consequences, so we can arbitrarily choose to be real (or in the case that is zero), leaving just two degrees of freedom: where is the physically significant relative phase. The possible quantum states for a single qubit can be visualised using a Bloch sphere (see picture). Represented on such a 2-sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where and are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state would lie on the equator of the sphere at the positive X-axis. In the classical limit, a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles. The surface of the Bloch sphere is a two-dimensional space, which represents the observable state space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles and . Mixed state A pure state is fully specified by a single ket, a coherent superposition, represented by a point on the surface of the Bloch sphere as described above. Coherence is essential for a qubit to be in a superposition state. With interactions, quantum noise and decoherence, it is possible to put the qubit in a mixed state, a statistical combination or “incoherent mixture” of different pure states. Mixed states can be represented by points inside the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles and , as well as the length of the vector that represents the mixed state. Quantum error correction can be used to maintain the purity of qubits. Operations on qubits There are various kinds of physical operations that can be performed on qubits. Quantum logic gates, building blocks for a quantum circuit in a quantum computer, operate on a set of qubits (a register); mathematically, the qubits undergo a (reversible) unitary transformation described by multiplying the quantum gates unitary matrix with the quantum state vector. The result from this multiplication is a new quantum state. Quantum measurement is an irreversible operation in which information is gained about