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arXiv:math-ph/0406010

A simple proof of the Jamiolkowski criterion for complete positivity of linear maps

Salgado, D., J. L. Sánchez‐Gómez, Miguel Ferrero

Published Jun 4, 2004• 0 citations
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Abstract

We give a simple direct proof of the Jamiolkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accesible for physicists than others found in the literature and provides a systematic method to give any set of Kraus matrices of its Kraus decomposition.
DOI
10.48550/arxiv.math-ph/0406010

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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 $A_i$, $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, such that $ α_1 \otimes α_2 \geq β$, factorizes as $β= γ\otimes α_2$ for some positive map $γ$. In particular we show that when $α_i \colon A_i \rightarrow B(\mathcal H_i)$ are completely positive (CP) maps for some Hilbert spaces $\mathcal H_i$ $(i=1,2)$, and $α_2$ is a pure CP map and $β$ is a CP map so that $α_1 \otimes α_2 - β$ is also CP, then $β= γ\otimes α_2$ for some CP map $γ$. We show that a similar result holds in the context of positive linear maps when $A_2 = C_2 = B(\mathcal H)$ and $α_2 = id$. 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 $τ$ from a unital C*-algebra $A$ to a C*-algebra $C$, if $τ\otimes id_k$ is decomposable for some $k \geq 2$, where $id_k$ is the identity map on the algebra $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $τ$ is completely positive.

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Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and, more generally, as witnesses for states having Schmidt number not greater than k. Such maps and witnesses are related to k-positive maps, and their properties may be investigated by making use of the Jamiołkowski isomorphism. In this article we review the properties of this isomorphism, noting that there are actually two related mappings bearing that name. As a new result, we give a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps and thus for the Jamiołkowski criterion for complete positivity. Another consequence is a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.
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.
Fractional kkk-positivity: a continuous refinement of the kkk-positive scale
We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and kkk-positivity for maps between matrix algebras. Starting from a compact family of ααα-admissible unit vectors (α∈[1,d]α\in[1,d]α∈[1,d]), we define closed cones Kα\mathsf K_αKα​ of bipartite positive operators that interpolate strictly between successive Schmidt-number cones, together with their dual witness cones. Via the Choi--Jamiołkowski correspondence this yields a matching filtration of map cones Pα\mathsf P_αPα​, recovering the usual kkk-positive/kkk-superpositive classes at integer parameters and complete positivity at the top endpoint. Two results show that the fractional levels capture genuinely new structure. First, we prove a \emph{fractional Kraus theorem}: ααα-superpositive maps are precisely the completely positive maps admitting a Kraus decomposition whose Kraus operators satisfy an explicit singular-value (Ky--Fan) constraint, extending the classical rank-kkk characterization. Second, for non-integer ααα the cones Pα\mathsf P_αPα​ fail stability under CP post-composition, highlighting a sharp structural transition away from the integer theory. Finally, we derive sharp thresholds on canonical symmetric families (including the depolarizing ray and the isotropic slice), turning familiar stepwise criteria into continuous, computable profiles.
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.