Entanglement and the States of Just One of the Qubits

A bit of history: Erwin Schrödinger invented the term entanglement. He noted that entanglement is the critical feature that distinguishes quantum mechanics from classical mechanics. Schrödinger’s point is that if the two-qubit system has an entangled state, then it is not legitimate to say that Alice’s qubit “has” a specific quantum state and my qubit “has” its own specific quantum state. If the system state is entangled, there is only the system state and not individual states of the individual qubits. The qubits are not independent, and therefore, cannot be said to have their own states.

Since this is such a key concept in quantum mechanics, let’s see how it plays out. The general two-qubit state in this equationEquation_9_16 is a good starting point. If we think back to the beginning of this chapter, we wrote a two-qubit state as a product state $|S⟩ = |A⟩ |B⟩$. If the system state can be factored in this way—a state vector for Alice’s qubit multiplied by a state vector for my qubit—then, and only then, can we say that Alice’s qubit is described by $|A⟩$ and mine by $|B⟩$. In the more general case shown in this equationEquation_9_16, we cannot factor the system state this way—a part associated with Alice’s qubit and a part associated with my qubit. In other words, for an entangled two-qubit state, we don’t have a definite state vector for Alice’s qubit or a definite state vector for my qubit.