  The asymmetry of quantum states is an important resource in quantum
information processing tasks such as quantum metrology and quantum
communication. In this paper, we introduce the notion of $asymmetry~weight$ ---
an operationally motivated asymmetry quantifier in the resource theory of
asymmetry. We study the convexity and monotonicity properties of asymmetry
weight and focus on its interplay with the corresponding semidefinite
programming (SDP) forms along with its connection to other asymmetry measures.
Since the SDP form of asymmetry weight is closely related to asymmetry
witnesses, we find that the asymmetry weight can be regarded as a
(state-dependent) asymmetry witness. Moreover, some specific entanglement
witnesses can be viewed as a special case of an asymmetry witness --- which
indicates a potential connection between asymmetry and entanglement. We also
provide an operationally meaningful coherence measure, which we term
$coherence~weight$, and investigate its relationship to other coherence
measures like the robustness of coherence and the $l_1$ norm of coherence. In
particular, we show that for Werner states in any dimension $d$, all three
coherence quantifiers, namely, the coherence weight, the robustness of
coherence, and the $l_1$ norm of coherence, are equal and are given by a single
letter formula.
