Quantum measurement ($p_m=\mathrm{Tr}(E_m\rho )$)

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.

The beauty of this basis dependence is that information invisible to one measurement becomes accessible to another. A $Z$ measurement of an ensemble estimates its quantum state populations, while $X$ probes one component of its quantum coherence. These are experiments on separate identically prepared copies. Sequentially measuring $Z$ and then $X$ on one copy instead probes the disturbed post-measurement state. Quantum state tomography combines suitable ensemble statistics to reconstruct the original state.

A rank-one projective measurement associated with an orthonormal basis $\{|b_j⟩{\}}_j$. For a density matrix $\rho$, outcome $j$ has probability $⟨b_j|\rho |b_j⟩$ and leaves the measured system in the pure state $|b_j⟩$ whenever that probability is nonzero. It can be implemented by mapping this orthonormal basis to the computational basis with a unitary transformation and performing a computational-basis measurement.^[ref]

For an ideal projective measurement with rank-one projectors $P_m=|m⟩⟨m|$, every initial state that can give outcome $m$ leaves the same conditional state $|m⟩⟨m|$. Every subsequent procedure using only that system and the recorded outcome therefore has the same statistics for all such initial hypotheses. It cannot improve the posterior about the preparation. This claim concerns additional measurements after the first collapse, not access to fresh copies or a record preserved before the measurement.

With $|V⟩=|0⟩$ and $|H⟩=|1⟩$, an analyzer oriented at angle $\varphi$ from vertical has positive-outcome state $\cos \varphi |0⟩+\sin \varphi |1⟩$. Subtracting the two orthogonal rank-one projection operators gives $M(\varphi )=\cos 2\varphi Z+\sin 2\varphi X$, where $X,Z$ are Pauli operators. Physical polarization angles are therefore half their corresponding Bloch sphere angles.

For a normalized quantum state $\psi$ and a time-independent Hamiltonian operator $H$, consider ideal projective measurements of the orthogonal projection $P=|\psi ⟩⟨\psi |$ at intervals $\Delta t=T/N$. Write $U=e^{-iH\Delta t/\hslash }$. A successful check leaves the post-measurement state $\psi$ up to global phase. Thus each successive success, conditional on earlier successes, has conditional probability $|⟨\psi ,U\psi ⟩{|}^2$, and multiplication gives the displayed formula. Equivalently,This is the probability of the entire sequence of successful checks. It underlies the quantum Zeno effect.

The projection step in the traditional measurement account. For outcome projectors $P_j$, the nonselective state is ${\sum }_j{P}_j\rho P_j$; conditional on outcome $j$ it is $P_j\rho P_j/\mathrm{Tr}(P_j\rho )$. Distinguishing these from the coherent state produced by quantum premeasurement avoids identifying a pure correlated projector with its dephased mixture.

Let self-adjoint orthogonal projections $P_i$ sum to the identity. For $v$ in the range of $P_j$,The $j$th term already equals $\Vert v{\Vert }^2$, so $P_{i}v=0$ for every $i\ne j$. Hence $P_i{P}_j=P_j{P}_i=0$ and the projective measurement operators commute.

// Target: quantum-theory.bigb

The orthonormal quantum states $|{\alpha }_s⟩=(|0⟩+(-1{)}^s{e}^{-i\alpha }|1⟩)/\sqrt{2}$, $s=0,1$, define an equatorial projective measurement. The measured observable is $\cos \alpha X-\sin \alpha Y$, with eigenvalue $(-1{)}^s$. Thus $\alpha =0$ measures $X$, and $\alpha =\pi /2$ measures $-Y$ with these outcome labels. The sign convention matters when comparing different definitions of equatorial bases.

For $|v_s(\eta )⟩=(|0⟩+(-1{)}^s{e}^{i\eta }|1⟩)/\sqrt{2}$, adding $\pi$ exchanges the two outcomes of the same projective measurement. At an angle $\varphi =j\pi /2$, the choices $\varphi$ and $-\varphi$ therefore specify the same unordered pair of projectors: their labels differ by $j\mathrm{mod}2$. An adaptive choice between these signs can be replaced by a fixed measurement and a classical label exchange. Together with Pauli-frame propagation along a measurement wire, this permits simultaneous measurements when all adaptive sign changes occur at these Clifford gate angles.

The Lüders rule specifies the ideal quantum instrument for a projective measurement. Conditional on an outcome with nonzero probability, it givesfor a density operator and a pure state, respectively. The second formula applies only to a pure state input. If the result is discarded, the output is the nonselective projective measurement ${\sum }_i{P}_i\rho P_i$.

The crucial feature for a degenerate eigenvalue is preservation of quantum coherence within its eigenspace. For a qutrit with orthonormal basis $|0⟩,|1⟩,|2⟩$, considerOn $|\chi ⟩=(|0⟩+|1⟩)/\sqrt{2}$, outcome zero has probability one and the Lüders rule leaves $|\chi ⟩$ unchanged. Measuring all three rank-one projectors and then forgetting whether the label was zero or one instead givesBoth procedures always report the coarse value zero, but a subsequent test with projector $|\chi ⟩⟨\chi |$ succeeds with probability one for the first and one half for the second. Resolving extra information destroyed coherence. This explains why a quantum nondemolition measurement of a degenerate quantity must specify what information the apparatus actually records.

Gerhart Lüders proposed this state-change rule in 1951, explicitly discussing degenerate quantities and distinguishing selected subensembles from the aggregate measured ensemble. Kirkpatrick's translation was first posted in 2004; equations (7) and (8) give the unnormalized selective update and the nonselective update. ^[ref].

Repeatability follows briefly from $P_j{P}_i={\delta }_{ij}{P}_i$: after outcome $i$, $\mathrm{Tr}(P_j{\rho }_i)={\delta }_{ij}$. This assumes a second application of the same ideal instrument without intervening dynamics. An instrument with the same effects can rotate or reset the output and need not satisfy this conclusion; see POVM does not determine the post-measurement state.

The post-measurement state is the conditional quantum state after a specified measurement outcome. For a projective measurement, it is given by the Lüders rule.

If the outcome of a projective measurement $\{P_i\}$ is discarded, its nonselective state update is the pinching channelIt removes coherence between distinct measurement subspaces.

For mutually orthogonal projectors summing to the identity, a pinching map discards coherences between their ranges while preserving each diagonal block. It is a quantum channel with Kraus operators $P_i$, and a unital quantum channel. A state is fixed by pinching exactly when it commutes with every projector.

In an orthonormal basis, rank-one dephasing is the quantum channel $\Delta (\rho )={\sum }_i|{\psi }_i⟩⟨{\psi }_i|\rho |{\psi }_i⟩⟨{\psi }_i|$. It removes off-diagonal entries while preserving the basis probabilities. Its output Von Neumann entropy is their Shannon entropy, and the relative-entropy identity for rank-one dephasing quantifies the entropy increase.

Because $\log \Delta (\rho )$ is diagonal, $\mathrm{Tr}(\rho \log \Delta (\rho ))=\mathrm{Tr}(\Delta (\rho )\log \Delta (\rho ))$. Hence $D_{\mathrm{rel}}(\rho \Vert \Delta (\rho ))=S(\Delta (\rho ))-S(\rho )$. Support inclusion under rank-one dephasing handles zero probabilities, and Klein's inequality makes the difference nonnegative. Equality holds exactly when the state was already diagonal.

If $p_i=⟨{\psi }_i|\rho |{\psi }_i⟩=0$, positivity gives $\sqrt{\rho }|{\psi }_i⟩=0$, so the corresponding basis vector is in $\ker \rho$. Therefore $\ker \Delta (\rho )\subseteq \ker \rho$, and $\mathrm{supp}\rho \subseteq \mathrm{supp}\Delta (\rho )$. This makes the quantum relative entropy of $\rho$ relative to its rank-one dephasing finite.

Quantum state tomography estimates a density operator from measurement frequencies on many independently and identically prepared systems. One fixed projective measurement reveals only its basis populations, so further measurement settings are generally needed. For a qubit, writeThe three Pauli matrices thus turn measured expectation values into the Bloch vector. This reconstruction uses the Bloch ball representation of arbitrary qubit density operators. ^[ref].

Divide the prepared ensemble into groups measured in the $X$, $Y$ and $Z$ bases. For example, positive-outcome frequencies $0.8,0.5,0.9$ give the estimate $\mathbf{r}=(0.6,0,0.8)$ andHere $|\mathbf{r}|=1$, so the reconstructed state is the pure state $(3|0⟩+|1⟩)/\sqrt{10}$. Its $Z$ frequencies alone also fit the diagonal mixed state with populations $0.9,0.1$; the $X$ data distinguish them. For the angular pure state in measurement in quantum mechanics, $r_x=\sin \theta \cos \varphi$, $r_y=\sin \theta \sin \varphi$ and $r_z=\cos \theta$, so $Y$ resolves the sign of the phase that $X$ alone cannot.

Finite samples introduce statistical uncertainty: with $N$ independent binary trials in one basis, the frequency has variance $p(1-p)/N$. At $p=1/2$ and $N=100$, its standard deviation is $0.05$. Fluctuations can place a naive Bloch vector estimate outside the Bloch ball; a physical reconstruction must enforce positivity of the density operator. Repeated measurements on one unprepared specimen sample its changing conditional state, rather than independent copies of its original state. Multiple bases are needed in this projective protocol; a single fixed informationally complete POVM, such as the tetrahedral qubit POVM, is an alternative. It still requires an ensemble of trials.

A polarizing beam splitter converts a photon polarization qubit to two paths. Rotating one arm's photon polarization to match the other and recombining at a balanced beam splitter gives $P_1(\theta )=1/2+\mathrm{Re}({\alpha }^{\ast }\beta e^{i\theta })$. Measurements at phases zero and $\pi /2$ recover the transverse Bloch vector components; direct counts in the two separated arms recover the population difference. All three components, rather than a fringe alone, determine an arbitrary photon polarization density matrix.

// Target: quantum-theory.bigb

To identify the physical channel $\rho \mapsto V\rho V^{†}$ of a single-qubit unitary gate, prepare each of the three positive Pauli matrix eigenstates and estimate each output Pauli expectation. Their nine means areThey determine the Bloch vector rotation $\mathbf{r}\mapsto R\mathbf{r}$, hence the channel on all states. If two gates have the same channel, their relative unitary commutes with all three Pauli matrices: commuting with $Z$ forces it diagonal and commuting with $X$ forces equal diagonal entries. Thus they differ by a scalar global phase, proving uniqueness up to that phase. This procedure needs no controlled gate or eigenstate of the unknown gate. Estimating its bounded measurement means to error $ϵ$ with fixed confidence uses order ${ϵ}^{-2}$ repetitions. It determines the relative eigenphases but cannot determine an overall eigenvalue phase from uncontrolled gate access alone.