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A two-dimensional color code is a topological stabilizer code commonly defined on a trivalent lattice whose faces admit three colors. Place a qubit at each vertex and assign both an X-type and a Z-type stabilizer generator to each face, acting on its boundary vertices.

In the usual allowed lattices, the relevant face overlaps have even size, making X- and Z-type checks commute. Violated checks identify an error syndrome, while operators spanning nontrivial paths or boundaries can act on encoded logical qubits. The geometry and boundary choices determine the number of logical qubits and code distance. Measuring local checks can reveal physical errors without measuring logical amplitudes; the full correction procedure must also address noisy check measurements and decoding.

The Steane code is a $[[7,1,3]]$ CSS code: seven physical qubits encode one logical qubit, and the distance of a stabilizer code is three. It can correct an arbitrary error on one physical qubit when the ideal syndrome measurements and recovery are implemented correctly.

One choice of binary check rows is $1111000$, $1100110$ and $1010101$. For each row, put Pauli matrices $X$ on its four marked positions to obtain an X-type generator, and use $Z$ on the same positions for a Z-type generator. Any two rows overlap in an even number of positions, so all six stabilizer generators commute. Their common positive eigenspace has dimension $2^{7-6}=2$.

A single-qubit Z error reverses the X-check outcomes indicated by its column, identifying its location; X errors are diagnosed by the Z checks. The recovery learns the error syndrome without reading the unknown logical amplitudes.

An error syndrome is the list of outcomes obtained by measuring a chosen ordered family of stabilizer generators. Write $s_j=0$ for eigenvalue $+1$ and $s_j=1$ for eigenvalue $-1$. For a definite Pauli error $E$ on a code state, the rule isThe commuting checks admit joint orthogonal projectorsFor density operator $\rho$, the Born rule gives outcome probability $\mathrm{tr}({\Pi }_s\rho )$ and conditional state ${\Pi }_s\rho {\Pi }_s/\mathrm{tr}({\Pi }_s\rho )$ when that probability is nonzero. A definite Pauli error has one deterministic error syndrome independent of the encoded amplitudes. A coherent linear combination of Pauli errors may occupy several sectors, so measuring the error syndrome selects a sector; recovery preserves the logical qubit when the entire error family is jointly correctable. ^[ref].

An error syndrome locates an error only relative to an assumed family and a recovery rule. Two Pauli errors differing by a logical operator can share an error syndrome; see stabilizer-syndrome coset. In the three-qubit bit-flip syndrome table, $X_3$ and $X_1{X}_2$ both give $01$. Recovering with $X_3$ corrects the first but converts the second into the logical operator $X_1{X}_2{X}_3$. Likewise, $Z_1$ has error syndrome $00$ yet changes the encoded relative phase.

The distance is the minimum weight of a Pauli operator that preserves the code space but acts nontrivially on its logical information.

In position representation,The radial derivatives cancel after substituting $P=-i\hslash \nabla$ into the Cartesian identity for $L^2$.

Let $g\in K[T]$ in characteristic $p$, let $f(T)=g(T^p)$, let $L$ split $g$ over $K$, and let $M$ split $f$ over $L$. If $y_i$ are the roots of $g$ and $x_i^p=y_i$, then the Frobenius endomorphism makes each $x_i$ unique andbecause $y_i=x_i^p$. Thus $M$ is already the splitting field of $f$ over $K$, and every root of $f$ is purely inseparable over $L$.

A finite extension $L/K$ is normal exactly when it is the splitting field over $K$ of some polynomial. For the forward direction, multiply the minimal polynomials of finitely many generators of $L$; normality puts all their roots in $L$. The reverse direction follows because a $K$-embedding permutes the roots of a polynomial and hence preserves its splitting field.

If $\alpha =\sqrt[3]{2}$ is the positive real root and ${\zeta }_3=e^{2\pi i/3}$, then the roots of $X^3-2$ areTheir splitting field over $\mathbb{Q}$ is $\mathbb{Q}(\alpha ,{\zeta }_3)$.

Every nonconstant polynomial $f\in K[X]$ has a splitting field of finite degree over $K$. It is constructed by adjoining a root of an irreducible factor and repeating over the enlarged field. Any two splitting fields of $f$ over $K$ are isomorphic by an isomorphism that fixes $K$. Inside one fixed algebraic closure, the field generated by all roots is unique as a subfield.

A spin coherent state is obtained by rotating the maximal-weight state $|s,s⟩$ so that it is an eigenstate of spin along a chosen direction with eigenvalue $s\hslash$.

In descending $S_z$ order, the spin-$3/2$ raising operator has superdiagonal entries $\hslash \sqrt{3},2\hslash ,\hslash \sqrt{3}$; $S_x$ and $S_y$ are its Hermitian real and imaginary combinations.

A spin ladder operator changes the magnetic quantum number within a fixed spin multiplet. For simultaneous eigenstates of $S^2$ and $S_z$, the labels satisfy $S^2|s,m⟩={\hslash }^{2}s(s+1)|s,m⟩$ and $S_z|s,m⟩=\hslash m|s,m⟩$.

The operators $S_{\pm }=S_x\pm i{S}_y$ satisfy

The angular momentum commutation relations give $S_{\mp }{S}_{\pm }=S^2-S_z^2\mp \hslash S_z$. Its expectation value is the squared norm of the raised or lowered state, fixing the square-root factors by normalization. For spin one half, $S_{+}|\frac{1}{2},-\frac{1}{2}⟩=\hslash |\frac{1}{2},\frac{1}{2}⟩$. At $m=s$ the spin raising operator gives zero. A spin ladder operator is therefore not a unitary operator or, by itself, a deterministic physical rotation. ^[ref]

The spin-$s$ irreducible representation has dimension $2s+1$, with basis $|s,m⟩$ for $m=-s,-s+1,\dots ,s$ and Casimir eigenvalue ${\hslash }^{2}s(s+1)$.

For $\mathbf{n}=(\sin \theta ,0,\cos \theta )$, the normalized eigenstates of $\mathbf{n}\cdot \mathbf{S}$ arewith eigenvalues $+\hslash /2$ and $-\hslash /2$.

For $N$ noninteracting units with energies $-ϵ,0,ϵ$, put $x=\beta ϵ$. Thenand

A vacuum cavity of radius $a(t)$ in an infinite inviscid incompressible fluid of density $\rho$ and far-field pressure $p_0$ obeysIf $a(0)=a_0$ and $\dot{a}(0)=0$, its collapsing branch satisfies

The convex hull of finitely many unit-circle points contains zero exactly when the points do not lie in any open semicircle, equivalently when their shortest containing arc has length at least $\pi$.

For $U_{\gamma }=\mathrm{diag}(1,e^{i\gamma })$ with $0\le \gamma <2\pi$,

A Laplacian term in the infected-population equation models unbiased random migration of infective individuals.

Mass-action transmission at unit nondimensional rate transfers population from susceptible to infected compartments at rate $SI$.