Kraus Rank in Quantum Channels

• Kraus rank is defined as the minimum number of operators required to represent a quantum channel via its completely positive trace-preserving map, equivalent to the Choi rank in finite dimensions.
• It underpins key quantum information techniques, including operator-sum representations, dilation theory, and practical implementations like process tomography and tomography ansatz size scaling.
• Kraus rank serves as a structural complexity parameter, influencing methods in channel compression, quantum learning, and determining the resource requirements in quantum process engineering.

Kraus rank is the minimum number of Kraus operators needed to represent a completely positive trace-preserving map in operator-sum form,

$$\mathcal N(X)=\sum_{i=1}^s K_i X K_i^\dagger,\qquad \sum_{i=1}^s K_i^\dagger K_i=\mathbb 1_A,$$

with the minimum taken over all such decompositions (Lancien et al., 2017). In finite dimensions, this coincides with the rank of the Choi matrix $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$ (Heightman et al., 2024). In the infinite-dimensional setting treated constructively in (Ende, 2023), the Kraus family is indexed by a general set $J$, which suggests viewing Kraus rank as the minimal cardinality of such an index set. Across quantum information, open-system dynamics, tomography, approximation, and learning, Kraus rank serves as a structural complexity parameter rather than merely a bookkeeping device (Ahmed et al., 2022, Kadri et al., 2020).

1. Definition and basic characterizations

For a quantum channel $\mathcal E:B(\mathcal H)\to B(\mathcal H)$, the action on a density matrix is written as

$$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$

In the standard finite-dimensional usage, the Kraus rank is “the minimum number of Kraus operators needed to represent the channel action” (Heightman et al., 2024). The same quantity is called the Choi rank in process-tomography language: “The Choi rank $r$ of a process is given by the minimum number of Kraus operators necessary to represent the process” (Ahmed et al., 2022).

The Choi formulation gives an equivalent characterization. With

$$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$

one has

$$\operatorname{rank}(J_{\mathcal E})=\text{Kraus rank of }\mathcal E$$

(Heightman et al., 2024). For a valid channel, $J_{\mathcal E}$ is Hermitian, positive semidefinite, and satisfies $\mathrm{Tr}_B(J_{\mathcal E})=\mathbb 1$ (Heightman et al., 2024). This makes Kraus rank simultaneously a statement about operator-sum representations and a matrix-rank invariant of the Choi state.

A concrete open-system example appears in the amplitude-damping derivation for a two-level atom interacting with the vacuum. The reduced dynamics is expressed by

$$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$0

with

$$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$1

(Maziero, 2015). In that representation the exhibited nonzero Kraus number is $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$2, illustrating the standard channel-theoretic notion in a microscopic derivation.

2. Nonuniqueness, minimality, and upper bounds

Kraus representations are not unique. If $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$3 and $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$4 are two $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$5-term Kraus decompositions of the same channel, then there exists a unitary $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$6 such that

$$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$7

(Lancien et al., 2017). In the system–environment picture this same freedom can be written as

$$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$8

(Maziero, 2015). The operator count in a displayed decomposition therefore need not equal the Kraus rank.

This distinction matters operationally. The paper on generalized Kraus operators for a one-qubit depolarizing channel explicitly presents four Kraus operators, but also notes that “four Kraus operators” does not automatically prove Kraus rank $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega}$$9, because Kraus decompositions need not be minimal (Arsenijevic et al., 2015). The same distinction underlies low-rank tomography, where the true rank $J$0 of the target process and the ansatz size $J$1 used in reconstruction are treated separately (Ahmed et al., 2022).

Finite-dimensional channels satisfy universal cardinality bounds. For a system of Hilbert-space dimension $J$2, one can always find a Kraus representation with at most $J$3 operators (Maziero, 2015). In the nonstandard GKSL derivation, any CP map on a $J$4-dimensional Hilbert space is recalled to admit a Kraus representation with at most $J$5 Kraus operators, and the infinitesimal channel $J$6 is taken with $J$7 Kraus operators before redundancy is removed at the generator level (Kuramochi, 2024). These are existence bounds, not generic minimality statements.

3. Relations to Choi theory, dilations, and environment size

Kraus rank is closely tied to dilation theory. Starting from a Kraus family $J$8, the constructive Stinespring result of (Ende, 2023) realizes the channel by choosing

$J$9

so that

$\mathcal E:B(\mathcal H)\to B(\mathcal H)$0

The same work states that “the smallest auxiliary system one can get this way has dimension ‘Kraus rank’ times two,” while the original Hellwig–Kraus construction uses an auxiliary system of dimension “Kraus rank plus one” (Ende, 2023). The extra factor $\mathcal E:B(\mathcal H)\to B(\mathcal H)$1 is catalytic in the construction: the qubit returns unchanged.

The operator-sum picture also arises directly from unitary system–environment dynamics. In the pedagogical derivation of (Maziero, 2015), with initial product state $\mathcal E:B(\mathcal H)\to B(\mathcal H)$2, Kraus operators are defined by matrix elements

$\mathcal E:B(\mathcal H)\to B(\mathcal H)$3

This identifies each Kraus operator with an environment branch relative to the chosen basis.

Beyond finite-dimensional channels with finite Kraus rank, the exact non-Markovian reservoir-damping construction of (Wonderen et al., 2018) gives a “Kraus map consisting of an infinite number of matrices.” The exact reduced dynamics is written as a sum over sectors $\mathcal E:B(\mathcal H)\to B(\mathcal H)$4, with continuously time-indexed multi-index Kraus matrices $\mathcal E:B(\mathcal H)\to B(\mathcal H)$5 (Wonderen et al., 2018). This shows that operator-sum descriptions and finite Kraus rank are not synonymous.

4. Kraus rank as a structural complexity parameter

Several modern works use Kraus rank as an explicit complexity parameter. In channel compression, any quantum channel mapping states on some input Hilbert space $\mathcal E:B(\mathcal H)\to B(\mathcal H)$6 to states on some output Hilbert space $\mathcal E:B(\mathcal H)\to B(\mathcal H)$7 can be approximated by one with order $\mathcal E:B(\mathcal H)\to B(\mathcal H)$8 Kraus operators, where $\mathcal E:B(\mathcal H)\to B(\mathcal H)$9; if the outputs are all very mixed, this improves to order $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$0 (Lancien et al., 2017). Here the exact Kraus rank of the original channel can be as large as $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$1, but approximation permits substantial reduction.

In quantum process tomography, the same quantity is a model-order hyperparameter. The gradient-descent QPT framework writes

$$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$2

treats the true minimal rank as $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$3, and uses $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$4 as the chosen ansatz size (Ahmed et al., 2022). The paper is explicit that $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$5 is a property of the target process, whereas $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$6 is a tunable rank parameter. Complete positivity is enforced by Kraus form, and trace preservation is enforced exactly through the stacked constraint $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$7 on a Stiefel manifold (Ahmed et al., 2022).

In matrix-valued regression, low Kraus rank plays the same role for completely positive maps $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$8. The hypothesis class with fixed Kraus rank $$\mathcal E(\rho)=\sum_{r=0}^{R} K_r \rho K_r^\dagger.$$9 satisfies

$r$0

and the paper interprets $r$1 as the regularization parameter controlling expressivity and statistical complexity (Kadri et al., 2020). The same source emphasizes that “low-rank” here means few Kraus operators, not low rank of the individual matrices $r$2, and not reduced-rank multivariate regression (Kadri et al., 2020).

A more recent generalization appears in channel-based knowledge-graph embedding. There, each relation channel has Kraus rank $r$3, and any exact realization of a relation matrix $r$4 must satisfy

$r$5

(Chaki, 11 May 2026). This suggests a broader interpretation: Kraus rank measures the number of concurrent transformation pathways required by the task.

5. Identifiability, observability, and effective operator counts

Kraus rank is not always directly observable from restricted data. Under one fixed projective measurement basis, (Heightman et al., 2024) shows that many different channels can yield identical output marginals for all inputs while having different Choi and Kraus ranks. Within such Type-II spoofing equivalence classes, a generic $r$6-dimensional channel can be lowered “from $r$7 to the theoretical minimum of $r$8” while preserving exactly the compatible projective marginals (Heightman et al., 2024). The same work formulates a Sinkhorn-like algorithm that searches the compatible Choi matrices for minimum admissible Kraus rank (Heightman et al., 2024).

This measurement-relativity contrasts with exact dynamical constructions. In (Wonderen et al., 2018), the exact non-Markovian reduced dynamics requires an infinite Kraus family, while perturbative factorization produces a finite hierarchy of perturbative Kraus matrices $r$9 that still preserves positivity and probability order by order (Wonderen et al., 2018). A plausible implication is that “effective Kraus rank” in approximations may reflect truncation strategy rather than a channel invariant.

A closely related caveat appears in the closed-form Kraus-map solution for GKSL dynamics under arbitrarily strong driving but linear order in the dissipator. That work derives one dominant operator $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$0 together with correction Kraus operators indexed by Riemann quadratures, but explicitly does not compute minimal Kraus rank, and the resulting linearized map is not exactly trace preserving (Chishti et al., 11 Mar 2026). The operator count in such constructions is therefore representation size, not necessarily Kraus rank in the strict minimal sense.

6. Constrained variants and common confusions

The standard channel-theoretic notion must be distinguished from constrained variants. In the resource theory of coherence, the relevant quantity is the smallest number of Kraus operators in a decomposition whose individual terms satisfy IO or SIO structural constraints. For qubit IO, every channel can be decomposed into four incoherent Kraus operators, improving the earlier bound of five; for qutrit IO the paper proves an upper bound of $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$1, improving $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$2; and for qutrit SIO it proves an upper bound of $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$3, improving $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$4 (Qiao et al., 2020). These are constrained Kraus-count questions, not ordinary unconstrained Kraus rank.

Several nearby notions are unrelated despite similar terminology. The “Kraus-Cirac number” of a two-qubit unitary counts the nonzero coefficients $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$5 in the decomposition

$$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$6

and is explicitly “not about the standard Kraus rank of a quantum channel” (Soeda et al., 2014). Likewise, “Kruskal rank” concerns linear independence of subsets of rows or columns of a matrix and arises in sparse linear regression, tensor decomposition, and latent variable models; it is unrelated to channel Kraus operators (Zhou, 6 Mar 2025).

A further source of confusion is low-rank state simulation. The Lindblad solver of (Appelo et al., 2024) is built from Kraus-form timestep maps, but its “low-rank” method refers to low rank of the density matrix $$J_{\mathcal E}=(\mathbb 1_d\otimes \mathcal E)\ket{\Omega}\bra{\Omega},\qquad \ket{\Omega}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{i\,i},$$7, not low Kraus rank of the channel. The paper explicitly does not define the term “Kraus rank” (Appelo et al., 2024).

Kraus rank is therefore best understood as a minimal operator-sum cardinality for completely positive maps, equal in finite dimensions to Choi rank, but frequently reinterpreted as a structural complexity parameter whose operational meaning depends on the surrounding representation, constraints, and measurement model (Heightman et al., 2024, Ahmed et al., 2022).