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This is from chabad
What is a jew?
Simple: A Jew is anyone who was born of a Jewish mother, or has undergone conversion to Judaism according to halachah (Jewish law). That’s the way it’s been since Biblical times and it’s also firmly established in the Code of Jewish Law.

And mysterious: You’ll never hear of an atheist Protestant or a Catholic Muslim, but a Jewish atheist, or even a Jew who converts to another religion, is still a Jew.1

The same applies to a convert. To provide an extreme case, let’s say a female convert changes her mind and reverts to her original religion. Any children she now has will be Jewish—because she is still Jewish.2

But don’t imagine that beliefs are irrelevant. It’s only through acceptance of all the beliefs, practices and ideology that a person becomes Jewish.

Not only that: A person who was not born Jewish and has not converted according to Jewish law may hold all the beliefs and keep all the laws and practices of Judaism and still not be a Jew.3 That makes Judaism seem more like a tribal identity than a religion.

And here’s perhaps the greatest mystery of Jewishness: Once in, there’s no way out. You can join the team, you can fumble the ball, but you can’t quit. No one can throw you out—not even G‑d.