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Quantum Superposition

A quantum system can exist in multiple states simultaneously until measured, a core concept underpinning many quantum optical phenomena.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space representing the state space of the quantum system. Assume H\mathcal{H} possesses a complete orthonormal basis set {∣ϕk⟩}k=1N\{|\phi_k\rangle\}_{k=1}^{N} (or N\mathbb{N} for infinite dimensions), where ⟨ϕk∣ϕj⟩=δkj\langle\phi_k|\phi_j\rangle = \delta_{kj}. A quantum state ∣ψ⟩∈H|\psi\rangle \in \mathcal{H} is said to be in a quantum superposition if it can be expressed as a linear combination of these basis states: ∣ψ⟩=∑k=1Nck∣ϕk⟩|\psi\rangle = \sum_{k=1}^{N} c_k |\phi_k\rangle where ck∈Cc_k \in \mathbb{C} are the complex probability amplitudes. The state must be normalized, satisfying the condition: ⟨ψ∣ψ⟩=∑k=1N∣ck∣2=1\langle\psi|\psi\rangle = \sum_{k=1}^{N} |c_k|^2 = 1 The measurement postulate dictates that the probability of observing the system in the state ∣ϕk⟩|\phi_k\rangle is Pk=∣ck∣2P_k = |c_k|^2. This linearity is guaranteed by the structure of the underlying quantum mechanical operators O^\hat{O} acting on H\mathcal{H}, such that O^(a∣ψ1⟩+b∣ψ2⟩)=aO^∣ψ1⟩+bO^∣ψ2⟩\hat{O}(a|\psi_1\rangle + b|\psi_2\rangle) = a\hat{O}|\psi_1\rangle + b\hat{O}|\psi_2\rangle for a,b∈Ca, b \in \mathbb{C}.