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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.

The 1999 experiment used an approximately $200\mathrm{fs}$, $394\mathrm{nm}$ pump pulse and a beta barium borate crystal. Two polarization-entangled photon pairs supplied four photons. A trigger and three other photodetectors selected a GHZ state through fourfold coincidence counting. Narrow interference filters lengthened the photon coherence time to approximately $500\mathrm{fs}$, exceeding the pump duration and suppressing timing information about which pair supplied which photon. The optical mode matching was essential to coherence, not merely to routing photons.^[ref]

The observed state was selected by all four detections, which also measured the photons. A trigger alone did not demonstrate a guaranteed surviving three-photon output. This distinction connects postselection to what an apparatus actually supplies, rather than identifying a fourfold coincidence with an unconditional source.^[ref]

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.

Start from three normalized Bell states ${\Phi }_{12}^{+}{\Phi }_{34}^{+}{\Phi }_{56}^{+}$. A polarizing beam splitter parity filter ${\Pi }_Z^{25}$ followed by $X$-basis outcome $++$, and a second filter ${\Pi }_X^{46}$ followed by $Z$-basis outcome $00$, have effective bras $⟨{\Phi }^{+}{|}_{25}/\sqrt{2}$ and $⟨{\Phi }^{+}{|}_{46}/\sqrt{2}$. Their contraction leaves ${\Phi }_{13}^{+}/8$. The herald probability is therefore $1/64$ per ideal three-pair input, or $1/16$ conditional on both parity-routing filters having succeeded. This distinction prevents renormalizing away the optical postselection cost.

Put $\delta =\pi /(2n)$. Between successive measurement bases, the squared overlap is $c={\cos }^2\delta$ for retaining the same label and $s={\sin }^2\delta$ for changing label. The first step obeys the same rule if we regard the initial state as label 0 at angle zero. Thus the recorded labels form a Markov chain with transition matrix $\left(\begin{array}{cc}c& s\\ s& c\end{array}\right)$.

Let $q_k$ be the probability of label 0 after step $k$. Then $q_0=1$ andSince $c-s=\cos (\pi /n)$, iteration yieldsThis is the total final-outcome probability, including histories that change label an even number of times. The smaller quantity ${\cos }^{2n}(\pi /(2n))$ counts only the history with label 0 at every step and is not the requested total. For example, at $n=3$ the total is $9/16$, whereas the uninterrupted history has probability $27/64$. At $n=1$, the final basis is $\{|1⟩,|0⟩\}$ up to signs, so the total probability of label 0 is zero, as the formula also gives. This distinguishes nonselective measurement-guided Zeno transfer from postselection on one record.

For heralded Bell-pair projection with two polarization parity filters, a Hadamard gate followed by computational-basis detection is a measurement in the diagonal polarization basis $|\pm ⟩=(|H⟩\pm |V⟩)/\sqrt{2}$. It can therefore be realized directly by a polarizing beam splitter at $45^{\circ }$ with a photodetector in each output: its transmitted and reflected clicks label $+$ and $-$. No separate Hadamard gate plate is required for this measurement.

Conditioning on one photon in each output of the first polarizing beam splitter implements the even $Z$-parity filterThe $45^{\circ }$ polarizing beam splitter similarly implements the even $X$-parity filter ${\Pi }_X^{46}=(I+X_4{X}_6)/2$. These are ideal coherent filters, with the optical phases chosen consistently. The all-zero photodetector record projects the upper outputs onto $⟨++{|}_{25}$ and the lower outputs onto $⟨00{|}_{46}$. The two effective bras areTherefore the unnormalized retained state isFor example, the simultaneous Bell contractions require the three pair bits to coincide, leaving only the $000$ and $111$ terms of the original pair expansion. Each surviving $|00{⟩}_{13}$ or $|11{⟩}_{13}$ amplitude is $1/(8\sqrt{2})$. ConsequentlyThis probability is conditioned on production of all three input pairs, but not yet on successful polarizing beam splitter routing. The routing probability itself is $⟨{\Pi }_Z^{25}{\Pi }_X^{46}⟩=1/4$: each nonidentity expectation vanishes on the relevant initial reduced state. Thus, if the question instead regards the routing postselection as already performed,These two denominators describe different conditioning. Neither includes the probability that the three crystals actually emit the required pairs, photodetector inefficiency or undesired multipair events.

Yes: a deterministic LOCC protocol exists for every allowed value of the parameter. Here is an explicit implementation, so existence does not depend on merely quoting a pure-state conversion theorem.

First convert the input Bell state to $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$ by local unitary operations:Write $c=\cos \varphi$ and $s=\sin \varphi$. Alice performs the two-outcome measurement in quantum mechanics with Kraus operatorsThey define a physical measurement in quantum mechanics becauseActing on $|{\Phi }^{+}⟩$, the unnormalized branch vectors areBoth branches have probability $1/2$. Alice communicates the outcome to Bob. If it is one, both apply the Pauli X gate; this turns the normalized second branch into $c|00⟩+s|11⟩$. If it is zero, neither applies this correction. Finally Bob applies a Pauli X gate in either branch, obtainingEvery outcome produces the same target pure state, so this is deterministic local operations and classical communication, not a successful branch selected by postselection. This two-outcome LOCC dilution of a Bell pair remains valid at $\varphi =0$, where the output is a product state, and at $\varphi =\pi /4$, where it is another Bell state.

The target Schmidt coefficients are $c,s$, while those of the input are both $1/\sqrt{2}$. For an interior value $0<\varphi <\pi /4$, local unitary operations alone could not change these Schmidt coefficients; the local measurement in quantum mechanics is the step that reduces the entanglement.

Expand an initial bipartite pure state in local orthonormal bases asBy the Schmidt decomposition, or equivalently the singular value decomposition of $C$, its Schmidt rank is $\mathrm{rank}C$. In a specified local measurement branch, the product Kraus operator $A\otimes B$ changes the coefficient matrix toIndeed the coefficient of $|k⟩|l⟩$ is ${\sum }_{i,j}{A}_{ki}{C}_{ij}{B}_{lj}$. The elementary matrix rank inequality givesNormalizing a nonzero branch multiplies $C^{\prime }$ by a scalar and therefore leaves its matrix rank unchanged.

Now refine a complete LOCC transcript to record every individual Kraus operator. Although later choices depend on the earlier classical outcomes, fixing a transcript fixes all those choices. Composing the local operations therefore gives a single product Kraus operator $A_{\tau }\otimes B_{\tau }$ in that branch. Each nonzero branch obeys the same Schmidt-rank contraction under product operators. Local auxiliary systems prepared independently are included by local isometries and do not add entanglement. Discarding a local auxiliary system is included by resolving its partial trace in an orthonormal basis; this again yields refined product Kraus operators.

If the protocol produces a specified pure state deterministically, its output density operator is a sum of positive rank-one branch density operators. Every nonzero branch vector must be proportional to that output vector: its squared overlap with any vector orthogonal to the output sums to zero, so each such overlap is separately zero. Thus the output Schmidt rank equals that of each surviving branch, andThis proves monotonicity of Schmidt rank under LOCC, even for a nonzero branch selected by postselection. If the classical transcript is instead discarded and the result is mixed, the refined branch vectors supply an ensemble with Schmidt ranks at most $r_{\mathrm{in}}$; by definition its Schmidt number is at most $r_{\mathrm{in}}$. This is the correct mixed-state extension, rather than a bound on the ranks of its reduced density matrices. In finite dimensions, any limiting exact protocol obeys the same pure-state bound because the set of coefficient matrices of rank at most $r_{\mathrm{in}}$ is closed, being defined by vanishing minors.

Let $U_1=e^{-iH(t-t_1)/\hslash }$ and $U_2=e^{-iH(t_2-t)/\hslash }$ describe unitary time evolution. For an ideal projective measurement with the Lüders rule, its unconditioned outcome probability is $p_j=\Vert P_j{U}_1|\psi ⟩{\Vert }^2$. After that outcome, the normalized state is $P_j{U}_1|\psi ⟩/\sqrt{p_j}$. The Born rule probability of successful postselection is then $|⟨{\psi }^{\prime }|U_2{P}_j{U}_1|\psi ⟩{|}^2/p_j$. Multiplying gives the joint probabilityConditional probability therefore gives the Aharonov-Bergmann-Lebowitz rule:The denominator must be positive; otherwise the selected subensemble does not occur. Define the forward-evolved ket $|a⟩=U_1|\psi ⟩$ and backward-evolved ket $|b⟩=U_2^{†}|{\psi }^{\prime }⟩$. The numerator becomes $|⟨b|P_j|a⟩{|}^2$, which is unchanged by interchanging $a,b$. Equivalently, with ${\rho }_a=|a⟩⟨a|$ and ${\rho }_b=|b⟩⟨b|$,This expresses the boundary-state symmetry explicitly. It follows from the ordinary time-asymmetric preparation, Born rule, and state update; it does not posit an additional backward dynamical collapse.

Restore the post-selected vector omitted entirely from the TeX aid by reading the original PDF. Write $D=N^2-N+1$ andIts norm is one because $N+(N-1{)}^2=D$. For the uniform prestate and $H=0$, put $c=1/\sqrt{(N+1)D}$. The individual transition amplitudes areIn experiment $E_i$, the complement amplitude is $c-c=0$. Thus for every $i\le N$,The successful postselection rate in this experiment is $c^2=1/((N+1)D)$.

In the fully resolved experiment $E_0$, the ABL rule squares the individual amplitudes before summing. Their squared sum is $D{c}^2=1/(N+1)$, givingThese probabilities sum to one. For $N=1$ they reduce to a single certain outcome in either quantum measurement. For $N\ge 2$, the N-box pre- and post-selection paradox is that each separate binary question can be answered affirmatively with certainty, although the fully resolved quantum measurement cannot give all those outcomes at once.

There is no inconsistency. In $E_i$, the unresolved complement preserves coherent cancellation between its basis contributions under the Lüders rule. In $E_0$, those alternatives are resolved, so their squared amplitudes add instead. The different projective measurements disturb the state differently and have different postselection success rates. Merely merging the recorded $E_0$ outcomes afterwards does not reproduce $E_i$. This is the context dependence of pre- and post-selected measurements; certainties in mutually alternative experiments do not describe simultaneous measurement-independent properties.

Apply $V^{-1}$ to the data and phase quantum registers, leaving the flag quantum ancilla untouched. This inverse needs no extra oracle assumption: the promised dyadic eigenvalues give $U^{2^n}=I$, hence $U^{-1}=U^{2^n-1}$. The inverse of each controlled unitary gate used in $V$ can therefore be built from repeated uses of the supplied controlled-$U$, and the known Hadamard gates and quantum Fourier transform gates can be reversed. Uncomputation is necessary to erase the eigenvalue label coherently. The resulting quantum state isA quantum measurement in the computational basis of the flag followed by postselection on one yieldsThis requires $A|b⟩\ne 0$. For a normalized input and nonnegative eigenvalues, the success probability of positive quantum spectral filtering satisfies $P_{\mathrm{success}}\ge {\lambda }_{\min }^2$. In the general inequality, equality holds precisely when the input is supported on the minimum-eigenvalue eigenspace. Omitting uncomputation and discarding the phase quantum register would instead leave a mixed state with diagonal weights proportional to $|{\beta }_j{|}^2{\lambda }_j^2$, rather than the desired coherent pure state.

The printed universal nonzero-success request needs a nonkernel-input hypothesis. An $n$-qubit Hermitian operator has $2^n$ eigenvalues, counted with multiplicity. All are distinct, and the printed dyadic grid contains exactly $2^n$ possible values. They therefore occupy the entire grid, including zero: this is a multiplicity-free complete dyadic spectrum, and ${\lambda }_{\min }=0$. Taking $n=1$, $A=\mathrm{diag}(0,1/2)$, and $|b⟩=|0⟩$ satisfies every printed spectral promise but gives $A|b⟩=0$. No normalized output vector exists, so no algorithm can deliver it with nonzero probability.

For every input outside the kernel, the procedure above has $P_{\mathrm{success}}>0={\lambda }_{\min }^2$, giving the requested strict bound on the meaningful domain. The corrected result is therefore exact quantum spectral filtering conditional on $A|b⟩\ne 0$, with the boxed probability. This does not require knowing the input probability amplitudes or making extra copies of an unknown quantum state. Arbitrarily small nonzero support outside the kernel gives arbitrarily small success probability, and a failed flag measurement disturbs the input; fresh independent trials cannot be assumed when only one unknown physical input is supplied.

A weak measurement couples an observable only weakly to a meter, so the pointer distributions associated with its different eigenvalues overlap strongly. One trial supplies little information and a small disturbance; an ensemble is needed to estimate a small mean shift. It is not an exact determination of a previously hidden eigenvalue with no disturbance.

For pointer position $q$ and conjugate momentum $p$, a short von Neumann measurement interaction has unitary $U=\exp (-igA\otimes p/\hslash )$. An eigenvalue $a$ translates the pointer by $ga$. The weak regime requires those translations to be small relative to the pointer width. Without postselection the first-order mean displacement is $g⟨A⟩$ for an initially uncorrelated, suitably centered meter.

If the system is prepared in $|i⟩$ and postselected in $|f⟩$, the unnormalized meter state isFor a real centered Gaussian pointer with no position–momentum correlation, the conditional position shift is $g\mathrm{Re}{A}_w$; the imaginary part affects the conjugate momentum distribution. The weak value can be complex or have a real part outside the observable's eigenvalue interval, because it is a conditional transition-amplitude ratio, not an eigenvalue.

An example apparatus uses photon polarization as the two-level system, a weak polarization-dependent transverse displacement as the interaction, and an output polarizer for postselection. Repeated position detections estimate the selected beam's displacement. Nearly orthogonal selection can make the conditional shift large, but also reduces the accepted sample and eventually invalidates the first-order expansion. The original weak-value proposal emphasized preselection, a weak coupling and postselection, rather than an anomalous result from an ordinary sharp spin measurement.^[ref]

Weak coupling limits quantum measurement backaction per trial; it does not abolish the information–disturbance tradeoff. Repeating weak observations can accumulate substantial backaction, and discarding rejected trials cannot automatically improve the information available per prepared system. Conditional pointer amplification and a statistically improved parameter estimate are distinct claims.

A postselection scheme in which two polarization-entangled photon pairs pass through polarizing beam splitters, a half-wave plate and a nonpolarizing beam splitter. One photon at each of a trigger and three output detectors selects a GHZ state in the three output polarizations, provided optical mode matching removes distinguishable pair labels. All four detections are needed for the selected event; the trigger alone does not guarantee the three other output occupations.

Let a uniformly random bit $A$ be copied into a three-state classical register $B$ with probability $\epsilon$, and otherwise replaced in $B$ by a flagged erasure independent of $A$. Then $I(A:B)=\epsilon$ bits. Projecting $B$ onto its two nonerased states and conditioning on success gives a perfectly correlated bit pair with mutual information one. The postselection branch is completely positive and trace decreasing, but renormalization after conditioning is not a deterministic quantum channel. This does not contradict data processing for quantum mutual information, which applies to trace-preserving operations or an unconditioned instrument including its outcome flag.

Let a Hermitian operator $A$ have dyadic eigenvalues ${\lambda }_j=c_j/2^t$ in $[0,1)$ for known $t$, with controlled access to the unitary operator $U=e^{2\pi iA}$. This interval ensures that different eigenvalues have different eigenphases; exact representability alone would not exclude phase aliasing, as $0$ and $1$ both give phase zero. Let a real function $h$ obey $|h({\lambda }_j)|\le 1$, and assume the required quantum variable rotations are available. Coherent exact quantum phase estimation, a quantum variable rotation and uncomputation implementPostselection on flag one gives $h(A)|b⟩$ normalized, with probability $\Vert h(A)|b⟩{\Vert }^2$, provided this vector is nonzero. Erasing the eigenvalue label by uncomputation is essential to preserve coherence between different eigenvectors. On this dyadic spectrum, $U^{2^t}=I$ gives $U^{-1}=U^{2^t-1}$, so reversing the phase-estimation gates is possible using forward controlled-$U$ calls. The choice $h(\lambda )=\lambda$ multiplies by $A$; a scaled reciprocal gives the different filter used in the HHL algorithm.

Let $c,s\ge 0$ and $c^2+s^2=1$. Starting from the Bell state $(|00⟩+|11⟩)/\sqrt{2}$, Alice applies the measurement in quantum mechanics with Kraus operators $M_0,M_1$ displayed above. Completeness follows from $M_0^{†}{M}_0+M_1^{†}{M}_1=I$. Each outcome has probability $1/2$. The normalized outputs are $c|00⟩+s|11⟩$ and $s|00⟩+c|11⟩$. Communicating the outcome allows both parties to apply a Pauli X gate in the second branch, making both outputs equal to the first. Thus this LOCC protocol deterministically converts one Bell pair into any two-qubit pure state with these Schmidt coefficients, after appropriate local unitary operations. It includes the product state endpoint without postselection.

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.