  We introduce two quantitative measures of the strength of causal relations in
general physical theories. These two measures called the maximum and minimum
causal effect, capture the maximum and minimum changes in a physical system
induced by changes in another system. In quantum theory, we show that both
measures possess important properties, such as continuity and faithfulness, and
can be evaluated through optimization over orthogonal pairs of input states.
For the maximum causal effect, we provide numerical lower bounds based on a
variational algorithm, which can be used to estimate the strength of causal
relations without performing a full quantum process tomography. To illustrate
our algorithm, we analyze two paradigmatic examples, the first involving a
coherent quantum superposition of direct cause and common cause, and the second
involving communication through a coherent quantum superposition of two
completely depolarizing channels.
