Its first nuclear test was in 2006, and successive blasts are believed to have been aimed at refining designs and reliability as well as increasing yield.
"It is absolutely unacceptable if North Korea did force another nuclear test, and we must protest strongly", Abe said.
South Korea's President is to chair a National Security Council meeting.
USA monitors measured a 6.3-magnitude tremor near the North's main testing site site at Punggye-ri.
Earlier in the day, the party's newspaper ran a front-page story showing photos of Kim examining what it said was a nuclear warhead being fitted on to the nose of an intercontinental ballistic missile.
The hydrogen bomb report by North Korea's official KCNA news agency comes amid heightened regional tension following Pyongyang's two tests of intercontinental ballistic missiles (ICBM) in July that potentially could fly about 10,000 km (6,200 miles), putting many parts of the mainland United States within range.
If the initial data holds, it would make it the most powerful weapon that the Democratic People's Republic of Korea, as the country is officially known, has ever tested.
The state news service cited Kim referring to the device as a "thermonuclear weapon with super explosive power".
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"It shows that the state has the long-term will and determination to do it despite the costs", says Daniel Pinkston, a Korea expert at Troy University in South Korea, referring to the sanctions.
The North conducted two nuclear tests a year ago alone.
Experts and outside governments are skeptical of the hydrogen claim, but it is nearly impossible to independently confirm North Korean statements about its highly secret weapons program. Talking is not the answer!"Trump said in a Twitter post".
She wrote on Twitter: "It doesn't need to be shaped like that on the outside, but they threw in a diagram, just so we would get the message".
"The bottom line is that they probably are going to do a thermonuclear test in the future. And the equations can to grip on things like depth or the geology of the area, and we don't know those kinds of facts", she said.
It came as USA and South Korean forces were nearing the end of the 10-day annual Ulchi Freedom Guardian joint exercises, which the North regards as a rehearsal for invasion.
There will be some scepticism about the claim from experts about Pyongyang's assertion that it has mastered hydrogen technology.
The United States and South Korea responded with a mock strike drill over the Korean peninsula.
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