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The scattering interpretation connects an incoming preparation to the relative frequency of outgoing directions and final atomic states. Max Born's 1926 paper treats a distant incoming Electron and a chemical atom, expands the outgoing wavefunction into final atomic states, and imposes conservation of energy. Its translated correction footnote specifies the square of the coefficient; the modern complex modulus notation below is ours.^[ref]

In his 11 December 1954 lecture, Max Born recalled the difficulty of interpreting bound Electrons alone. He instead described an incident Electron wave generating an outgoing spherical wave, with direction-dependent squared amplitude determining relative scattering probabilities. Distinct final atomic states describe excitation with corresponding Electron energy loss. This is a retrospective account of an interpretation, rather than the report of a new apparatus.^[ref]

The distinction from the modern Born rule matters. The historical question was which collision outcome occurs: a direction of motion and a final atomic state. The general projective measurement formula applies to arbitrary observables, including degenerate outcomes and mixed states; a positive operator-valued measure describes still more general detector statistics. Those formulations should not be read back as the explicit contents of the short 1926 scattering paper. Nor does the scattering interpretation itself supply a rule for the conditional post-measurement state.

Actual experimental settings are described in the Davisson–Germer experiment and the Franck-Hertz experiment. The former scans collected electric current as an Electron beam scatters from prepared nickel; changing direction, accelerating voltage and surface preparation probes electron diffraction. The latter uses acceleration through mercury vapor and a retarding collector to infer discrete energy losses from current minima, with vapor density and retarding settings as controls. Angular resolution and surface backgrounds limit the first inference; initial energy spread and collision positions limit the second. These linked experiments supply the preparation, apparatus and readout; the examples below are original calculations, not Born's measured data.

Uniformly sample one positive local term and measure the effect $h_j$. The averaged energy-flag effect on a witness is $H/M$, where $M$ is the number of terms. Repetition with a threshold halfway between the promised flag probabilities uses $O(M^2/(b-a{)}^2)$ witness registers for constant error. A spectral tensor-product analysis, as in QMA parallel repetition with entangled witnesses, proves soundness against entangled witnesses. Constant-locality positive operator-valued measures are efficiently implementable using one ancilla qubit.

Represent any universally quantum device, including its auxiliaries and arbitrary final readout, by a positive operator-valued measure $\{E_i\}$ on the input. Perfect identification would requireSince a positive operator has a positive square root, zero expectation implies $E_i|{\varphi }_j⟩=0$ for $j\ne i$. Also $I-E_i\ge 0$ and its expectation in ${\varphi }_i$ is zero, so $E_i|{\varphi }_i⟩=|{\varphi }_i⟩$. ConsequentlyThis proves that perfect discrimination of pure states requires orthogonality. Any nonzero overlap contradicts perfect identification, independently of whether the device disturbs its input.

Equivalently, a unitary operator coupling the input to a readout apparatus must preserve inner products. Distinct perfectly readable records have zero inner product, so their corresponding input vectors must already be orthogonal. Nonorthogonal candidate states cannot be identified with certainty in one use. Probabilistic unambiguous quantum state discrimination can have an inconclusive outcome, which is excluded by the guaranteed-identification requirement.

A measurement in quantum mechanics with outcomes $m$ is specified, in the single-operator-per-outcome description, by linear operators $M_m$ satisfying ${\sum }_m{M}_m^{†}{M}_m=I$. Its effects $F_m=M_m^{†}{M}_m$ form a positive operator-valued measure. The two measurement postulates are the Born rule for outcomes and the conditional state update:For a density operator the corresponding formulas are $p_m=\mathrm{Tr}(M_m\rho M_m^{†})$ and ${\rho }_m=M_m\rho M_m^{†}/p_m$. More general quantum instruments have multiple operators for each outcome and sum these terms. For a projective measurement of an observable $A={\sum }_m{a}_m{P}_m$, take $M_m=P_m$; the outcome value is $a_m$ and the postmeasurement state lies in its eigenspace. A POVM alone specifies probabilities, whereas the measurement operators or instrument also specify the backaction.

For quantum teleportation, Alice and Bob initially share the Bell pair $|{\Phi }^{+}{⟩}_{AB}=(|00⟩+|11⟩)/\sqrt{2}$. Alice also holds the input $|\psi {⟩}_C=\alpha |0⟩+\beta |1⟩$, with $|\alpha {|}^2+|\beta {|}^2=1$. The shared entanglement is a resource; the two transmitted classical bits alone could not transfer an arbitrary unknown quantum state.

Alice applies a controlled-NOT gate with $C$ as control and $A$ as target, then a Hadamard gate to $C$. Starting from $|\psi {⟩}_C|{\Phi }^{+}{⟩}_{AB}$, direct expansion givesFor example, the four receiver vectors are $\alpha |0⟩+\beta |1⟩$, $\beta |0⟩+\alpha |1⟩$, $\alpha |0⟩-\beta |1⟩$, and $-\beta |0⟩+\alpha |1⟩$. Thus the gate calculation fixes the ordering of the Pauli corrections, including the last branch's sign.

Alice measures $C,A$ in the computational basis, obtaining $(a,b)$. The Born rule gives each outcome probability $1/4$, independent of the input amplitudes. Bob's normalized conditional state is $X^b{Z}^a|\psi ⟩$. Alice sends the two outcome bits. Bob applies the inverse, $Z^a{X}^b$, and henceThe same circuit is a Bell-basis measurement on Alice's two qubits, expressed using ordinary gates followed by computational-basis measurements. It restores the state without Alice or Bob knowing $\alpha ,\beta$.

By linearity, the calculation also holds when the input is entangled with a reference: each outcome has branch map $|\psi ⟩\mapsto X^b{Z}^a|\psi ⟩/2$, and after correction every branch is one-half of the identity map. Summing the four corrected density operators gives exactly the original input density operator, with all reference correlations preserved. This is teleportation as an identity channel on a reference. Without the two classical bits Bob must average the four uncorrected states, giving $\frac{1}{4}{\sum }_{a,b}{X}^b{Z}^a\rho Z^a{X}^b=I/2$, so the process cannot signal before the message arrives. The Bell pair is consumed and Alice retains no copy of the unknown state, in accordance with the no-cloning theorem.

Use the optimal three-to-one qubit random access code. For Alice's bits $(b_1,b_2,b_3)$, put $s_j=(-1{)}^{b_j}$ and send the pure state with Bloch vectorThese are the eight vertices of a cube inscribed in the Bloch sphere. If an explicit ket is wanted, choose $\cos {\theta }_b=s_3/\sqrt{3}$ and $e^{i{\phi }_b}=(s_1+i{s}_2)/\sqrt{2}$, and prepare $\cos ({\theta }_b/2)|0⟩+e^{i{\phi }_b}\sin ({\theta }_b/2)|1⟩$.

Bob wanting bit $j$ performs a Pauli measurement of ${\sigma }_x,{\sigma }_y,{\sigma }_z$ respectively, reporting zero for eigenvalue $+1$ and one for $-1$. Its success for every input and every requested bit isOnly one chosen bit is extracted; the protocol does not recover all three simultaneously and uses no shared entanglement.

To prove that no one-qubit protocol gives a greater guaranteed probability, allow Alice arbitrary mixed encoding vectors $|r_b|\le 1$ and Bob any binary positive operator-valued measure. Write his decision difference as $D_j=E_{j,0}-E_{j,1}=t_{j}I+v_j\cdot \sigma$. Its eigenvalues lie in $[-1,1]$, implying $|v_j|\le 1-|t_j|\le 1$. On input $b$, success for bit $j$ is $[1+s_j(t_j+v_j\cdot r_b)]/2$. Average over all eight strings and three choices. The terms in $t_j$ cancel, leavingThe Cauchy-Schwarz inequality and cancellation of cross terms over all sign choices giveHence $p_{\mathrm{avg}}\le p_{\ast }$. Any worst-case guarantee is no greater than this average; the cube protocol attains it uniformly, proving optimality both for guaranteed success and for the uniform-input average. Biased prior information about the bits would define a different optimization problem.

In the generalized measurement postulate, outcome $m$ is described by a Kraus operator $M_m$, with ${\sum }_m{M}_m^{†}{M}_m=I$. On a density operator $\rho$ its probability and conditional output, for nonzero probability, areFor a pure input the conditional vector is $M_m|\psi ⟩/\sqrt{p_m}$. The effects $E_m=M_m^{†}{M}_m$ are positive and sum to the identity, defining a positive operator-valued measure.

More generally an outcome can have several Kraus operators $M_{m\alpha }$. Then sum over $\alpha$ in both the numerator and the probability, and require ${\sum }_{m,\alpha }{M}_{m\alpha }^{†}{M}_{m\alpha }=I$.

The usual projective measurement with the Lüders rule is recovered when the measurement operators are mutually orthogonal projections, $M_m=P_m=P_m^{†}=P_m^2$, $P_m{P}_n=0$ for $m\ne n$, and ${\sum }_m{P}_m=I$. Projector-valued effects alone give projective outcome statistics; to obtain the Lüders state update as well, the instrument must implement $\rho \mapsto P_m\rho P_m$, without an additional conditional state change.

No measurement on the supplied system alone distinguishes ensembles with the same density operator. A completely general measurement in quantum mechanics has POVM effects $E_r\ge 0$ with ${\sum }_r{E}_r=I$. The outcome probability for a quantum ensemble isIf both ensembles have the same $\rho$, every outcome probability is identical. This proves the claim for all positive operator-valued measures, and includes projective measurements as a special case. An quantum ancilla in a fixed independent state does not help: the input density operators remain identical after tensoring with that ancilla, and the same argument applies to any joint measurement.

The premise concerns access only to the prepared state. An accessible preparation label or a correlated purification of a density operator is additional information outside that premise and can distinguish global preparations. Without such side information, different ensemble decompositions do not define distinguishable states.

Use an $n$-qubit state as the witness. Uniformly choose a term $h_j$ and perform the two-outcome positive operator-valued measure $\{I-h_j,h_j\}$. Regard the second outcome as an energy flag. For witness $\rho$, its probability isThis is an efficient local-energy measurement verification: a unitary operator on the $k$ relevant qubits and one ancilla qubit can implement the measurement using the isometryThe local dimension is constant; the term matrices and the measurement can be approximated to inverse-polynomial accuracy. Uniform choice of $j$ and the classical threshold comparison also have polynomial overhead.

SetIn the nontrivial case $0\le a<b\le M$, $\delta$ is inverse-polynomial. Cases outside this energy range are decided directly from $0\le H\le MI$.

Ask for $r$ witness registers, measure an independently sampled local term on each, and accept if the mean number of energy flags is at most $\theta$. An honest product ground state on a YES instance has flag probability at most $p_{\mathrm{y}}$. The Hoeffding inequality gives failure at most $e^{-r{\delta }^2/2}$. For completeness, the required Bernoulli concentration estimate follows by bounding the centered log moment-generating function $f(\lambda )=\ln \mathbb{E}{e}^{\lambda (Y-p)}$: $f(0)=f^{\prime }(0)=0$ and $f^{\prime \prime }(\lambda )$ is a tilted Bernoulli variance, at most $1/4$. Hence $f(\lambda )\le {\lambda }^2/8$; exponential Markov's inequality optimized at $\lambda =4u$ yields $\mathrm{Pr}(\overline{Y}-p\ge u)\le e^{-2r{u}^2}$, and the lower-tail version follows in the same way.

Soundness must also cover entangled witness registers. The averaged flag effect is $R=H/M$. All its eigenvalues are at least $p_{\mathrm{n}}$ on a NO instance. The repeated verifier's acceptance effect is a polynomial in the commuting tensor-factor operators $R_j$ and $I-R_j$. In their product eigenbasis its eigenvalues are lower-tail probabilities of independent Bernoulli variables whose parameters are each at least $p_{\mathrm{n}}$. Their lower tail is bounded by the identical-parameter tail at $p_{\mathrm{n}}$, using the same uniform-variable coupling as in QMA parallel repetition with entangled witnesses. Therefore its operator norm is at most $e^{-r{\delta }^2/2}$, which bounds acceptance for any entangled witness.

Choosinggives completeness at least $2/3$ and soundness at most $1/3$. A slightly larger constant absorbs implementation errors; for example, approximate each flag effect in norm to at most $\delta /8$, retain a gap at least $3\delta /4$, and increase $r$ by a constant factor. The total witness length $rn$ and circuit size are polynomial. Thus

The generalized measurement postulate specifies operators $\{M_i\}$ satisfyingOn state $\rho$, outcome $i$ has probabilityand, when $p_i>0$, conditional stateThe effects $E_i=M_i^{†}{M}_i$ form a positive operator-valued measure.

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.

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.

A measurement in quantum mechanics 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].