Can a Future Choice Affect a Past Measurement's Outcome?

An EPR experiment is studied where each particle within the entangled pair undergoes a few weak measurements (WMs) along some pre-set spin orientations, with the outcomes individually recorded. Then the particle undergoes one strong measurement along an orientation chosen at the last moment. Bell-inequality violation is expected between the two final measurements within each EPR pair. At the same time, statistical agreement is expected between these strong measurements and the earlier weak ones performed on that pair. A contradiction seemingly ensues: (i) Bell's theorem forbids spin values to exist prior to the choice of the orientation measured; (ii) A weak measurement is not supposed to determine the outcome of a successive strong one; and indeed (iii) Almost no disentanglement is inflicted by the WMs; and yet (iv) The outcomes of weak measurements statistically agree with those of the strong ones, suggesting the existence of pre-determined values, in contradiction with (i). Although the conflict can be solved by mere mitigation of the above restrictions, the most reasonable resolution seems to be that of the Two-State-Vector Formalism (TSVF), namely, that the choice of the experimenter has been encrypted within the weak measurement's outcomes, even before the experimenters themselves know what their choice will be.


Introduction
Bell's theorem [1] has dealt the final blow on all hopes to explain the EPR correlations [2] as previously determined. Bell proved that these cosinelike correlations also depend on the two particular spin-orientations chosen for each measurement. As these choices can be made at the last moment, the resulting combinations of measurement outcomes, being mutually exclusive, could not co-exist in advance. Consequently, nonlocal effects between the particles have been commonly accepted as the only remaining explanation.
A variation of the EPR experiment is hereby presented, however, that suggests a simpler local explanation, namely allowing causation to be timesymmetric in the quantum realm. Then, what appears to be nonlocal in space turns out to be perfectly local in spacetime. This account's gist is given in Fig. 2.
The outline of this paper is as follows. Sec. 1 introduces the foundations of the Two-State-Vector Formalism (TSVF) and 2 weak measurement (WMs). 3 describes a combination of strong and weak measurements on a single particle illustrating a prediction of TSVF. In 4 we proceed to the EPR-Bell version of this experiment. Secs. 5-6 discuss and summarize the predicted outcomes' bearings.

A Particle's State between Two Noncommuting Measurements
Consider a particle undergoing two consecutive strong (i.e., projective) measurements, along the co-planar spin orientations α and β (the strongweak distinction will be further discussed in Sec. 2). The correlation between their outcomes depends on their relative angle θ αβ : Also, by the uncertainty relations between spin operators, these two measurements disturb each other's outcomes: If, e.g., the α measurement is repeated after the β, when the two directions are orthogonal, then the initial value of the spin-α measurement may flip to the opposite value with probability of 1/2.
Aharonov, Bergman and Lebowitz (ABL) [3] argued that, at any time between the two measurements, the state of the particle is equally thus having a definite value which agrees with both outcomes due to two state-vectors, one evolving from the past and the other from the future: creating the two-state-vector which holds for every intermediate moment in the evolution of the quantum system. This combination of forward and backward-evolving wave states taken from two Hilbert spaces, is argued to better describe a quantum system between two strong measurements. It is also the one which gives rise to the so called "weak value" [10][11][12][13][14].
TSVF accords with earlier physical models which sought to explain the spatiotemporal oddities of quantum interactions, such as [4][5][6][7]. Even the Wheeler-Feynman "absorber theory" [8] originally proposed to explain classical electromagnetic interaction, was later generalized by Cramer [9] into a comprehensive model for all quantum interactions. Admittedly, TSVF is not acknowledged as superior to more conservative, one-vector formulations of quantum mechanics. Neither has its approach to the measurement problem [15] been universally accepted. In what follows, however, we stress its rigor, elegance and simplicity, and in a consecutive papers [16,17] we present novel predictions that appear more natural with two-time vectors.

Weak measurements
TSVF, however, is unique among these models in that it has derived several predictions that, although fully consistent with the standard formalism (see Appendix 2), seem surprising and more acutely opposed to classical laws. In addition, TSVF has produced a unique technique for observing these predictions, namely WM [10][11][12][13][14]. For a simple and up-todate introduction to WM's underlying principles and its rigor in cases where projective measurement proves inadequate, see [11].
In brief (taking spin measurement as an example), WM is best performed on a large ensemble of particles, weakly coupling (through a unitary interaction) each spin of the ensemble to a macroscopic device.
After the quantum particle weakly interacts with the quantum pointer according to Eq. 6, the pointer's movement is amplified and classically recorded. That is, the pointer undergoes a "strong measurement", but the outcome tells us very little about the individual state of any of the N particles.
The weakness of the measurement (and consequently its strength in preserving the wavefunction of the particles) is due to the small factor 4 5 / N  , inversely proportional to the square root of the ensemble's size, where λ is a constant. When the N particles have different states, e.g., spins, WM correctly gives their average. But when they all share the same ↑ or ↓ spin value along the same orientation, WM indicates the entire ensemble's state. As pointed out in [14]: where the subscript w denotes weak values, and n enumerates the measured particles. Eq. 8 states that the weak value of A approaches the expectation value of A operating on  .
The slicing method [11], presented below, enables isolating such samestate particles even within random ensembles. For such homogenous subensembles, WM's rigor approaches that of strong measurement.

Combining Strong and Weak Measurements
We are now in a position to propose an experimental demonstration of the ABL-TSVF main argument ( Fig.1), namely: A particle's state between two strong measurements carries both the past and future outcomes. 4 Weakness of 1/N is sufficient in this case where one measuring apparatus is used, but for the cases considered in the next sections we chose interaction strength. See also [14].  arbitrary non-parallel directions. For the sake of later purposes we will assume that the angles α, β and γ are those which maximize the violation of Bell's inequality. Her measurements are similarly individual, each particle measured in its turn, and the measuring device calibrated before the next measurement. To make the results more striking, she repeats this series 3 times, total 9 weak measurements per each particle. All lists of outcomes 6 7 are then publically recorded, e.g., engraved on stone (see Fig. 1), along 9 rows with each outcome's position along the row being equal to that of Bob's list. Summing up her α-measurements (whether α (1) , α (2) , α (3) separately or all 3N together), she finds the spin distribution to be approximately 50%↑-50%↓. Similarly for β and γ. In other words nothing unusual.
c) In the evening Bob, oblivious of Alice's noon outcomes, again strongly measures all N particles, this time along the β orientation.
Again he draws a binary line as in (a). d) Bob then gives Alice the two binary lines, without disclosing to her whether "above/below line" refers to ↑/↓ or whether the orientation chosen for the morning/evening series was α, β or γ. e) Alice slices her data, according to Bob's divisions. In terms of Fig.   2, she merely shifts each of Bob's lines across her 9 rows of outcomes carved on stone. Each of the N sequences is thereby split into two approximately N/2 "above/below line" sub-rows (to the overall of 18 subgroups). She then re-sums each half separately. 9 Each row is thus sliced twice, first according to Bob's morning and then evening line.

Predictions
Upon Alice's re-summing up each of her sliced lists, a) Out of the 9 sliced rows of the WM outcomes, 3 immediately stand out with maximal correlation with Bob's above/below list (see differently, the WM's "bias", invoked above by the one-vector account, is viewed within the TSVF as affecting both past and future strong measurements, granting the measured particles well-defined properties (the weak values) in between the two strong measurements. Clearly the one-vector account lacks the TSVF's simplicity and elegance.
We will return to this comparison between the one-and two-vector accounts in Sec. 5.

Strong and Weak Measurements in the EPR-Bell Experiment
We can now demonstrate the weak outcomes' possible anticipation of a future human free choice. Consider an EPR-Bell experiment [1,2] on an ensemble of N particle pairs. Then c) Bob sends Alice his two binary lines, again not telling her whether "above/below line" refers to ↑/↓ or whether he has chosen α, β or γ for the right-and left-side particles. d) Based on Bob's binary lines, Alice slices her data, carved on stone since morning, shifting each of the lines across her rows, as in Sec.

Predictions
Calculating the separate averages of each sub-ensemble, QM obliges the following (a statement about a WM refers to its overall outcome): a) Bob's pairs of strong outcomes exhibit the familiar Bell-inequality violations [1], minus the slight O(λ 2 ) spoiling [11] due to the intermediate WMs, indicating that the particles were mostly superposed prior to his measurements.
Bob1 t x Fig. 2. An EPR pair. For each particle in a pair, several weak measurements are performed by Alice, followed by one strong measurement by Bob.
Evening Each particle undergoes one strong measurement along either α, β or γ, chosen at the last moment.

Morning
Each particle undergoes three weak measurements along α, β and γ. Alice1 13 b) Alice's weak outcomes strictly agree with those of the strong measurements8 6 7 at the level of each sub-ensemble; c) with the following twist: The strong measurement of each spin determines also that of the other particle as if it has occurred in its past, i.e., as in the experiments described in Sec. 3 with single particles, with the ↑/↓ sign inverted. This occurs regardless of the two measurements' actual timing [7].

What Kind of Causality?
The above experiment is designed to test the alleged time-symmetric quantum causality. What is the significance of the predicted results?
We begin with the main apparent contradiction. i) Bell's nonlocality proof states in effect that the particle pair could not possess pre-determined values. ii) In our experiment the WMs yield outcomes that turn out to match those of the strong measurements chosen later, thereby suggesting pre-existing weak values.
As TSVF and traditional quantum theory are equivalent, obliging one-and two-vector explanations to be equally valid, this contradiction can be resolved in two ways. Either i) the weak measurements have somehow anticipated the outcomes of the strong ones, chosen later (two-vector account), or ii) they have merely affected them in a subtle way (one-vector account). Obviously ii) is the more convenient option, especially with the alternative being retrocausal in character. The one-vector account, however, also comes with heavy conceptual cost: 7 One could make the results even more dramatic by adding here another feature used in Sec.3, namely repeating the three weak measurements three times, revealing their remarkable robustness.
a) It relies on "weak biases" like those invoked in Sec. 3 for the single- appropriate post-selection, TSVF predicts that the particles will possess even more surprising outcomes [16][17][18][19]. Again, these predictions, while consistent with standard quantum theory, are extremely unlikely to have emerged within the standard formalism.
Alternatively, one can turn to the Many-Worlds Interpretation, where there is no "action at a distance" [20], but then the principle of simplicity is 15 severely harmed. Finally, regardless of the one-or two-vector account one opts for, the following offshoots of our experiment merit attention:  [10,11]. Moreover, in [16,17] we discuss the issue of "odd" weak values (those that exceed the spectrum of the measured operator) which emerge naturally in these setups and lend further support to the TSVF. 3. Subtler inequalities? By being continuous rather than discrete, the weak pointer outcomes obtained by Alice, call for a new kind of Bell-like inequality.

Summary
We explored an apparent contradiction between two well-established findings: a) The EPR-Bell experiment proves that one particle's spin outcome depends on the choice of the spin-orientation to be measured on the other particle, and its outcome thereof. Relativistic locality is not violated in this experiment due to the reciprocity between the two measurements, allowing either Alice's choices to affect Bob's, or vice versa.
b) Such reciprocity, however, is challenged for a combination of measurements of which one is strong and the other weak. The latter affects the former to a much lesser extent, i.e. all the weak outcomes do not oblige the strong ones, but the strong outcomes do determine the weak values. Moreover, the strong correlation between past and future outcomes, suggests the appearance of a subtle local hidden variablethe future state vector.
Therefore, when a weak measurement precedes a strong one within an EPR experiment, the weak-strong agreements between past and future measurement outcomes can be interpreted in two ways. One may adhere to the one-vector non-local explanation and ascribe it to the slight biasing of the weak on the strong measurements. Simplicity and elegance, we suggest, favors the local two-vector account, where the future choice plays a crucial role within the noisy weak outcomes carved on the rock. Torino. Results will be published in due course.