Optimal convex approximations of qubit states under Pauli distance

Abstract Utilizing the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> norm as the quantum state distance metric, we consider the Pauli distance of arbitrary target quantum states, that is, the optimal convex approximation of any qubit mixed states relative to the available state of pairs of eigenstates in three Pauli matrices. We get the complete analytical expressions for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> -distance, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> -distance and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> -distance. We determine the parameter range for the optimal approximation of each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> -distance in eight distinct scenarios. Furthermore, some trade-off relations about <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:math> -distances are also studied.