How does the many-worlds interpretation look like in bra ket notation? If I understand correctly the many worlds interpretation says the universe is continously splitting into multiple branches and quantum measurements occur when decoherence causes a quantum state to select one of its possible outcomes. Each branch gets its own possible outcome and likely outcomes occur in more branches. I added this paragraph because I'm still a bit unsure about this so you can correct me if I'm wrong.
The problem is I don't understand how this would translate to quantum states this so I can't really appreciate the full glory of this theory which is sad because I think it sounds really compelling. Do multiple branches live in superposition of each other? So do you have to add the different states together? Or do you have to tensor product them together somehow?
As an example consider Schrödinger's cat. Let $U(t_2,t_1)$ be an unitary time evolution operator that takes a state from time $t_1$ to $t_2$. The poison to kill Schrödinger's cat is triggered by the decay of an atom. At time $t_1$ the atom and cat are in the state $|\text{not decayed}\rangle\otimes|\text{cat alive}\rangle$. At $t_2$ the atom that would have  a 50% chance of being decayed. At $t_3$ the poison has had enough time to do its things (poor cat). Would then the following equation be an example of many worlds theory?
\begin{align}
U(t_3,t_2)\Big(U(t_2,t_1)\big(\,|\text{not decayed}\rangle\otimes|\text{cat alive}\rangle\,\big)\Big)&=\\
U(t_3,t_2)\big(\,\frac{1}{\sqrt 2}(|\text{not decayed}\rangle+|\text{decayed}\rangle)\otimes |\text{cat alive}\rangle\,\big)
&=\\
\frac{1}{\sqrt 2}(|\text{not decayed}\rangle\otimes|\text{cat alive}\rangle+|\text{decayed}\rangle\otimes|\text{cat dead}\rangle)
\end{align}
So what would be the best way to describe many worlds? Better examples are also welcome.
 A: The Many-Worlds Interpretation (MWI) is very nice in bra-ket notation! Before diving in, what you wrote for the state of Schrödinger's cat as (no decay $\otimes$ alive cat) + (decay $\otimes$ dead cat) is correct...at least as a toy model of both the radiation-induced poisoning mechanism and cats.
Anyway, it will be convenient to consider the standard Bell state,
$$ \left| {\rm Bell} \right\rangle \, \equiv \, \frac{1}{\sqrt{2}} \left( \left| 00 \right\rangle + \left| 11 \right\rangle \right) \, ,$$
in the computational ($Z$) basis, in which $Z \left| 0 \right\rangle = \left| 0 \right\rangle$ and $Z \left| 1 \right\rangle = -\left| 1 \right\rangle$.
Now, suppose we give one qubit to Alice and the other to Bob, like always, and suppose that both Alice and Bob intend to measure $Z$ on their respective qubits. This measurement requires the use of an apparatus, and let's suppose that the apparatus works by checking if the qubit is in the excited state $\left| 1 \right\rangle$ or not. This is how measurements actually work, e.g. in trapped-ion implementations of qubits. Intuitively, we basically try to induce stimulated emission in a way that only works if the ion is in the excited state that corresponds to  $\left| 1 \right\rangle$ in the $Z$ basis. If we detect a photon (the "click" result), then we measured the particle to be in the state $\left| 1 \right\rangle$, and if we do not detect a photon (the "default" result), then we know the particle is in the state $\left| 0 \right\rangle$. In practice, we send lots of photons so that the wait time is negligible (this is not possible in all experimental settings, however).
Now, MWI tells us that the reason measurements seem weird in the Copenhagen picture is that quantum mechanics (QM) only really makes sense in closed systems. However, a system being measured is open by construction! So, if we extend the physical system to include the measurement apparatus, we once again recover unitary quantum mechanics! In this case, when Alice measures $Z$ on her physical qubit (labelled $a$), the measurement channel $\mathsf{M}$ acts on the physical qubit $a$ and also on the state of the detector (labelled $A$). The state of the detector will be $\left| 0 \right\rangle$ by default if no photon is detected, but "flips" to $\left| 1 \right\rangle$ if a photon is detected, captured by the operator $\widetilde{X}^{\,}_A$.
The measurement channel corresponding  to Alice's measurement of $Z$ on her qubit (labelled $a$) using the measurement apparatus (labelled $A$) is given by
$$ \mathsf{M} \, = \, \frac{1}{2} \left( \mathbb{1}^{\,}_a + Z^{\,}_a \right) \otimes \widetilde{\mathbb{1}}^{\,}_A + \frac{1}{2} \left( \mathbb{1}^{\,}_a - Z^{\,}_a \right) \otimes \widetilde{X}^{\,}_A \, ,$$
where the tildes denote operators that act on the apparatus (or "ancilla") qubit $A$ that stores the outcome. We can also write this as $\mathsf{M} = P^{(0)}_a \, \widetilde{\mathbb{1}}^{\,}_A + P^{(1)}_a \,\widetilde{X}^{\,}_A$, which is just a CNOT gate with the physical qubit $a$ the control qubit and the measurement device $A$ the target qubit! Importantly, the measurement channel is unitary!
This representation of measurements can also be generalized beyond qubits (see: arXiv:2210.07256, e.g.) and to other types of measurements. What I've described above is  the "strong" case of "projective measurements", but nearly identical (and much older) ideas apply to continuous, weak, and generalized measurements: e.g., in the context of cavity-QED experiments coupled to qubits, or in scenarios where "no click" doesn't automatically imply the "default" outcome $\left| 0 \right\rangle$, and you have to wait.
These weaker cases are well described by von Neumann's pointer-particle formulation of measurement (see, e.g., these notes or Preskill's notes), which reduces to what I've written above in the limit that you do the measurement very quickly.
In all cases, measurements are simply an entangling interaction between the physical system and some measurment apparatus in a particular basis. That basis always corresponds to the eigenbasis of the observable being measured (for the physical system), while the basis for the state of the measurements apparatus corresponds to something that is classically stable (i.e., won't decohere), such as the bases corresponding to the number of real photons emitted/absorbed, the charge of some ion or particle, or more generally, some symmetry-related basis.
Now, since both Alice and Bob intend to measure $Z$, we can rewrite the Bell state above to include the states of the two measurement apparati:
$$ \left| {\rm Bell} \right\rangle \, \to \, \frac{1}{\sqrt{2}} \left( \left| 00 \right\rangle + \left| 11 \right\rangle \right)^{\,}_{ab} \otimes \left| 00 \right\rangle^{\,}_{AB} \, = \, \frac{1}{\sqrt{2}} \left( \left| 0000 \right\rangle + \left| 1100 \right\rangle \right)^{\,}_{abAB} \, ,$$
where the qubits $abAB$ correspond to the physical qubit for Alice, the physical qubit for Bob, the measurement qubit for Alice, and the measurement qubit for Bob, respectively.
When we apply $\mathsf{M}^{\,}_A$ (corresponding to Alice's measurement of $Z$ on her qubit $a$), the state becomes
$$ \left| {\rm Bell} \right\rangle \, \to \, \frac{1}{\sqrt{2}} \left( \left| 000 \right\rangle + \left| 111 \right\rangle \right)^{\,}_{abA} \otimes \left| 0 \right\rangle^{\,}_B \, ,$$
and if we next apply $\mathsf{M}^{\,}_B$ (corresponding to Bob's measurement of $Z$ on his qubit $b$), the state becomes
$$ \left| {\rm Bell} \right\rangle \, \to \, \frac{1}{\sqrt{2}} \left( \left| 0000 \right\rangle + \left| 1111 \right\rangle \right)^{\,}_{abAB} \, ,$$
and several comments are in order:

*

*First, note that the order of the $Z$ measurements does not matter. One way to see this is to note the invariance under $a \leftrightarrow b$ ($A \leftrightarrow B$). The best way to see this is to note that the two channels commute: $\left[ \, \mathsf{M}^{\,}_A \, , \, \mathsf{M}^{\,}_B \, \right]=0$. This is always true of nonoverlapping measurements.

*The measurement channels $\mathsf{M}^{\,}_{A,B}$ are completely local, each acting only on $a,A$ \emph{or} $b,B$. Importantly, the $A$ and $a$ qubits are always nearby one another (and likewise for the $b$ and $B$ qubits). But nothing acts simultaneously on $a$ (and/or $A$) and $b$ (and/or $B$). Hence,there is nothing nonlocal about the measurement operation.

*There is no "active" branching! The two possible branches (i.e., "both 0" and "both 1") were always part of the Bell state from the outset. The measurements don't change (or even affect) the state of the physical qubits ($a$ and $b$)...they only ensure that the state of each measurement apparatus agrees with the state of the measured qubit.

*There is no information communicated by the measurements. There is no change to the state of the physical qubits upon either measurement, nor is there any "disturbance". Moreover, there is no way for Alice or Bob to extract information about the other's qubit from any local operation on their own qubit. Now, if they know in advance that they share this Bell state and they know that they both intend to measure $Z$, then it's true they will know that the other party's measurement outcome will be the same as their own. But there is nothing either Alice or Bob can do to determine whether the other has measured yet. The reason is that no information is sent upon measurement.

*Both outcomes occur (0 and 1), which agrees with real life. Consider, e.g., the double-slit experiment: When we don't try to determine the particle's path (i.e., which slit was traversed), we see an interference pattern between the two paths, even for a single particle. What this means is that as long as we don't prematurely entangle ourselves to a quantum system, we see evidence that both outcomes of a binary "experiment" indeed occur. It's only when we entangle ourselves to the system in a way that determines the path (i.e., if we observe an outcome) that we no longer see both paths. More on this below.

Now, I should also point out that I did not assume MWI to be true in any way to derive this formulation of measurements, nor anything written above really. Rather, this formulation of measurement follows from the Stinespring Dilation Theorem which says that all quantum channels can be represented as isometries, including measurements. One then realizes that all isometries can be embedded in unitaries. When thinking about what form the measurement unitaries ought to take, one considers actual experiments, and arrives at the above description (at least for projective measurements). That description happens to be many-worlds! Contained within this description the Copenhagen interpretation (whose wavefunction means something different and is anthropocentric): Basically, the Copenhagen wavefunction $\left| \psi \right\rangle$ recovers from the Stinespring/MWI wavefunction $\left| \Psi \right\rangle$ upon projecting onto a particular measurement outcome (via an operation on the apparatus' state) and renormalizing. Finally, by Choi's Theorem, the Stinespring/MWI representation of measurements is equivalent to CPTP maps aka Kraus operator channels.
As a final example using the Bell state, let's consider a slightly different scenario in which Alice still measures $Z$ on qubit $a$, but Bob instead measures $X$ on qubit $b$. In this case, the two outcomes are completely independent and have probability 1/2 each. It'll be more convenient if we write the Bell state in the $Z$ basis for qubit $a$ and $X$ basis for qubit $b$, where we define $\left| 0 \right\rangle_x = \left( \left| 0 \right\rangle_z + \left| 1 \right\rangle_z \right)/\sqrt{2}$ and $\left| 1 \right\rangle_x = \left( \left| 0 \right\rangle_z - \left| 1 \right\rangle_z \right)/\sqrt{2}$, so that
$$ \left| {\rm Bell} \right\rangle \, = \, \frac{1}{2} \left( \left| 00 \right\rangle + \left| 01 \right\rangle + \left| 10 \right\rangle - \left| 11 \right\rangle \right) \, ,$$
in the $Z^{\,}_a \otimes X^{\,}_b$ basis (i.e., the measurement basis). As long as we work in the measurement basis, the measurement channels $\mathsf{M}$ are the same as before (measuring a Pauli operator $A$ with $A^2 = \mathbb{1}$ no a qubit always acts as a CNOT gate where the projectors for the control qubit correspond to the eigenbasis of the operator $A$ being measured). Upon measurement of both qubits, we have:
$$ \left| {\rm Bell} \right\rangle \, \to \, \frac{1}{2} \left( \left| 00 \right\rangle \otimes \left| 00 \right\rangle + \left| 01 \right\rangle \otimes \left| 01 \right\rangle + \left| 10 \right\rangle \otimes \left| 10 \right\rangle - \left| 11 \right\rangle \otimes \left| 11 \right\rangle \right) \, ,~$$
where again, the qubits correspond to $abAB$. As before, there is no "active" branching upon measurement: The branches are already encoded in the state as long as we work in the measurement basis! The other points above (e.g., that measurements are local) still hold.
Moreover, we see that if Alice and Bob don't discuss which operator the other intends to measure ahead of time, then even if they know when the other intends to measure, there is no action they can perform on their qubit alone to determine which operator the other measured. That's because information isn't communicated by measurements alone, and the Stinespring/MWI picture makes this very clear (see also arxiv:2206.09929).
Now, let's think more about "branching". Basically, the reason that we experience wavefunction collapse is not due to our being conscious, nor human, nor in any way special. It's merely because we are classical. A defining feature of classical systems is that their degrees of freedom are more entangled with the rest of the universe than they are internally (i.e., with each other). In fact, it is now understood that entanglement is what leads to the emergence of thermodynamics as well, and microscopic quantum entanglement entropy is the thermodynamic entropy. This means that classical systems are maximally entangled with their environments (since their extensive entropies scale as the volume of the system).
Associated with this entanglement is the process of quantum decoherence. Basically, there are preferred basis states (as mentioned above, they correspond to symmetries), and states that are superpositions in this basis are "unstable". Basically, the chaotic quantum dynamics that entangles classical systems with their environments (via interactions) causes scrambling kills off any quantum coherence (hence the term "decoherence"). At least, as far as other classical objects (whose experience is restricted to the preferred basis) are concerned. I'll get to the "why" momentarily.
What this means in practice is that if you try to prepare a classical system in a superposition of two classically distinct basis states (e.g., corresponding to different charges of an object, different numbers of photons, etc), then decoherence will absorb that superposition into the state of the entire universe very rapidly. As a result, the coherently superposed state will appear to decay (or collapse) into a single state.
So basically, if I try to prepare a box with Schrödinger's poor cat in it with the radiation-activated poison, the fact that the cat is maximally entangled with the rest of the universe actually means that there's not much point in putting the cat in the box lol. But suppose for argument's sake that I prepare the cat's state in isolation, and maintain this isolation while I put the cat in the box. As I start the experiment, I allow interactions with the rest of the universe again. Now, in perfect isolation (i.e., without any interactions with the rest of the universe), sure, the cat can be in a superposition of being both alive and dead. But the second we view the cat's state—or, more accurately, the second any degree of freedom in the entire rest of the universe interacts with any degree of freedom in the cat—the superposition of the two states "dead" and "alive" is absorbed into the full state of the universe. The universe then appears to contain two classical realities: one in which the cat is alive, and one in which it is dead. But we can experience only one of these realities (this is an observational fact only), and hence see the cat as alive or dead. If we try to prepare a state in which it's both, decoherence just recollapses that state. Now let's understand why.
What I've written above is not merely conjecture, but is actually how thermodynamics is now understood to emerge from microscopic quantum mechanics! Importantly, chaotic systems (i.e., dynamical systems that thermalize / relax to thermal equilibrium / obey thermodynamics at late times) forget as much information about their initial conditions as possible! But quantum systems can't actually "forget" anything, because unitary evolution is reversible. What happens is that the evolution of the whole universe (a closed system) indeed retains all information, but the information about a local subsystem is smeared out throughout the universe under dynamics with entangling interactions, so that it can no longer be accessed locally. However, information about symmetries cannot be forgotten—at least, not as easily—since there are associated charges to conserve. This is also why hydrodynamics is dominated by conserved charges. It is a key feature of chaotic dynamics and relaxation.
Hence, chaotic dynamics privilege certain operators corresponding to conserved charges (along with operators that commute with the charge operators). Correspondingly, chaotic dynamics "prefers" the eigenbases of these charge operators. Moreover, operators that couple between different values of the conserved charge are not conserved themselves, and their expectation values quickly decay to zero. Perhaps more rigorously, the probabilities associated with these transition go to zero as chaotic dynamics bring the system to a steady state. That steady state looks like a superposition (or mixed state) of different charge sectors, with operators that couple between these sectors heavily suppressed.
Why is this meaningful? Well, if we think about measuring an observable in a quantum state, the von Neumann pointer-particle formalism tells us that we need to couple some "reference" system (i.e., the apparatus) to the system being measured. But we need to be able to "read off" the result of the experiment (i.e., the state of the apparatus) without requiring another measurement, without messing up the state of the apparatus, and while ensuring that the state of the apparatus doesn't appear to overlap with multiple values. This is most easily accomplished if there exists some "preferred" basis that is classically accessible to us...as it happens, such a basis exists! It's the classical basis I described above, which is related to symmetries. So personally, I wouldn't say that collapse is an integral feature of measurements at all—after all, a measurement is merely an entangling interaction. Rather, the collapse comes from the fact that the only way we can "read off" an outcome from such an interaction is if we find a way to store the outcome in a basis that collapses itself. And that's what we do...in fact, that's why pretty much all quantum measurements use excitations of atoms and photon detection! Because these correspond to the charge/photon number bases, which are classically "preferred".
Now, back to the branches. Crucially, the measurement apparatus is a classical system (or strongly coupled to one), and thus cannot be observed by other classical systems to realize a superposition of macroscopically distinct states (in the clasically preferred basis). This also makes the measurement apparatus more robust and more useful. But the measurement operation (i.e., the channel $\mathsf{M}$) clearly prepares such a superposition (i.e., in the Bell case, the two apparati are basically absorbed into GHZ-like states above). What happens (in our perception as classical objects) is that the superposition of the apparatus' state is absorbed into the state of the universe as a whole, so that the universe's state is a superposition of states corresponding to distinct classical realities!
In the case where Alice and Bob both measure $Z$, the two "realities" correspond to "both 0" and "both 1". Information about the reality spreads throughout the universe and can be made available to all classical entities, all of whom will agree on the outcome by the nature of classical reality (we all experience ourselves as participating in the same one, owing to our entanglement). We only observe one classical reality, and the different realities are merely different states in some preferred basis (the one we experience), all of which contribute to the true state of the universe. Importantly, there is only one universe, and it has only one state.
Now, where does the information about the reality originate? Well for one measurement, it originates at the location of measurement. In the case where Alice measured $Z$ and Bob measured $X$, there are two outcomes needed to specify the precise reality, since there are four "realities" total. Now, before you know that Bob measured, you don't care which Bob reality we're all on, and likewise for Alice's measurement. So basically, the particular reality is independently determined by the two measurement outcomes, and information about that reality may be sent independently from Alice and Bob's measurements. More interesting is the case where they both measure $Z$. In that case, the fact that the outcomes agree is guaranteed by the state itself, and there are only two realities. Information about which reality we're on originates from both measurement locations (or either), and spreads locally from there.
Even more importantly: The state of the universe evolves deterministically and locally. Thus, Einstein was correct that God does not roll dice, at last as far as the state of any closed system (including the universe) is concerned. It is only our experience that seems random, but nature doesn't have any obligation to care about our experience! There is also no "collapse" in this picture, avoiding the well-known "measurement problem" entirely. Moreover, this picture of measurement makes it clear that there is nothing to worry about in the "paradox" put forth by Einstein, Podolsky, and Rosen (EPR): Measurements of an entangled state cannot send any quantum information nor achieve a useful quantum task on their own. However, by combining such a measurement with classical signals (indicating the measurement outcomes), it is possible to transfer quantum information. But that classical signal obeys relativity, and therefore causality / locality. What this means is that there is nothing nonlocal about measurements, and quantum physics is local (the only meaningful notion of locality is the notion that information must propagate at finite speed $v \leq c$, which certainly holds for quantum measurements, as proven in arxiv:2206.09929). So all is well!
Lastly, all of the interpretations basically agree on purely unitary dynamics of closed quantum systems, which is why I focused on measurements above. However, the point of MWI is that measurements are merely another example of an entangling interaction, which appear weird because we consider an open system, but appears to be regular old unitary QM if we correctly identify the closed system by including the measurement apparatus. The rest of what I wrote explains that measurements are completely local (in the sense that information can be traced and never exceeds the speed $c$), that the evolution of the universe is deterministic (not random), that there is a sense of realism (i.e., the outcomes are known before the measurement), and that "branching" is not some active process, but the branches exist all along (measurements merely "unearth" them). By the way, the Born rule comes from $p_n = \left\langle \widetilde{P}^{(n)}_{\rm meas} \right\rangle$ is just the expectation value of finding the measurement device in the state $n$, contrary to what people claim (that MWI doesn't explain the probability thing). The only "downside" to MWI is that it can't predict which outcome we'll observe. But it turns out, no interpretation can—that's just a feature of quantum mechanics and really the weirdness comes from the fact that we are classical.
I hope this answered your question (albeit late), and let me know if there's anything I could clarify, or if there are typos!
A: MWI is simply standard quantum mechanics without any additional axioms. The Copenhagen interpretation involves making up more machinery (projection onto an eigenstate when we make a measurement). So in bra-ket notation, MWI just looks like standard quantum mechanics.
For Schrodinger's cat, I don't think you're going to get a satisfactory notation using a finite-dimensional vector space. It's an irreversible exponential decay process, and you can't describe such a process using two states. You need decay into an infinite continuum of states.
You might think, OK, then let's work with something like a  Schrodinger's cat experiment based on the output of a Stern-Gerlach spectrometer. Then the problem is that you presumably want to work out how the measurement process looks. But again, measurement is an irreversible process, and you can't mock up an irreversible process using a finite-dimensional vector space. Measurement is irreversible precisely because the measuring device has so many degrees of freedom.
So for a really satisfying description using only standard quantum mechanics, you really want the full description in terms of decoherence.

If I understand correctly the many worlds interpretation says the universe is continously splitting into multiple branches

Not really. The stuff about branching is not present at the level of the basic formalism of MWI. Branching is really just a story that we conscious observers tell ourselves in order to make sense of why decoherence prevents us from simultaneously experiencing two different realities, or observing interference between two cat states. If branching was literally going to be present in the formalism, it would be something like a non-Hausdorff manifold, which isn't what any actual formalism of quantum mechanics uses.
