Choi Rank in Quantum Channels

• Choi rank is defined as the rank of the Choi matrix or equivalently the minimal number of nonzero Kraus operators needed for a quantum channel representation.
• It serves as a universal invariant, linking channel-state duality, minimal environment dimensions in Stinespring dilations, and bounds in mixed-unitary and entanglement-breaking decompositions.
• Choi rank has an operational impact in entanglement-assisted tasks like k-state exclusion, where it sets concrete limits on deterministic decoding performance.

Choi rank is the rank of the Choi matrix of a quantum channel and, equivalently, the minimum number of nonzero Kraus operators required in an operator-sum representation. For a channel $\Phi:M_m\to M_n$, this invariant is also the rank of the associated Choi state/operator and, in the operational formulation used for channel-state duality, it equals the minimal environment dimension in a Stinespring dilation. Recent work uses Choi rank as a structural invariant for channel decompositions, a parameter in constrained channel classification, and an operational limit in entanglement-assisted communication tasks (Girard et al., 2020, Li et al., 2017, Stratton et al., 2024).

1. Definition and basic equivalences

For a quantum channel $\Phi:M_n\to M_n$ with Kraus representation

$\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$

the Choi rank is the minimum possible number of nonzero Kraus operators in such a representation. In the standard matrix formulation, the Choi matrix may be written as

$$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$

or equivalently, with the tensor factors ordered oppositely,

$C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$

The Choi rank is then

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$

and this equals the minimum number of nonzero terms in any Kraus form (Girard et al., 2020, Li et al., 2017).

The same invariant appears in the channel-state picture. For a channel $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$, the Choi-Jamiolkowski isomorphism identifies $\Phi$ with $J(\Phi)$, and for a channel $n^{-1}J(\Phi)$ is a density matrix. In the normalized state language, the Choi rank is simply the matrix rank of that bipartite state. In the operational formulation of the Choi state used for dense-coding-type tasks,

$\Phi:M_n\to M_n$0

and the paper on $\Phi:M_n\to M_n$1-state exclusion defines

$\Phi:M_n\to M_n$2

That paper also states that $\Phi:M_n\to M_n$3 iff $\Phi:M_n\to M_n$4 is unitary (Kribs et al., 2022, Stratton et al., 2024).

These equivalences make Choi rank the intrinsic “Kraus length” of a channel. In finite dimensions, it is the channel invariant that simultaneously packages Kraus minimality, Choi-matrix rank, and the state-space support size of the associated Choi state.

2. Choi-Jamiolkowski conventions and what rank does not depend on

The numerical value of Choi rank is stable under the standard variants of the Choi-Jamiolkowski correspondence, even though the positivity interpretation depends on which convention is used. One formulation reviewed in "Variations on the Choi-Jamiolkowski isomorphism" writes the basis-independent Jamiołkowski operator as

$\Phi:M_n\to M_n$5

while the standard Choi form is

$\Phi:M_n\to M_n$6

The inverse of the latter uses basis-dependent transposition,

$\Phi:M_n\to M_n$7

The paper emphasizes that these versions differ by transposition, adjoint-twisted bookkeeping, and identification of $\Phi:M_n\to M_n$8 with $\Phi:M_n\to M_n$9, but not in the numerical rank of the resulting operator (Frembs et al., 2022).

This convention-stability should be distinguished from a different rank statement that appears in "Choi matrices revisited". That paper studies when a generalized tensor

$\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$0

can replace the standard maximally entangled tensor in the Choi construction while preserving the correspondence

$\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$1

Its main characterization is that the generalized tensor must be a positive rank-one matrix whose range vector has full Schmidt rank, equivalently $\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$2 with $\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$3 of full Schmidt rank, or equivalently the associated map must be $\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$4 for some nonsingular $\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$5. In the basis formulation, the Choi correspondence is preserved exactly when the basis has the form

$\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$6

Here the relevant rank-one condition concerns the auxiliary tensor $\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$7, not the channel invariant usually called Choi rank (Kye, 2022).

3. Relations to other decomposition lengths

Choi rank is often the smallest decomposition length among several natural channel representations, but it does not generally determine the minimal length of more restricted decompositions.

For mixed-unitary channels, the mixed-unitary rank $\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$8 is the minimum number of unitary conjugations in a convex decomposition

$\Phi(X)=\sum_{k=1}^r A_k X A_k^*,$9

If $$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$0 denotes the Choi rank, then

$$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$1

because every mixed-unitary decomposition is also a Kraus representation. The main structural bound proved in "On the mixed-unitary rank of quantum channels" is

$$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$2

and more generally

$$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$3

when $$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$4 and

$$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$5

The same paper proves that $$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$6 when $$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$7, and that if $$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$8 then $$J(\Phi)=\sum_{1\le j,k\le n}\Phi(E_{j,k})\otimes E_{j,k},$$9. It also gives the first explicit examples with $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$0: there exist mixed-unitary channels having Choi rank $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$1 and mixed-unitary rank $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$2 for infinitely many positive integers $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$3, including every prime power $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$4 (Girard et al., 2020).

For entanglement-breaking channels, the corresponding invariant is the entanglement breaking rank, the least number of rank-one Kraus operators in a decomposition

$C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$5

The general inequality is

$C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$6

because a rank-one Kraus decomposition is still a Kraus decomposition. "Entanglement Breaking Rank via Complementary Channels and Multiplicative Domains" identifies a rigid equality case: if $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$7 is a rank-$C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$8 projection and $C_\Phi=\sum_{i,j=1}^n E_{ij}\otimes \Phi(E_{ij}).$9 is entanglement breaking, then

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$0

The same paper shows that such channels are exactly, up to unitary equivalence, the complements of Schur product channels. It also stresses that this equality fails when the Choi matrix is only a scalar multiple of a projection; the Werner-Holevo channel provides a counterexample with

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$1

(Kribs et al., 2022).

These comparisons place Choi rank at the base of a hierarchy: it is the minimal Kraus length, a universal lower bound on more structured decomposition lengths, and sometimes—under exact projection structure—the structured length itself.

4. Attainable Choi ranks under output constraints

Choi rank can be analyzed through bipartite-state marginal problems. Let

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$2

If

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$3

then "Ranks of quantum states with prescribed reduced states" proves that there exists a minimum rank $$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$4 with $$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$5, and that a state $$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$6 has rank $$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$7 if and only if

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$8

Thus the attainable ranks form the full interval

$$\operatorname{Choi\text{-}rank}(\Phi)=\operatorname{rank}(J(\Phi)),$$9

The same paper gives the rank-one compatibility criterion: there exists a rank-one $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$0 iff $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$1 and $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$2 have the same nonzero eigenvalues with multiplicities (Li et al., 2017).

Specializing to channels, a channel $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$3 satisfies

$\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$4

iff its Choi matrix lies in

$\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$5

Therefore the possible Choi ranks of channels with fixed output on the maximally mixed state are exactly the attainable ranks of that marginal set. The paper states that the possible Choi ranks are

$\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$6

where $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$7 is the minimum rank from the state problem. In the unital case,

$\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$8

and every Choi rank from $\Phi:M_n(\mathbb C)\to M_m(\mathbb C)$9 to $\Phi$0 occurs. The minimum rank can be determined algorithmically through Klyachko-type eigenvalue inequalities, using the polyhedral sets $\Phi$1 and $\Phi$2 introduced in Proposition 2.10 and Algorithm 2.9 (Li et al., 2017).

This marginal viewpoint is significant because it replaces an abstract channel-classification problem by a constrained bipartite-rank problem. The result is unusually complete: for the fixed-output and unital settings, the set of possible Choi ranks is not merely bounded but exactly classified as an interval of integers.

5. Operational interpretation through $\Phi$3-state exclusion

"Operational Interpretation of the Choi Rank Through $\Phi$4-State Exclusion" gives Choi rank a direct communication-theoretic meaning. In the protocol, Alice and Bob share a maximally entangled state of local dimension $\Phi$5, Alice encodes one of $\Phi$6 bit-strings using unitary channels $\Phi$7, sends her subsystem through a channel $\Phi$8, and Bob attempts not to identify the message but to exclude $\Phi$9 candidate messages with certainty. The encoded output states all have the same rank, namely the Choi rank $J(\Phi)$0, and the paper proves the universal bound

$J(\Phi)$1

In the super-dense-coding setting $J(\Phi)$2, this reduces to

$J(\Phi)$3

If Bob wants conclusive $J(\Phi)$4-state exclusion, which is equivalent to deterministic identification in that setting, then one must have

$J(\Phi)$5

For $J(\Phi)$6, this means

$J(\Phi)$7

so only a unitary channel allows perfect deterministic decoding (Stratton et al., 2024).

The same paper gives concrete channel examples. For a depolarizing channel,

$J(\Phi)$8

hence

$J(\Phi)$9

so no bit-string can be excluded with certainty. For the $n^{-1}J(\Phi)$0-dimensional dephasing channel,

$n^{-1}J(\Phi)$1

and in the dense-coding setting

$n^{-1}J(\Phi)$2

The paper states that this bound is tight: using Heisenberg-Weyl operators and Bell-basis measurement, Bob can perform conclusive weak $n^{-1}J(\Phi)$3-state exclusion (Stratton et al., 2024).

This operational interpretation is unusually sharp. Rather than describing only implementation cost or dilation size, Choi rank becomes a universal obstruction to certainty in an entanglement-assisted exclusion task.

6. Nearby notions, non-rank uses of Choi matrices, and terminological distinctions

Although Choi rank is a central invariant, several papers use the Choi matrix in ways that are explicitly not rank-based. "Assessing non-Markovian dynamics through moments of the Choi state" represents intermediate dynamics by a Choi matrix

$n^{-1}J(\Phi)$4

and defines moments

$n^{-1}J(\Phi)$5

Its witness for CP-divisibility breaking is the moment inequality

$n^{-1}J(\Phi)$6

whose violation certifies CP-indivisibility. The paper states explicitly that it does not formulate a Choi-rank criterion such as “rank change implies non-Markovianity”; the test is a moment-based spectral constraint on the Choi matrix. Likewise, "Detecting Non-Markovianity via Linear Entropy of Choi State" uses the linear entropy

$n^{-1}J(\Phi)$7

as a witness, and the paper emphasizes that negative linear entropy indicates a breakdown of positivity of the Choi operator, not a rank criterion. "Choi-Defined Resource Theories" also makes the renormalized Choi matrix central, but states that rank is not a central structural criterion for those theories. "Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States" introduces a pseudo-Choi state analogous to a Choi state and states that it does not define or analyze the usual Choi rank of a quantum channel (Mallick et al., 2023, Zheng et al., 2019, Zanoni et al., 2024, Castaneda et al., 2023).

A further source of confusion is nomenclature. "Chow rank" is an algebraic-geometric invariant: for a homogeneous form $n^{-1}J(\Phi)$8, it is the least $n^{-1}J(\Phi)$9 such that

$\Phi:M_n\to M_n$00

and it is governed by secant varieties of the Chow variety $\Phi:M_n\to M_n$01; it is unrelated to Choi rank despite the near-homophony (Torrance, 2015). Similarly, the "Choi map" is a specific positive linear map on $\Phi:M_n\to M_n$02, studied for extremality and optimality. The papers "Notes on extremality of the Choi map" and "Optimality of generalized Choi maps in $\Phi:M_n\to M_n$03" concern extremal rays, optimal witnesses, and generalized Choi maps, not the Kraus-number invariant called Choi rank (Ha, 2013, Scala et al., 2023).

Taken together, these distinctions clarify the scope of the term. Choi rank refers specifically to the rank of the Choi matrix of a channel, with equivalent Kraus and dilation meanings. Many nearby uses of Choi matrices—moment inequalities, entropy witnesses, resource-theoretic free-state characterizations, pseudo-Choi encodings, the Choi map, and Chow rank—are mathematically adjacent but conceptually separate.