Quantum theory

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].

An effective generator of evolution over a period of a time-periodic Quantum Hamiltonian, with the choice of quasienergy branch stated.

The normalized Hadamard transform maps a vector indexed by binary strings through entries $2^{-n/2}(-1{)}^{x\cdot y}$, where the dot product is evaluated modulo two. It is its own inverse and an orthogonal transformation, hence also a unitary operator on complex input vectors. The Hadamard gate is the $n=1$ quantum case, and the Hadamard basis is the image of the computational basis. The Walsh-Hadamard transform gives the same tensor product construction.

For two qubits, applying one Hadamard gate to each prepares the uniform quantum superpositionEach probability amplitude is $1/2$, so each computational-basis measurement outcome has probability $1/4$. Applying the Hadamard transform again reconstructs $|00⟩$: signed contributions cancel at the other three outputs. Its usefulness lies in combining simple local quantum gates into an operation on exponentially many amplitudes. Extracting useful information still requires an appropriate subsequent transformation and measurement.

The Hadamard gate is the unitary operatorwhere $I$ is the identity operator. It maps the computational basis to the Hadamard basis:

These outputs have equal computational-basis measurement probabilities, but their relative quantum phase differs. Applying another Hadamard gate sends them to different definite outputs. Calling this gate a classical coin toss would discard precisely the information that makes this possible. ^[ref].

Starting from $|0⟩$, compare two quantum circuits, reading matrix products from right to left:Here $Z$ is the Pauli Z gate. After the first Hadamard gate, the probability amplitudes are $(1/\sqrt{2},1/\sqrt{2})$. Inserting $Z$ changes them to $(1/\sqrt{2},-1/\sqrt{2})$ without changing their squared magnitudes. At the final gate, the contributions to output zero cancel and those to output one add. Without $Z$, cancellation occurs at output one instead. This is controlled interference, with the quantum phase determining which answer emerges.

The same calculation works for any phase gate $P_{\varphi }=\mathrm{diag}(1,e^{i\varphi })$:For example, $\varphi =\pi /3$ gives $P(1)=1/4$, while $\varphi =\pi /2$ gives $P(1)=1/2$. The output-one amplitude at $\varphi =\pi /2$ is $(1-i)/2$: its two complex contributions add before the Born rule is applied. A phase gate alone leaves the populations unchanged; recombination converts its relative quantum phase into observable probabilities.

If quantum decoherence completely removes the computational basis coherence between the gates, the intermediate density matrix becomes $I/2$. Both the phase gate and final Hadamard gate leave $I/2$ unchanged, so $P(1)=1/2$ for every $\varphi$. The disappearing fringe explains why preserving quantum coherence matters: it enables the gate sequence to carry phase information into the final measurement.

For $|\pm i⟩=(|0⟩\pm i|1⟩)/\sqrt{2}$, direct matrix multiplication by the Hadamard gate gives coefficients $(1\pm i,1\mp i)/2$, proving the displayed identities. This exchanges the two Pauli eigenstates of the Pauli Y gate and agrees with $HYH=-Y$. The phases in the formula are global phases for each separate state; the relative phase inside each state is essential.

For $P_{\alpha }=\mathrm{diag}(1,e^{i\alpha })$, conjugation by the Hadamard gate turns a relative quantum phase into an $x$-axis Bloch vector rotation. In the convention $\rho =(I+\mathbf{P}\cdot \mathbf{\sigma })/2$, the map is $(P_x,P_y,P_z)\mapsto (P_x,P_y\cos \alpha -P_z\sin \alpha ,P_y\sin \alpha +P_z\cos \alpha )$. An $x$-polarized input is unchanged, whereas a $z$-polarized input gives a sinusoidal population difference. A maximally mixed input remains maximally mixed under every unitary time evolution.

A flat reference over $k$ adjacent Fock states has overlap $r_k$ with a one-photon shift and $s_k$ with a two-photon shift. These overlaps, obtained by counting shared number states, determine the reduced qubit channel of a number-conserving Hadamard interaction. For $k\ge 2$, its squared fidelity with the desired pure output is $1-(1+xz)/(2k)$ for input Bloch vector $(x,y,z)$. Its worst-case infidelity is $3/(4k)$ because $|xz|\le 1/2$. At $k=1$ the two-shift overlap is zero, not $-1$, and the fidelity is $1/2+(y^2-xz)/4$. Broad coherent number support, rather than merely a large fixed photon number, suppresses distinguishability of the outgoing field states.

The uniform superposition of all $n$-bit strings is

On $n$ qubits, the Walsh-Hadamard transform satisfieswhere the dot product is evaluated modulo two.

This is the Hadamard transform acting on the computational basis of $n$ qubits.

The qubit Hadamard basis consists of $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$.

Minimal coupling introduces a gauge potential through a covariant derivative, or replaces canonical momentum by $p-qA$. The associated scalar potential supplies the electric coupling. Magnetic minimal coupling is the magnetic-only nonrelativistic case.

For a particle of charge $q$ in a vector potential $\mathbf{A}$, the canonical momentum in the free Hamiltonian is replaced by $\mathbf{p}-q\mathbf{A}$, giving

A quantum fluctuation is a nonzero variance of an observable in a quantum state, even when its mean vanishes. A vacuum field mode has such zero-point variance. In an inflationary background, the expanding geometry transfers short-wavelength vacuum field fluctuations into primordial perturbations as their wavelengths cross the Hubble radius.

A vacuum fluctuation is a nonzero quantum variance in a vacuum state. For a single electromagnetic normal mode, dimensionless optical quadratures $a+a^{†}$ and $-i(a-a^{†})$ have zero means and unit variances in the vacuum state, despite zero photon number. The mode's zero-point energy is $\hslash \omega /2$.

For a particle of charge $-e$, the changes $A\mapsto A+\nabla f$ and $\varphi \mapsto \varphi -{\partial }_{t}f$ leave the minimally coupled Schrödinger equation invariant when $\psi \mapsto e^{-ief/\hslash }\psi$.

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].

Prepare $I/2$, measure a Pauli observable $N=\mathbf{n}\cdot \mathbf{\sigma }$, apply a fixed unitary operator $U$, then measure $N$ again. If the first and second eigenvalue labels are $j,k\in \{+1,-1\}$, their joint probability is $\frac{1}{2}|⟨n_k|U|n_j⟩{|}^2$. ConsequentlyThe first outcome has probability $1/2$ and its conditional state is $|n_j⟩$ by the Lüders rule. Summing $jk$ times the joint probabilities and using $N={\sum }_{j}j|n_j⟩⟨n_j|$ proves the trace formula. This temporal correlation differs from the spatial correlation $⟨\psi |N\otimes N|\psi ⟩$ of two simultaneously measured qubits.

A readout in which one logical quantum state is coupled to an optical cycling transition and scatters many photons, while another is dark. An ideal bright/dark record implements a projective measurement in the logical basis. Neglecting that record removes their relative quantum coherence. Missed photons, off-resonant excitation and pumping out of the logical subspace qualify the ideal model.

After a computational-basis projective measurement, an observer who lacks the result assigns a diagonal density matrix. If the observer's preparation prior has mean $q=\mathbb{E}|\alpha {|}^2$, the assignment is the displayed matrix. No outcome record and no stated prior do not logically force $q=1/2$; that value follows from a symmetric or maximum-entropy ignorance convention. An observer who knows the preparation instead uses $q=|\alpha {|}^2$.

The first moment of the energy measurement distribution. For discrete energy eigenstates it is ${\sum }_n{E}_n|c_n{|}^2$ when finite; a normalizable state need not have a finite value.

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.

In 1990, Wayne M. Itano, Daniel J. Heinzen, J. J. Bollinger and David J. Wineland reported inhibited transitions in about 5,000 ${}^9{\mathrm{Be}}^{+}$ ions held in a cylindrical Penning trap. Trapped magnesium ions, illuminated at 280 nm, kept the beryllium ions cold. A 320.7 MHz radio frequency field coupled two ground-state hyperfine structure sublevels during a 256 ms pulse adjusted for complete transfer. Up to 64 intervening 313 nm pulses addressed an optical transition from level 1: scattering distinguished it from level 2. Afterward, calibrated fluorescence counts over 100 ms measured the final quantum state population. This ensemble readout did not select ions that remained in the initial quantum state at every check; departure followed by return also contributes. The apparatus made inhibition by repeated quantum measurements experimentally accessible.^[ref]

Start in $a$, drive resonantly with Hamiltonian operator $\hslash \Omega X/2$, and perform $n$ ideal basis measurements separated by $T/n$. An interval changes either basis state with probability $p={\sin }^2(\Omega T/(2n))$. Ignoring measurement records leaves a two-state Markov chain whose population difference is multiplied by $1-2p=\cos (\Omega T/n)$ each step. This proves the displayed final-population formula.

For a pi pulse, $\Omega T=\pi$, it is ${\pi }^2/(4n)+O(n^{-2})$. The repeated projective-measurement survival probability instead counts records in which every result is $a$, giving $P_{\mathrm{all}\mathrm{a}}={\cos }^{2n}(\pi /(2n))$. Final return after an earlier departure separates the two finite-$n$ events.

Successively measure in bases whose first vector rotates from $|0⟩$ to $|1⟩$ in $N$ equal angular steps. The probability to remain on that branch throughout is the displayed product of squared overlaps. Without postselection, the final target probability is $[1+{\cos }^N(\pi /N)]/2$. Both approach one for an initial $|0⟩$, but a general input loses its relative phase between the two measurement branches; the process does not implement a coherent Pauli X gate on unknown inputs.

The coherent interaction that correlates a system observable with detector records before selecting an outcome. An ideal coupling maps $|s_j⟩|d_0⟩$ to $|s_j⟩|d_j⟩$. Linearity then maps a superposition to ${\sum }_j{c}_j|s_j{d}_j⟩$. The full state can remain pure and entangled, whereas a nonselective projective measurement removes coherence between different record sectors. Conditioning on one observed record is a third, selective operation.

A measurement with small information and disturbance per trial, often implemented by coupling an observable weakly to a meter so that its eigenvalue-dependent pointer distributions overlap. Repeated observations can still accumulate substantial measurement backaction. A weak value additionally requires specified preparation and postselection; a weak measurement without postselection need not involve one.

For preselected $|i⟩$ and nonorthogonal postselected $|f⟩$, the displayed transition-amplitude ratio controls the first-order conditional pointer response to a von Neumann measurement interaction. It can be complex or outside an observable's eigenvalue range. Expanding $⟨f|\exp (-igA\otimes p/\hslash )|i⟩$ to first order gives $⟨f|i⟩[1-ig{A}_{w}p/\hslash ]$, proving that interpretation within the expansion's validity regime.^[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.

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]

With two Hadamard gates as mixers, an unobstructed zero-phase arm gives only port 0 for upper-port incidence. An ideal absorber in arm 1 acts on the intermediate superposition $(|0⟩+|1⟩)/\sqrt{2}$. It absorbs with probability $1/2$; its no-absorption branch is the unnormalized state $|0⟩/\sqrt{2}$. Recombination gives amplitude $1/2$ at each surviving output, so the dark-port success probability is $1/4$. The result assumes a calibrated transparent alternative and an ideal absorber; a dark-port event alone does not establish those apparatus assumptions.

For an Elitzur-Vaidman bomb tester using a rotation and its inverse, let $p$ be the probability of entering the absorbing path and $q=1-p$. Conditional on a live absorber, one trial has explosion probability $p$, successful safe identification probability $pq$, and inconclusive probability $q^2$. The inconclusive projective measurement resets the input, so summing a geometric series gives the displayed success probability after at most $N$ trials. Its limit is $q/(1+q)$, approaching one half as $p$ tends to zero through positive values. Exactly zero coupling produces no information.

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.

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.

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.

In the 2007 cavity quantum electrodynamics experiment of Christine Guerlin and collaborators, including Sébastien Gleyzes, Michel Brune, Jean-Michel Raimond and Serge Haroche, an injected coherent state occupied a 51.1 GHz microwave cavity with niobium-coated mirrors exhibiting superconductivity. Velocity-selected rubidium chemical atoms were prepared as circular Rydberg atoms in levels with principal quantum numbers 50 and 51. Repeated atomic readouts progressively narrowed the inferred photon number to a Fock state; later records revealed jumps. The cavity damping time was $T_c=0.130\mathrm{s}$. Empty-cavity Ramsey interferometry scans fixed the analysis-phase origin; weak injected fields refined the response calibration. Finite fringe contrast required calibrated likelihoods; photon losses could interrupt convergence.^[ref]

The functional sequence is preparation, first Ramsey interferometry zone $R_1$, dispersive cavity $C$, second zone $R_2$, and state-selective detection. The first $\pi /2$ microwave pulse prepares an atomic quantum superposition; dispersive atom-cavity coupling adds a photon number-dependent relative quantum phase while suppressing energy exchange. The second pulse converts that quantum phase into atomic populations; state-selective field ionization reads $e$ or $g$. Successive chemical atoms carry the meter records away while the same field remains available.^[ref]

This arrangement makes quantum nondemolition measurement useful for following a field repeatedly and preparing a Fock state by conditioning. The photon number is inferred from a sequence of physical records; it is not a directly visible label attached to a photon. Each atomic probe makes a weak, partial measurement in quantum mechanics of photon number, and the accumulated records can approach a resolved count. A single binary meter need not identify one among many possible Fock states, as perfect discrimination of pure states requires orthogonality.

Consider an original balanced-readout model, rather than a fit to the experiment. Let $n\in \{0,\dots ,N_{\max }\}$, with a normalized prior probability $p_0(n)\ge 0$ and ${\sum }_n{p}_0(n)=1$. A detected chemical atom has outcome $s\in \{e,g\}$. Define the conditional probabilitiesHere $\chi$ is the accumulated quantum phase per photon, $\varphi$ the controlled analysis quantum phase, and $C$ the fringe contrast. The two probabilities sum to one for every $n$. Actual calibration may also require an offset and asymmetric detection errors; $C$ alone does not specify a quantum instrument.

Assume no photon number change during the record and fresh independently prepared atomic meters. For observed outcomes $s_1,\dots ,s_k$ at recorded settings ${\varphi }_1,\dots ,{\varphi }_k$, the Born rule and Bayes' theorem giveThe posterior probability is defined only for $Z_k>0$. Summing its numerator over $n$ proves normalization. This is also the update of diagonal elements of a field density operator under a measurement in quantum mechanics; it does not assume that an initial quantum superposition was an unknown pre-existing integer. The observed settings are part of the record, especially when selected adaptively.

For an exact finite example take $p_0=(1,1,1,1)/4$, $C=4/5$, and $\chi =\pi /2$. An $e$ outcome at $\varphi =0$ has likelihood $(9,5,1,5)/10$, giving $p_1=(9,5,1,5)/20$. Another $e$ at $\varphi =-\pi /2$ gives $p_2=(9,9,1,1)/20$. A subsequent $g$ at $\varphi =0$ givesThus three atomic records raise the posterior probability of $n=1$ from $1/4$ to $45/68$, while leaving finite uncertainty. No one of these outcomes identifies $n$ with certainty.

The periodic likelihood function imposes an identifiability limit. If $\chi =2\pi /q$ for an integer $q>0$, then $L_s(n+q;\varphi )=L_s(n;\varphi )$ for every setting: the posterior odds between $n$ and $n+q$ never change. Varying $\varphi$ can resolve a cosine reflection ambiguity within a period, but cannot remove this exact period. Changing $\chi$ can supply additional information. A finite cutoff is therefore a physical assumption to justify, not evidence that excluded high photon numbers were absent. At $C=0$ no outcome changes the prior probability; at $0<C<1$ neither binary outcome has an exactly zero likelihood.

The no-loss assumption also has a time limit. In an original zero-temperature loss model, let $\kappa =1/T_c$. A Fock state $|n⟩$ loses its first photon at rate $n\kappa$, so its survival probability is $\exp (-n\kappa t)$ and its mean lifetime is $T_c/n$ for $n>0$. Several records must be acquired within that lifetime to identify a plateau. When losses matter, first predict $p_k^{-}(n)={\sum }_m{T}_{nm}(\Delta t)p_{k-1}(m)$ using a normalized transition matrix, then apply the atomic Bayesian inference update to $p_k^{-}$. Thermal excitation requires upward transitions too. A delayed or broadened inference after a jump need not indicate that an atom caused the jump.

An explicit original ideal meter model explains how photon number survives while field quantum coherence changes. Let $\hat{N}$ be the number operator and prepare an atomic qubit in $|+⟩=(|e⟩+|g⟩)/\sqrt{2}$. The dispersive unitary operatorcommutes with $\hat{N}\otimes I$. For a field quantum superposition ${\sum }_n{c}_n|n⟩$, it produces ${\sum }_n{c}_n|n⟩|a_n⟩$, where $|a_n⟩=(e^{-in\chi /2}|e⟩+e^{in\chi /2}|g⟩)/\sqrt{2}$. The atomic meter states encode $n$ through relative quantum phase and can become entangled with the field. The second Ramsey interferometry pulse and detector turn those correlations into the outcome probabilities above when $C=1$.

Tracing out the atom multiplies a field density matrix entry ${\rho }_{nm}$ by $⟨a_m|a_n⟩=\cos ((n-m)\chi /2)$. Hence the nonselective quantum measurement preserves every photon number population ${\rho }_{nn}$ while changing off-diagonal quantum coherence. For $\chi =\pi /2$, coherence between $|0⟩$ and $|1⟩$ is multiplied by $1/\sqrt{2}$ per fresh meter, even with no photons absorbed. Field quantum phase information can therefore be disturbed by an energy-preserving probe. No universal self-adjoint phase operator or naive canonical number–phase commutator is needed for this conclusion.

One possible finite-contrast quantum instrument makes the distinction sharper. Define diagonal Kraus operators $K_e=\cos ((\varphi +\chi \hat{N})/2)$ and $K_g=\sin ((\varphi +\chi \hat{N})/2)$. After their ideal readout, flip the reported label with probability $r=(1-C)/2$. The unnormalized selective maps areTheir traces reproduce $L_e,L_g$ on a Fock state, and their sum reproduces the displayed dephasing quantum channel, independently of $r$. At $C=0$ the recorded labels carry no information although the physical meter interaction can still cause measurement backaction. This is an illustrative instrument, not a reconstruction of all laboratory imperfections; equal likelihoods alone do not determine the disturbance.

Choose a shared random axis uniformly from $x,y,z$, perform its entanglement-assisted nondemolition parity measurement with one Bell pair, and accept the anticorrelated parity. The spin singlet state always passes unchanged. The averaged acceptance operator is $\Omega ={\sum }_j(I-{\sigma }_j\otimes {\sigma }_j)/6=P_s+(I-P_s)/3$, so an orthogonal triplet state passes with probability one third. This is a statistical verifier with a nonzero acceptance gap, rather than an exact single-shot singlet-versus-triplet decision. The Bell-pair cost of exact nondemolition singlet verification concerns the stronger zero-error task.

Exact singlet verification accepts the spin singlet state with certainty, rejects every orthogonal state, and preserves the singlet on acceptance. Each nonzero accepting Kraus operator is then a scalar multiple of the singlet projector, whose operator Schmidt rank is four. The local Kraus rank bound from an entangled resource excludes a single shared Bell pair, which has Schmidt rank two. Two shared Bell pairs suffice by a Bell-state nondemolition measurement followed by reporting whether its label is the singlet. This protocol may disturb triplet coherence. The rank bound concerns the resource Schmidt rank, and is not an entropy lower bound of two bits for arbitrary resource states.

Share a Bell state meter pair, apply local system-to-meter CNOT gates, and measure each meter in the computational basis. The parity of the local meter records measures system $Z_A{Z}_B$ while preserving all coherence within either parity sector. The complete tuple of local records has Kraus operator $P_{u\oplus v}/\sqrt{2}$. Each local record is uniform, and the nonlocal result requires local operations and classical communication to compare records. A local Hadamard gate conjugation supplies the corresponding $X_A{X}_B$ measurement.

A finite family of quantum states is antidistinguishable if a quantum measurement has labelled outcomes such that each outcome has zero probability for its corresponding state, while the outcomes form a complete quantum measurement. It can exclude a state without identifying which state was prepared. The PBR exclusion measurement for zero and plus supplies a four-state projective measurement example.

Postselection conditions a quantum measurement on a specified outcome. If its branch acts by a linear operator $K$ on a normalized input $|b⟩$, its probability is $\Vert K|b⟩{\Vert }^2$, and its normalized output is $K|b⟩/\Vert K|b⟩\Vert$. This conditional state is defined only when the probability is nonzero. Physically unsuccessful trials are discarded; postselection is not a deterministic implementation of the nonlinear normalization map.

For an ideal projective measurement with the Lüders rule, condition on both an initial quantum state preparation and a final successful postselection. Its outcome probabilities are proportional to $|⟨b|P_j|a⟩{|}^2$, where $a$ is the forward-evolved initial state and $b$ the backward-evolved final state. The normalization must be nonzero. The expression is symmetric in these two boundary states, although its derivation uses the ordinary Born rule and conditional state update.

In the ABL rule, probabilities depend on the full intermediate quantum measurement instrument. A binary Lüders rule quantum measurement preserves coherence within its unresolved complement, whereas a fully resolved projective measurement removes it. Summing fine-grained probabilities after quantum measurement does not generally reproduce the coarse-grained experiment. Different postselection success rates are part of the distinction.

For a uniform prestate in $N+1$ dimensions and poststate proportional to ${\sum }_{i=1}^N|i⟩-(N-1)|N+1⟩$, each binary question $P_i$ versus $I-P_i$ has conditional probability one for $i\le N$. A fully resolved quantum measurement instead assigns $1/D$ to each of the first $N$ outcomes and $(N-1{)}^2/D$ to the last. The context dependence of pre- and post-selected measurements explains why these alternative certainties do not imply simultaneous properties.

A measurement interaction correlates a measured system's alternatives with distinguishable states of an apparatus. Reading the apparatus then implements the associated quantum measurement.

A coupling between system observable $A$ and meter momentum $p$ that translates a pointer coordinate by $ga$ for an $A$ eigenvalue $a$. For a weak measurement, these translations are small compared with the initial pointer width. Larger separated pointer distributions instead support a sharp readout of the coupled observable.

A generalized quantum measurement is specified by operators $M_i$ satisfying ${\sum }_i{M}_i^{†}{M}_i=I$. On input $\rho$, outcome $i$ has probability $\mathrm{Tr}(M_i^{†}{M}_i\rho )$ and conditional state $M_i\rho M_i^{†}/\mathrm{Tr}(M_i^{†}{M}_i\rho )$.

A nonselective measurement in quantum mechanics averages over outcomes that are not read or not communicated. Its quantum channel is $\rho \mapsto {\sum }_i{A}_i\rho A_i^{†}$, with ${\sum }_i{A}_i^{†}{A}_i=I$. The completeness condition makes the channel trace preserving. The average equals ${\sum }_i{p}_i{\rho }_i$, where ${\rho }_i$ are the conditional states from a selective quantum measurement. A nonselective projective measurement is the special case in which the operators are orthogonal projectors.

A selective measurement in quantum mechanics conditions on a known outcome. For Kraus operator $A_i$, the Born rule gives $p_i=\mathrm{Tr}(A_i^{†}{A}_i\rho )$ and the conditional density operator is $A_i\rho A_i^{†}/p_i$ when $p_i>0$. Conditioning changes the state and can change the conditional reduced density matrix of a correlated remote system. That conditional state is available as such only to an observer with the outcome information.

A positive operator-valued measure is a family of positive operators $E_i$ satisfying ${\sum }_i{E}_i=I$. Each $E_i$ is called an effect and determines a probability $p_i=\mathrm{Tr}(E_i\rho )$ for input density operator $\rho$. The effects need not be orthogonal projections or commute. This permits measurements with more outcomes than the dimension of the Hilbert space, such as the trine qubit POVM. Coupling to a quantum ancilla and measuring the larger system supplies a physical route to generalized measurements. ^[ref].

For an elementary unsharp $Z$ example on a qubit, takeOn the eigenstate $|0⟩$, $p_{\pm }=(1\pm \eta )/2$; at $\eta =1/2$ they are $3/4$ and $1/4$. At $\eta =0$ the outcome is a fair coin independent of the input, and at $\eta =1$ these effects specify a sharp $Z$ projective measurement. Choosing $K_{\pm }=\sqrt{E_{\pm }}$ specifies one quantum instrument, but $U_{\pm }\sqrt{E_{\pm }}$ for outcome-dependent unitary operators has the same effects and different state updates. Thus POVM does not determine the post-measurement state.

If every allowed input density operator has support of a positive operator inside a subspace with orthogonal projector $P$, completeness need only equal $P$ on that subspace. To define a measurement on the full Hilbert space, one can append $M_{\perp }=I-P$ when the original operators act only on the subspace. The added outcome has zero probability on supported inputs. The effects $M_r^{†}{M}_r$ form a POVM; the operators themselves additionally specify state updates.

A POVM $\{E_i\}$ in dimension $d$ corresponds to a quantum state ensemble with $p_i=\mathrm{Tr}(E_i)/d$ and ${\rho }_i=E_i/\mathrm{Tr}(E_i)$, omitting zero effects. Its average is the maximally mixed state $I/d$. Conversely any ensemble with that average defines a POVM by $E_i=d{p}_i{\rho }_i$. These transformations are inverse. A pure POVM corresponds exactly to a pure-state ensemble decomposition of $I/d$.

A POVM whose nonzero effects have rank one, $E_i=w_i|{\psi }_i⟩⟨{\psi }_i|$, with normalized pure states and $w_i>0$. Completeness is ${\sum }_i{w}_i|{\psi }_i⟩⟨{\psi }_i|=I$. This use of pure concerns effect rank, and does not mean an extremal point in the convex set of all POVMs.

Choose three pure qubit states whose unit Bloch vectors are coplanar, sum to zero and have pairwise angle $120$ degrees. The three effects $E_j=\frac{2}{3}|{\varphi }_j⟩⟨{\varphi }_j|$ sum to the identity, giving a POVM. If an input is the state orthogonal to ${\varphi }_1$ or to ${\varphi }_2$, outcome one excludes the first input and outcome two excludes the second. Each conclusive outcome has probability one half on its corresponding input; the third outcome is inconclusive with probability one half on either input. This gives unambiguous quantum state discrimination, rather than identification on every trial.

A POVM is informationally complete if its outcome probabilities determine every input density operator uniquely. Its effects must span the real space of Hermitian operators. For a qubit, normalization leaves at most $m-1$ independent probabilities from $m$ outcomes; an arbitrary Bloch vector has three real parameters, so one fixed informationally complete measurement requires at least four outcomes. Using several differently oriented binary measurements is a different experimental setting and can also perform quantum state tomography.

Choose the four unit Bloch vectors at the vertices of a regular tetrahedron, so ${\sum }_i{\mathbf{n}}_i=0$ and ${\sum }_i{\mathbf{n}}_i{\mathbf{n}}_i^T=4I/3$. The displayed four positive rank-one effects sum to $I$. On input $(I+\mathbf{r}\cdot \mathbf{\sigma })/2$, their probabilities are $p_i=(1+\mathbf{r}\cdot {\mathbf{n}}_i)/4$, giving the reconstruction $\mathbf{r}=3{\sum }_i{p}_i{\mathbf{n}}_i$. This realizes the smallest fixed qubit informationally complete POVM.

Two positive rank-one effects summing to $I$ in dimension two must be complementary orthogonal projections. Write the first as $\alpha |\psi ⟩⟨\psi |$ with normalized $\psi$. The second has eigenvalues $1-\alpha$ and $1$, so its rank-one condition forces $\alpha =1$. The two unit Bloch vectors are antipodal, and this POVM is a projective measurement.

A projective measurement has self-adjoint orthogonal projections $P_i$ satisfying $P_i{P}_j={\delta }_{ij}{P}_i$ and ${\sum }_i{P}_i=I$. Measuring an observable with discrete spectral decomposition $A={\sum }_i{a}_i{P}_i$ reports eigenvalue $a_i$ with Born rule probability $p_i=\mathrm{Tr}(P_i\rho )$. A projector can have rank greater than one: the outcome then identifies an eigenspace, rather than a particular eigenvector inside it. Projectors specify the outcome statistics; the usual ideal conditional update is the additional Lüders rule.

In the qubit example from measurement in quantum mechanics, $Z=|0⟩⟨0|-|1⟩⟨1|$. At $\theta =\pi /3$, the zero and one probabilities are $3/4$ and $1/4$, regardless of the relative quantum phase $\varphi$. Under the Lüders rule, obtaining zero prepares $|0⟩$; obtaining one prepares $|1⟩$, up to global phase. Repeating the same measurement returns the same label with certainty, provided no intervening evolution or operation changes the measured sector. Repeatability concerns the conditional output, and does not mean the first outcome was predetermined.

Change to $X$ and its Pauli X eigenstates $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$. Adding the two probability amplitudes before squaring givesAt $\theta =\pi /3$ and $\varphi =0$, $p_{+}=(2+\sqrt{3})/4\simeq 0.9330$ and $p_{-}\simeq 0.0670$. At $\varphi =\pi /2$, both are $1/2$; at $\varphi =\pi$, the first pair is exchanged. The $Z$ statistics remain $3/4,1/4$ throughout. This is the same addition-and-cancellation principle as wave interference: relative quantum phase changes how amplitudes reinforce or cancel in another basis.