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A bipartite pure state is entangled when it is not a product state. Its Schmidt decomposition is $|\psi ⟩={\sum }_i\sqrt{{\lambda }_i}|i_A{i}_B⟩$, where ${\lambda }_i\ge 0$ 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 ${\sum }_j{p}_j{\rho }_A^{(j)}\otimes {\rho }_B^{(j)}$, 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: $|\psi ⟩\to |\varphi ⟩$ is possible precisely when ${\lambda }^{\psi }\prec {\lambda }^{\varphi }$. 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 $|\psi ⟩=\sqrt{\lambda }|00⟩+\sqrt{1-\lambda }|11⟩$ with $1/2\le \lambda <1$, Alice's successful operator $F=\mathrm{diag}(\sqrt{(1-\lambda )/\lambda },1)$ 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.

The nonselective channel of a hypothetical spacelike nonlocal measurement in quantum mechanics must not allow a distant input change to alter local output statistics. For a rank-one measurement in the rotated even and odd two-qubit bases with angle $\theta$, Alice's output satisfiesBob can change the last correlation using only his local Pauli Z gate while Alice keeps the input $|+⟩$. The resulting signal excludes $0<\theta <\pi /4$. Degenerate observables that do not distinguish this basis are not covered by the rank-one argument. Locally recording a jointly encoded result, which is decoded later by local operations and classical communication, is compatible with quantum no-signalling.

Combine the single-copy Werner filtering normalization bound with concurrence scaling under invertible local filters. Every complete classical record of local operations and classical communication has a product filter $A\otimes B$; singular branches are separable. Thus every retained branch has concurrence at most that of the input. Coarse-graining successful records preserves the bound by convexity. Since concurrence of a two-qubit Werner state increases strictly with $F$, no successful single-copy protocol can output a Werner state with larger $F$. Several input copies permit collective distillation and are outside this statement.

Alice holds the first two qubits and Bob the third of a three-vertex graph-state wire. Alice measures at $\alpha$ and $(-1{)}^r\beta$, with outcomes $r,s$, and sends these two bits. Bob applies $Z^r{X}^s$, the inverse of the known Pauli frame $X^s{Z}^r$, obtaining $U(\beta )U(\alpha )|+⟩$ up to global phase. Bob need not know the angles. Every branch succeeds, with branch probability $1/4$, and no postselection is needed. Alice's measurements and Bob's corrections constitute local operations and classical communication on the previously shared entangled resource.