A shared resource enables deterministic exact transmission of an arbitrary unknown qubit using only local operations and classical communication if and only if it can produce a shared Bell pair deterministically by such operations. Sufficiency follows from quantum teleportation. For necessity, prepare a Bell pair locally at the sender and apply the purported transmission protocol to one half: exact identity-channel action leaves a shared Bell pair between the retained reference and the receiver. A separable shared resource cannot satisfy this condition because local operations and classical communication preserve separability.

For odd $n=2k+1$, a symmetric quantum teleportation resource can encode an unknown qubit into $E(\alpha |0⟩+\beta |1⟩)=\alpha |D_k^n⟩+\beta |D_{k+1}^n⟩$. Begin with $|D_{k+1}^{n+1}⟩$, apply a controlled-NOT gate from the input to the sender's resource qubit, apply a Pauli X gate to that resource qubit and a Hadamard gate to the input, and measure those two qubits. The four unnormalized remaining vectors are $E|\psi ⟩/2$, $EX|\psi ⟩/2$, $EZ|\psi ⟩/2$ and $EXZ|\psi ⟩/2$. Each outcome has probability $1/4$. Since $X^{\otimes n}$ interchanges $|D_k^n⟩$ and $|D_{k+1}^n⟩$, and $Z^{\otimes n}$ acts as logical $Z$ up to the common phase $(-1{)}^k$, local Pauli corrections make the output independent of the result. The one-qubit reduction of Dicke-state superpositions givesFor $n>1$ this distributes imperfect copies, as required by the no-cloning theorem. For $n=1$ it reduces to exact quantum teleportation.

For common classical weights, the matrix logarithm on each positive-weight block is $\log (p_i{\rho }_i)=(\log p_i)I+\log {\rho }_i$ on its support. The two logarithms of $p_i$ cancel in the quantum relative entropy, giving the displayed identity. Zero-weight blocks are omitted, including if their unweighted relative entropy is infinite. If a positive-weight block violates support inclusion, both sides are infinite. Applying the data-processing inequality for quantum relative entropy to the partial trace over the flag gives joint convexity of quantum relative entropy.

For an energy eigenstate with instantaneous energy eigenvalue $E(t)$, the dynamical phase isIt comes from the Schrödinger equation and depends on the time spent evolving. In an ideal cycle under the adiabatic theorem, a periodic choice of instantaneous energy eigenstate gives $|\Psi (T)⟩=e^{i(\delta +\gamma )}|\Psi (0)⟩$, with Berry phase $\gamma$.

A geometric phase is a quantum phase associated with the geometry of a path of state rays, after separating the dynamical phase. The Berry phase is its cyclic case governed by the adiabatic theorem; more general cyclic evolution can also have a geometric phase.

For a zero-mean Gaussian bosonic input, $d{B}^{†}d{B}^{†}=M^{\ast }dt$ completes the displayed table, with $N\ge 0$ and $|M{|}^2\le N(N+1)$. The vacuum has $N=M=0$; an unsqueezed thermal input has $M=0$. Products with $dt$ and higher differentials vanish in the first-order Ito equation. The difference between the first two products preserves the bosonic commutator.

A local quantum operation on one tensor-factor subsystem acts as ${\Phi }_A\otimes {\mathrm{id}}_B$, with ${\Phi }_A$ completely positive. An individual outcome branch can decrease trace, while the sum over outcomes of a normalized instrument preserves trace. The unselected operation consequently leaves the remote reduced state unchanged. In a local field algebra, Kraus operators inside one region and commuting with a distant observable give an analogous no-signalling statement. Locality of the operation must be established; an arbitrary globally defined projection is not automatically a physically local measurement.

A POVM fixes outcome probabilities, but a quantum instrument additionally fixes the conditional output state. For computational-basis effects $E_y=|y⟩⟨y|$, the operators $M_y=|y⟩⟨y|$ leave the measured eigenstate as output, whereas $N_y=|0⟩⟨y|$ reset every outcome to $|0⟩$. Both have $M_y^{†}{M}_y=N_y^{†}{N}_y=E_y$. Thus a specification of effects alone cannot select a unique joint state that retains the measured quantum system.

For Kraus operators satisfying ${\sum }_i{A}_i^{†}{A}_i=I$, the map $V|\varphi ⟩={\sum }_i{A}_i|\varphi ⟩\otimes |i⟩$ is an isometry. Implement it by a unitary operator on the system and a quantum ancilla, initially in an extra unused state $|0⟩$. Projecting the output quantum ancilla onto $|i⟩$ gives joint unnormalized state $A_i\rho A_i^{†}\otimes |i⟩⟨i|$. The outcome probability is $\mathrm{Tr}(A_i\rho A_i^{†})$, and the conditional system state is $A_i\rho A_i^{†}$ divided by this probability. The unused level can be merged into the first orthogonal projection without changing any outcome on the prepared state, giving a complete projective measurement with exactly the original outcome labels.

// Target: quantum-theory.bigb

A nondegenerate quantum code is perfect at correction radius $t$ if its distinct correctable Pauli error images exhaust the physical Hilbert space. Each image has dimension $2^k$, and their orthogonal direct sum is the whole $2^n$-dimensional space precisely when the quantum Hamming bound is an equality. This perfectness concerns unknown-location errors of weight at most $t$; it does not imply correction of every higher-weight error.

The symmetric quantum integer is the displayed Laurent polynomial, with $[0{]}_q=0$. It obeys $[n+1{]}_q=q^{-1}[n{]}_q+q^n$ and $[r{]}_q[s+1{]}_q-[s{]}_q[r+1{]}_q=[r-s{]}_q$. At a primitive odd $\ell$th root of unity, $[n{]}_q=0$ exactly when $\ell$ divides $n$.

If $Ev=0$ and $Kv=\lambda v$, induction using the defining commutator gives the displayed identity. When $q$ is not a root of unity, a finite string of length $n+1$ in a simple module has $\lambda =ϵq^n$, $ϵ=\pm 1$. Its distinct $K$ eigenvalues make the string vectors independent, and the nonzero raising coefficients prove simplicity.

At a primitive cube root of unity, the cubic homogeneous module has basis $x^3,x^{2}y,x{y}^2,y^3$. Its only simple modules occurring as submodules are the one-dimensional subspaces of $⟨x^3,y^3⟩$, on which $K=1$ and $E=F=0$. Its quotient by this two-dimensional submodule is simple, but it has no simple complement: any lift of the highest $K$ weight space contains $x^{2}y$, whose $E$ image is $x^3\ne 0$. Thus the cubic module is not a semisimple module.

For one canonically normalized real scalar field and $V^{\prime \prime }(\phi )>0$, the modified minimal subtraction scheme gives the displayed one-loop term. The Gaussian fluctuation approximation produces $\frac{1}{2}\int d^d{k}_E(2\pi {)}^{-d}\log (k_E^2+V^{\prime \prime }(\phi ))$. Its derivative in $V^{\prime \prime }$ is half the Euclidean massive loop integral with $n=1$. Integrating and subtracting its pole yields the formula, up to a field-independent constant. Other finite counterterms alter the scheme-dependent nonlogarithmic part. Negative curvature requires an analytic continuation and can yield an imaginary part indicating instability of that background.

For a bipartite pure state with finite Von Neumann entropy, the joint Von Neumann entropy is zero, so its quantum conditional entropy is $-S({\rho }_A)$. The Schmidt decomposition makes the nonzero eigenvalues of ${\rho }_A$ the squared Schmidt coefficients. Their Shannon entropy vanishes exactly when there is one coefficient, equivalently the joint pure state is a product state. Thus negative quantum conditional entropy detects exactly the entangled bipartite pure states; it is only a sufficient criterion for entanglement of general mixed density operators.

The quantile function is finite at every probability strictly between zero and one, but need not be finite at one. For any unbounded-above normal distribution, $F(x)<1$ for every real $x$, so its generalized inverse at one is $+\infty$. Inverse transform sampling still produces a real random variable almost surely because a continuous uniform distribution puts zero mass at that endpoint. A real-valued implementation may discard endpoint draws or assign an arbitrary finite value there without changing the sampled law.

For observations $X_1,\dots ,X_n$ and $0<p<1$, the sample quantile in the generalized-inverse convention iswhere ${\hat{F}}_n$ is the empirical distribution function and the observations are ordered with ties retained. For five observations, the sample median in this convention is the third order statistic. Interpolated sample-quantile conventions can give different finite-sample answers.

For the asymptotic statement, assume the observations are independent and identically distributed with cumulative distribution function $F$. Let $q_p=F^{-1}(p)$ be its unique $p$-quantile, and suppose $F$ has a density $f$ continuous and strictly positive at $q_p$. ThenFor a proof, $\{{\hat{q}}_p\le x\}$ is exactly the event that at least $\lceil np\rceil$ observations are at most $x$. At $x=q_p+t/\sqrt{n}$, the count is binomial with success probability $F(x)=p+f(q_p)t/\sqrt{n}+o(n^{-1/2})$. The binomial central limit theorem gives the displayed normal limit. A larger density at the target quantile makes the quantile more precisely estimated. The independence and local-density hypotheses are needed; the formula cannot be applied unchanged to arbitrary dependent observations or an atom.

The probable error is the half-width of an interval around a specified central value containing probability $1/2$. For a normal distribution $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, its quantile function giveswhere $\Phi$ is the standard normal cumulative distribution function and $\sigma$ is the standard deviation. The factor is specific to the normal distribution. The centered Cauchy distribution with parameter $\gamma >0$ has probable error $\gamma$, despite having no finite variance. Thus a central probability width need not require a finite second moment.

In units $c=1$, a slowly moving isolated source has trace-free mass quadrupole moment $Q^{ij}=\int T^{00}(x^i{x}^j-{\delta }^{ij}|\mathbf{x}{|}^2/3)d^{3}x$. The retarded linearized gravity solution and the stress-energy quadrupole identity give the displayed physical spatial strain, with $\mathbf{n}=\mathbf{x}/r$ and the transverse-traceless projector $\Lambda$. If the spatial metric is written $g_{ij}=-{\delta }_{ij}-2E_{ij}$ for metric signature $(+---)$, its trace-free Lorenz-gauge coefficient before the transverse projection is $E_{ij}=G{\ddot{Q}}_{ij}(t-r)/r$. Thus $H^{\mathrm{TT}}=2E^{\mathrm{TT}}$. Far distance alone does not permit dropping source-dependent retardation: the quadrupole approximation also requires the source size to be small compared with the radiation wavelength.

For a quadrilateral with side lengths $a,b,c,d$, semiperimeter $s$, area $K$, and opposite angles $A,C$,The second term vanishes exactly when $A+C=\pi$, which is the cyclicity condition for a nondegenerate convex quadrilateral.