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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.

For a specified correctable set of distinct Pauli operators modulo phase, a nondegenerate code has mutually orthogonal error images $E\mathcal{X}$. Equivalently, the matrix in the Knill--Laflamme condition is diagonal for that Pauli error set. Each error therefore requires a separate syndrome subspace of dimension equal to the logical code dimension. Degenerate codes can have distinct errors with the same action on the code and do not satisfy this orthogonality counting argument.

A code with orthogonal projector $P$ detects an error $E$ when its compression to the code is scalar. A projective measurement of $P$ then either flags departure from the code or, on a successful code-space outcome, leaves the logical state unchanged up to normalization. This includes harmless scalar errors, even though they need not produce a nonzero error syndrome. Detectability extends by linearity to the span of a set of errors. Every Pauli operator of weight less than the quantum code distance is detectable.

The five-qubit code encodes one logical qubit into five physical qubits and corrects an arbitrary error on one physical qubit. In a nondegenerate realization, the identity and fifteen one-qubit Pauli operators take the two-dimensional code into sixteen mutually orthogonal syndrome subspaces, filling the 32-dimensional physical Hilbert space.

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A concatenated quantum code replaces each physical qubit of an outer code by a logical qubit of an inner code. Correction proceeds from inner blocks to outer blocks. For independent errors and ideal correction, a logical-error map $f$ iterates as $p_{\ell +1}=f(p_{\ell })$. This recursion describes the code/channel model and does not by itself include errors in encoding and recovery gates.

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A logical qubit is a two-dimensional encoded quantum subsystem with chosen orthonormal logical basis states $|0_L⟩,|1_L⟩$. An encoding linear isometry sends $\alpha |0⟩+\beta |1⟩$ to $\alpha |0_L⟩+\beta |1_L⟩$. In a concatenated quantum error-correcting code, one level's logical qubit supplies one qubit of the next outer code.

A quantum error-correcting code is a subspace encoding a logical Hilbert space, together with a specified family of noise operators that a common recovery quantum channel reverses on every encoded state. A K-dimensional code carries ${\log }_{2}K$ logical qubits. For a code projector $P$, correctability is characterized by the Knill--Laflamme condition; code distance is defined by the smallest undetectable operator weight, not merely by the number of generators.

The quantum measurement problem asks how a unitary superposition of system and apparatus outcomes gives rise to one definite observed outcome. Quantum decoherence explains the suppression of interference between outcome branches, but by itself does not select one branch.

For $\dot{\rho }=-i[H_0+f{H}_1,\rho ]$ and ${\dot{\rho }}_d=-i[H_0,{\rho }_d]$, the Hilbert-Schmidt distance gives $V=\Vert \rho -{\rho }_d{\Vert }^2/2$. The choice $f=\mathrm{Tr}({\rho }_d[-i{H}_1,\rho ])$ yields $\dot{V}=-f^2$. Decrease alone does not prove convergence: orthogonal pure eigenstates of a diagonal $H_0$ give zero input for a transverse Pauli control even though their distance remains nonzero. Such a law can be simulated to produce an open-loop control.

Hamiltonian engineering designs accessible interactions or their effective time average to implement desired quantum states or unitary operators. The available control inputs, Quantum Hamiltonian generators and laboratory constraints determine which transformations can actually be achieved. Model-based open-loop control computes a waveform before running it; adaptive experiments instead evaluate its performance on the physical system.

Design Lindblad operators and a Quantum Hamiltonian so that desired density operators are stationary and unwanted perturbations decay. A dissipative interaction can change the spectrum and purity of a density operator, reaching targets unavailable to purely coherent Hamiltonian engineering. The engineered stationary state must be physical, and uniqueness alone must be distinguished from an explicit proof of asymptotic convergence.

In coherent quantum feedback, a quantum controller interacts with the controlled system without first converting the control signal into a classical measurement record. Optical feedback through a beam splitter is one example. The input-output interconnection must preserve the physical quantum dynamics, including noise and commutation relations; a classical state-feedback formula does not automatically define a valid quantum controller.