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arXiv:1611.02838 · International Mathematics Research Notices

There Are Many More Positive Maps Than Completely Positive Maps

Igor Klep, Scott McCullough, Klemen Šivic, Aljaž Zalar

Published Aug 8, 2017• 12 citations
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Abstract

Abstract A $\ast$-linear map $\Phi$ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations $I_n\otimes \Phi$ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.

DOI
10.1093/imrn/rnx203

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The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.

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arXiv:2607.13251•Jul 14, 2026•0 citations
Abstract A ∗\ast∗-linear map Φ\PhiΦ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations In⊗ΦI_n\otimes \PhiIn​⊗Φ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.
Several investigations have been done on positive maps on their algebraic structures with more emphasis on completely positive maps. In this study, we have described the structure of the Choi matrices for 2-positive maps on positive semidefinite matrices and the conditions for complete positivity of positive linear maps from n to n+1. The motivation behind these objectives is work done by Majewski and Marciniak on the structure of positive maps ф from Mn to Mn + 1(2 ≥ 2) between matrix algebras.
A factorization property of positive maps on C∗C^*C∗-algebras
The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let AiA_iAi​, CiC_iCi​ be unital C*-algebras and let αiα_iαi​ be positive linear maps from AiA_iAi​ to Ci,C_i,Ci​, i=1,2i=1,2i=1,2. We obtain conditions under which any positive map βββ from the minimal C*-tensor product A1⊗minA2A_1 \otimes_{min} A_2A1​⊗min​A2​ to C1⊗minC2C_1 \otimes_{min} C_2C1​⊗min​C2​, such that $ α_1 \otimes α_2 \geq β$, factorizes as β=γ⊗α2β= γ\otimes α_2β=γ⊗α2​ for some positive map γγγ. In particular we show that when αi ⁣:Ai→B(Hi)α_i \colon A_i \rightarrow B(\mathcal H_i)αi​:Ai​→B(Hi​) are completely positive (CP) maps for some Hilbert spaces Hi\mathcal H_iHi​ (i=1,2)(i=1,2)(i=1,2), and α2α_2α2​ is a pure CP map and βββ is a CP map so that α1⊗α2−βα_1 \otimes α_2 - βα1​⊗α2​−β is also CP, then β=γ⊗α2β= γ\otimes α_2β=γ⊗α2​ for some CP map γγγ. We show that a similar result holds in the context of positive linear maps when A2=C2=B(H)A_2 = C_2 = B(\mathcal H)A2​=C2​=B(H) and α2=idα_2 = idα2​=id. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map τττ from a unital C*-algebra AAA to a C*-algebra CCC, if τ⊗idkτ\otimes id_kτ⊗idk​ is decomposable for some k≥2k \geq 2k≥2, where idkid_kidk​ is the identity map on the algebra Mk(C)M_k(\mathbb {C} )Mk​(C) of k×kk\times kk×k matrices, then τττ is completely positive.
The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.
A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre's type of semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated biquadratic form over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.
Building on recent works that investigate positivity preserving matrix products, we {examine} the class of \JCP (\jcp) matrix products. A bilinear map on the Cartesian product of the space of n by n matrices with itself into m by m matrices is a \jcp matrix product if the natural linear map it induces on the tensor product of the space of n by n matrices with itself into m by m matrices is completely positive. In particular, a matrix product is \jcp if and only if its naturally associated Choi matrix is positive semidefinite. Similarly, a matrix product is \jcp if and only if it admits a Choi-Kraus representation. We use the Choi-Kraus representation of \jcp matrix products to study various basic properties, including positivity lower bounds, commutativity, units, causality, and separability. As examples, we apply our results to the Schur (Hadamard) product and the convolution product.