2-Superpositive Maps: Definition and Choi Correspondence

• 2-superpositive maps are completely positive maps with a Kraus representation whose operators have rank at most 2, placing them between entanglement-breaking and arbitrary completely positive maps.
• Their Choi matrices are positive operators with Schmidt number at most 2, linking low-rank Kraus decompositions to bounded bipartite entanglement and providing a practical characterization.
• The cone of 2-superpositive maps is dual to the cone of 2-positive maps and remains stable under completely positive composition, with applications to tensor products, mapping cones, and quantum-channel semigroups.

2-superpositive maps are completely positive maps between finite-dimensional matrix algebras that admit a Kraus decomposition in which every Kraus operator has rank at most $2$. They form the second level of the superpositivity hierarchy, lying between entanglement-breaking maps and arbitrary completely positive maps. Under the Choi correspondence, $2$-superpositivity is equivalent to the Choi matrix having Schmidt number at most $2$; under cone duality, the dual class is the cone of $2$-positive maps. These equivalences connect Kraus-rank constraints, bipartite entanglement structure, ampliation positivity, block positivity, tensor products, and mapping-cone theory (Kye, 2022).

1. Definition and position in the hierarchy

Let $M_A=M_a(\mathbb C)$ and $M_B=M_b(\mathbb C)$, with finite $a,b$. For an $a\times b$ matrix $s$, viewed as a map $\mathbb C^B\to\mathbb C^A$, define the elementary completely positive map

$2$0

The cone of $2$1-superpositive maps is

$2$2

Consequently, a map $2$3 is $2$4-superpositive precisely when

$2$5

The coefficients in a finite nonnegative sum can be absorbed into the matrices $2$6, so the definition is equivalently the existence of a completely positive Kraus representation with rank-at-most-two Kraus operators.

Writing $2$7, the superpositive cones form the increasing hierarchy

$2$8

where $2$9 denotes the cone of completely positive maps $2$0. The endpoint equality holds because every $2$1 matrix has rank at most $2$2.

The first cone is the entanglement-breaking cone: $2$3 Thus

$2$4

Every entanglement-breaking map is $2$5-superpositive, every $2$6-superpositive map is completely positive, and every completely positive map is $2$7-positive. In general, the reverse implications fail when $2$8. If $2$9, the rank restriction is automatic and

$2$0

The condition is an existence condition: a map belongs to $2$1 if at least one Kraus representation has all Kraus operators of rank at most $2$2. It does not assert that every Kraus representation has this property.

2. Choi matrices and Schmidt number

For a linear map $2$3, its Choi matrix is

$2$4

where

$2$5

For a matrix

$2$6

let

$2$7

be its vectorization. Direct computation gives

$2$8

The central rank relation is

$2$9

Therefore, if $M_A=M_a(\mathbb C)$0, then

$M_A=M_a(\mathbb C)$1

Define

$M_A=M_a(\mathbb C)$2

Then

$M_A=M_a(\mathbb C)$3

For a positive bipartite operator, membership in $M_A=M_a(\mathbb C)$4 means that it admits a decomposition into rank-one positive operators generated by vectors of Schmidt rank at most $M_A=M_a(\mathbb C)$5. Equivalently, its Schmidt number is at most $M_A=M_a(\mathbb C)$6. Hence

$M_A=M_a(\mathbb C)$7

This establishes the precise correspondence between Kraus rank and Choi-state entanglement dimensionality. At level $M_A=M_a(\mathbb C)$8, the Choi matrix is separable and the map is entanglement breaking. At level $M_A=M_a(\mathbb C)$9, entangled Choi matrices are permitted, provided their Schmidt number does not exceed $M_B=M_b(\mathbb C)$0 (Han et al., 2024).

3. Distinction from complete and 2-positivity

A map $M_B=M_b(\mathbb C)$1 is $M_B=M_b(\mathbb C)$2-positive if

$M_B=M_b(\mathbb C)$3

is positive. Complete positivity requires positivity of every ampliation. In finite dimensions, complete positivity is equivalent to positivity at the largest relevant ampliation and, by Choi’s theorem, to

$M_B=M_b(\mathbb C)$4

The relevant hierarchy for $M_B=M_b(\mathbb C)$5 is

$M_B=M_b(\mathbb C)$6

where $M_B=M_b(\mathbb C)$7 denotes the cone of $M_B=M_b(\mathbb C)$8-positive maps.

The two middle conditions have different meanings:

• $M_B=M_b(\mathbb C)$9-superpositivity is a restricted Kraus condition, equivalently a Choi-Schmidt-number condition.
• $a,b$0-positivity is an ampliation-positivity condition.
• Complete positivity is positivity of all ampliations, equivalently positivity of the entire Choi matrix.

A $a,b$1-superpositive map is completely positive and therefore $a,b$2-positive. A $a,b$3-positive map need not be completely positive, and a completely positive map need not be $a,b$4-superpositive when the dimensions permit Kraus operators of rank greater than $a,b$5.

Under the ordinary Choi map,

$a,b$6

where $a,b$7 is the cone of $a,b$8-block-positive operators: $a,b$9 for every vector $a\times b$0 of Schmidt rank at most $a\times b$1. A $a\times b$2-block-positive Choi matrix need not be positive semidefinite on all vectors. Thus $a\times b$3-positivity does not imply complete positivity, whereas $a\times b$4-superpositivity always implies complete positivity.

The distinction is especially transparent through Choi matrices: $a\times b$5

4. Dual cones and equivalent criteria

On the real vector space of Hermiticity-preserving maps, the Choi or Hilbert–Schmidt pairing gives the duality relations

$a\times b$6

For $a\times b$7,

$a\times b$8

Thus $a\times b$9 is $s$0-superpositive if and only if

$s$1

for every $s$2-positive map $s$3. On the state side, the corresponding duality is

$s$4

The cone duality yields several equivalent formulations of $s$5-superpositivity. For a Hermiticity-preserving map $s$6, the following conditions are equivalent:

1. $s$7 has a Kraus representation

$s$8

2. Its Choi matrix satisfies

$s$9

3. For every $\mathbb C^B\to\mathbb C^A$0-positive map $\mathbb C^B\to\mathbb C^A$1, both

$\mathbb C^B\to\mathbb C^A$2

are completely positive.

1. The ampliation $\mathbb C^B\to\mathbb C^A$3 sends every positive matrix to an operator of Schmidt number at most $\mathbb C^B\to\mathbb C^A$4.
2. For every $\mathbb C^B\to\mathbb C^A$5-positive $\mathbb C^B\to\mathbb C^A$6,

$\mathbb C^B\to\mathbb C^A$7

in the appropriate dimensions.

1. For every $\mathbb C^B\to\mathbb C^A$8-positive $\mathbb C^B\to\mathbb C^A$9,

$2$00

is positive.

2. The Choi matrix is recovered as

$2$01

The amplification criterion generalizes the entanglement-breaking characterization. At $2$02, $2$03 sends every positive input to a separable output. At $2$04, separability is replaced by Schmidt number at most $2$05.

The main tensor-composition identity is

$2$06

With $2$07, this becomes

$2$08

It translates cone duality and composition properties into tensor-product positivity statements.

5. Mapping cones, composition, and generalized Choi correspondences

A closed convex cone $2$09 is a mapping cone when it is stable under completely positive composition on both sides: $2$10 The cone $2$11 is a mapping cone. Therefore, if

$2$12

then

$2$13

This follows from Kraus ranks. If $2$14 has rank-at-most-two Kraus operators and $2$15 have arbitrary Kraus operators, each composite Kraus operator is a product containing the rank-at-most-two factor, and therefore has rank at most $2$16.

A rank-two elementary generator suffices to generate the cone as a mapping cone. If $2$17, then

$2$18

The same structure relates $2$19-positivity to a single rank-two ampliation: $2$20 for one, equivalently every, rank-two matrix $2$21. By singular-value decomposition, this is equivalent to

$2$22

Generalized Choi identifications have also been studied. If

$2$23

for a linear isomorphism $2$24 on the tensor-product matrix space, simultaneous preservation of all Schmidt-number cones is obtained from compositions of local congruences

$2$25

transposition on both tensor factors when dimensions agree, and the flip when the tensor factors have equal dimensions. Such transformations preserve the $2$26 and $2$27 correspondences when considered within the theorem’s simultaneous all-$2$28 setting (Han et al., 2024).

The classification is subject to an important qualification: it establishes the stated family for simultaneous preservation of the entire hierarchy. It does not prove an unconditional classification of every linear isomorphism satisfying only

$2$29

The fixed-$2$30 automorphism problem, including $2$31, is formulated as a conjectural extension in general.

Partial transpose must likewise be treated carefully. It preserves the separable cone $2$32, but is not generally an automorphism of $2$33. Consequently, replacing the ordinary Choi matrix by a partially transposed Choi-like matrix can alter the $2$34-superpositive/Schmidt-number correspondence.

6. Semigroups and generator criteria

The Schoenberg correspondence characterizes generators of semigroups that remain inside cones of $2$35-positive, $2$36-superpositive, or $2$37-entanglement-breaking maps (Bhat et al., 2023).

A non-unital semigroup is a continuous family $2$38 satisfying

$2$39

without requiring $2$40. The semigroup law implies

$2$41

so $2$42 is an idempotent. This is necessary for a semigroup contained entirely in $2$43 when the identity is not $2$44-superpositive, as occurs for dimensions greater than $2$45.

Let $2$46 be an idempotent and let $2$47 satisfy

$2$48

The $2$49-exponential is

$2$50

The Schoenberg correspondence states that

$2$51

if and only if $2$52 is $2$53-conditionally positive on the dual cone $2$54: $2$55

The analogous conditions hold separately for the cone $2$56 and for the cone $2$57. In the unital case, $2$58 belongs to $2$59 but, for dimensions greater than $2$60, does not belong to $2$61. Consequently, the direct unital semigroup criterion applies to $2$62-positive semigroups, not to semigroups that are $2$63-superpositive at every time.

For a generator written as

$2$64

the unital $2$65-positivity criterion is

$2$66

This is a rank-two-sensitive conditional-positivity condition. It characterizes preservation of $2$67-positivity, not $2$68-superpositivity.

For $2$69, the cones of $2$70-positive and $2$71-superpositive maps both become the completely positive cone, and the same correspondence yields the Lindblad–Gorini–Kossakowski–Sudarshan theorem. Thus the GKLS characterization is the complete-positivity endpoint of the broader cone-theoretic framework.

7. Relations to PPT maps, entanglement breaking, and fractional hierarchies

$2$72-superpositivity is distinct from the PPT property. A map is PPT when it is both completely positive and completely copositive, equivalently when

$2$73

A $2$74-superpositive map has a Choi matrix of Schmidt number at most $2$75, but that Choi matrix need not have positive partial transpose. Conversely, PPT states can have Schmidt number greater than $2$76 in sufficiently high dimensions.

The inclusions relevant to the map side include

$2$77

where

$2$78

The PPT-square conjecture asks whether

$2$79

that is, whether the composition of two PPT maps is entanglement breaking. This is a statement about $2$80, not merely $2$81.

A related enlargement is

$2$82

The class $2$83 strictly contains $2$84 and can contain maps that are not $2$85-positive. Recent work proves

$2$86

Consequently,

$2$87

so every PPT $2$88-superpositive map has entanglement-breaking index at most $2$89. The result applies to the larger class $2$90 and does not establish the PPT-square conjecture for arbitrary PPT maps. More generally, every PPT map has finite entanglement-breaking index, although no dimension-independent bound is established for all PPT maps (Park, 13 Aug 2026).

$2$91-entanglement-breaking maps are another distinct class. A map belongs to $2$92 when it is $2$93-positive and its two-fold ampliation sends positive matrices to separable matrices. The inclusion

$2$94

is recorded in the semigroup framework, but the two cones are not identified in general. In contrast, $2$95 is defined by the Schmidt number of the Choi matrix.

Finally, fractional $2$96-positivity provides a continuous refinement of the integer hierarchy (Kian, 13 Feb 2026). For an integer $2$97, the fractional admissibility condition reduces exactly to Schmidt rank at most $2$98, and the corresponding cones satisfy

$2$99

The fractional Kraus theorem reduces to the rank-at-most-two Kraus characterization. Integer levels retain stability under completely positive pre- and post-composition, whereas noninteger positive cones can fail stability under completely positive post-composition. Thus $2$00-superpositive maps represent an integer, mapping-cone-stable level within a broader hierarchy interpolating between separability and complete positivity.