Positive tensor products of maps and <i>n</i> -tensor-stable positive qubit maps

Abstract We analyze positivity of a tensor product of two linear qubit maps, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mstyle> <mml:mo>⊗</mml:mo> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:mstyle> </mml:math> . Positivity of maps <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mstyle> </mml:mstyle> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:mstyle> </mml:math> is a necessary but not a sufficient condition for positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mstyle> <mml:mo>⊗</mml:mo> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:mstyle> </mml:math> . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mstyle> </mml:mstyle> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mstyle> </mml:mstyle> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mstyle> </mml:math> that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊗</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mstyle> </mml:mstyle> </mml:math> 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.