You have to pay California tax on the income you earn for working in California, whether or not you have an apartment there. You have to file a California nonresident tax return. You also have to pay Arizona tax on all of your income, no matter where the income is from. But you will get a credit on your California tax return for the tax that you pay to Arizona on your California income.
(The more usual arrangement when you live in one state and work in another is to get the credit from the state that you live in. But Arizona and California are "reverse credit states.")
