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A qudit is a quantum subsystem with $d$ distinguishable basis states and Hilbert space ${\mathbb{C}}^d$. A qubit is the special case $d=2$. Tensor products of qudits give a natural setting for local Hamiltonians and finite-dimensional quantum circuits.

Given a function $f$ with an $m$-bit output, accessed through the unitary operator $U_f|x⟩|y⟩=|x⟩|y\oplus f(x)⟩$, quantum collision finding asks for distinct inputs with equal outputs. Here $\oplus$ is bitwise exclusive or on the $m$-bit answer quantum register. The distinctness requirement excludes the uninformative pair $(x,x)$. For a two-to-one function on $N$ inputs, Grover search algorithm methods yield a cube-root quantum query complexity, despite a square-root cost for finding a partner of just one preselected input.

Let a Hermitian operator $A$ have dyadic eigenvalues ${\lambda }_j=c_j/2^t$ in $[0,1)$ for known $t$, with controlled access to the unitary operator $U=e^{2\pi iA}$. This interval ensures that different eigenvalues have different eigenphases; exact representability alone would not exclude phase aliasing, as $0$ and $1$ both give phase zero. Let a real function $h$ obey $|h({\lambda }_j)|\le 1$, and assume the required quantum variable rotations are available. Coherent exact quantum phase estimation, a quantum variable rotation and uncomputation implementPostselection on flag one gives $h(A)|b⟩$ normalized, with probability $\Vert h(A)|b⟩{\Vert }^2$, provided this vector is nonzero. Erasing the eigenvalue label by uncomputation is essential to preserve coherence between different eigenvectors. On this dyadic spectrum, $U^{2^t}=I$ gives $U^{-1}=U^{2^t-1}$, so reversing the phase-estimation gates is possible using forward controlled-$U$ calls. The choice $h(\lambda )=\lambda$ multiplies by $A$; a scaled reciprocal gives the different filter used in the HHL algorithm.

Choose eigenphase representatives $0\le {\varphi }_j<1$ for a unitary operator $U$ and a positive integer $M$. Its positive-phase fractional power acts on each eigenstate by $e^{2\pi i{\varphi }_j/M}$. This branch uses arguments in $[0,2\pi )$ and can differ from the usual complex principal branch using $(-\pi ,\pi ]$. For exactly dyadic phases, exact quantum phase estimation stores each label in a coherent phase register; bitwise phase gates multiply it by the required phase, and inverse phase estimation removes the labels. Keeping the phase register unmeasured preserves arbitrary eigenstate superpositions and resets the ancilla qubits. With only controlled-$U$ and controlled-$U^{-1}$ primitives, the direct method uses $2(2^n-1)$ oracle calls for $n$ exact phase bits.

In quantum order finding, a phase $s/r$ produces amplitude $Q^{-1}{\sum }_{x=0}^{Q-1}{e}^{2\pi ix(s/r-j/Q)}$ after an inverse quantum Fourier transform. Its nearest grid point has probability at least $4/{\pi }^2$, by bounding the sine ratio of this finite geometric Fourier amplitude. The initial modular orbit state is an equal-weight superposition of the $r$ phase eigenstates, so the phase label is uniform.

Discarding a subsystem cannot decrease unsquared quantum fidelity: $F({\rho }_A,{\sigma }_A)\ge F({\rho }_{AB},{\sigma }_{AB})$. Maximizing purifications of the joint states are also candidates for purifications of the marginals, with the discarded subsystem included in the reference. Uhlmann's theorem then proves the inequality. Information loss makes the two states at least as similar.

In the unsquared convention, $F({\sum }_x{p}_x{\rho }_x,{\sum }_x{p}_x{\sigma }_x)\ge {\sum }_x{p}_{x}F({\rho }_x,{\sigma }_x)$. Use Uhlmann's theorem to choose component purifications with nonnegative maximizing overlaps, and then construct matching flagged purifications of a quantum ensemble. Their overlap is the weighted sum on the right; the fidelity of the mixtures maximizes over all such overlaps.

Replace the derivative in the Dirac action by the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$. The interaction is then $-q{A}_{\mu }\bar{\Psi }{\gamma }^{\mu }\Psi$. With $\Psi \mapsto e^{-iq\alpha }\Psi$ and $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, the covariant derivative transforms in the same way as the spinor, proving local gauge invariance.

In the large-color limit of QCD with $g_s^2{N}_c$ fixed, baryon masses scale as $N_c$ while meson masses remain of order one. A Skyrme model action with coefficients of order $N_c$ has soliton mass and rotational moment of inertia of order $N_c$, and rotor splittings of order $1/N_c$. This motivates a semiclassical mesonic description of baryons; it does not uniquely determine every term in a finite-color effective theory.

For $k$ equiprobable quantum code states decoded by a POVM $\{E_m\}$, testing whether the classical message label agrees with the decoded output gives a joint binary POVM with $F_0={\sum }_m|m⟩⟨m|\otimes E_m$. Its acceptance probability is the decoding success $1-ϵ$ on the joint classical-quantum state and $1/k$ on the product of its marginals. Applying quantum relative entropy monotonicity to this test gives the one-shot classical-quantum coding converse.

For an odd prime $p$, the least quadratic nonresidue $n(p)$ is the smallest positive integer whose nonzero residue class is a quadratic nonresidue modulo $p$.

Two square matrices $A,C$ are congruent when $C=P^{T}AP$ for an invertible matrix $P$. This is the transformation of the matrix of a bilinear form under change of basis, and also transforms a quadratic form under an invertible linear map on its coordinates. The matrices need not be symmetric. It differs from matrix similarity, which uses $P^{-1}AP$ for a linear operator. For real symmetric matrices, completing the square diagonalizes the corresponding quadratic form by matrix congruence.

For a field of characteristic different from two, adjoining a square root of a nonsquare gives every quadratic extension. If two such extensions are isomorphic over $K$, write $\sqrt{a}=x+y\sqrt{b}$; the vanishing of its cross coefficient forces $x=0$, giving $a/b=y^2$. This explains why one subtracts the identity class when counting extensions from the square-class group.

A resistive velocity term proportional to the speed squared removes energy at rate proportional to $|\omega {|}^3$. Its derivative at zero velocity vanishes, so it does not appear in the linearization of a dynamical system at rest. Nonlinear energy arguments can still prove asymptotic stability.

A verifier with clean ancillas and a designated output measurement induces an effect $0\le M\le I$ on the witness register. Its acceptance probability is $\mathrm{Tr}(M\rho )$, and the largest possible acceptance is $\Vert M\Vert$. Spectral bounds on $M$ make the soundness quantifier over all witnesses, including entangled repeated witnesses, explicit.

Independent verifier circuits on $r$ witness registers induce a threshold acceptance effect built from tensor products of $M$ and $I-M$, where $M$ is the quantum verifier acceptance operator. In a product eigenbasis of $M$, the effect's eigenvalues are threshold probabilities for independent Bernoulli random variables with parameters given by the selected eigenvalues of $M$. Bounding every parameter on a NO instance bounds the effect's operator norm, even if the actual submitted witness is entangled. Independence is used only for the spectral calculation, not assumed for the witness.

Starting from completeness $2/3$ and soundness $1/3$, run an odd number $r$ of parallel copies and take majority. The QMA parallel repetition with entangled witnesses argument proves soundness for arbitrary repeated witnesses. Exponential Markov's inequality gives majority error at most $(2\sqrt{2}/3{)}^r$. Hence $O(\log (1/ϵ))$ copies suffice for error $ϵ$, with polynomial overhead when the target error is inverse-polynomial or exponentially small in input length.

For quantum witness embedding $J$, verifier unitary $U$ and accepting projector $\Pi$, the witness acceptance operator is the positive contraction $F=J^{†}{U}^{†}\Pi UJ$. The maximum acceptance over normalized quantum witnesses is ${\lambda }_{\max }(F)$ by the Rayleigh quotient. Restricting to basis quantum witnesses only tests its diagonal entries and need not find that maximum.

For $\rho ={\sum }_x{p}_x{\rho }_x$, choose component purifications $|u_x{⟩}_{AR}$ and an orthonormal flag register. The vector $|U⟩={\sum }_x\sqrt{p_x}|u_x⟩|x⟩$ is normalized and reduces to $\rho$ after tracing out $R$ and the flag. Orthogonal labels remove the cross terms. This construction turns mixture identities into pure-state overlap arguments and proves joint concavity of quantum fidelity.

A strategy is strictly dominated when another strategy gives a strictly larger payoff against every opposing strategy. A strictly dominated strategy cannot receive positive probability in a Nash equilibrium, because moving that probability to its dominator increases expected payoff.