The trade is the latest among a series of deals the Padres have made in the past two months as they move to rebuild.
Last week, they traded All-Star left-hander Drew Pomeranz to the Boston Red Sox for top right-handed prospect Anderson Espinoza. The team also sent starting pitcher James Shields to the Chicago White Sox for starter Erik Johnson and minor league shortstop Fernando Tatis Jr., and sent closer Fernando Rodney to the Florida Marlins for right-handed prospect Chris Paddack.
Kemp, 31, has struggled in two seasons since the Padres acquired him in a trade with the Los Angeles Dodgers. This season, he is hitting .265 with 23 home runs and 69 RBIs, but has posted a .285 on-base percentage.
He signed an eight-year, $160 million deal with the Dodgers in 2012.
Olivera, a 31-year-old Cuban defector, has played just 30 games since signing a six-year, $62.5 million contract with the Dodgers before being traded to Atlanta last summer.
He played six games this season, batting .211 with two RBIs, before being hit with an 82-game suspension without pay for violating the league's domestic-violence policy. The suspension is set to end Monday.