= Solution
A bipartite pure state is entangled when it is not a <product state>. Its <Schmidt decomposition> is $|\psi\rangle=\sum_i\sqrt{\lambda_i}|i_Ai_B\rangle$, where $\lambda_i\ge0$ and $\sum_i\lambda_i=1$. More than one nonzero coefficient means entanglement. The two <reduced density matrices> have the same nonzero spectrum, and their <Von Neumann entropy> $-\sum_i\lambda_i\log_2\lambda_i$ is the pure-state <entanglement entropy>. A maximally entangled pair of qubits has one bit of this entropy.
For mixed states, classical correlations are not sufficient: a <separable quantum state> is a convex mixture $\sum_jp_j\rho_A^{(j)}\otimes\rho_B^{(j)}$, and entanglement means that no such decomposition exists. A reduced entropy alone no longer measures mixed-state entanglement. An <entanglement witness> or a suitable separability criterion can detect it.
<Local unitary transformations> preserve the Schmidt spectrum. More general <local operations and classical communication> can discard entanglement but cannot freely create it from a separable input. <Nielsen's pure-state conversion theorem> gives the exact deterministic criterion: $|\psi\rangle\to|\phi\rangle$ is possible precisely when $\lambda^\psi\prec\lambda^\phi$. In decreasing order, every partial sum for the input is no larger than the corresponding output sum, with equal totals. The direction matters: a uniform spectrum can be converted to a concentrated one; a product state cannot deterministically become a Bell pair.
A probabilistic local filter can concentrate a partially entangled pair. For $|\psi\rangle=\sqrt\lambda|00\rangle+\sqrt{1-\lambda}|11\rangle$ with $1/2\le\lambda<1$, Alice's successful operator $F=\operatorname{diag}(\sqrt{(1-\lambda)/\lambda},1)$ produces equal amplitudes. The outcome is a Bell pair with probability $2(1-\lambda)$; a complementary operator supplies a complete measurement. Failure is heralded, so there is no deterministic creation of a more entangled pair. This illustrates <entanglement concentration>.
Many-copy <entanglement distillation> and <entanglement dilution> connect noisy or nonuniform resources with standard Bell pairs. Entanglement enables <quantum teleportation> and other protocols, but classical communication is still required; the <no-communication theorem> excludes message transmission merely by choosing a distant local measurement.
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