"提供证据之责任" can be translated into English as "the burden of producing evidence" or, more formally in legal contexts, "the onus of adducing evidence".
"Burden of producing evidence" is a commonly used phrase that clearly conveys the idea of the responsibility to present evidence.
"Onus of adducing evidence" is a more technical and formal legal term, where "onus" means responsibility or burden, and "adducing evidence" refers to the act of presenting or introducing evidence in a legal proceeding.
Both expressions are accurate, but the choice between them may depend on the specific context and level of formality required.