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Quantum Entanglement

A phenomenon where two or more particles become linked in such a way that they share the same fate, no matter how far apart they are, a cornerstone of quantum optics.
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The statement of the theorem

Let HA\mathcal{H}_A and HB\mathcal{H}_B be finite-dimensional Hilbert spaces representing two subsystems AA and BB, respectively. The total system Hilbert space is H=HA⊗HB\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B. A pure state ∣Ψ⟩∈H|\Psi\rangle \in \mathcal{H} is defined as entangled if and only if it cannot be factorized into a product state, i.e., ∣Ψ⟩≠∣ψA⟩⊗∣ϕB⟩|\Psi\rangle \neq |\psi_A\rangle \otimes |\phi_B\rangle for any ∣ψA⟩∈HA|\psi_A\rangle \in \mathcal{H}_A and ∣ϕB⟩∈HB|\phi_B\rangle \in \mathcal{H}_B. Equivalently, the reduced density matrix ρA=TrB(ρ)\rho_A = \text{Tr}_B (\rho) and ρB=TrA(ρ)\rho_B = \text{Tr}_A (\rho) do not satisfy the condition ρ=ρA⊗ρB\rho = \rho_A \otimes \rho_B. Specifically, for a state ρ=∣Ψ⟩⟨Ψ∣\rho = |\Psi\rangle \langle \Psi|, entanglement is detected if the Schmidt rank of ∣Ψ⟩|\Psi\rangle is greater than one, meaning the Schmidt decomposition requires more than one non-zero Schmidt coefficient λk\lambda_k: ∣Ψ⟩=∑k=1Rλk∣ak⟩⊗∣bk⟩,where R>1 and ∑k=1Rλk=1.|\Psi\rangle = \sum_{k=1}^{R} \sqrt{\lambda_k} |a_k\rangle \otimes |b_k\rangle, \quad \text{where } R > 1 \text{ and } \sum_{k=1}^{R} \lambda_k = 1.