It is usually assumed that the register consists of qubits. It is also generally assumed that registers are not
density matrices, but that they are
pure, although the definition of "register" can be extended to density matrices. An n size quantum register is a quantum system comprising n
pure qubits. The
Hilbert space, \mathcal{H}, in which the data is stored in a quantum register is given by \mathcal{H} = \mathcal{H_{n-1}}\otimes\mathcal{H_{n-2}}\otimes\ldots\otimes\mathcal{H_0} where \otimes is the
tensor product. The number of dimensions of the Hilbert spaces depends on what kind of quantum systems the register is composed of.
Qubits are 2-dimensional
complex spaces (\mathbb{C}^2), while
qutrits are 3-dimensional complex spaces (\mathbb{C}^3), etc. For a register composed of
N number of
d-dimensional (or
d-
level) quantum systems we have the Hilbert space \mathcal{H}=(\mathbb{C}^d)^{\otimes N} = \underbrace{\mathbb{C}^d \otimes \mathbb{C}^d \otimes \dots \otimes \mathbb{C}^d }_{N\text{ times}} \cong \mathbb{C}^{d^N}. The registers
quantum state vector \psi of this d^N-dimensional Hilbert space can in the
bra-ket notation be written as a linear combination of some set of orthogonal basis vectors labeled |0\rangle to |d^N-1\rangle, as |\psi\rangle = \sum_{k=0}^{d^N-1} a_k|k\rangle = a_0|0\rangle + a_1|1\rangle + \dots + a_{d^N-1}|d^N-1\rangle. Such linear combinations are in quantum mechanics called
superpositions and the values a_k are
probability amplitudes. Because of the
Born rule and the
2nd axiom of probability theory, \sum_{k=0}^{d^N-1} |a_k|^2 = 1, so the possible
state space of the register is the surface of the
unit sphere in \mathbb{C}^{d^N}.
Examples: • The quantum state vector of a 5-qubit register is a
unit vector in \mathbb{C}^{2^5}=\mathbb{C}^{32}. • A register of four qutrits similarly is a unit vector in \mathbb{C}^{3^4}=\mathbb{C}^{81}. == Quantum vs. classical register ==