Additivity of coherence measures

Abstract In this paper, we investigate the additivity of coherence measures. We demonstrate that all known additive conditions for coherence measures under incoherent operations can be reduced to a simple criterion: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>⊗</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . Utilizing this result, we present the necessary and sufficient conditions for the additivity of superadditive convex roof coherence measures. Furthermore, we examine whether the set of additive coherence measures can provide the necessary and sufficient conditions for state transformations. We find that additive coherence measures are not sufficient for identifying state transformations under incoherent operations.