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.

An ideal measurement preserving the measured eigenspaces. In the projective setting, the selective instrument $\rho \mapsto P_j\rho P_j$ preserves every vector within outcome $j$, including coherence inside a degenerate eigenspace. Repeated measurements agree in the absence of intervening dynamics changing the measured quantity. Measuring a finer observable can destroy degeneracy coherence and need not realize the same Lüders rule instrument.

A physical measurement interaction must correlate the system with a meter; in Haroche cavity photon nondemolition experiment that meter is a chemical atom. For a measured observable $A$, a useful sufficient ideal condition is $[A,H_{\mathrm{int}}]=0$ together with $[A,H_0]=0$, where $H_{\mathrm{int}}$ is the interaction Hamiltonian operator and $H_0$ the free Hamiltonian operator. These conditions protect its eigenvalues during probing and between probes. They do not protect every other observable, nor prevent conditional quantum state changes. In particular, preserving photon number can disturb the field's quantum phase.

The single-band Hubbard model balances electron hopping between localized spatial orbitals against local interaction. A common form isThe first sum is over neighbouring unordered site pairs. The parameter $t$ sets hopping, while $U_H>0$ penalizes double occupancy. At $U_H=0$ electrons occupy independent energy bands; strong repulsion suppresses local charge fluctuations and can favour local-moment behaviour. This competition was the motivation for the model's treatment of narrow-band electron correlation.^[ref]

At one electron per site and $U_H\gg |t|$, virtual double occupancy yields the antiferromagnetic scale $J_{\mathrm{ex}}=4t^2/U_H$. The two-site Hubbard dimer makes this concrete: the singlet state is lowered by $(\sqrt{U_H^2+16t^2}-U_H)/2$, while triplet states have no such shift. Its small-hopping limit gives Hubbard kinetic exchange.

Interaction conventions must match. For dimensionless spin-half operators, $-2U({\mathbf{S}}^2-3n/4)=3U{n}_{\uparrow }{n}_{\downarrow }$, so that spin-square convention has $U_H=3U$, not $U_H=U$.

Geometrical frustration occurs when an interaction graph prevents all local preferred configurations from being satisfied simultaneously. For classical equal-length vector spins with positive antiferromagnetic exchange, a bipartite graph permits opposite vectors on its two sublattices, minimizing every bond. A triangle cannot make all three pairs antiparallel.

For three classical vectors of length $S$ on an equal-coupling triangle,The minimum has zero vector sum, hence a coplanar $120$-degree arrangement. For Ising spins on the same triangle, at least one bond remains unsatisfied. The allowed spin space matters.

The vector picture is classical. Three quantum spin-half degrees of freedom cannot have total spin zero; their frustrated states require a Quantum Hamiltonian treatment. Geometric frustration can enhance competition and low-energy ground-state degeneracy, but does not by itself prove a quantum spin liquid or a particular ordered state.

Classical communication transmits a distinguishable classical record, such as measurement bits, between parties. In quantum teleportation it tells the receiver which local correction to apply. Quantum no-signalling prevents shared entanglement alone from transmitting the record. A classical message cannot propagate across a spacelike separation in a relativistic protocol.

The harmonic oscillator transition kernel is the position-space unitary time evolution amplitudeFor $0<T<\pi /\omega$,Here $\hslash$ is the reduced Planck constant. The interval fixes the initial square-root branch; extension beyond focusing times requires its continued phase.

In the path integral, write a path as the classical oscillator path plus a fluctuation vanishing at both endpoints. The quadratic action splits exactly into the classical action and the endpoint-independent fluctuation action. The Dirichlet oscillator determinant ratio gives the prefactor, while the classical boundary action gives the exponent. ^[ref]

As $\omega \to 0$ with $T$ fixed, the expression reduces to the free-particle propagatorAt $T=n\pi /\omega$, the coordinate kernel is distributional because classical paths refocus. The Feynman i-epsilon prescription fixes the continuation through these caustics. Integrating the kernel against an initial wavefunction gives its later wavefunction.

A Fourier sample is a computational-basis measurement outcome after applying a quantum Fourier transform $F$ to a quantum state $|\psi ⟩$. Its distribution is given by the squared Fourier amplitudes. For a uniform periodic coset of spacing $r\mid N$, samples are the multiples $sN/r$ with uniform $s\in {\mathbb{Z}}_r$. Samples need not individually reveal the whole period: a greatest common divisor can remove factors from a single reduced denominator, motivating two-sample exact period recovery.

An observer at constant proper acceleration in the Minkowski vacuum detects a thermal response at this temperature. The associated Rindler coordinates separate the observer's accessible wedge from a horizon. This is an observer-dependent particle/detector response, not an assertion that inertial observers measure a thermal bath.

Positive-frequency modes in stationary asymptotic past and future regions can differ by a Bogoliubov transformation mixing positive and negative frequency solutions. The conserved Klein-Gordon inner product gives its canonical normalization. The out annihilation operator then contains in creation operators, yielding the displayed number expectation in the initial vacuum. A nonzero mixing coefficient changes the particle content relative to final observers without violating deterministic field evolution.

A device qubit couples diagonally to a finite bath of spins. Conditional bath states acquire opposite phase rotations, and their conditional environment overlap multiplies the device coherence. An initially aligned bath leaves the overlap's modulus at one. Initially equatorial spins yield $z_N(t)={\prod }_k\cos (2g_{k}t)$. Global unitary time evolution remains reversible, and a finite spin-bath coherence recurrence prevents a strict zero long-time limit at finite bath size.

No deterministic quantum operation can make two perfect copies of every unknown pure state. For a proposed unitary operator acting with a fixed blank and environment, preservation of inner products gives $c=c^{2}e$, where $c$ is the input overlap and $|e|\le 1$ is the overlap of final environment states. Nonorthogonal distinct states have $0<|c|<1$, contradicting this equality. The no-cloning theorem for two pure states is its elementary two-state form.

The quadratic congruence $x^2\equiv -1(\mathrm{mod}p)$ is soluble exactly when $p=2$ or $p\equiv 1(\mathrm{mod}4)$. At $p=2$, take $x=1$.

For an odd prime $p$, any solution is nonzero modulo $p$. Fermat's little theorem givesso $(p-1)/2$ is even and $p\equiv 1(\mathrm{mod}4)$. Conversely, put $m=(p-1)/2$ when $p\equiv 1(\mathrm{mod}4)$. Pairing opposite factors in the factorial givesWilson's theorem therefore gives $(m!{)}^2\equiv -1(\mathrm{mod}p)$, constructing a solution. For example, at $p=13$ the construction gives $6!\equiv 5(\mathrm{mod}13)$ and $5^2\equiv -1(\mathrm{mod}13)$. The criterion is the first supplementary law for quadratic reciprocity expressed as existence of a square root.

Cantor space is the binary sequence space with the product topology of discrete two-point factors. Here the Bernoulli-space synonym refers to this topology, without selecting a particular probability measure. The metric $d(x,y)={\sum }_{j\ge 1}{2}^{-j}|x_j-y_j|$ induces the topology. A diagonal subsequence argument gives compactness. Fixing a finite prefix gives a clopen cylinder set. The map $x\mapsto {\sum }_{j}2x_j{3}^{-j}$ is a homeomorphism to the usual Cantor set.

For a metric space with a probability measure $\mu$, the concentration function is $\alpha (\epsilon )={\sup }_{\mu (A)\ge 1/2}(1-\mu (A_{\epsilon }))$, where $A_{\epsilon }$ is the metric neighbourhood of radius $\epsilon$. A sequence exhibits concentration of measure if these functions tend to zero for every fixed positive radius. The metric scale is part of this definition.

A random matrix is a matrix whose entries are random variables. Its eigenvalues form a random configuration, and its empirical spectral measure records their distribution with equal mass on each eigenvalue. Entry assumptions and normalization determine the spectral regime.

A point process records random locations by a random measure that counts them. Examples include arrival times of a queue, positions of stars, and the finite eigenvalue configuration of a random matrix. The count in an observation window is a random variable; its expected value is supplied by the intensity measure of a point process, while its fluctuations and relationships with other windows require more information.

For a precise spatial convention, take $E={\mathbb{R}}^d$ with its Borel sigma-algebra and a probability space $(\Omega ,\mathcal{F},\mathbb{P})$. A locally finite point process $\Pi$ is a measurable function from $\Omega$ to the space ${\mathcal{N}}_{\mathrm{lf}}(E)$ of locally finite point measures, equipped with the sigma-algebra generated by evaluations $\nu \mapsto \nu (A)$ for Borel sets $A$. Each realization has the formwhere $J(\omega )$ is finite or countable and repeated locations are allowed; ${\delta }_x$ is a Dirac measure. Here a point measure is a counting measure with possible multiplicities, rather than the counting measure of a set of distinct locations.^[ref]

Local finiteness means ${\Pi }_{\omega }(B)<\infty$ for every bounded Borel set $B$, on a single almost sure event. In ${\mathbb{R}}^d$ it suffices to check the countable collection of balls of integer radius centered at the origin, and this is equivalent to finiteness on every compact set. On a general measurable space there is no notion of boundedness without additional structure: state a metric or an explicit family of observation windows. Another convention on locally compact spaces requires finiteness on compact sets; this need not coincide with finiteness on all bounded sets outside ${\mathbb{R}}^d$. Local finiteness does not require finitely many points in all of $E$.^[ref]

For every integer $n>1$, there is a prime number strictly between $n$ and $2n$. This supplies cyclic prime moduli comparable to an integer interval's length in additive combinatorics.

The region bounded by the dividing streamlines of a two-dimensional point source superposed on uniform flow. Its stagnation point is $-Q/(2\pi U)$ upstream and its far downstream width is $Q/U$. The boundary encloses fluid originating at the source rather than a rigid obstacle; the upstream stagnation point is reached only asymptotically by the upstream source trajectory.

For steady incompressible flow with no stagnant volume, partition the layer into streamtubes of volume flux $dQ$. The pore volume in a streamtube is $\tau dQ$, so $\int \tau dQ=V_{\mathrm{pore}}$. This gives the mean residence time $V_{\mathrm{pore}}/Q$ even when individual path times vary widely. Streamfunction labels measure flux in two-dimensional flow, so they supply the correct weighting.

Immiscible fluids share pore space, with each phase's Darcy flux determined by its pressure, viscosity and relative permeability. Fluid saturation specifies how much pore space each phase occupies. A capillary pressure couples their pressures, while their individual mass conservation laws determine displacement and saturation evolution.