Axiom 1:

The state space of an n-qubit quantum computer is defined as the n-fold tensor product space ⨂𝐶². The pure states in this system are represented by the rays of ⨂𝐶ⁿ. Thus, these states are expressed as elements of the quotient space ⨂𝐶ⁿ/~.

Example for two qubits:

– H = 𝐶² ⨂ 𝐶²

This shows that the overall space is a 4-dimensional complex space (2² = 4). The pure states for a 2-qubit system are represented as unit vectors in this space, such as:

• ∣00⟩ (both qubits are 0)
• ∣01⟩ (first qubit is 0, second qubit is 1)
• ∣10⟩ (first qubit is 1, second qubit is 0)
• ∣11⟩ (both qubits are 1)

The vectors representing these pure states are:

– ∣00⟩: [1 0 0 0]ᵀ
– ∣01⟩: [0 1 0 0]ᵀ
– ∣10⟩: [0 0 1 0]ᵀ
– ∣11⟩: [0 0 0 1]ᵀ
These four vectors form the basis vectors of the space 𝐶² ⨂ 𝐶², represented as unit vectors in 𝐶⁴.

Axiom 2:

The observables of a quantum system are associated with Hermitian operators defined on the space ⨂𝐶²ⁿ. These operators characterize the state of the system.

Axiom 3:

When a measurement is performed on an observable O and the system is in the state ∣ψ⟩ ∈ ⨂𝐶²ⁿ, the post-measurement state is determined by the normalization of the projection P∣ψ⟩. Here, P is the orthogonal projection onto the subspace of ⨂𝐶²ⁿ formed by the eigenstates of O that are part of the linear superposition of ∣ψ⟩ consistent with the measurement outcome.

The probability of obtaining a measurement outcome is expressed as ∥P∣ψ⟩∥².

Example with a 2-qubit quantum state:

– ∣ψ⟩ = (1/√2) ∣00⟩ + (1/√2) ∣11⟩
| (1/√2) |
| 0 |
| 0 |
| (1/√2) |

The eigenstates are:

– Eigenstate 1: ∣00⟩

– Eigenstate 2: ∣11⟩

The matrix representation of the observable O could be:

| 1 0 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
| 0 0 0 1 |

The projection operator P can be defined as:

P = ∣00⟩⟨00∣ + ∣11⟩⟨11∣
| 1 0 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
| 0 0 0 1 |

Applying P to ∣ψ⟩:

P∣ψ⟩ = P[(1/√2) ∣00⟩ + (1/√2) ∣11⟩]
| (1/√2) |
| 0 |
| 0 |
| (1/√2) |

Normalization and probability calculation:

– ∥P∣ψ⟩∥² = ⟨P∣ψ⟩ ∣ P∣ψ⟩⟩ = 1

The probability of measuring ∣00⟩ is:

P00 = (1/√2)² = 1/2

The probability of measuring ∣11⟩ is:

P11 = (1/√2)² = 1/2

Axiom 4:

The computation steps in quantum computing are performed using unitary operators, allowing for the transformation of the system’s state.