What does it look like to split an EPR pair?

Question

I am reading Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, and I am confused about a concept presented in Section 1.3.7: Quantum Teleportation. The book writes "While together, Alice and Bob generated an EPR pair, each taking one qubit of the EPR pair when they separated." For example, if the EPR pair in question is:

$|\beta_{00}\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}$

How can we write mathematically what Alice and Bob each have in their possession?

This idea comes up again when quantum teleportation is applied to superdense coding in Section 2.3, where again Alice and Bob "share a pair of qubits in the entangled state $|\beta_{00}\rangle$.

Thanks in advance!

Answer 1

The whole point of an EPR pair is that you cannot write (without losing some information) "This is what Alice has" and "This is what Bob has". Partial descriptions can be given using reduced density matrices.

However, if you want to identify which bits Alice and Bob each have, we often use a notation like $$ (|0\rangle_A|0\rangle_B+|1\rangle_A|1\rangle_B)/\sqrt{2} $$ which helps to emphasise which ket corresponds to the qubits being held by each party.