The ket B_12 in front is not ignored. This is the state of the teo qubits 12 in Alice’s lab.
It is simply not written explicitly in the next few lines who’s purpose is to calculate the state of Bob’s qubit 3.
In this calculation the identity matrix acts trivially on the qubit 3. Once it has ‘acted’, well you dont see the matrix anymore. The tensor sumbol is not written just for convenience: its like when you write axb or a.b or ab. Its a matter of convenience of notation. The subscript 1,2,3 are all at the correct places: this matters if you have many qubits to treat to know what acts on what. Matrices and bras myst multiply kets in their ‘own’ Hilbert space (dont mix apples and oranges).
Hope this helps.
Best
It is simply not written explicitly in the next few lines who’s purpose is to calculate the state of Bob’s qubit 3.
In this calculation the identity matrix acts trivially on the qubit 3. Once it has ‘acted’, well you dont see the matrix anymore. The tensor sumbol is not written just for convenience: its like when you write axb or a.b or ab. Its a matter of convenience of notation. The subscript 1,2,3 are all at the correct places: this matters if you have many qubits to treat to know what acts on what. Matrices and bras myst multiply kets in their ‘own’ Hilbert space (dont mix apples and oranges).
Hope this helps.
Best
