North Korea said it carried out the test on January 6.
If confirmed it would be North Korea’s fourth nuclear test, and its first of the more powerful H-bomb.
The UN Security Council has also agreed to start drawing up new measures against North Korea.
However, skepticism remains over whether North Korea really did conduct such a test.
Experts have said the seismic activity generated by the blast was not large enough for it to have been a full thermonuclear explosion.
The White House said President Barack Obama had spoken separately to South Korea’s President Park Geun-Hye and Japan’s PM Shinzo Abe.
They “agreed to work together to forge a united and strong international response to North Korea’s latest reckless behavior”, it said in a statement.
PM Shinzo Abe told reporters: “We agreed that the provocative act by North Korea is unacceptable… We will deal with this situation in a firm manner through the cooperation with the United Nations Security Council.”
He added that Japan may take unilateral action, saying it is “considering measures unique to our nation”.
South Korea’s presidential office said in a statement that Presidents Geun-hye and Barack Obama had agreed to closely co-operate and that the international community “must make sure that North Korea pays the corresponding price” for the nuclear test, reported Yonhap news agency.
The UN Security Council held an emergency session on January 6 and condemned the test claim as “a clear threat to international peace and security.”
Japan’s ambassador to the UN, Motohide Yoshikawa, called for a swift and robust new UN resolution, insisting: “The authority and credibility of the Security Council will be put in question if it does not take these measures.”
However, the UN ambassador for Russia, which has been developing warmer relations with Pyongyang, said it would be going “too far” to say Moscow supported further sanctions.
Meanwhile, South Korea has begun limiting entry to the Kaesong industrial park in North Korea, jointly run by both countries. Only those directly involved in operations there will be allowed to enter from the South, said Seoul’s Unification Ministry.
South Korea has also said it will restart propaganda broadcasts across the border on January 8, an act which North Korea strongly opposes. The broadcasts were stopped in 2015 as part of a deal with North Korea to ease tensions that had escalated sharply in the summer.
The nuclear test came days before North Korean leader Kim Jong-un’s 33rd birthday, which falls on January 8 and is expected to be marked by celebrations.
Hydrogen bombs are more powerful and technologically advanced than atomic weapons, using fusion – the merging of atoms – to unleash massive amounts of energy.
Atomic bombs, like those that devastated two Japanese cities in World War Two, use fission, or the splitting of atoms.
South Korea’s intelligence agency also told politicians that the estimated power of the blast fell far short of what would be expected from a hydrogen bomb.
Some analysts have suggested it is possible Pyongyang tested a “boosted” atomic bomb, which uses some fusion fuel to increase the yield of the fission reaction.
The United States and nearby countries are thought to be carrying out atmospheric sampling, hoping to find leaked radioactive material, which would give clues as to what kind of device was tested.
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