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An antiunitary operator between complex Hilbert spaces is a surjective antilinear map satisfying the displayed inner product identity. It preserves norms but conjugates scalars. In a chosen orthonormal basis it is $UK$, with $U$ a unitary operator and $K$ coefficientwise complex conjugation. Conversely any such $UK$ is antiunitary. Antilinearity by itself does not preserve norms.

A block encoding of a matrix $A$ is a unitary operator whose designated block equals $A/\alpha$ for a known normalization factor $\alpha$. It gives a quantum circuit coherent access to matrices that need not themselves be unitary.

A real-linear map between complex vector spaces that commutes with multiplication by $i$. On $\mathbb{C}$, its real matrix is $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$, representing multiplication by $a+ib$. On $\mathbb{C}$, such a map with real determinant one is a rotation. In higher complex dimension, determinant one does not force the map to be a unitary operator.

An eigenphase of a unitary operator is a real number $\theta$ modulo one specifying an eigenvalue $e^{2\pi i\theta }$. Every unitary eigenvalue has modulus one. Quantum phase estimation records the eigenphase corresponding to an input eigenvector; a superposition retains the corresponding labels coherently before measurement.

For Hermitian matrices $H_1,\dots ,H_m$ and $t\ge 0$, let $S(t)={\prod }_{j=1}^m{e}^{-it{H}_j}$. ThenHere every norm is the spectral norm. A short proof starts with two summands. Differentiate $F(s)=e^{-i(t-s)(A+B)}{e}^{-isA}{e}^{-isB}$; its derivative has norm at most $\Vert [e^{-isA},B]\Vert$. Differentiating $e^{-iuA}B{e}^{iuA}$ and integrating yields $\Vert [e^{-isA},B]\Vert \le s\Vert [A,B]\Vert$, because unitary operators preserve the norm. Integrating $s$ from zero to $t$ gives $t^2\Vert [A,B]\Vert /2$. Inductively separate $H_1$ from the remaining sum, use the triangle inequality on their commutator, and apply the telescoping bound for products of operators to obtain the displayed many-term bound.

Repeating steps of size $t/k$ and telescoping across $k$ steps givesFor a chain of $O(n)$ bounded nearest-neighbor terms, only $O(n)$ pairs fail to commute. Consequently first-order product-formula Hamiltonian simulation has error $O(n{t}^2/k)$ and uses $O(nk)$ constant-size gates. The general bound is also given in Proposition 9 of Childs and collaborators' analysis of Trotter error.

For bounded matrices $A_j$, the Lie product formula gives operator norm convergenceFor two summands, expanding their matrix exponentials gives the one-step difference below. For Hermitian matrices multiplied by $-i$, the factors are unitary operators and the first-order unitary product-formula error bound gives an explicit error estimate.

The matrix 2-norm is the operator norm induced by the Euclidean norm:It equals the largest singular value of $A$ and is invariant under multiplication by orthogonal or unitary matrices.

The maximally mixed density operator on an $N$-dimensional Hilbert space assigns equal weights to every vector of any orthonormal basis. It is invariant under every unitary operator. Its purity is $\mathrm{Tr}({\rho }_{\ast }^2)=1/N$, the minimum possible because the nonnegative eigenvalues sum to one and obey the Cauchy-Schwarz inequality. For $N>1$ it is a mixed state; for $N=1$ it is the sole pure state. A rank-$k$ projector has probability $k/N$ in this state.

For $h:{\mathbb{Z}}_N\to {\mathbb{Z}}_M$, its modular-addition oracle is the unitary operatorFor each fixed $x$, addition by $h(x)$ permutes the answer register's computational basis. This preserves inner products, even when $h$ is not injective.

Let $S_M|y⟩=|-y\mathrm{mod}M⟩$. The modular-addition quantum oracle obeysThe three steps replace $y$ by $-y$, then $-y+h(x)$, then $y-h(x)$. Thus a query to the inverse costs one forward query and two known unitary operators. The negation operator is a permutation matrix with $S_M^2=I$.

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.

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 deterministic finite-dimensional universal programmable quantum gate cannot implement every unitary exactly: programs for physically distinct unitaries would have to be mutually orthogonal, but a finite-dimensional program register contains only finitely many mutually orthogonal states.

Let $P$ be a rank-$k$ projector with $0<k<N$. No unitary operator can map every pure state whose $P$ probability equals $k/N$ to a state with the same fixed smaller probability. Average $\sqrt{k/N}|g⟩\pm \sqrt{1-k/N}|b⟩$ over good and bad basis vectors and both signs. The average density operator is the maximally mixed state, so every unitary preserves its average $P$ probability. This contradicts a strictly smaller probability for all states in the ensemble. A preparation tailored to one known starting state is a different resource and is used in exact amplitude amplification.

For positive wavenumber $k$, the odd channel of the symmetric double-delta potential hasFor real $k$ its numerator and denominator are complex conjugates, making the channel unitary. Its poles are the zeros of $k{e}^{-ika}+U_0\sin (ka)$.

For a unitary operator $U$, the orthogonal complement of $\mathrm{Ran}(I-U)$ is $\ker (I-U^{\ast })=\ker (I-U)$. On the fixed space, Cesaro averages act as the identity. On $(I-U)g$ they telescope to $(g-U^{N}g)/N$. Their uniform norm bound extends convergence to the closure of the range, proving the Von Neumann mean ergodic theorem.

For a parity-symmetric one-dimensional potential, the even and odd parity states diagonalize the One-dimensional S-matrix:For a real potential, unitarity further gives $|S_{++}|=|S_{--}|=1$ at real positive energy.