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Journal of Mathematical Physics

Tensor powers of 2-positive maps

Erling Størmer

Published Oct 1, 2010• 4 citations
Publisher

Abstract

By studying tensor powers ϕ⊗n of a positive map ϕ of B(K) into B(H) for finite dimensional Hilbert spaces K and H, we give conditions for a map ϕ, such that ϕ⊗n 2-positive for all n to be nondistillable and completely positive.
DOI
10.1063/1.3497628

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We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.
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.
Abstract We analyze positivity of a tensor product of two linear qubit maps, Φ 1 ⊗ Φ 2 . Positivity of maps Φ 1 and Φ 2 is a necessary but not a sufficient condition for positivity of Φ 1 ⊗ Φ 2 . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ 1 and Φ 2 are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for n -tensor-stable positive qubit maps, i.e. such qubit maps Φ that Φ ⊗ n is positive. Particular cases of 2- and 3-tensor-stable positive qubit maps are fully characterized, and the decomposability of 2-tensor-stable positive qubit maps is discussed. The case of non-unital maps is reduced to the case of appropriate unital maps. Finally, n -tensor-stable positive maps are used in characterization of multipartite entanglement, namely, in the entanglement depth detection.
Using the natural duality between linear functionals on tensor products of C∗C^*C∗-algebras with the trace class operators on a Hilbert space HHH and linear maps of the C∗C^*C∗-algebra into B(H)B(H)B(H), we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF C∗C^*C∗-algebras.