Quantum States and State Space

How do we describe the state of a qubit? One of the fundamental assumptions of quantum mechanics is that quantum states are represented by vectors in an abstract state space. As we mentioned in the State Vectors lesson, these vectors, like ordinary force or position vectors, have both a magnitude (length) and a direction. However, unlike ordinary vector quantities, quantum state vectors live in an abstract state space. These kinds of vector spaces have been thoroughly studied by mathematicians, but for our purposes, we will not need to know most of those details.

The abstract space contains the state vectors and comes equipped with other mathematical objects, called operators (or, in QC language, quantum gates), that allow us to change one state vector into another. These state vectors are not themselves directly observable, but they allow us to predict, as mentioned previously, the probability of the outcomes of measurements made on the qubits.