Unlike for Certified ScrumMaster® (CSM) there is no test nor exam to gain the Certified Scrum Product Owner® (CSPO) qualification. Evidently many people starting on their Scrum journey want some kind of assessment to see where they stand relative to others in their embedded knowledge from the coursework they’ve taken.
There are some tests that can be found, but all are created by people outside of the Scrum Alliance® with one exception.
Our friends, notably CSTs® at Applied Frameworks have created a Scrum Product Owner (CSPO) Exam . While this isn’t affiliated with Scrum Alliance, it is created by Certified Scrum Trainers with decades of experience in product management. The exam consists of 25 questions to be answered in 30 minutes with a pass mark of 76%. The test itself is designed to be a learning experience for CSPOs as well as for validation and so you can retake the test as you wish.
There is a full rationale as to why the Scrum Product Owner exam test was created.