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For complex solutions of the Klein-Gordon equation,The associated current is conserved, so the inner product is independent of the Cauchy hypersurface when boundary flux vanishes.

A stabilizer code is the common positive eigenspace of a commuting subgroup of the Pauli group.

A code projector $P$ corrects errors $\{E_a\}$ exactly when $P{E}_a^{†}{E}_{b}P=C_{ab}P$ for every pair.

Quantum recoherence is a later restoration of phase coherence. It can occur when a finite environment revisits a state having substantial overlap with its initial conditional state.

If a joint pure state is $\alpha |0⟩|E_0⟩+\beta |1⟩|E_1⟩$, the off-diagonal entry of the system's reduced density matrix is multiplied by the environmental overlap $⟨E_1|E_0⟩$. Independent repeated interactions multiply these overlaps.

If a device perfectly identifies which member of a known finite state family was supplied, its classical output can control state-preparation unitaries that prepare any requested number of fresh copies. The discrimination may destroy the supplied system.

If a device produces arbitrarily many copies of either of two distinct pure-state rays, their $N$-copy overlap isThe Helstrom measurement on the copies therefore distinguishes the alternatives with success probability tending to one.

No unitary operator $U$ can satisfyfor two distinct nonorthogonal pure states. Taking the inner product of these two equations would requirewhich is impossible when $0<|⟨c_0|c_1⟩|<1$.

A propagator is a time-ordered two-point function and a Green function for the quadratic kinetic operator with an $iϵ$ boundary prescription.

Intrinsic parity is the phase by which a field transforms under spatial inversion after its spacetime argument is reflected. For a Hermitian scalar field it is $+1$ or $-1$.

A Fock state has specified occupation numbers. Applying creation operators to the vacuum constructs multiparticle momentum eigenstates.

A free field is expanded in positive- and negative-frequency plane waves multiplied by annihilation and creation operators.

For an odd prime $p$,

For $\omega =e^{2\pi i/p}$ and the Legendre symbol $\chi$, the quadratic Gauss sum isIt equals $\sqrt{p}$ for $p\equiv 1(\mathrm{mod}4)$ and $i\sqrt{p}$ for $p\equiv 3(\mathrm{mod}4)$ with the standard choice of square root.

A nonzero class modulo an odd prime is a quadratic nonresidue when it is not a square.

For an odd prime $p\ne 3$, quadratic reciprocity givesThus $-3$ is a square modulo $p$ exactly when $p\equiv 1(\mathrm{mod}3)$.

The static Green function of ${\nabla }^2-m^2$ in three dimensions is proportional to $e^{-mr}/r$. It describes an interaction screened beyond range $m^{-1}$ and tends to the Coulomb potential as $m\to 0$.

Taking a divergence and using antisymmetry of $F^{\mu \nu }$ and current conservation gives $m^2{\partial }_{\mu }{A}^{\mu }=0$. For nonzero mass, the Lorenz condition follows from the field equation rather than from a gauge choice.

Dividing a centered sum by the square root of its total variance gives unit variance and is the natural central-limit scaling.

The characteristic function of a real random variable $X$ isIt determines the probability distribution of $X$.