arXiv:2609.07260

A Family of Completely PPT Maps

Kennett L. Dela Rosa, Iana Angela C. Fajardo, Cedie B. Reyes

Published Sep 7, 2026• 0 citations

Abstract

Given $α\in \mathbb R$ and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $α$ and $B_1,\ldots,B_q$ are given that guarantee $Φ_{α,\mathfrak{B}}^\pm$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.

DOI: 10.48550/arxiv.2609.07260

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arXiv:2609.07260

A Family of Completely PPT Maps

Kennett L. Dela Rosa, Iana Angela C. Fajardo, Cedie B. Reyes

Published Sep 7, 2026• 0 citations

arXiv HTML PDF

Abstract

Given

α\in \mathbb R

and

B_1,\ldots,B_q\in M_{m,n}

, consider the map [Φ^{\pm}{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.] In this study, necessary and sufficient conditions on

α

and

B_1,\ldots,B_q

are given that guarantee

Φ_{α,\mathfrak{B}}^\pm

k

-positive,

k

-copositive, and

k

-PPT, respectively. As application, known results about the map

α\mbox{tr}(X)I_n\pm X

are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.

DOI: 10.48550/arxiv.2609.07260

Related papers

Linear maps of positive partial transpose matrices and singular value inequalities

Xiaohui Fu, Pan-Shun Lau, Tin-Yau Tam

Linear maps : M n M k are called m -PPT if [(A i j )] m i, j=1 are positive partial transpose matrices for all positive semi-definite matrices

Mathematical Inequalities & ApplicationsJan 1, 20202 citations

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arXiv:1009.5809Pacific Journal of MathematicsOct 29, 20116 citations

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Mehmet Gumus, Jianzhen Liu, Samir Raouafi +1 more

This paper mainly focuses on the class of $2 \times 2$ block PPT matrices. The relationship between PPT matrices and the norm inequalities is further explored. Some properties of a non-PPT matrix in terms of its eigenvalues are investigated. Moreover, a number of useful sufficient conditions for a matrix to be PPT are provided.This paper mainly focuses on the class of $2 \times 2$ block PPT matrices. The relationship between PPT matrices and the norm inequalities is further explored. Some properties of a non-PPT matrix in terms of its eigenvalues are investigated. Moreover, a number of useful sufficient conditions for a matrix to be PPT are provided.

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The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps $Φ_{A,B,C}$ and $Φ_{D,E,F}$ on $M_d$ , the entrywise bound $(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}$ for all $i,j$ guarantees that $Φ_{A,B,C}\circΦ_{D,E,F}$ is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with $a\ne1$ is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps $Φ_{A,B,C}$ and $Φ_{D,E,F}$ on $M_d$, the entrywise bound $(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}$ for all $i,j$ guarantees that $Φ_{A,B,C}\circΦ_{D,E,F}$ is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with $a\ne1$ is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.

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Frederik vom Ende, Sumeet Khatri, Sergey Denisov

We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.

arXiv:2508.21348Open Systems & Information DynamicsDec 1, 20251 citation

Given

α\in \mathbb R

and

B_1,\ldots,B_q\in M_{m,n}

, consider the map [Φ^{\pm}{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.] In this study, necessary and sufficient conditions on

α

and

B_1,\ldots,B_q

are given that guarantee

Φ_{α,\mathfrak{B}}^\pm

k

-positive,

k

-copositive, and

k

-PPT, respectively. As application, known results about the map

α\mbox{tr}(X)I_n\pm X

are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.

This paper mainly focuses on the class of

2 \times 2

block PPT matrices. The relationship between PPT matrices and the norm inequalities is further explored. Some properties of a non-PPT matrix in terms of its eigenvalues are investigated. Moreover, a number of useful sufficient conditions for a matrix to be PPT are provided.

k

-positivity are better than those derived from the spectral conditions considered by Chruściński and Kossakowski. We further look with special attention at the case where we give the precise bound for the regions of decomposability.

Φ_{A,B,C}

and

Φ_{D,E,F}

M_d

, the entrywise bound

(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}

for all

i,j

guarantees that

Φ_{A,B,C}\circΦ_{D,E,F}

is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with

a\ne1

is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.