Can we use entanglement for cascaded measurements

Question

My question is as follows: Suppose we have two entangled states, for example, a GHZ state where $|000\rangle$ and $|111\rangle$ are entangled Now, suppose the $|111\rangle$ state is entangled with another state, say $|110\rangle$ (correct me if I make wrong assumptions along the way, for example, concerning the states, however, I am asking for the general concept) Wouldn’t measuring one of the qubits tell us the state of the other two qubits, and by knowing the state of the other 2 qubits we now know the state of the 4th and 5th qubit?

How would we represent this mathematically if possible?

For context (e.g.: there are three entangled qubits and one of these qubits is entangled with another 2 $\to$ also, would that make them all entangled? )

Answer 1

As explained in the comments, your comment about the $|111\rangle$ state being entangled with another state seems to suggest that you wish to violate the monogamy of entanglement. If a qubit is maximally entangled with some other qubits then it cannot be entangled with any other qubit.

But as suggested by the title of your question and further clarified in the comments you wish to know what happens when, given a state such as $\frac{1}{\sqrt 2}(|00\cdots 0\rangle+|11\cdots 1\rangle)$, we measure only one of the provided qubits. In this case if one of the qubits measures $|0\rangle$ in the computational basis then all other qubits will be $|0\rangle$. Contrariwise if the qubit measures $|1\rangle$ then all other qubits measure $|1\rangle$ as well.

Often such generalized GHZ states have been referred to as "cat states" by analogy to Schrodinger's cat. There are multiple qubits in the cat-in-the-box and observing only one of the qubits lets us know whether poor kitty is alive or dead.