Mirrored entanglement witnesses for multipartite and high-dimensional quantum systems

Abstract Entanglement witnesses (EWs) are a versatile tool to detect entangled states and characterize related properties of entanglement in quantum information theory. The verification of entangled states via EWs relies on the fact that separable states form a convex set; this also means that the framework presented by EWs generally applies to other quantum resources where free resources contain the convexity. A witness W corresponds to an observable satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Tr</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>W</mml:mi> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>sep</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mtext>⩾</mml:mtext> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for all separable states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>sep</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> ; entangled states are detected once the inequality is violated. Recently, mirrored EWs have been introduced by showing that there exist non-trivial upper bounds to EWs, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mtable columnalign="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>W</mml:mi> </mml:msub> <mml:mtext>⩾</mml:mtext> <mml:mrow> <mml:mi>Tr</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>W</mml:mi> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>sep</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mtext>⩾</mml:mtext> <mml:mn>0.</mml:mn> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> An upper bound to a witness W signifies the existence of the other one M , called a mirrored EW, such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>+</mml:mo> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>W</mml:mi> </mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">I</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">I</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . The framework of mirrored EWs shows that a single EW can be even more useful, as it can detect a larger set of entangled states by lower and upper bounds. In this work, we develop and investigate mirrored EWs for multipartite qubit states and also for high-dimensional systems, to find the efficiency and effectiveness of mirrored EWs in detecting entangled states. We provide mirrored EWs for n -partite GHZ states, graph states such as two-colorable states and tripartite bound entangled states. We also show that optimal EWs can be reflected with each other. For bipartite systems, we present mirrored EWs for existing optimal EWs and also construct a mirrored pair of optimal EWs in dimension three. Finally, we generalize mirrored EWs such that a pair of EWs can be connected by another EW, i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>+</mml:mo> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is also an EW. Our results enhance the capability of EWs to detect a larger set of entangled states in multipartite and high-dimensional quantum systems.