Kraus Operator Sum Representation

• Kraus operator-sum representation is a canonical framework for completely positive, trace-preserving maps derived from unitary system-environment interactions.
• It underpins practical methodologies like quantum process tomography and noise modeling by decomposing quantum channels into minimal and structured operator sets.
• The formalism bridges operator theory with applications in master equations and non-Markovian dynamics, offering actionable insights for quantum measurement and experimental reconstructions.

The Kraus operator-sum representation provides a canonical, operator-theoretic form for the most general quantum evolution—completely positive, trace-preserving maps—on a finite or infinite-dimensional quantum system. Originating from the physical scenario of a system interacting unitarily with an environment and then discarding the environment, the Kraus representation undergirds the mathematical formalism of open-system quantum dynamics, noise models in quantum information, process tomography, and the structural analysis of quantum channels.

1. Mathematical Formulation and General Properties

A linear map $\Phi$ on the operators $\mathcal{B}(H)$ of a finite-dimensional Hilbert space $H$ is completely positive and trace-preserving (CPTP) if and only if it admits a Kraus decomposition

$$\Phi(\rho) = \sum_{k=1}^N K_k \rho K_k^\dagger, \qquad \sum_{k=1}^N K_k^\dagger K_k = I,$$

with $\{K_k\}$ a finite set of Kraus operators acting on $H$, and $N \leq d^2$, where $d = \dim H$ (Kuramochi, 6 Jun 2024, Maziero, 2015, Ende, 2023). This representation encodes both complete positivity (CP) and trace preservation (TP): CP is manifest termwise, as each $K_k\,\cdot\,K_k^\dagger$ is CP; TP is guaranteed by the completeness relation.

The generalization to infinite-dimensional spaces involves convergence in the trace-norm topology and may require a countably infinite family $\{K_j\}$ satisfying $\sum_j K_j^\dagger K_j = I$ in the strong-operator sense (Ende, 2023).

2. Physical Origin and Derivation

The Kraus form originates from the unitary evolution of a system-plus-environment composite, followed by a partial trace over the environment: $$\rho_S(t) = \operatorname{Tr}_E\left[ U_{SE}(t) (\rho_S(0) \otimes |\psi_E\rangle\langle\psi_E|) U_{SE}^\dagger(t) \right],$$ where $|\psi_E\rangle$ is the initial environment state. By choosing an orthonormal basis $\{|e_l\rangle\}$ for the environment $E$, the reduced state admits the sum

$$\rho_S(t) = \sum_l M_l \rho_S(0) M_l^\dagger, \qquad M_l = \langle e_l| U_{SE}(t) |\psi_E\rangle$$

(Maziero, 2015, Ivan et al., 2010). This construction shows that all CPTP maps can emerge from system-environment unitary dynamics.

The Stinespring dilation theorem formalizes this result for arbitrary (even infinite-dimensional) spaces, asserting that every CP map can be realized as a unitary on a larger Hilbert space followed by a partial trace (Ende, 2023).

3. Structure, Freedom, and Minimality

The set of Kraus operators is not unique. Any alternative set $\{\tilde K_m\}$ defined via a unitary or isometric mixing on the index set—a result of the freedom in environmental basis choice—will yield the same CP map as long as

$\tilde K_m = \sum_k U_{mk} K_k$

with $U$ unitary (Arsenijevic et al., 2015, Maziero, 2015). The minimal number of Kraus operators required, called the Choi rank, equals the rank of the associated Choi matrix $C_\Phi$ (Arsenijevic et al., 2015).

Table: Minimal Kraus number for common channels

Channel type	Minimal Kraus number
$d$-dim unitary	1
depolarizing/Pauli (qubit)	4
amplitude damping (qubit)	2
general $d$-dim channel	$\leq d^2$

A diagonalization of the Choi matrix determines a canonical minimal set, but in practice, analytic diagonalization is only possible for low-dimensional systems due to the Abel-Galois theorem (Omkar et al., 2012).

4. Connection to Quantum Master Equations and Lindblad Form

For continuous-time quantum dynamical semigroups, the infinitesimal-Kraus representation underpins the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) structure of quantum Markovian generators. By evaluating the Kraus operators for the channel $\exp(\delta t L)$ at an infinitesimal time step and decomposing their identity and traceless parts,

$$L_j(\delta t) = x_j I + M_j,\qquad \operatorname{tr} M_j = 0,$$

one obtains order estimates

$M_j = \mathcal{O}(\sqrt{\delta t}),$

leading in the limit $\delta t \to 0$ to the canonical Lindblad (GKSL) generator: $$L(\rho) = -i[H,\rho] + \sum_j \left( W_j \rho W_j^\dagger - \tfrac{1}{2} \{ W_j^\dagger W_j, \rho \} \right)$$ where $W_j = \mathrm{st}(M_j/\sqrt{\delta t})$ and $H$ emerges from the diagonal part (Kuramochi, 6 Jun 2024). This construction, using nonstandard analysis, provides a direct route from Kraus sums to the GKSL master equation.

5. Applications: Noise, Quantum Information, and Process Tomography

Kraus representations classify all physical error models in quantum information, including amplitude damping, phase damping, depolarizing, and generalized Pauli channels (Maziero, 2015, Arsenijevic et al., 2015, Mahdian et al., 2013). For example, the amplitude-damping (qubit) channel has

$$K_0 = \begin{pmatrix} 1 & 0 \ 0 & \sqrt{1-p} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{p} \ 0 & 0 \end{pmatrix},$$

with $0 \leq p \leq 1$ (Maziero, 2015). In multi-qubit noise or entanglement-breaking channels, the number and structure of Kraus operators reflect physical constraints such as locality, symmetry, or environment-induced decoherence (Ivan et al., 2010, Arsenijevic et al., 2015, Omkar et al., 2012).

Quantum process tomography protocols recover the set of Kraus operators of an unknown channel either by reconstructing the full Choi matrix (standard process tomography) or by compressed-sensing and gradient-descent approaches exploiting low Choi rank (Ahmed et al., 2022). Direct experimental characterization of individual Kraus operator matrix elements is possible in scenarios where full process tomography is intractable; such protocols operate via adaptive measurement settings on extended probe-system-environment setups (Sahil et al., 8 Oct 2025).

6. Non-Markovian Dynamics and Extensions

In non-Markovian open-system dynamics, the Kraus sum extends to continuous or infinite sums, and the associated hierarchy of Kraus matrices can be constructed via recurrence relations or continued-fraction expansions depending on the reservoir structure (Wonderen et al., 2018, Reimer et al., 2018). Truncations of the infinite Kraus hierarchy produce completely positive, trace non-increasing maps, with trace preservation restored only upon completing the sum (Reimer et al., 2018).

The Keldysh real-time diagrammatics reveal a physical interpretation: each Kraus operator corresponds to a set of interaction histories (involving jumps or quantum trajectories), and the full reduced dynamics are synthesized by tracing over all such histories. This approach ensures explicit preservation of complete positivity at every stage of approximation.

7. Kraus Representation, Entanglement, and Measurement

The operator sum form precisely delineates which channels are entanglement breaking—those channels that admit a decomposition into rank-one Kraus operators (Ivan et al., 2010). Connections to generalized measurements (POVMs) are explicit: for a quantum measurement with outcomes $j$, the POVM element $E_j = \sum_i K_{j,i}^\dagger K_{j,i}$, but the post-measurement dynamics depend on the specific choice of $\{K_{j,i}\}$, not just $E_j$ (Sahil et al., 8 Oct 2025). Multiple inequivalent sets of Kraus operators may correspond to the same observable statistics, but only the full set encodes the dynamical map’s action.

Experimental strategies for directly reconstructing Kraus operators, without full state or process tomography, have been developed using interferometric or optical schemes, revealing the feasibility of direct access to channel structure at the level of operator matrix elements (Sahil et al., 8 Oct 2025, Morazotti et al., 2021).

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In summary, the Kraus operator-sum representation is the definitive structural characterization of CPTP maps and thus of physically admissible quantum evolutions under open-system conditions. It provides conceptual, analytical, and computational tools for modeling noise, analyzing master equations, designing quantum protocols, and connecting the formalism of reduced dynamics to fundamental theorems in operator theory and quantum measurement (Kuramochi, 6 Jun 2024, Maziero, 2015, Ende, 2023, Reimer et al., 2018).