Choi-Rank Separation Criterion

• Choi-Rank Separation Criterion is a framework that employs Choi matrices, states, and polynomials to differentiate structural classes in matrix algebras and quantum channels.
• It integrates various mechanisms including rank conditions, kernel–ideal separations, and positivity tests to characterize complete positivity and separability.
• The criterion underpins practical applications in quantum information, such as verifying entanglement, channel decomposition, and detecting non-Markovian dynamics.

The expression “Choi-Rank Separation Criterion” is not used uniformly in the literature cited here. A plausible umbrella usage is to treat it as an Editor’s term for a family of criteria in which a Choi matrix, Choi state, Choi polynomial, or Choi rank separates one structural class from another. In the standard matrix-algebra setting, the Choi matrix of a linear map $\Phi:M_n\to M_m$ is

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

and the Choi rank of a quantum channel is the rank of $C(\Phi)$ (Kye, 2022, Li et al., 2017). Across operator algebras and quantum information, the associated separation mechanisms include complete-positivity tests, rank-one Kraus criteria for separability, product-vector criteria for low-rank PPT states, entropy- and moment-based witnesses for non-Markovianity, and inequalities comparing Choi rank with other operational ranks.

1. Terminological scope and basic Choi constructions

The basic Choi correspondence identifies complete positivity with positivity of a Choi object. For a linear map $\Phi:M_n\to M_m$, Choi’s theorem states

$$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$

with $C_\Phi$ defined using the standard matrix units $\{e_{ij}\}$ (Kye, 2022). The same paper studies generalized Choi constructions of the form

$C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$

for an alternative basis $B=\{b_{ij}\}\subset M_n$, and also the tensor-level formulation

$C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$

where $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$0 is fixed (Kye, 2022).

In quantum information, the same channel-state duality appears as the Jamiołkowski–Choi construction. For a dynamical map $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$1, the Choi state is written as

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$2

with $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$3 maximally entangled; complete positivity is equivalent to $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$4 (Mallick et al., 2023). For bipartite states, the duality is expressed by associating to $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$5 a completely positive map $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$6 satisfying

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$7

so that separability can be reformulated as a property of the associated map (Antipin, 2019).

A recurrent misconception is that every “Choi-based separation” is literally a rank criterion. The cited works do not support that identification. Some criteria are genuinely rank-theoretic, such as Choi-rank interval theorems for channels or rank-one Kraus conditions for separability (Li et al., 2017, Antipin, 2019). Others are positivity, kernel, or product-vector criteria phrased through Choi objects rather than through matrix rank itself (Prunaru, 2013, Zheng et al., 2019).

2. Choi–Effros multiplication and kernel–ideal separation

In operator-algebraic form, the Choi–Effros theorem concerns a completely positive projection rather than a finite-dimensional Choi matrix. If $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$8 is a $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$9-algebra and $C(\Phi)$0 is completely positive, contractive, and idempotent, then there exist a $C(\Phi)$1-algebra $C(\Phi)$2 and a complete order isomorphism

$C(\Phi)$3

such that

$C(\Phi)$4

(Prunaru, 2013). The induced product on the range is the Choi–Effros multiplication

$C(\Phi)$5

The short proof in “On the Choi-Effros multiplication” identifies the decisive structural separator not as a rank condition but as a kernel–ideal criterion. Let $C(\Phi)$6 be the closed right ideal generated by

$C(\Phi)$7

The key technical statement is

$C(\Phi)$8

(Prunaru, 2013). The inclusion $C(\Phi)$9 is obtained using the Kadison–Schwarz inequality

$\Phi:M_n\to M_m$0

while the reverse inclusion is established by induction on products $\Phi:M_n\to M_m$1 with $\Phi:M_n\to M_m$2, showing that

$\Phi:M_n\to M_m$3

Once $\Phi:M_n\to M_m$4 is recognized as a bilateral ideal, the quotient $\Phi:M_n\to M_m$5 is a $\Phi:M_n\to M_m$6-algebra, and $\Phi:M_n\to M_m$7 inherits associativity and $\Phi:M_n\to M_m$8-compatibility from the quotient model (Prunaru, 2013). The paper explicitly states that it does not discuss “Choi-rank separation criteria.” The closest relevant criterion is the ideal characterization of $\Phi:M_n\to M_m$9 by multiplicative defects. This suggests an operator-algebraic notion of separation by failure of multiplicativity on the range rather than by matrix rank.

3. Generalized Choi correspondences and the rank-one/full-Schmidt-rank condition

A more literal separation theorem appears in the generalized Choi correspondence studied by Kye. If $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$0 and $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$1 for a linear map $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$2, then the following are equivalent: $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$3 satisfies the Choi correspondence; $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$4 is a complete order isomorphism; $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$5 for some nonsingular $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$6; and $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$7 is a positive rank-one matrix whose range vector has full Schmidt rank (Kye, 2022). In this setting, the generalized criterion

$$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$8

holds for every output dimension $$\Phi \text{ is completely positive } \iff C_\Phi\ge 0,$$9 exactly under those equivalent conditions.

This theorem turns the generalized Choi problem into a precise separation between faithful and non-faithful test tensors. The tensor $C_\Phi$0 must be rank one, positive, and supported on a vector of full Schmidt rank; otherwise positivity of $C_\Phi$1 does not characterize complete positivity for all $C_\Phi$2 (Kye, 2022). The equivalence with $C_\Phi$3 identifies the relevant geometric symmetry as a complete order isomorphism.

The basis-change version sharpens an earlier sufficient condition of Paulsen and Shultz into a necessary-and-sufficient characterization. For a basis $C_\Phi$4 of $C_\Phi$5, the criterion

$C_\Phi$6

for every $C_\Phi$7 holds if and only if

$C_\Phi$8

for a nonsingular $C_\Phi$9, equivalently if and only if there exists a basis $\{e_{ij}\}$0 of $\{e_{ij}\}$1 such that

$\{e_{ij}\}$2

(Kye, 2022). Thus only bases arising as matrix units relative to some vector basis preserve the Choi positivity criterion.

The paper also gives a dual-cone interpretation: the bilinear pairing

$\{e_{ij}\}$3

makes the cone of completely positive maps self-dual, and the generalized Choi criterion is equivalent to a cone identity

$\{e_{ij}\}$4

This is a separation statement at the level of cone automorphisms rather than at the level of spectral rank alone (Kye, 2022).

4. Separability, entanglement breaking, and product-vector criteria

For bipartite states, the strongest rank-based separation mechanism in the cited literature is the channel-state reformulation of separability. Given $\{e_{ij}\}$5, one defines

$\{e_{ij}\}$6

and obtains an operator-sum representation

$\{e_{ij}\}$7

(Antipin, 2019). The central lemma states that $\{e_{ij}\}$8 is separable if and only if the operators $\{e_{ij}\}$9 can be transformed by

$C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$0

to rank-one operators $C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$1 (Antipin, 2019). Equivalently, the Choi matrix of the associated map is separable exactly when the map is entanglement-breaking, and this is equivalent to the existence of a rank-one Kraus decomposition.

The same paper derives spectral consequences from this rank-one structure. If

$C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$2

is a spectral decomposition and $C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$3 are the eigenvalues of the reduced state of $C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$4, then

$C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$5

and

$C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$6

are necessary conditions for separability (Antipin, 2019). For isotropic states, applying the criterion yields the threshold $C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$7, reproducing the known separability boundary for that family (Antipin, 2019).

A distinct low-rank separability criterion appears for multipartite PPT states of rank at most four. Theorem 28 states that a multipartite PPT state of rank $C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$8 is separable, while a rank-four PPT state is entangled if and only if its range $C_B=\sum_{i,j=1}^n b_{ij}\otimes \Phi(b_{ij})$9 is a completely entangled subspace, i.e.

$B=\{b_{ij}\}\subset M_n$0

(Chen et al., 2013). In the $B=\{b_{ij}\}\subset M_n$1 and $B=\{b_{ij}\}\subset M_n$2 cases, the product-vector question is reduced to the vanishing of the Chow form in the Plücker coordinates of the range, giving an explicit polynomial test (Chen et al., 2013).

These results show that “rank separation” in separability theory can mean at least three different things: rank-one operator-sum form for the Choi-dual map, low-rank restrictions on the global state, and absence or presence of product vectors in the relevant support subspace. The cited papers do not collapse these into a single theorem, but they place them in a common Choi/Jamiołkowski framework (Antipin, 2019, Chen et al., 2013).

5. Choi polynomials, positive maps, and extremality

A polynomial version of Choi-based separation is developed through the Choi polynomial

$B=\{b_{ij}\}\subset M_n$3

for a linear map $B=\{b_{ij}\}\subset M_n$4 (Ho et al., 29 Apr 2026). The paper gives the identity

$B=\{b_{ij}\}\subset M_n$5

so the Choi polynomial is the expectation of the Choi matrix on product vectors (Ho et al., 29 Apr 2026). Positivity of the map is equivalent to $B=\{b_{ij}\}\subset M_n$6 for all $B=\{b_{ij}\}\subset M_n$7, or equivalently to block positivity of $B=\{b_{ij}\}\subset M_n$8. Complete positivity is equivalent to $B=\{b_{ij}\}\subset M_n$9, and complete copositivity to $C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$0 (Ho et al., 29 Apr 2026).

The same work introduces the decomposable Gram cone

$C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$1

and states that decomposability of the map is equivalent to membership of the associated Gram matrix in $C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$2 (Ho et al., 29 Apr 2026). Its principal separation mechanism is a kernel/product-vector condition: for a decomposable $C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$3, minimality is equivalent to

$C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$4

Under this condition, subtraction of a small multiple of the identity preserves positivity on product vectors while breaking decomposability, yielding indecomposable positive maps and PPT entanglement witnesses (Ho et al., 29 Apr 2026).

A related issue is extremality of the Choi map. The note “Notes on extremality of the Choi map” proves that the Choi map generates an extreme ray in the cone $C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$5 of positive linear maps on $C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$6 (Ha, 2013). The paper also corrects two misclaims: extremality of the associated real biquadratic form does not imply extremality in the full complex cone, and the correspondence between positive semidefinite biquadratic forms and positive maps does not extend trivially from real symmetric matrices to all of $C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$7 (Ha, 2013). This controversy matters because separation arguments based only on the real symmetric restriction can fail to establish full complex extremality.

6. Choi-state positivity and moment criteria for non-Markovianity

In open-system dynamics, the relevant separation is between CP-divisible Markovian evolution and non-Markovian evolution. One entropy-based witness uses the Choi state of the short-time map

$C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$8

and the normalized linear entropy

$C_\Phi=(\operatorname{id}_n\otimes \Phi)(E),$9

(Zheng et al., 2019). The theorem stated in the paper is that the dynamics is non-Markovian if

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$00

The underlying logic is that CP divisibility implies positivity of the Choi state, whereas violation of positivity allows negative eigenvalues and therefore negative values of the entropy formula (Zheng et al., 2019).

A moment-based version replaces entropy by spectral moments

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$01

and proves that Markovian dynamics must satisfy

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$02

equivalently

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$03

(Mallick et al., 2023). Violation,

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$04

witnesses CP indivisibility and hence non-Markovianity (Mallick et al., 2023).

Both papers explicitly support a Choi-based separation viewpoint but do not formulate a literal Choi-rank criterion. The separator is failure of positivity or of a moment inequality for the intermediate-map Choi state, not a theorem about the rank of the Choi matrix itself (Zheng et al., 2019, Mallick et al., 2023).

7. Choi rank as a quantitative separator for channels

The most direct use of Choi rank as a separating invariant occurs in the study of channels with prescribed marginals. For density matrices $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$05 and $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$06, let

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$07

If $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$08, Theorem 2.1 states that there exists a minimum achievable rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$09, and a rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$10 occurs in $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$11 if and only if

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$12

(Li et al., 2017). The minimum rank is determined by a finite algorithm based on Klyachko/Horn inequalities (Li et al., 2017).

This translates directly to channels because

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$13

and a channel $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$14 with $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$15 corresponds exactly to a state in $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$16 (Li et al., 2017). Hence the possible Choi ranks of such channels are precisely the integers

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$17

For unital channels on $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$18, every rank

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$19

occurs (Li et al., 2017).

A different separation problem compares Choi rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$20 with mixed-unitary rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$21. For every mixed-unitary channel,

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$22

because every mixed-unitary decomposition is also a Kraus decomposition (Girard et al., 2020). The paper proves the universal upper bound

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$23

and the special case $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$24 for $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$25 (Girard et al., 2020). It also gives the first known examples with strict separation $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$26: there exist mixed-unitary channels with Choi rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$27 and mixed-unitary rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$28 for infinitely many positive integers $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$29, including every prime power $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$30 (Girard et al., 2020).

The structural mechanism behind that separation is the operator system

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$31

If

$C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$32

then $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$33 has mixed-unitary rank $C_\Phi=\sum_{i,j=1}^n e_{ij}\otimes \Phi(e_{ij}),$34 and a unique mixed-unitary decomposition (Girard et al., 2020). A direct-sum construction then produces channels whose mixed-unitary rank doubles while the Choi rank increases by only one. This is an explicit quantitative realization of Choi-rank separation in channel theory.