The quantum rotor with angle period $2\pi$ has both the spectral kernel $(2\pi {)}^{-1}{\sum }_n{e}^{in\Delta -i\hslash n^{2}T/(2I)}$ and the displayed winding representation. The Poisson summation formula converts one into the other by a Gaussian integral. Each winding sector has a constant-velocity classical path and action $I(\Delta +2\pi m{)}^2/(2T)$. Quadratic fluctuations give the same prefactor in every sector, so this is exact. Real time is understood by $T\to T-i0$, which fixes the square-root branch and convergence prescription.

A projective measurement on one half of a Bell state prepares the complex-conjugate basis on the other half, conditional on the outcome. Measuring Alice in the columns of $U^{\ast }$ prepares Bob in the columns of $U$. Ignoring the equally likely outcomes leaves Bob's reduced density matrix equal to $I/2$ in either basis.

If a positive operator-valued measure distinguishes normalized pure states $|u⟩,|v⟩$ with certainty, the effect $E$ for the first answer satisfies $⟨u|E|u⟩=1$ and $⟨v|E|v⟩=0$. Positivity of $E$ and $I-E$ implies $E|v⟩=0$ and $E|u⟩=|u⟩$, because a positive operator with zero expectation annihilates the vector. Consequently $⟨u|v⟩=⟨u|E|v⟩=0$. Conversely, orthogonal states can be measured with their rank-one linear projections. A common unitary operator preserves this inner product and cannot remove the obstruction.

Measurement backaction is the effect of a measurement in quantum mechanics on later predictions for the measured system. A quantum instrument with one Kraus operator $K_m$ per recorded outcome hasThe conditional density operator ${\rho }_m$ exists only when $p_m>0$. Its nonlinear normalization expresses conditioning; the unnormalized map $\rho \mapsto K_m\rho K_m^{†}$ is linear. More generally, an unresolved outcome can contain several Kraus operators: replace the numerator by ${\sum }_a{K}_{ma}\rho K_{ma}^{†}$ and take its trace for $p_m$. ^[ref].

Keeping the outcome gives a selective quantum measurement. Discarding it gives a nonselective quantum measurement with quantum channel $\rho \mapsto {\sum }_m{p}_m{\rho }_m$. The latter describes the whole measured population without its labels, whereas each ${\rho }_m$ describes a selected subpopulation. Neither description requires the measured quantity to have possessed its eventual value before the interaction.

For a concrete test, prepare the Pauli X eigenstate $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. Its $X=+1$ probability is one. An ideal $Z$ projective measurement gives zero or one with equal probability, leaving $|0⟩$ or $|1⟩$ under the Lüders rule. Discarding the result leaves the maximally mixed state $I/2$. A subsequent $X$ measurement now gives either sign with probability one half, even if the $Z$ result was kept. This observable change in statistics is backaction. Measuring $X$ on the original $|+⟩$ instead leaves it unchanged: disturbance depends on both the preparation and the instrument. Measurement-based quantum feedback uses the recorded outcome to choose a subsequent control operation, so both parts of the instrument matter.

A photon in a quantum superposition of a safe path and an absorbing path can sometimes certify a live absorber without being absorbed. Recombination is chosen so that one output is impossible if the absorber is absent. A detection at that output is then an unambiguous successful test. The measurement back-action of a surviving live-absorber trial changes the interference pattern.

In 1993, Avshalom Elitzur and Lev Vaidman asked how to identify a working bomb whose sensor would be triggered by even one photon. Their theoretical proposal places the sensor in one arm of a Mach-Zehnder interferometer, adjusted so that destructive interference keeps one detector dark when both arms are unobstructed. A click there can identify a working bomb while leaving it unexploded. The motivation is to obtain information about a fragile object without destroying it; the bomb makes that operational claim explicit without committing to an interpretation of quantum mechanics.^[ref]

An ideal position measurement of a normalized wavefunction $\psi$ assigns a probability to each spatial region $A$ through the Born rule: $P(A)={\int }_A|\psi (x){|}^{2}dx$. This specifies outcome statistics; it does not by itself specify the post-measurement state. A physical detector has finite spatial resolution, which must be included in its measurement model.

An ancilla bit is an auxiliary bit providing workspace in reversible computation. In a quantum circuit, an ancilla qubit provides quantum workspace; a general quantum ancilla need not be a single bit.

A Werner state is a bipartite density operator invariant under $U\otimes U$ for every local unitary operator $U$. An isotropic quantum state instead has invariance under $U\otimes \overline{U}$; these are different families.

Quantum state discrimination uses a measurement in quantum mechanics to identify which candidate density operator was prepared. Minimum-error and unambiguous discrimination impose different objectives; quantum binary hypothesis testing is the two-state case.

For density operators $\rho ,\sigma$ on the same finite-dimensional Hilbert space, quantum fidelity measures how close their quantum states are. This article uses the unsquared convention:Each square root is the positive square root of an operator, and $\Vert \cdot {\Vert }_1$ is the trace norm. Quantum fidelity is symmetric in its inputs; it is one exactly when $\rho =\sigma$, and zero exactly when $\rho \sigma =0$, meaning that the support of a positive operator for one state is orthogonal to that for the other. Larger quantum fidelity means greater similarity. The separate squared quantum fidelity is $F_{\mathrm{sq}}=F^2$. ^[ref]

For two normalized pure states, quantum fidelity is the magnitude of their inner product, $F=|⟨\psi |\phi ⟩|$. Thus $|0⟩$ compared with itself has $F=1$, while $|0⟩$ and $|1⟩$ have $F=0$. For $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, comparison with $|0⟩$ gives $F=1/\sqrt{2}$, $F_{\mathrm{sq}}=1/2$, and trace distance $D=1/\sqrt{2}$ by the pure-state trace distance and fidelity identity.

Why compare density operators rather than their entries? A change of coordinates should not make the same physical preparations look more or less alike. Unitary invariance of trace distance and fidelity makes quantum fidelity independent of a common basis choice. Multiplying a pure state vector by a global phase leaves its density operator unchanged, so its quantum fidelity with the original state is one. These features make quantum fidelity useful for checking whether quantum channels, state preparations, and Schumacher compression preserve a target quantum state.

The connection to Uhlmann's theorem supplies another beautiful interpretation: quantum fidelity is the largest absolute inner product between pure vectors obtained from a purification of a density operator for each state, using a sufficiently large common reference space. Uhlmann's 1976 paper extended transition probability to algebraic states, including those of a C-star algebra; for density operators its transition probability corresponds to our squared quantum fidelity. The measurement characterization of quantum fidelity gives a complementary description directly in terms of observable outcome statistics. ^[ref]

An encoding stores $n$ classical bits in $m$ qubits so that a receiver who later chooses a bit index can recover that bit with a specified success probability. The sender cannot tailor the encoding to the unknown requested index. This task permits approximate recovery of any one bit rather than simultaneous recovery of all bits, and must specify whether additional communication, shared randomness, or entanglement is allowed.

Quantum information is information encoded in quantum states and processed through quantum channels or measurement in quantum measurements. A qubit supplies a two-dimensional state space; quantum phase and entanglement affect which operations and outcome statistics are possible. The Born rule connects the state description with the probabilities of classical measurement records.

The continuously varying probability amplitudes of a qubit do not supply a readable list of arbitrarily many classical bits. A measurement produces a classical record with state-dependent probabilities. Quantum state tomography estimates the state from many identically prepared systems. For a classical label encoded into one transmitted qubit, with no additional shared resource, Holevo's theorem bounds the mutual information available from a measurement by one bit. This bound concerns information about the label, rather than the number of possible outcome labels a measuring device can print. ^[ref]

Maxwell–Boltzmann statistics describes a dilute classical gas whose single-particle occupation factors have the displayed exponential form. It is the low-occupation classical limit of Bose--Einstein and Fermi--Dirac statistics. Integrating this factor over momentum gives the nonrelativistic Maxwell--Boltzmann number density when the particle energy is $m{c}^2+p^2/(2m)$.

A squeezed coherent state is obtained from the vacuum by a squeeze operator and a displacement operator. Its quadrature variances can differ while saturating the uncertainty product. With zero displacement it is a squeezed vacuum state.

ConsiderThe perturbation connects the oscillator ground state only to the number state $|r⟩$. Its squared matrix element is $(\hslash \omega {)}^{2}r!$, and the energy denominator is $-r\hslash \omega$. Thus second-order nondegenerate perturbation theory gives the formal shift $-{\lambda }^2\hslash \omega (r-1)!$. A finite formal coefficient does not establish that the full Hamiltonian has a ground state.

For $r>2$, a normalized coherent state with label $R{e}^{i\theta }$ hasFor nonzero $\lambda$, choose the phase so that the last term is negative. It dominates as $R\to \infty$, proving that the Hamiltonian is unbounded below on these trial states. The perturbative vacuum branch cannot be an exact lowest-energy state.

For $r=2$, put $Q=(a+a^{†})/\sqrt{2}$ and $P=(a-a^{†})/(i\sqrt{2})$, so $[Q,P]=i$. ThenIt is bounded below for $|\lambda |\le 1/2$. A positive-frequency oscillator and a normalizable ground state require the strict inequality $|\lambda |<1/2$, giving frequency $\omega \sqrt{1-4{\lambda }^2}$. At $\lambda =1/2$ only $Q^2$ remains; at $\lambda =-1/2$ only $P^2$ remains. Their zero infimum is not attained by a normalizable vector. This is the zero-mode obstruction to bosonic Bogoliubov diagonalization, rather than unboundedness below.

For $r=1$, completing the displacement gives $H_1=\hslash \omega [(a^{†}+\lambda )(a+\lambda )+1/2-{\lambda }^2]$, which has a normalizable displaced oscillator ground state for every real $\lambda$.

Quantum gravity with a two-dimensional spacetime or worldsheet. In the string formulation, one sums over surface metrics and topologies with matter fields on each surface. This worldsheet theory is distinct from the target-space gravity described by the graviton in the closed-string spectrum.

Write the integer label in most-significant-bit-first order as $l={\sum }_j{b}_j{2}^{n-j}$. The phase factors as $e^{i\varphi l}={\prod }_j{e}^{i\varphi b_j{2}^{n-j}}$, giving the displayed product state. For dyadic Fourier phases this explains why the Fourier basis states are unentangled even though a general quantum Fourier transform can create entanglement.

In most-significant-bit-first order, let $x=x_1\cdots x_n$ and $0.x_j\cdots x_n={\sum }_{l=j}^n{x}_l{2}^{-(l-j+1)}$. On a computational-basis state, process wire $j$ with a Hadamard gate, then controlled phase gates $R_{l-j+1}$ controlled by unprocessed wire $l$ for each $l>j$. Wire $j$ becomes $(|0⟩+e^{2\pi i0.x_j\cdots x_n}|1⟩)/\sqrt{2}$. Reversing the final wire order yields the positive-exponent quantum Fourier transform. There are $n$ Hadamard gates and $n(n-1)/2$ controlled phase gates. Physical reversal costs $\lfloor n/2\rfloor$ swap operators, each expressible as three controlled-NOT gates, themselves implemented with Hadamard gates and a Controlled-Z gate. Thus the entire construction uses $O(n^2)$ gates from the specified family.

Suppose fixed unitary encoding and recovery correct each error $E_j$ by mapping every logical input $|\psi ⟩$ to $|\psi ⟩\otimes |{\eta }_j⟩$. The syndrome vector can be taken independent of the logical input: applying linearity to basis inputs and their equal superposition forces their syndrome vectors to agree. Hence an error $E={\sum }_j{c}_j{E}_j$ is mapped to $|\psi ⟩\otimes {\sum }_j{c}_j|{\eta }_j⟩$. If $E$ is unitary, this final syndrome vector is normalized and the circuit corrects $E$. Orthogonality of the syndrome states is not needed for the linear-span argument; it is a separate condition concerning distinguishability of errors.

An $[[n,k,d]]$ quantum error correction code is a $2^k$-dimensional subspace of $n$ qubits. Its distance $d$ is the minimum weight of a Pauli operator $E$ for which $PEP$ is not a scalar multiple of the code projector $P$. Thus every operator supported on fewer than $d$ qubits has scalar compression to the code, by expansion in the Pauli basis. This definition includes degenerate codes; a low-weight operator acting as a scalar does not reduce the distance.