dimension state space tensor product vector space Bell state entangled correlated qubit vector vectors ion qubits probability amplitudes bit computer bits
Each additional qubit doubles the dimension of the state space.
As an example, the vector 1/√2|00⟩ + 1/√2|01⟩ represents a two-qubit state, a tensor product of the qubit |0⟩ with the qubit 1/√2|0⟩ + 1/√2|1⟩.
This vector inhabits a four-dimensional vector space spanned by the basis vectors |00⟩, |01⟩, |10⟩, and |11⟩.
The Bell state 1/√2|00⟩ + 1/√2|11⟩ is impossible to decompose into the tensor product of two individual qubits—the two qubits are entangled because their probability amplitudes are correlated.
In general, the vector space for an n-qubit system is 2-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2 classical values.
