dimension equal to the Kraus rank ( $\le d^2$ ), usually far smaller than the astronomically large real one. So the dilation certifies that the channel is physical — unitary evolution on a larger space — and is a computational convenience, not a claim about which degrees of freedom the system truly couples to; different environments can yield the same channel, mirroring the Kraus non-uniqueness.

How do I recover Kraus operators from experimental data?

Through process tomography and the Choi–Jamiołkowski isomorphism. You prepare input states, apply the unknown channel, measure the outputs, and reconstruct the Choi matrix $J(\mathcal{E})$ . Because noise can make the raw estimate slightly non-positive, you project it onto the nearest matrix satisfying $J\succeq0$ and trace preservation — a convex optimization. Diagonalizing the physical $J=\sum_k\lambda_k|v_k\rangle\langle v_k|$ and setting $K_k=\sqrt{\lambda_k}\,\mathrm{unvec}(|v_k\rangle)$ gives a canonical minimal Kraus set; the count of nonzero eigenvalues is the effective Kraus rank of your device's noise.