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Pre- and post-selection, weak values, and contextuality

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Jeff Tollaksen

Quantum Physics · Submitted 28 Feb 2006 · 50 references

Pith's verdict: Elegant weak-value examples and a plausible strengthening of Leifer-Spekkens, but the central theorem is a placeholder until the simultaneous weak-measurement derivation is supplied.

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arXiv IDquant-ph/0602226 (version 3)
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CitationJeff Tollaksen. "Pre- and post-selection, weak values, and contextuality." arXiv:quant-ph/0602226 (2006). https://pith.science/paper/2DLNC3KN
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The paper

The paper claims

Every definite pre/post-selection paradox is a contextuality proof

The reading

This paper argues that contextuality—the Bell–Kochen–Specker obstruction to treating quantum observables as carrying pre-existing values—is not merely a formal no-go theorem but an effect that can be probed experimentally. The route is pre- and post-selection: prepare a state, post-select a later state, and look at what can be said about measurements in between. The claim is that every logical PPS-paradox whose intermediate outcomes are definite, such as the three-box paradox, forces contextuality once the counterfactual intermediate measurements are performed weakly, because weak measurements realize all of them at once without disturbing the pre- and post-selection. The measurable signature is an "eccentric" weak value outside the eigenvalue spectrum—minus one particle in box C in the three-box example—which cannot be reproduced by any classical-like hidden-variable theory. If the claim is right, contextuality stops being a purely abstract constraint and becomes an accessible phenomenon, with consequences for how non-classical correlations are used in quantum information.

What carries the argument

The load-bearing objects are weak values and the diagonal-PPS mechanism. A weak value of an observable $\hat{A}$ between a pre-selected state $\lvert \Psi_{\mathrm{in}}\rangle$ and a post-selected state $\langle \Psi_{\mathrm{fin}}\rvert$ is $A_w = \langle \Psi_{\mathrm{fin}}\rvert \hat{A}\rvert \Psi_{\mathrm{in}}\rangle / \langle \Psi_{\mathrm{fin}}\vert \Psi_{\mathrm{in}}\rangle$; it is the pointer shift of a measurement too weak to disturb the pre/post-selection. Two theorems carry the argument: the weak value of a sum is the sum of the weak values, and if an ideal measurement would give a definite outcome, the weak value equals that outcome. The troublesome intermediate observables in a PPS-paradox are "diagonal-PPS": their definite values are inferred from both the forward-propagating pre-selected vector and the backward-propagating post-selected vector, so an ideal measurement of one destroys the information needed by another, even when the observables commute. Weak measurements avoid this disturbance, so all counterfactual alternatives can be probed at once, and the eccentric weak values they produce witness contextuality.

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the counterfactual measurements appearing in a PPS-paradox are not merely logical alternatives: performed weakly, they are all realized at the same time, and their weak values match the definite outcomes that ideal measurements would give individually. In the three-box example, with pre-selection $(\lvert A\rangle+\lvert B\rangle+\lvert C\rangle)/\sqrt{3}$ and post-selection $(\lvert A\rangle+\lvert B\rangle-\lvert C\rangle)/\sqrt{3}$, an ideal check of box A would certainly find the particle and an ideal check of box B would too, yet total particle number is one. Weak measurements then give $(\hat{P}_A)_w=1$ and $(\hat{P}_B)_w=1$, so by linearity $(\hat{P}_C)_w=-1$: a negative number of particles in box C. The same machinery is applied to Mermin's four-dimensional Bell–Kochen–Specker nonet, where the pair-occupation projector has weak value $(\hat{N}_{--})_w=-1/2$, to EPR-type states, and to the GHZ state. In each case the violation of the product rule—the weak value of a product is not the product of the weak values—is presented as contextuality made empirically visible. The paper also offers a physical reason why two ideal measurements of commuting observables can still interfere: the definite assignments rely on information from both boundary vectors ("diagonal-PPS"), and the first ideal measurement destroys that shared information.

Load-bearing premise

The argument assumes that several weak measurements can be performed at the same time between the same pre- and post-selected states and that their pointer shifts combine into the individual weak values; if simultaneous weak measurements disturb one another or fail to reproduce the counterfactual ideal outcomes, the step from definite PPS predictions to demonstrated contextuality does not go through.

Editorial extensions

If this is right

  • A logical PPS-paradox with definite predictions is not a curiosity: it is an experimental demonstration of contextuality, because weak measurements realize all its counterfactual intermediate measurements simultaneously.
  • Replacing ideal measurements by weak measurements in the three-box paradox yields a directly observable negative occupation weak value $(\hat{P}_C)_w=-1$, so "eccentric" weak values are testable signatures of non-classicality.
  • In Mermin's 4D nonet, the product rule fails: $\hat{\sigma}_x^1\hat{\sigma}_y^2$ and $\hat{\sigma}_x^2\hat{\sigma}_y^1$ each have weak value $+1$ while their product has weak value $-1$, and the pair-occupation weak value $(\hat{N}_{--})_w=-1/2$ can be measured, giving Bell–Kochen–Specker an experimental handle.
  • The isomorphism between weak values in BKS nonets and in EPR-type entangled states suggests that contextuality and entanglement are two faces of the same non-classical correlation, not separate phenomena.
  • Because weak measurements do not disturb the pre/post-selection, the attempt to explain PPS-paradoxes away by measurement disturbance is closed off; any classical-like hidden-variable theory would have to reproduce eccentric weak values, which a positive-definite distribution over eigenvalues cannot.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the simultaneous-composition assumption survives experimental test, the same reasoning should extend to state-independent contextuality proofs: a large finite set of pre/post-selected ensembles could weakly register the Kochen–Specker contradiction directly, without needing a special entangled state.
  • The negative weak values look like a quantifiable resource: in the same way entanglement moved from a foundational puzzle to a computational and cryptographic resource, contextuality witnessed by eccentric weak values could be used for tasks such as state discrimination, metrology, or randomness certification.
  • A systematic search over GHZ-type states could test the paper's closing conjecture that every Bell-inequality or BKS violation admits a post-selection yielding non-classical weak values, giving a universal experimental signature for non-classicality.
  • One could interpolate continuously between the three-box and Mermin settings to see whether the size of the negative weak value tracks the strength of the contextuality proof, providing a quantitative measure of non-classicality.
Original abstract

By analyzing the concept of contextuality (Bell-Kochen-Specker) in terms of pre-and-post-selection (PPS), it is possible to assign definite values to observables in a new and surprising way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated including a proof that every PPS-paradox with definite predictions implies contextuality. Certain results of these measurements (eccentric weak values with e.g. negative values outside the spectrum), however, cannot be explained by a "classical-like" hidden variable theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,665 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.74 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.24 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

By analyzing the concept of contextuality (Bell-Kochen-Specker) in terms of pre-and-post-selection (PPS), it is possible to assign definite values to observables in a new and surprising way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated including a proof that every PPS-paradox with definite predictions implies contextuality. Certain results of these measurements (eccentric weak values with e.g. negative values outside the spectrum), however, cannot be explained by a "classical-like" hidden variable theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 By analyzing the concept of contextuality (Bell-Kochen-Specker) in terms of pre-and-post-selection (PPS), it is possible to assign definite values to observables in a new and surprising way. contribution 0.88 0.64 0.51 hedged possibility 0.74
s1 Physical reasons are presented for restrictions on these assignments. contribution 0.53 0.72 0.07 asserted 0.94
s2 When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated including a proof that every PPS-paradox with definite predictions implies contextuality. result 0.74 0.70 0.45 asserted 0.97
s3 Certain results of these measurements (eccentric weak values with e.g. negative values outside the spectrum), however, cannot be explained by a "classical-like" hidden variable theory. result 0.57 0.59 0.21 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.82.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. a) pre-selected vector |Ψini = 1/ √ 3 (|Ai + |Bi + |Ci) propagates forwards in time from tin to t1, and post-selected vector |Ψfini = 1/ √ 3 (|Ai + |Bi − |Ci) propagating backwards in time from tfin to t2. b) IM of PˆA at t1 and of Pˆ B at t2. 2.1 Weak Measurements WMs can be quantified in the quantum measurement theory developed by von Neu￾mann [38]: to measure an observable Aˆ of the system, one may use an interaction Hamiltonian of the form Hint = −λ(t)QˆAˆ where Qˆ is an observable of the measuring device (MD) and λ(t) is a coupling constant which is non-zero only during a short
Figure 2. 4-D BKS example ✦ Individually, each of the 9 observables depicted in fig. 2 has eigenvalues ±1. A noncontextual-HVT requires an assignment of values that is ±1. In addition, eq. 1.1 requires that the value assigned to the product of all three observables in any row or column must obey the same identities that the observables themselves satisfy, i.e. the product of the values assigned to the observables in each oval yields a result of +1 except in the last column which gives −1. (The value assignments are given by Vψ~(ˆσ 1 x ) = hσˆ 1 x N I 2 i, Vψ~(ˆσ 2 x ) = hI 1 N σˆ 2
Figure 3. PPS states for Mermin example. it is easy to see that we can ascertain with certainty any one of the following values σˆ 1 x = 1, ˆσ 2 x = 1, ˆσ 1 y = 1 or ˆσ 2 y = 1. We also know that we will obtain definite values of +1 if we measure any one of the following products of observables: ˆσ 1 xσˆ 2 x = 1 and σˆ 1 yσˆ 2 y = 1, so we must also obtain ˆσ 1 xσˆ 2 xσˆ 1 yσˆ 2 y = +1. hσˆ 1 xσˆ 2 xσˆ 1 yσˆ 2 y i = hσˆ 1 y = 1|hσˆ 2 y = 1|σˆ 1 xσˆ 2 xσˆ 1 yσˆ 2 y i|σˆ 1 x = 1i|σˆ 2 x = 1i = +1 (3.25) In addition, we obtain the same results if we
Figure 4. Time sequence of PPS measurements for Mermin example. time t2 is performed and therefore cannot inform the later ˆσ 2 x measurement at time t3. In other words, with the particular PPS given in fig. 3.a and 4.a, the operator, σˆ 1 xσˆ 2 yσˆ 2 xσˆ 1 y depends on information from both the pre-selected vector ˆσ 1 x = 1, ˆσ 2 x = 1 and the post-selected vector ˆσ 1 y = 1, ˆσ 2 y = 1 in a diagonal-PPS sense. We call this diagonal-PPS because a line connecting ˆσ 1 x (t1) with ˆσ 2 x (t3) will be diagonal or will cross the line connecting ˆσ 2 y (t2) with ˆσ 1 y (t4), where tin < t1 <…
Figure 5. a) Measurement of ˆσ 1 xσˆ 2 yσˆ 2 xσˆ 1 y is diagonal, b) measurement of ˆσ 1 xσˆ 2 xσˆ 1 yσˆ 2 y is diag￾onal
Figure 6. Products of observables that are not disturbed, a) give
Figure 7. Generalized State: superpositions of 2-vectors given by
Figure 8. Generalized state for BKS nonets As Mermin proved, all 4 components of this generalized state are necessary (i.e. we only determine a Bell-state at tfin on the ancilla rather than make a projection onto any given component f µ ) to ascertain a definite answer to any one of the individual observables. 3.2.4 A physical reason for restrictions on these assignments We have suggested a physical reason based on TSQM and PPS for the 2 different values for ˆσ 1 zσˆ 2 z . This points to a physical reason why no 2 measurements can be
Figure 9. Weak Value density for 2 particles concerning WMs are completely general. The results of this article could be used to further exploit the differences between classical and quantum information: while the study of the non-classical aspects of entanglement (Einstein-Podolsky-Rosen/Bohm) started as a foundational examina￾tion of HVTs, it was subsequently probed experimentally and used as a resource for quantum computation and communication. Similarly, using TSQM [12], En￾glert demonstrated a new form of cryptography [19] and experiments based on the optical version of this problem [18]…

Figures reproduced from arXiv:quant-ph/0602226 by the authors.

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