---
title: '2-Superpositive Maps: Definition and Choi Correspondence'
url: https://www.emergentmind.com/topics/2-superpositive-maps
type: topic
---

# 2-Superpositive Maps: Definition and Choi Correspondence

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 [2204.02516].

## 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
\[
\operatorname{Ad}_s(x)=s^*xs.
\]
The cone of $k$-superpositive maps is
\[
SP_k=\operatorname{cone}\{\operatorname{Ad}_s:\operatorname{rank}s\leq k\}.
\]

Consequently, a map $\phi:M_A\to M_B$ is $2$-superpositive precisely when
\[
\phi(x)=\sum_i s_i^*xs_i,
\qquad
\operatorname{rank}(s_i)\leq2.
\]
The coefficients in a finite nonnegative sum can be absorbed into the matrices $s_i$, so the definition is equivalently the existence of a completely positive Kraus representation with rank-at-most-two Kraus operators.

Writing $r=\min\{a,b\}$, the superpositive cones form the increasing hierarchy
\[
SP_1\subseteq SP_2\subseteq\cdots\subseteq SP_r=CP_{AB},
\]
where $CP_{AB}$ denotes the cone of completely positive maps $M_A\to M_B$. The endpoint equality holds because every $a\times b$ matrix has rank at most $r$.

The first cone is the entanglement-breaking cone:
\[
SP_1=EB.
\]
Thus
\[
SP_1\subseteq SP_2\subseteq CP_{AB}.
\]
Every entanglement-breaking map is $2$-superpositive, every $2$-superpositive map is completely positive, and every completely positive map is $2$-positive. In general, the reverse implications fail when $r>2$. If $r\leq2$, the rank restriction is automatic and
\[
SP_2=CP_{AB}.
\]

The condition is an existence condition: a map belongs to $SP_2$ if at least one Kraus representation has all Kraus operators of rank at most $2$. It does not assert that every Kraus representation has this property.

## 2. Choi matrices and Schmidt number

For a linear map $\phi:M_A\to M_B$, its Choi matrix is
\[
C_\phi=\sum_{i,j}|i\rangle\langle j|\otimes\phi(|i\rangle\langle j|)
=(\operatorname{id}_A\otimes\phi)(|\omega\rangle\langle\omega|),
\]
where
\[
|\omega\rangle=\sum_i|i\rangle\otimes|i\rangle.
\]

For a matrix
\[
s=\sum_{i,j}s_{ij}|i\rangle\langle j|,
\]
let
\[
|\widetilde{s}\rangle=\sum_{i,j}s_{ij}|i\rangle\otimes|j\rangle
\]
be its vectorization. Direct computation gives
\[
C_{\operatorname{Ad}_s}
=
|\widetilde{s}\rangle\langle\widetilde{s}|.
\]
The central rank relation is
\[
\operatorname{rank}(s)
=
\operatorname{Schmidt\,rank}(|\widetilde{s}\rangle).
\]

Therefore, if $\phi\in SP_2$, then
\[
C_\phi=\sum_i|\widetilde{s_i}\rangle\langle\widetilde{s_i}|,
\qquad
\operatorname{Schmidt\,rank}(\widetilde{s_i})\leq2.
\]
Define
\[
S_2
=
\operatorname{cone}
\{|\zeta\rangle\langle\zeta|:
\operatorname{Schmidt\,rank}(\zeta)\leq2\}.
\]
Then
\[
\phi\in SP_2
\quad\Longleftrightarrow\quad
C_\phi\in S_2.
\]

For a positive bipartite operator, membership in $S_2$ means that it admits a decomposition into rank-one positive operators generated by vectors of Schmidt rank at most $2$. Equivalently, its Schmidt number is at most $2$. Hence
\[
\boxed{
\phi\in SP_2
\quad\Longleftrightarrow\quad
C_\phi\geq0
\text{ and }
\operatorname{SN}(C_\phi)\leq2.
}
\]

This establishes the precise correspondence between Kraus rank and Choi-state entanglement dimensionality. At level $1$, the Choi matrix is separable and the map is entanglement breaking. At level $2$, entangled Choi matrices are permitted, provided their Schmidt number does not exceed $2$ [2410.13120].

## 3. Distinction from complete and 2-positivity

A map $\phi$ is $k$-positive if
\[
\operatorname{id}_k\otimes\phi
\]
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
\[
C_\phi\geq0.
\]

The relevant hierarchy for $k=2$ is
\[
SP_1\subseteq SP_2\subseteq CP_{AB}\subseteq P_2\subseteq P_1,
\]
where $P_k$ denotes the cone of $k$-positive maps.

The two middle conditions have different meanings:

- **$2$-superpositivity** is a restricted Kraus condition, equivalently a Choi-Schmidt-number condition.
- **$2$-positivity** is an ampliation-positivity condition.
- **Complete positivity** is positivity of all ampliations, equivalently positivity of the entire Choi matrix.

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

Under the ordinary Choi map,
\[
\phi\in P_2
\quad\Longleftrightarrow\quad
C_\phi\in B_2,
\]
where $B_2$ is the cone of $2$-block-positive operators:
\[
\langle\zeta|\rho|\zeta\rangle\geq0
\]
for every vector $|\zeta\rangle$ of Schmidt rank at most $2$. A $2$-block-positive Choi matrix need not be positive semidefinite on all vectors. Thus $2$-positivity does not imply complete positivity, whereas $2$-superpositivity always implies complete positivity.

The distinction is especially transparent through Choi matrices:
\[
\begin{aligned}
\phi\in SP_2
&\Longleftrightarrow C_\phi\geq0
\text{ with Schmidt number at most }2,\\
\phi\in P_2
&\Longleftrightarrow C_\phi
\text{ is }2\text{-block-positive}.
\end{aligned}
\]

## 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
\[
SP_k^\circ=P_k,
\qquad
P_k^\circ=SP_k.
\]
For $k=2$,
\[
\boxed{
SP_2^\circ=P_2,
\qquad
P_2^\circ=SP_2.
}
\]

Thus $\phi$ is $2$-superpositive if and only if
\[
\langle\phi,\psi\rangle\geq0
\]
for every $2$-positive map $\psi$. On the state side, the corresponding duality is
\[
S_2^\circ=B_2,
\qquad
B_2^\circ=S_2.
\]

The cone duality yields several equivalent formulations of $2$-superpositivity. For a Hermiticity-preserving map $\phi$, the following conditions are equivalent:

1. $\phi$ has a Kraus representation
   \[
   \phi=\sum_i\operatorname{Ad}_{s_i},
   \qquad
   \operatorname{rank}(s_i)\leq2.
   \]

2. Its Choi matrix satisfies
   \[
   C_\phi\in S_2.
   \]

3. For every $2$-positive map $\psi$, both
   \[
   \psi^*\circ\phi
   \quad\text{and}\quad
   \phi\circ\psi^*
   \]
   are completely positive.

4. The ampliation $\operatorname{id}_A\otimes\phi$ sends every positive matrix to an operator of Schmidt number at most $2$.

5. For every $2$-positive $\psi$,
   \[
   (\operatorname{id}_A\otimes\psi)(C_\phi)\geq0
   \]
   in the appropriate dimensions.

6. For every $2$-positive $\psi$,
   \[
   \phi\otimes\psi
   \]
   is positive.

7. The Choi matrix is recovered as
   \[
   (\operatorname{id}_A\otimes\phi)(C_{\operatorname{id}_A})
   =
   C_\phi\in S_2.
   \]

The amplification criterion generalizes the entanglement-breaking characterization. At $k=1$, $\operatorname{id}_A\otimes\phi$ sends every positive input to a separable output. At $k=2$, separability is replaced by Schmidt number at most $2$.

The main tensor-composition identity is
\[
(\phi_1\otimes\phi_2)(C_\sigma)
=
C_{\phi_2\circ\sigma\circ\phi_1^*}.
\]
With $\phi_1=\operatorname{id}_A$, this becomes
\[
(\operatorname{id}_A\otimes\phi)(C_\sigma)
=
C_{\phi\circ\sigma}.
\]
It translates cone duality and composition properties into tensor-product positivity statements.

## 5. Mapping cones, composition, and generalized Choi correspondences

A closed convex cone $K\subset H(M_A,M_B)$ is a mapping cone when it is stable under completely positive composition on both sides:
\[
CP_{BB}\circ K\circ CP_{AA}\subseteq K.
\]
The cone $SP_2$ is a mapping cone. Therefore, if
\[
\phi\in SP_2,\qquad
\alpha\in CP_{AA},\qquad
\beta\in CP_{BB},
\]
then
\[
\beta\circ\phi\circ\alpha\in SP_2.
\]

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

A rank-two elementary generator suffices to generate the cone as a mapping cone. If $\operatorname{rank}(s)=2$, then
\[
\bigl(CP\circ\{\operatorname{Ad}_s\}\circ CP\bigr)^{\circ\circ}=SP_2.
\]
The same structure relates $2$-positivity to a single rank-two ampliation:
\[
\phi\text{ is }2\text{-positive}
\quad\Longleftrightarrow\quad
\operatorname{Ad}_s\otimes\phi
\text{ is positive}
\]
for one, equivalently every, rank-two matrix $s$. By singular-value decomposition, this is equivalent to
\[
\operatorname{id}_2\otimes\phi\geq0.
\]

Generalized Choi identifications have also been studied. If
\[
\mathcal T_\Theta(\Phi)=\Theta(C_\Phi)
\]
for a linear isomorphism $\Theta$ on the tensor-product matrix space, simultaneous preservation of all Schmidt-number cones is obtained from compositions of local congruences
\[
\operatorname{Ad}_s\otimes\operatorname{Ad}_t,
\]
transposition on both tensor factors when dimensions agree, and the flip when the tensor factors have equal dimensions. Such transformations preserve the $SP_2\leftrightarrow S_2$ and $P_2\leftrightarrow B_2$ correspondences when considered within the theorem’s simultaneous all-$k$ setting [2410.13120].

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
\[
\Theta(S_2)=S_2.
\]
The fixed-$k$ automorphism problem, including $k=2$, is formulated as a conjectural extension in general.

Partial transpose must likewise be treated carefully. It preserves the separable cone $S_1$, but is not generally an automorphism of $S_2$. Consequently, replacing the ordinary Choi matrix by a partially transposed Choi-like matrix can alter the $2$-superpositive/Schmidt-number correspondence.

## 6. Semigroups and generator criteria

The Schoenberg correspondence characterizes generators of semigroups that remain inside cones of $k$-positive, $k$-superpositive, or $k$-entanglement-breaking maps [2301.10679].

A non-unital semigroup is a continuous family $(T_t)_{t\geq0}$ satisfying
\[
T_sT_t=T_{s+t},
\]
without requiring $T_0=\operatorname{Id}$. The semigroup law implies
\[
T_0^2=T_0,
\]
so $T_0$ is an idempotent. This is necessary for a semigroup contained entirely in $SP_2$ when the identity is not $2$-superpositive, as occurs for dimensions greater than $2$.

Let $T_0\in SP_2$ be an idempotent and let $L$ satisfy
\[
LT_0=T_0L=L.
\]
The $T_0$-exponential is
\[
\exp_{T_0}(tL)
=
T_0+\sum_{r=1}^{\infty}\frac{t^r}{r!}L^r.
\]
The Schoenberg correspondence states that
\[
\exp_{T_0}(tL)\in SP_2
\qquad\text{for every }t\geq0
\]
if and only if $L$ is $T_0$-conditionally positive on the dual cone $SP_2^\circ=P_2$:
\[
\varphi(T_0)=0,\quad
\varphi\in P_2
\quad\Longrightarrow\quad
\varphi(L)\geq0.
\]

The analogous conditions hold separately for the cone $P_2$ and for the cone $EB_2$. In the unital case, $T_0=\operatorname{Id}$ belongs to $P_2$ but, for dimensions greater than $2$, does not belong to $SP_2$. Consequently, the direct unital semigroup criterion applies to $2$-positive semigroups, not to semigroups that are $2$-superpositive at every time.

For a generator written as
\[
L=T_W,
\qquad
W=\sum_iA_i\otimes B_i,
\]
the unital $2$-positivity criterion is
\[
\operatorname{rank}(V)\leq2,\quad
\operatorname{Tr}(V)=0
\quad\Longrightarrow\quad
\sum_i
\operatorname{Tr}(A_iV^*)
\operatorname{Tr}(B_iV)\geq0.
\]
This is a rank-two-sensitive conditional-positivity condition. It characterizes preservation of $2$-positivity, not $2$-superpositivity.

For $k=n$, the cones of $k$-positive and $k$-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

$2$-superpositivity is distinct from the PPT property. A map is PPT when it is both completely positive and completely copositive, equivalently when
\[
C_\phi\geq0,
\qquad
(C_\phi)^\Gamma\geq0.
\]
A $2$-superpositive map has a Choi matrix of Schmidt number at most $2$, but that Choi matrix need not have positive partial transpose. Conversely, PPT states can have Schmidt number greater than $2$ in sufficiently high dimensions.

The inclusions relevant to the map side include
\[
SP_1\subseteq PPT\subseteq CP\subseteq DEC\subseteq P_1,
\]
where
\[
CCP=CP\circ t,
\qquad
DEC=CP\vee CCP,
\qquad
PPT=CP\wedge CCP.
\]
The PPT-square conjecture asks whether
\[
PPT\circ PPT\subseteq SP_1,
\]
that is, whether the composition of two PPT maps is entanglement breaking. This is a statement about $SP_1$, not merely $SP_2$.

A related enlargement is
\[
DSP_k=\{\Psi_1+\top\circ\Psi_2:\Psi_1,\Psi_2\in SP_k\}.
\]
The class $DSP_2$ strictly contains $SP_2$ and can contain maps that are not $2$-positive. Recent work proves
\[
PPT\circ DSP_2\circ PPT\subseteq EB.
\]
Consequently,
\[
\Phi\in PPT\cap SP_2
\quad\Longrightarrow\quad
\Phi^3\in EB,
\]
so every PPT $2$-superpositive map has entanglement-breaking index at most $3$. The result applies to the larger class $PPT\cap DSP_2$ 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 [2608.13551].

$2$-entanglement-breaking maps are another distinct class. A map belongs to $EB_2$ when it is $2$-positive and its two-fold ampliation sends positive matrices to separable matrices. The inclusion
\[
SP_2\subseteq EB_2
\]
is recorded in the semigroup framework, but the two cones are not identified in general. In contrast, $SP_2$ is defined by the Schmidt number of the Choi matrix.

Finally, fractional $k$-positivity provides a continuous refinement of the integer hierarchy [2602.12729]. For an integer $\alpha=2$, the fractional admissibility condition reduces exactly to Schmidt rank at most $2$, and the corresponding cones satisfy
\[
\mathsf{SP}_2=SP_2,
\qquad
\mathsf P_2=P_2.
\]
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 $2$-superpositive maps represent an integer, mapping-cone-stable level within a broader hierarchy interpolating between separability and complete positivity.

Source: https://www.emergentmind.com/topics/2-superpositive-maps