Yes. However, I would write "should have" in full. For me, "should've" is a lower-grade contraction that I would normally avoid even in relatively informal writing.
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tenjingHere should've has been used to show expectation in the future. Is it correct?Not quite. The expectation is not, strictly speaking, in the future.
GPYFor me, "should've" is a lower-grade contractionIt's often spoken that way, so in writing a casual conversation, it is OK.