As we have seen, even a single qubit is a surprisingly interesting object. However, the real power of quantum information processing begins with systems of two or more qubits. Before studying these in detail we need to expand our notation a little.

Consider a system of two qubits, labeled a and b, each of which has two basis states, |0⧽ and |1⧽. The whole system then has four basis states, which can be written as |0a0b⧽, |0a1b⧽, |1a0b⧽ and |1a1b⧽, and can be found in any general superposition of these states, so that it occupies a four-dimensional Hilbert space. In the same way, a system of three qubits inhabits an eight-dimensional Hilbert space, and so on. This exponential increase in the size of the Hilbert space with a linear increase in the number of qubits underlies the power of quantum computers.

Direct products

The size of the Hilbert spaces involved can also be a huge problem, however, making it difficult to describe states of systems with many qubits. A partial solution is to note that some states can be described in a simpler way, using the concept of direct products. These states, in which the individual qubits can in principle be discussed separately, make up a tiny minority of the states accessible to a multi-qubit system, but include many important states, most notably the basis states. States of this kind are said to be separable, and states which are not separable are said to be entangled. Entangled states are much more interesting than separable ones, but it is wise to begin with the simpler case.