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Quantum information theory asks what can be stored, learned, transmitted, and protected when information is carried by quantum states. Its basic objects are density operators, measurement in quantum measurements, and quantum channels; its resources include qubits, entanglement, and classical communication. Specifying the task matters: preserving an unknown quantum state, recovering a classical bit, and estimating a probability amplitude are different demands on the same physical system.

A particularly beautiful feature is that knowing the whole need not mean knowing its parts. A Bell state is a completely specified pure state, yet either qubit alone has a maximally uncertain reduced density matrix. Conversely, two systems can share perfectly matching measurement records without entanglement. The worked Bell correlations versus a classical mixture comparison separates these possibilities using only two choices of Pauli measurement.

This viewpoint makes limitations useful design rules. Schumacher compression asks how much quantum memory a source needs; Holevo's theorem limits the classical information extractable from an encoding; quantum error correction protects states against noise. Quantum teleportation uses shared entanglement and classical communication to transfer an unknown state. Its resource accounting respects both quantum no-signalling and the no-cloning theorem.

A quantum measurement turns a quantum state preparation into a classical outcome, such as a detector label. Two questions must be distinguished: how likely is each outcome, and what quantum state remains conditional on that outcome? A positive operator-valued measure answers the first; a quantum instrument answers both. This distinction is useful whenever a measured system participates in a later experiment, including quantum state tomography and measurement-based quantum feedback.

For a density operator $\rho$, the Born rule gives $p_m=\mathrm{Tr}(E_m\rho )$, where the $E_m$ are positive operators satisfying ${\sum }_m{E}_m=I$. Here $I$ is the identity operator, and $\mathrm{Tr}$ denotes the trace. Positivity makes every $p_m$ nonnegative; completeness makes their sum one. For a pure state, $\rho =|\psi ⟩⟨\psi |$ and $p_m=⟨\psi |E_m|\psi ⟩$. A projective measurement is the special case in which the $E_m$ are orthogonal projections onto mutually orthogonal subspaces.

For a qubit, choose the computational basis and write a normalized pure state, up to global phase, asThe angle $\varphi$ is a relative quantum phase; at the poles it does not distinguish states. Measuring the Pauli matrix $Z$ reports $+1$ for label zero and $-1$ for label one, withThese probabilities describe repeated identical preparations. They do not assign a hidden, pre-existing value to every possible measurement basis.

The Born rule grew out of Max Born's 1926 work on quantum scattering: squared probability amplitudes determine the relative frequencies of scattering outcomes. Born explained this statistical interpretation again in his 1954 Nobel lecture. This probability law does not by itself select a post-measurement state. ^[ref]; ^[ref].

Quantum mechanics describes a quantum system using a quantum state, with observables represented by self-adjoint operators. For a single particle in one dimension, a pure state can be described by a wavefunction $\psi (x,t)$: its squared modulus gives the probability density for position. The Schrödinger equation predicts how this probability amplitude evolves between interventions; measurement in quantum mechanics connects it to experimental outcomes through the Born rule. A wavefunction therefore predicts a distribution of possible positions rather than a single trajectory as in classical mechanics.

The infinite square well makes three central ideas visible with elementary sine functions. Confinement selects discrete energy eigenvalues; wavefunction normalization converts a shape into quantitative probabilities; and a quantum superposition can have a changing probability density even when each constituent energy eigenstate has a stationary one. This combination explains why quantum mechanics uses both linear algebra and differential equations: eigenfunctions describe the building blocks, while quantum phases determine how they interfere. The same tools help explain the discrete energies of chemical atoms and molecules, although their Quantum Hamiltonians require different potentials and often many particles.

In his first 1926 paper, Erwin Schrödinger formulated quantization as an eigenvalue problem for a hydrogen atom in nonrelativistic quantum mechanics. On its opening page he compared the emergence of integers with the number of nodes of a standing wave on a vibrating string. The infinite square well below realizes that appealing idea especially transparently: the allowed integers follow from the boundary conditions. This historical analogy concerns the origin of discrete energies; the Born rule supplies the statistical interpretation of the modern wavefunction. ^[ref].

A verification scheme using quantum states as part of a banknote or token. Secret-basis qubits can make extra copies hard to produce because their encodings are nonorthogonal. Security claims must specify the adversary's information, access to original tokens and the verification procedure; the no-cloning theorem alone does not fix every forging probability.

An ideal measurement preserving the measured eigenspaces. In the projective setting, the selective instrument $\rho \mapsto P_j\rho P_j$ preserves every vector within outcome $j$, including coherence inside a degenerate eigenspace. Repeated measurements agree in the absence of intervening dynamics changing the measured quantity. Measuring a finer observable can destroy degeneracy coherence and need not realize the same Lüders rule instrument.

A physical measurement interaction must correlate the system with a meter; in Haroche cavity photon nondemolition experiment that meter is a chemical atom. For a measured observable $A$, a useful sufficient ideal condition is $[A,H_{\mathrm{int}}]=0$ together with $[A,H_0]=0$, where $H_{\mathrm{int}}$ is the interaction Hamiltonian operator and $H_0$ the free Hamiltonian operator. These conditions protect its eigenvalues during probing and between probes. They do not protect every other observable, nor prevent conditional quantum state changes. In particular, preserving photon number can disturb the field's quantum phase.

A quantum number is a discrete label, usually an observable's eigenvalue, used to distinguish quantum states.

Quantum optics studies light using quantum states of the electromagnetic field and their interaction with matter. Photon statistics asks how detections fluctuate; the quantum theory of optical coherence relates those detections to field correlations. A coherent state, thermal light, and a Fock state can have the same mean photon number yet different pair counts. This distinction makes photon counting useful for testing sources intended to emit one photon at a time.

Multiplying a quantum state by one global complex phase does not change any measurement probability; relative phases between components can affect interference.

The global phase is common to the whole state; relative phases control interference between state components.

A quantum register is a named tensor product factor of a quantum circuit's Hilbert space, usually a collection of qubits. Registers separate data, control, output and workspace roles; uncomputation returns workspace to a fixed quantum state without destroying coherence in the data.

A time-dependent unitary operator $R(t)$ defines transformed quantum states by $|{\psi }_{\mathrm{r}}⟩=R^{†}|\psi ⟩$. Substituting $|\psi ⟩=R|{\psi }_{\mathrm{r}}⟩$ into the Schrödinger equation gives the displayed transformed Hamiltonian operator, in units $\hslash =1$. If $R=e^{-it{H}_0}$ for constant $H_0$, this becomes $H_{\mathrm{r}}=R^{†}HR-H_0$. Removing a common nuclear Zeeman energy with $H_0=\omega (ZI+IZ)/2$ leaves an isotropic Heisenberg model coupling unchanged, because it commutes with the generator of the common rotation. Removing two unequal nuclear frequencies instead makes their exchange terms oscillate at the difference frequency.

A quantum state describes a preparation through its predicted measurement in quantum mechanics probabilities. In the Hilbert space formulation, a pure state is a quantum state ray: a normalized vector representative $|\psi ⟩$ satisfies $⟨\psi |\psi ⟩=1$, and representatives differing by a global phase describe the same quantum state. ^[ref]

Both pure states and mixed states are represented by a density operator on the chosen complex Hilbert space:Thus $\rho$ is Hermitian, positive semidefinite and of unit trace; in infinite dimension it must also be a trace-class operator. For a normalized pure state, $\rho =|\psi ⟩⟨\psi |$ is a rank-one orthogonal projection. A mixed state has no single such vector representative. Its density matrix is the matrix of $\rho$ in a chosen orthonormal basis. ^[ref] For a discrete POVM with effects $E_a\ge 0$ and ${\sum }_a{E}_a=I$, the Born rule readsThese constraints and Born rule probabilities describe arbitrary finite-dimensional preparations. ^[ref]

Randomly choosing normalized pure states with probabilities $w_j$ and forgetting the label givesThis quantum state ensemble averages probabilities; a quantum superposition adds probability amplitudes. Several ensembles can give the same density operator. Quantum entanglement can make a reduced density matrix mixed while the joint quantum state is pure. ^[ref]

The preparation, the choice of basis, and the measurement are separate ingredients. A wavefunction $\psi (x)=⟨x|\psi ⟩$ is the position representation of a pure state; a general mixed state requires a density operator. For a closed quantum system, a unitary operator evolves it as $\rho \mapsto U\rho U^{†}$, preserving its eigenvalues and purity of a density operator. ^[ref] The Bloch sphere and Bloch ball give the qubit geometry; quantum state tomography connects repeated preparations to an estimated density matrix; projective measurement treats the conditional state after an outcome.

For a fixed Hilbert space, a bounded self-adjoint Hamiltonian operator with a finite step $H_i\to H_f$ at $t=0$ does not deliver an instantaneous rotation of the quantum state. Integrating the Schrödinger equation across a short interval givesFor a normalized quantum state and $\Vert H(t)\Vert \le M$ in operator norm, the difference has norm at most $2ϵM/\hslash$, which tends to zero. The time derivative can jump even though the quantum state is continuous. Here $\hslash$ is the reduced Planck constant. An instantaneous replacement of the eigenbasis changes coordinates, not the vector being represented.

For a real switch occupying $0\le t\le \tau$, remove any real scalar contribution $a(t)I$ that produces only a global phase, and write $|\tilde{\psi }(t)⟩=e^{i{\int }_0^{t}a(s)ds/\hslash }|\psi (t)⟩$. The same integral argument gives the sufficient bounded-model estimateThus $\epsilon \ll 1$ controls the sudden approximation up to a global phase. For any normalized final energy eigenstate, its probability amplitude differs from the instantaneous-switch value by at most $\epsilon$, and its Born rule probability differs by at most $2\epsilon$. These are sufficient error bounds, not necessary criteria. For the fixed-gap two-level sudden quench, any switching path $H(t)=-\Delta \mathbf{n}(t)\cdot \mathbf{\sigma }/2$ with a unit vector $\mathbf{n}(t)$ has operator norm $\Delta /2$, giving $\epsilon \le \Delta \tau /(2\hslash )$.

This estimate is deliberately for bounded Hamiltonian operators. An unbounded energy spectrum does not supply a finite global $M$; relevant quantum states, operator domains and excited energy eigenvalues require separate control. The adiabatic theorem also requires hypotheses about the path and its spectral gaps; there is no universal switch-duration criterion covering all spectra. The cold-atom collapse and revival after a lattice quench illustrates how a single switch can be sudden for one degree of freedom and slow for another.

A quantum superposition is a linear combination of quantum state vectors in a complex Hilbert space. In a specified orthonormal basis, a normalized pure state has the displayed expansion, where $a_j$ are probability amplitudes. A nonzero linear combination can always be normalized; an arbitrary sum of normalized vectors need not already have norm one. Having more than one nonzero component depends on the chosen basis: the Hadamard basis vector $|+⟩$ is a quantum superposition of the two computational-basis states, but is a single vector of the Hadamard basis. ^[ref].

The distinction from classical ignorance is quantitative. Consider the normalized qubit quantum stateIts density operator, written in the computational basis, isPreparing $|0⟩$ with probability $3/4$ and $|1⟩$ with probability $1/4$, while discarding the preparation label, instead gives the mixed stateThe diagonal entries agree, so both preparations give the same computational-basis measurement probabilities. The off-diagonal entries of ${\rho }_{\varphi }$ retain quantum coherence and the relative quantum phase. In the Hadamard basis, the Born rule givesThe density matrix representation retains this distinction between a pure state and a mixed state. ^[ref].

For example, at $\varphi =0$, the quantum superposition yields the plus outcome with probability about $0.9330$, whereas the mixed state yields it with probability $0.5$. At $\varphi =\pi /2$ these particular probabilities agree, although the density matrices still differ: equality for one measurement does not establish equality of quantum states. The probability amplitude figure shows the comparison throughout a full cycle of relative quantum phase.

This is why a quantum superposition is useful: quantum gates act linearly on its probability amplitudes, and unitary operators preserve the full pure state information, including relative quantum phases. A Hadamard gate can turn otherwise hidden quantum coherence into different outcome probabilities. The beautiful feature is that the same basis weights can support different predictions when amplitudes recombine. The expansion specifies a preparation and its measurement predictions; it does not assign every unmeasured outcome a simultaneous classical value. See projective measurement for the separate question of the conditional state after a measurement.

A quantum system is modeled by a complex Hilbert space; its quantum states describe preparations, while observables describe measurable quantities.

Quantum theory supplies a common language for quantum mechanics, measurement in quantum mechanics, and quantum information: prepare a quantum state, transform it, and ask which experimental outcomes occur. In a finite-dimensional idealization, a pure state is a normalized vector of complex numbers. Its entries are probability amplitudes; the Born rule turns their squared magnitudes into probabilities. Between measurements, an isolated system evolves by a unitary operator, preserving its norm and inner products. A quantum circuit makes this evolution explicit as a sequence of quantum gates.

The extra information carried by quantum phases is essential. Relative quantum phases affect how probability amplitudes add when different contributions reach the same output. Their interference can enhance one outcome and cancel another. A global phase, multiplying the entire quantum state by the same complex factor of magnitude one, changes neither these relative phases nor observable predictions. The Hadamard gate below provides a small experiment in which this distinction becomes a deterministic change of output.

Quantum coherence describes this capacity to interfere relative to a specified orthonormal basis. In a density matrix, it appears in off-diagonal entries; a diagonal mixed state retains the populations but lacks that basis coherence. Thus matching the probabilities of one measurement need not match the predictions of a later experiment. Baumgratz, Cramer, and Plenio made this basis-dependent viewpoint precise in their resource framework for quantum coherence. ^[ref].

This makes quantum information an operational question: which distinctions can a preparation, an available unitary operator, and measurement in quantum mechanics reveal? Amplitude amplification illustrates how carefully arranged interference converts a weakly represented answer into a likely observable result. Preserving the relevant quantum coherence throughout the computation is part of the physical task. ^[ref].

Transport of particles or charge when quantum mode structure, transmission and occupation of reservoir states determine the measured current.

A quantum witness is a polynomial-size quantum state supplied to a verifier as evidence for a YES instance. The verifier fixes its own ancilla qubits independently. Pure quantum witnesses suffice when maximizing a linear acceptance functional on density operators. Quantum witnesses are not generally restricted to basis vectors or product states.

Inhibition of a transition by increasingly frequent ideal projective measurements that check whether a quantum system remains in its initial quantum state. For a normalized vector $\psi$, a fixed Hamiltonian operator $H$ and the orthogonal projection $P=|\psi ⟩⟨\psi |$, the probability of passing every check at $N$ equally spaced times up to fixed time $T$ isFor a finite-dimensional Hilbert space, the quadratic short-time quantum survival probability implies ${\mathbb{P}}_N(T)\to 1$. This is an ideal limit with instantaneous measurements and unitary time evolution between them.