Classifying coherence with a finite set of witnesses

Abstract Coherence is a fundamental resource in quantum information processing, which can be certified by a coherence witness. In order to detect all the coherent states, we introduce a useful concept of coherence witness and structure the set of coherence witnesses <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> . We present necessary and sufficient conditions of detecting quantum coherence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> </mml:math> dimensional quantum states based on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> . Moreover, we show that each coherent state can be detected by one of the coherence witnesses in a finite set. The corresponding finite set of coherence witnesses is presented explicitly, which detects all the coherent states.