Write a source and target in Schmidt decomposition with squared coefficients $(1-s,s)$ and $(1-t,t)$, where $0<s,t\le 1/2$. If $s\ge t$, Nielsen's pure-state conversion theorem allows deterministic conversion. If $s<t$, the Schmidt-tail entanglement monotone gives $p\le s/t$. Alice attains this bound with a local two-outcome measurement in quantum mechanics whose success Kraus operator is $K_s=\mathrm{diag}(\sqrt{s(1-t)/(t(1-s))},1)$ and whose failure operator is $K_f=\mathrm{diag}(\sqrt{1-s(1-t)/(t(1-s))},0)$. They obey $K_s^{†}{K}_s+K_f^{†}{K}_f=I$. The successful unnormalized state is $\sqrt{s/t}$ times the target, and failure leaves a product state. A classical message tells Bob which branch occurred. Known local changes of Schmidt bases allow the same protocol for arbitrary two-qubit pure states with these coefficients.

For bipartite pure source and target states with positive target entanglement entropy, the optimal rate of LOCC conversion with vanishing output error is the ratio of their entanglement entropies. Here vanishing error means that the trace distance between the output and $|\varphi ⟩⟨\varphi {|}^{\otimes m}$ tends to zero; a recorded failure probability must also tend to zero. Entanglement concentration first provides $nS({\rho }_A^{\psi })-o(n)$ Bell pairs, and entanglement dilution needs $mS({\rho }_A^{\varphi })+o(m)$ such pairs. This achieves the ratio. Conversely, refine the output into pure branches. Average monotonicity of pure-state entanglement entropy bounds their average entropy by the input entropy; continuity bound for quantum conditional entropy, with trivial conditioning, makes the output entropy loss per target tend to zero. This forbids a larger rate. The convention permits approximations for finite blocks: exact deterministic conversion of every finite block can obey stricter majorization constraints. The reversible concentration construction is given in the original concentration paper.

Let $|{\Phi }_d⟩=d^{-1/2}{\sum }_{j=0}^{d-1}|j,j⟩$ and $1\le m\le d$. It can be converted deterministically by LOCC to $|{\Phi }_m⟩$. For each $m$-element subset $S$ of the local basis labels, let ${\Pi }_S$ be its projector and let Alice's Kraus operator be $K_S={\Pi }_S/\sqrt{\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right)}$. Every label occurs in $\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right)$ subsets, so ${\sum }_S{K}_S^{†}{K}_S=I$. Each outcome has probability $m/[d(\genfrac{}{}{0ex}{}{d-1}{m-1})]=1/\left(\genfrac{}{}{0ex}{}{d}{m}\right)$ and leaves the maximally entangled state $m^{-1/2}{\sum }_{j\in S}|j,j⟩$. Alice reports $S$, and both parties relabel these $m$ basis states to $0,\dots ,m-1$ by local unitary operations.

For $d=6$, $m=4$, only three outcomes are needed: partition the six labels into disjoint pairs $T_0,T_1,T_2$ and use $K_r=(I-{\Pi }_{T_r})/\sqrt{2}$. Each label is retained twice, proving completeness, and each outcome has probability $1/3$. Every branch leaves four equal Schmidt coefficients, hence two independent Bell pairs after binary relabelling.

Let $c,s\ge 0$ and $c^2+s^2=1$. Starting from the Bell state $(|00⟩+|11⟩)/\sqrt{2}$, Alice applies the measurement in quantum mechanics with Kraus operators $M_0,M_1$ displayed above. Completeness follows from $M_0^{†}{M}_0+M_1^{†}{M}_1=I$. Each outcome has probability $1/2$. The normalized outputs are $c|00⟩+s|11⟩$ and $s|00⟩+c|11⟩$. Communicating the outcome allows both parties to apply a Pauli X gate in the second branch, making both outputs equal to the first. Thus this LOCC protocol deterministically converts one Bell pair into any two-qubit pure state with these Schmidt coefficients, after appropriate local unitary operations. It includes the product state endpoint without postselection.

A local minimum is strict when every other point of the domain of a function in some neighbourhood has a larger value. Strictness is local: two separated strict local minima can have the same value, as $x^4-2x^2$ at $\pm 1$ demonstrates. Negating the function gives a strict local maximum.

For a complete discrete valuation ring $R$, a simple root of a polynomial modulo the maximal ideal lifts uniquely to a root in $R$.

The bounded Lipschitz norm combines the Lipschitz constant and supremum norm of a bounded real-valued Lipschitz function on a metric space. Both summands satisfy the triangle inequality; the supremum term makes the sum definite even on constant functions. Its unit ball requires the sum of the two bounds to be at most one, a stronger requirement than bounding each separately by one.

A bi-Lipschitz map between metric spaces satisfies the displayed bounds for all input pairs, with constants $0<m\le M<\infty$. The lower bound makes it injective; its inverse on its image is Lipschitz continuous with bound $1/m$, while the map itself has Lipschitz bound $M$. Thus it controls both stretching and compression. It is a homeomorphism onto its image, preserves Hausdorff dimension, and has bi-Lipschitz distortion at most $M/m$. Uniformly multiplying every distance by the same positive factor has distortion one, as explained in metric embedding.

If an entire function $f=u+iv$ has $uv$ bounded above, then $f$ is constant. Indeed $F=\exp (-i{f}^2)$ is entire and $|F|=\exp (2uv)$ is bounded, so the Liouville theorem makes $F$ constant. Differentiating gives $f{f}^{\prime }=0$. The derivative of $f^2$ is therefore zero, and continuity on the connected complex plane makes $f$ itself constant. No separate boundedness of $u$ or $v$ is required.

If an entire function $g$ satisfies $\mathrm{Re}(cg)\ge b$ for constants $c\ne 0$ and real $b$, then $e^{-cg}$ is entire and has modulus at most $e^{-b}$. The Liouville theorem makes it constant, and differentiating gives $g^{\prime }=0$ since the exponential never vanishes. The conclusion also applies after rotating and translating any containing half-plane; separate bounds on the real and imaginary parts are unnecessary.

For an irrational $x$ with continued fraction convergents $p_n/q_n$, one has $|x-p_n/q_n|\le 1/(q_n{q}_{n+1})\le 1/(a_{n+1}{q}_n^2)$. The Liouville approximation theorem for a quadratic irrational gives a positive lower bound $C/q_n^2$ instead. Thus unbounded partial quotients preclude quadratic irrationality, and in particular imply $x^2$ is irrational.

For an oriented closed curve, geometric writhe is the signed crossing number averaged over generic viewing directions. For an individual link diagram, writhe of a link diagram is its signed crossing sum. Both depend on geometry or projection rather than only the link isotopy class.

The Borromean rings are a three-component link whose components are unknotted, which is nontrivial although removing any one component leaves an unlink. Its complement has a complete finite-volume hyperbolic three-manifold structure obtained from two regular ideal octahedra.

A split link admits an embedded sphere in its complement with nonempty portions of the link on both sides. A split diagram admits a simple closed curve in the projection sphere separating nonempty portions of the diagram. A split diagram always represents a split link; the converse need not hold for an arbitrary diagram.

A link invariant assigns the same value to links related by link isotopy. The Jones polynomial applies to oriented links, while its bracket precursor retains a framing dependence. An invariant need not distinguish all link types.

The area of a symmetry sphere in the weak-field isotropic metric is $4\pi {\rho }^2(1+2GM/\rho )$. Equating it to $4\pi r^2$ defines the areal radius and gives the displayed change. To first order $dr=d\rho$, and replacing $1/\rho$ by $1/r$ changes the perturbation only at second order. This explains why the isotropic angular correction disappears in the linearized Schwarzschild metric without discarding spatial curvature.

The first-order null-orbit equation $(u^{\prime }{)}^2=r_S^2{E}^2/h^2-u^2(1-u)$ formally has a circular solution at $u=2/3$, with $|h/E|=3\sqrt{3}{r}_S/2$. Its second-order equation gives a perturbation $\delta u^{\prime \prime }=\delta u$, hence instability. However, $u=2/3$ is outside the weak-field expansion. The exact Schwarzschild spacetime has a corresponding unstable photon sphere, but that physical result needs the exact geometry; extrapolating the linearized metric is not a controlled proof.

For signature $(+---)$ and the curvature convention giving $\square {\bar{h}}_{ik}=-16\pi G{ϵ}^{-1}{T}_{ik}$, take $u=t-x$, ${\ell }_i=(1,-1,0,0)$ and $T_{ik}=ϵ\rho (u){\ell }_i{\ell }_k$. A trace-free Lorenz-gauge particular solution has the displayed form. The vector ${\ell }^i=(1,1,0,0)$ is null, and ${\Gamma }^i{}_{jk}{\ell }^j{\ell }^k=0$, so co-propagating test photons are undeflected by this beam field. For a transversely uniform source, $H=4\pi G\rho (u)(y^2+z^2)$ is one particular solution. A function of $u$ alone has vanishing wave operator and cannot source nonzero $\rho$; homogeneous gravitational perturbations and boundary conditions remain separate data.

The effective boundary coefficient is $\tilde{\Gamma }=-(2k-ic)b^{♯}(-k)/(a\mathcal{D})$. All its ingredients come from initial scattering and the Robin boundary condition. Genuine zeros produce pole data; common zeros in numerator and denominator can instead be removable. It is incorrect to assert absence of boundary poles for every real Robin parameter.

Here $\Gamma =B^{♯}/(ad)$ and $\Delta =\mathcal{D}/(2k+ic)$. The difference follows algebraically from the half-line NLS global relation and the Robin symmetry. Multiplication by the spatial-temporal jump factor gives $e^{2ikx-4i{k}^2(T-t)}$, decaying in the second quadrant for $t<T$. A triangular analytic change of the Riemann-Hilbert problem removes this finite-horizon defect while retaining the reconstruction and transferring all genuine pole information to the known determinant. No unproved large-time decay of the boundary values is needed.