Maximum and Minimum Causal Effects of Physical Processes

We introduce two quantitative measures of the strength of causal relations in quantum theory and more general physical theories. These two measures, called the maximum and minimum causal effect, quantify the maximum and minimum changes in the output of a quantum process induced by changes in its input. The maximum and minimum causal effect possesses useful properties, such as continuity and data-processing inequality. In quantum theory, they have close connections with quantum information tasks. The maximum quantum causal effect can be used to detect quantum channels with nonzero capacity for transmitting classical information. The minimum causal effect can be used to guarantee the recoverability of quantum information: every quantum process with a high value of the minimum quantum causal effect can be approximately inverted. Moreover, we show that the quantum causal effects satisfy a duality relation: if the minimum causal effect of a quantum system <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>A</a:mi></a:math> on another quantum system <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>B</c:mi></c:math> is high, then <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>A</e:mi></e:math> must have a low value of the maximum causal effect on any other quantum system <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>B</g:mi><g:mo>′</g:mo></g:msup></g:math> that is spacelike separated with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>B</i:mi></i:math>. This duality implies a monogamy relation for quantum causal effects, and represents a fundamental difference between quantum and classical causal relations. We illustrate the application of the maximum causal effect to the analysis of two paradigmatic examples, the first involving a coherent superposition of direct cause and common cause, and the second involving a coherent superposition of multiple quantum processes.