How did Claudia come back from the dead in Dark?

There are more than two Claudias

Okay. Let's assume there are two Claudias in Netflix's Dark. One from Jonas's world and another from Eva's world. Jonas' Claudia kills Eva's Claudia and she takes her place, collecting information from both the world. So far so good.

You're forgetting that there are other ways of there being alternate versions of the same person, e.g. that there are also two Jonases who are both from Jonas' world:

• The Jonas who gets saved from the apocalypse by alt-Martha, and ends up being killed by one of the alt-Martha's in Eve's headquarters. (Let's call this world Adam-A)
• The Jonas who saves himself from the apocalypse by running into the basement and will eventually grow up to become Adam. (Let's call this world Adam-B)

The point here is that there are alternate Schrödinger-like universes.

There are more than two Claudias. At the very least, there are three of them: Adam-A, Adam-B and Eve. It's unclear to me if Eve's world has a similar A/B split. If it does, that makes four Claudias.

Claudia has knowledge sources that don't reset when a cycle repeats

How can she remember all that? Because it's proven that one reality's incidents are not remembered by other realities' incidents, that's why Jonas can't remember when he was saved by Martha from Eva's world.

Claudia has two unusual sources of information: her own notebook, and conversations with her older self.

In the Adam-B world, Jonas spends 33 years of his life, from the 2019 apocalypse to 2052, together with young Noah and middle age Claudia, trying to invent time travel.

During this time, old Claudia repeatedly meets middle-age Claudia and passes information along.

Claudia also keeps a notebook where she analyzes the entire chain of events, which is passed inbetween time and therefore is able to transfer knowledge from an "older" person to a "younger" one, which means that this notebook avoids the death/rebirth cycle that people go through, which causes them to forget what happened in the last cycle.

Both of these knowledge sources, because they can be handed down from person to person, are able to avoid the fact that people from the next cycle don't remember what happened in the previous cycle (or the alternate Schröidinger universe).

You already mention that the Adam-Claudia who kills Eve-Claudia takes her place, which answers exactly how Adam-Claudia managed to figure out the information from both Adam and Eve's worlds.

How did Claudia come back from the dead in Dark?

She didn't.

• Eve-Claudia is dead, killed by one of the Adam-Claudias.
• One of the Adam-Claudias is dead, killed by Noah
• The remaining Adam-Claudia was never killed.

I'm unaware if there's a way to know if Adam-A-Claudia or Adam-B-Claudia was the one who survives - but regardless which one it is, there is one Adam-Claudia who never died. This is the Claudia who ends up talking to Adam about finally breaking the cycle.

Okay I must say this first, Claudia is not looping like we see in movies like Groundhog Day or The Edge Of Tomorrow. She is not an exception. Her memory is not cumulative. She ages like all other other characters in the series. Claudia lives her life once just like everyone else. While Dark is a causal loop, this is not an infinite loop where the looper lives at the same age till the loop is broken. I wanted to put this theory to rest as many sites are talking about this.

That said. Lets get to your question:

There are two worlds, so there are two Claudias. No confusion there, as you mentioned.

Also, Eve's World's Claudia died well before she could have learnt about Quantum Entanglement, so Eve' World has only one Claudia.

Eve's World's Claudia dies at the hands of Adam's World's Claudia. Therefore Eve's Claudia never grows old beyond her 40s.

We're now left to believe that there was only one Claudia left operating. But there wasn't. There were two. Claudia-1 goes back in time to the point of the apocalypse and uses the moment when time stops to change the path of her younger self and result in Claudia-2. This is referred to as Quantum Entanglement in the series. Like you mentioned, Jonas are three as a result of Quantum Entanglement. Remember, a finite number, three.

Similarly, we have Claudia-1 and Claudia-2. A finite number of them, two. Each Claudia lives one path of her life. They possibly took turns doing the time-traveling and scouting for information and learning about the nature of the two worlds over 33 years. They would have had to regularly meet in secret to exchange information and notes to keep up-to-date.

But no one knew there were two Claudia's in Adam's world.

In the end, Claudia-1 accepts her fate and dies at the hands of Noah. But Claudia-2 stays hidden. She waits. Finally when Adam kills Martha, she reveals herself and explains that she used quantum entanglement to be there at that point.

I've spent 33 years looking for answers in your world and in hers. I've tried to put together the pieces of the puzzle.

She has been doing this for a finite period of time, 33 years.

During the apocalypse, it stood still for a fraction of a second. And that threw everything out of balance. But when time stands still, the chain of cause and effect is also momentarily broken. One can change things. Eva knows that. She uses the loophole in your world to send her younger self off in one direction or another, in order to preserve the cycle. And I used it to send myself in another direction, too. To be here today.

This dialogue here adds to what I'm talking about. There were two Claudias in Adam's World thanks to Quantum Entanglement.

How did Claudia come back from the dead in Dark?

She didn't.

how did she return finally in old age when we know that the old Claudia is killed by Noah in the first place?

In Claudia's timeline and perspective, she talked to Adam (S03E08 - series finale) before getting killed by Noah (S02E03). It's not the other way around. Notice that after her conversation with Adam in S03E08, in the same episode, we see Claudia take the portable time machine made by H. G. Tannhaus while talking to her younger self, showing that she's about to take the final series of trips back to 1987 and 1953 (Season 2).

This is supported by Claudia's timeline as stated in the official Dark website. I'm quoting the text sorted in order, chronologically (in Claudia's perspective), from the official Dark website, starting from her visit to Adam in the series finale (S03E08):

SEASON 3 EPISODE 8
CLAUDIA EXPLAINS ALL TO ADAM

Claudia visits Adam in 2053 in the ruins of the nuclear power plant. He's just killed Martha, who was pregnant, and is beginning to realize that his plan to wipe out the world hasn't worked. Claudia explains that Eva has used quantum entanglement, meaning there is another pregnant Martha who is still alive. Eva has always been one step ahead of him, causing Adam to fall short each time, but this is the first time Claudia and Adam are having this conversation.

SEASON 3 EPISODE 8
THE ORIGINAL WORLD

Claudia tells Adam about the existence of a third world - the original world - which has been broken into two parts due to an unfortunate event. She tells him that it's possible to undo the event, ensuring Adam and Eva's worlds would never exist. Claudia is sharing this information in her own interest, because the original world is the only one in which Regina doesn't get cancer.

SEASON 2 EPISODE 2
VISITING HERSELF

Claudia visits her younger self in the nuclear power plant and tells her about the existence of time travel. She claims there is a time machine buried in her yard and that the clockmaker can explain how it works.

SEASON 2 EPISODE 3
CLAUDIA'S LAST TRIP

Claudia travels to 1954 and buries her time machine on a building plot where her future house will stand. She gives the clockmaker the book he will later write, apologizes to her father Egon, and discusses Sic Mundus and her mother Doris with Agnes Nielsen.

SEASON 2 EPISODE 3
DEATH IN THE WOODS

On the same day, Agnes tells Noah that Claudia has the final pages from the leather notebook. Noah finds Claudia in the woods and kills her with a rifle. She calmly faces her own death - after all, she was expecting it.

Notice that in Claudia's timeline and perspective, S03E08 happens before S02E02 and S02E03. And that S02E03 is described as Claudia's last trip, not S03E08.

Source: Claudia Tiedemann's timeline - official Dark website

Even if we assume that there are two realities possible for 'entanglement'. In one, Claudia gets killed by Noah and in another, she isn't.

As shown in the timeline above, in the final cycle, there is only one Claudia still in play. The Claudia that talked to Adam in the series finale and the one that got killed in S02E03 is the same Claudia.

How can she remember all that?

She didn't have to. She has the leather, Triquetra-adorned notebook written by the Unknown. That's why she had foreknowledge of her death at Noah's hands.