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Null weak values and the past of a quantum particle
Abstract
Non-destructive weak measurements (WM) made on a quantum particle allow to extract information as the particle evolves from a prepared state to a finally detected state. The physical meaning of this information has been open to debate, particularly in view of the apparent discontinuous trajectories of the particle recorded by WM. In this work we investigate the properties of vanishing weak values for projection operators as well as general observables. We then analyze the implications when inferring the past of a quantum particle. We provide a novel (non-optical) example for which apparent discontinuous trajectories are obtained by WM.\ Our approach is compared to previous results.
... Such is, for example, the "three box paradox" [2]- [5], claiming that a particle can be, at the same time, at two different locations "with certainty". A similarly "paradoxical" suggestion that a photon could, on its way to detection, have visited the places it had "never entered, nor left, was made in [6]- [7], and further discussed in [8]- [11]. In the discussion of the Hardy's paradox [12]- [15] the particle is suspected of simultaneously "being and not being" at the same location [13]. ...
... as a valid solution of the first three equations in (12). It is, however, impossible to find a set 0 ≤ p i ≤ 1, i = 1, .., 8, which would satisfy all four equations in (11). Indeed, adding the last three equations, and subtracting the result from the first one, we have p 3 + p 7 = −1/8 < 0. We can, however, reproduce quantum results classically, if we change the rules of the experiment. ...
... At the point I, a photon in injected in a wave packet state |ψ I and ends up being split into three final states, |d m , m = 1, 2, 3, leaving system in three different ways, as shown in Fig. 7. The system is tuned in such a way that no part of the wave function travels through the fibre F (dashed), and we are interested in detecting the photon by means of a detector D. The subject of the ongoing discussion, often touching on the topic of "weak measurements (WM)", is the past of the detected photon [6]- [11]. We will briefly return to the WM in Sect. ...
We consider a finite-dimensional quantum system, making a transition between known initial and final states. The outcomes of several accurate measurements, which {\it could be} made in the interim, define virtual paths, each endowed with a probability amplitude. If the measurements are {\it actually made}, the paths, which may now be called "real", acquire also the probabilities, related to the frequencies, with which a path is seen to be travelled in a series of identical trials. Different sets of measurements, made on the same system, can produce different, or incompatible, statistical ensembles, whose conflicting attributes may, although by no means should, appear "paradoxical". We describe in detail the ensembles, resulting from intermediate measurements of mutually commuting, or non-commuting, operators, in terms of the real paths produced. In the same manner, we analyse the Hardy's and the "three box" paradoxes, the photon's past in an interferometer, the "quantum Cheshire cat" experiment, an the closely related subject of "interaction-free measurements". It is shown that, in all these cases, inaccurate "weak measurements" produce no real paths, and yield only limited information about the virtual paths' probability amplitudes
... It does, however, forbid to make assumptions about the system's past where the scenarios one wishes to distinguish continue to interfere. Nothing in the Principle stops Bob from learning the values of either the amplitudes (18), or of their various combinations (17). The question is, what conclusions is he allowed to draw from the "weak values" once they have been measured? ...
... Finally, and at first glance most reasonably, is simply to say that if the WV of a projector is 0 (the weak pointer has not moved), the system has definitely not taken the chosen path [11], [17]. Conversely, if the pointer has moved, the system must have been present in the corresponding pathway. ...
Quantum mechanics, in its orthodox version, imposes severe limits on what can be known, or even said, about the condition of a quantum system between two observations. A relatively new approach, based on so-called "weak measurements", suggests that such forbidden knowledge can be gained by studying the system's response to an inaccurate weakly perturbing measuring device. It goes further to propose revising the whole concept of physics variables, and offers various examples of counterintuitive quantum behaviour. Both views go to the very heart of quantum theory, and yet are rarely compared directly. A new technique must either transcend the orthodox limits, or just prove that these limits are indeed necessary. We study both possibilities, and find for the orthodoxy.
... Indeed, it can be obtained as a mean reading of an accurate pointer, ⟨ f (F i )⟩ by using Eq. (20). Calculating the same average for a weakly coupled pointer, the authors of [14] obtained a somewhat similar expression, ...
... Finally, this seems to be a safe option (for more discussion see [20]), since ⟨B⟩ W in Eq. (5) can vanish only if the amplitude A(F i ← b 1 ← I) is itself zero. With only one path, F i ← b 2 ← I, leading to the final state, there is no interference to destroy. ...
Feynman famously recommended accepting the basic principles of quantum mechanics without trying to guess the machinery behind the law. One of the corollaries of the Uncertainty Principle is that the knowledge of probability amplitudes does not allow one to make meaningful statements about the past of an unobserved quantum system. A particular type of reasoning, based on weak values, appears to do just that. Has Feynman been proven wrong by the more recent developments? Most likely not.Quanta 2023; 12: 180–189.
... (16) was first obtained in [20], where the complex valued fraction in brackets was called "the weak value (WV) of the opera-torQ 2 ". Written in this way, a WV looks like a physical variable of a new kind [21], whose physical significance is still discussed in the literature (see, for example [22], [23]). However the first expression in the r.h.s. of eq. ...
... Two spins-1/2 are prepared in an entangled state(22). Alice and Bob measure their spins along the directions n and n , respectively. Of the sixteen virtual paths, passing through states in Eqs.(23) (dots), only four (solid lines) have non-zero amplitudes. ...
We analyse a proposition which considers quantum theory as a mere tool for calculating probabilities for sequences of outcomes of observations made by an Observer, who him/herself remains outside the scope of the theory. Predictions are possible, provided a sequence includes at least two such observations. Complex valued probability amplitudes, each defined for an entire sequence of outcomes, are attributed to Observer's reasoning, and the problem of wave function's collapse is dismissed as a purely semantic one. Our examples include quantum "weak values", and a simplified version of the "delayed quantum eraser".
... It is noteworthy that there are debates on the meaning of null/vanishing weak values [18][19][20][21][22]. Here we just consider the weak values to be 0 or 1. Operationally speaking, when the quantum pointer is left untouched in the case of interaction with the system and the post-selection is successful, the property of the operator is "not there" [18]. ...
... It is noteworthy that there are debates on the meaning of null/vanishing weak values [18][19][20][21][22]. Here we just consider the weak values to be 0 or 1. Operationally speaking, when the quantum pointer is left untouched in the case of interaction with the system and the post-selection is successful, the property of the operator is "not there" [18]. That is, a zero weak value of a projector indicates a zero probability. ...
Quantum Cheshire cats represent the violation of the product rule in terms of weak values being either 0 or 1. We explore the generalized fatter quantum Cheshire cats living in the larger Hilbert spaces. Specifically, two even-dimensional quantum systems are exploited. As for the physical implications, given that no disturbance is made in the weak measurements, quantum Cheshire cats reveal an alternative way of defying EPR elements of reality. In addition, we discuss the inconsistency between the Wigner's friend problem and quantum Cheshire cats.
... It can hardly be maintained that the current density at a particular space-time point does not characterize a partial property of the system at that particular space-time point. We have amply discussed elsewhere [24][25][26]32] the case of null weak values. In the case of a projector A = |a k a k | , a null weak value A w = 0 means that the property represented by A cannot be registered by the weakly coupled pointer for the given post-selection. ...
... This implementation of the three-box paradox[2] has been described in details elsewhere[23,24]. ...
We investigate in this work the meaning of weak values through the prism of property ascription in quantum systems. Indeed, the weak measurements framework contains only ingredients of the standard quantum formalism, and as such weak measurements are from a technical point of view uncontroversial. However attempting to describe properties of quantum systems through weak values—the output of weak measurements—goes beyond the usual interpretation of quantum mechanics, that relies on eigenvalues. We first recall the usual form of property ascription, based on the eigenstate-eigenvalue link and the existence of “elements of reality”. We then describe against this backdrop the different meanings that have been given to weak values. We finally argue that weak values can be related to a specific form of property ascription, weaker than the eigenvalues case but still relevant to a partial description of a quantum system.
... A deeper understanding of the dynamics of these weak values is offered by "mirage particles," momentary particles springing from the initial particle during the above "betweenmeasurements" interval. This concept was already alluded in earlier works of ours [14,[19][20][21][22] and colleagues [23,24]. Among these mirage particles there are some whose very presence has a minus sign, implying that, upon a weak enough interaction, their properties, including mass and charge, reverse their sign [15,25,26]. ...
While quantum reality can be probed through measurements, the Two-State-Vector formalism (TSVF) reveals a subtler reality prevailing between measurements. Under special pre- and post-selections, odd physical values emerge. This unusual picture calls for a deeper study. Instead of the common, wave-based picture of quantum mechanics, we suggest a new, particle-based perspective: Each particle possesses a definite location throughout its evolution, while some of its physical variables (characterized by deterministic operators, some of which obey nonlocal equations of motion) are carried by "mirage particles" accounting for its unique behavior. Within the time-interval between pre- and post-selection, the particle gives rise to a horde of such mirage particles, of which some can be negative. What appears to be "no-particle," known to give rise to Interaction-Free Measurement, is in fact a self-canceling pair of positive and negative mirage particles, which can be momentarily split and cancel out again. Feasible experiments can give empirical evidence for these fleeting phenomena. In this respect, the Heisenberg ontology is shown to be conceptually advantageous compared to the Schr\"odinger picture. We review several recent advances, discuss their foundational significance and point out possible directions for future research.
... For a strong measurement of a projection operator, the outcome eigenvalue 1 means you have found the system in the corresponding state, and eigenvalue 0 means you have found it in an orthogonal state. It is a hotly contested assumption [5,24,25,26,27,28,29,30,31,32,33,34,35,36,37] that measuring the weak value of a projector to be 1 corresponds to finding every element in the measured ensemble to actually have the physical properties associated with that state, or for a weak value of 0, to actually have the physical properties associated with an orthogonal state. This assumption is motivated in several ways. ...
The Quantum Cheshire Cat experiment showed that when weak measurements are performed on pre- and post-selected system, the counterintuitive result has been obtained that a neutron is measured to be in one place without its spin, and its spin is measured to be in another place without the neutron. A generalization of this effect is presented with a massive particle whose mass is measured to be in one place with no momentum, while the momentum is measured to be in another place without the mass. The new result applies to any massive particle, independent of its spin or charge. A g e d a n k e n experiment which illustrates this effect is presented using a nested pair of Mach-Zehnder interferometers, but with some of the mirrors and beam splitters moving relative to the laboratory frame. The titular interpretation of this experiment is extremely controversial, and rests on several assumptions, which are discussed in detail. An alternative interpretation using the counterparticle model of Aharonov et al. is also discussed.
... The slight deviation from the original interpretation of discontinuous trajectories [22] is in the sense that the weak trace becomes zero as a result of the pre-selected and post selected wavefunctions being orthogonal at mirrors E and F rather than destructive interference which makes the entire wavefunction disappear. To get a more reliable past of a particle, we could look at all the possible observables associated with the particle along with the spatial projection operator [41,42]. ...
We consider weak values in the Feynman propagator framework, to gain a broader understanding of their interpretation in terms of path integrals. In particular, we examine the phenomenon of seemingly discontinuous paths that particles take in nested Mach-Zender interferometer experiments. We extend on existing path integral approaches for weak values by deriving expressions to model a sequence of weak measurements, and study the probe shifts across the different branches of a weak value interferometer. We apply this to scrutinise two scenarios of interest, one which treats photons as measurement apparatus via their spatial projection operators, and the second treating mirrors as probes.
... As a result, it is hard to tell whether or not A has been measured when the weak value is zero. Because of this, it is crucial to have nonzero weak values which carry the information about the observable A. Null weak values have recently been given a useful interpretation [43]: if a successful postselection occurs with a null weak value, then the property represented by the observable A cannot be detected by the weakly coupled quantum pointer. In other words, the pointer state remains unchanged when the weak value is zero (see Introduction section). ...
In this work, we derive Robertson-Heisenberg–type uncertainty relation for two incompatible observables in a pre- and postselected (PPS) system. The newly defined standard deviation and the uncertainty relation in the PPS system have physical meanings which we present here. We demonstrate two unusual properties in the PPS system using our uncertainty relation. First, for commuting observables, the lower bound of the uncertainty relation in the PPS system does not become zero even if the initially prepared state, i.e., preselection, is the eigenstate of both the observables when specific postselections are considered. This implies that for such case, two commuting observables can disturb each other's measurement results which is in fully contrast with the Robertson-Heisenberg uncertainty relation. Second, unlike the standard quantum system, the PPS system makes it feasible to prepare sharply a quantum state (preselection) for noncommuting observables (to be detailed in the main text). Some applications of uncertainty and uncertainty relation in the PPS system are provided: (i) detection of mixedness of an unknown state, (ii) stronger uncertainty relation in the standard quantum system, (iii) “purely quantum uncertainty relation” that is, the uncertainty relation which is not affected (i.e., neither increasing nor decreasing) under the classical mixing of quantum states, (iv) state-dependent tighter uncertainty relation in the standard quantum system, and (v) tighter upper bound for the out-of-time-order correlation function.
... As a result, it is hard to tell whether or not A has been measured when the weak value is zero. Because of this, it is crucial to have nonzero weak values which carry the information about the observable A. Null weak values have recently been given a useful interpretation [43]: if a successful post-selection occurs with a null weak value, then the property represented by the observable A cannot be detected by the weakly coupled quantum pointer. In other words, the pointer state remains unchanged when the weak value is zero (see introduction section). ...
Till date, no uncertainty relation has been reported for two incompatible observables in a pre- and post-selected (PPS) system which can express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring those observables. Here, we derive such an uncertainty relation for a PPS system by defining an uncertainty (standard deviation) of an observable measured in the given pre- and post-selected states. We provide here physical interpretations of the newly defined standard deviation and the uncertainty relation. Zero uncertainty condition, which is also the condition for optimized Fisher information in quantum metrology using the PPS system, is derived. It is shown that joint sharp preparation of a quantum state for non-commuting observables is possible when the standard quantum system is transformed into a PPS system with certain conditions, an impossible task in standard quantum system. We provide here applications of uncertainty and uncertainty relation in the PPS system. Namely, (i) detection of mixedness of the given pre-selection using two different definitions of the uncertainty, (ii) stronger uncertainty relation in the standard quantum system (i.e., the uncertainty relation can not be made trivial or the lower bound can not be null for almost all possible choices of initially prepared systems) using the uncertainty relation in the PPS system, (iii) genuine quantum mechanical uncertainty relation can be found using the first definition of uncertainty when the pre-selection is a mixed state.
... However, the null transition amplitudes in Eq. (12) show that the pointer states remain unshifted after the post-selection of Ψ f j i. Thus, the null transition amplitudes may be interpreted as the absence of the particle or the physical property 17,39 . ...
One of the common conceptions of nature, typically derived from the experiences with classical systems, is that attributes of the matter coexist with the substance. In the quantum regime, however, the quantum particle itself and its physical property may be in spatial separation, known as the quantum Cheshire cat effect. While there have been several reports to date on the observation of the quantum Cheshire cat effect, all such experiments are based on first-order interferometry and destructive projection measurement, thus allowing simple interpretation due to measurement-induced disturbance and also subject to trivial interpretation based on classical waves. In this work, we report an experimental observation of the quantum Cheshire cat effect with noninvasive weak quantum measurement as originally proposed. The use of the weak-measurement probe has allowed us to identify the location of the single photon and that of the disembodied polarization state in a quantum interferometer. The weak-measurement probe based on two-photon interference makes our observation unable to be explained by classical physics. We furthermore elucidate the quantum Cheshire cat effect as quantum interference of the transition amplitudes for the photon and the polarization state which are directly obtained from the measurement outcomes or the weak values. Our work not only reveals the true quantum nature of Cheshire cat effect but also sheds light on a comprehensive understanding for the counter-intuitive quantum phenomena.
... Let us emphasise that the physical meaning of vanishing weak values in a current context remains disputable [34][35][36][37] . Most of controversies originate, however, from highly counter-intuitive conclusions provided in Ref. 1 indicating possibility of discontinuous trajectories followed by a particle passing trough Vaidman interferometer. ...
We study weak traces of particle passing Vaidman’s nested Mach–Zehnder interferometer. We investigate an effect of decoherence caused by an environment coupled to internal degree of freedom (a spin) of a travelling particle. We consider two models: pure decoherence leading to exact results and weak coupling Davies approximation allowing to include dissipative effects. We show that potentially anomalous discontinuity of particle paths survives an effect of decoherence unless it affects internal part of the nested interferometer.
... The faint trace will be on all parts of this line, but also in a separate loop disconnected from the line. The prediction of this closed loop, which does not start at the source and does not end at the detector, generated a hot discussion [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] which only intensified [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] after the first experiment with photons [22] confirmed discontinuous trajectories. ...
Recent experiments with identically tuned nested Mach-Zehnder interferometers which attempted to observe the location of particles inside these interferometers are analyzed. In spite of claims to the contrary, it is argued that all experiments support the same surprising picture according to which the location of the particles inside the interferometers is not described by continuous trajectories.
... Ce chapitre est consacré à l'annulation d'une valeur faible et se réfère à [100] et [101]. L'examen de cette valeur particulière est motivé par le fait qu'elle apparait lorsque les états pré/post-sélections sont tels que des valeurs faibles paradoxales émergent comme dans le paradoxe des trois boîtes ou le chat du Cheshire (chapitre suivant). ...
La théorie de la mesure, basée sur la mesure projective, constitue un aspect fondamental de la mécanique quantique. La mesure faible diffère de la mesure projective traditionnelle sur laquelle les axiomes élémentaires de la physique quantique sont bâtis. Bien que définies dans le cadre de la théorie quantique standard, les mesures faibles sont encore mal comprises. Le travail de thèse s'inscrit dans une large démarche qui vise à comprendre les implications conceptuelles et pratiques d'une telle mesure et à la comprendre dans le cadre de la théorie quantique.Le chapitre 1 est une introduction détaillée à la mesure faible et à la valeur faible. Ensuite, nous étudierons les "trajectoires faibles" dans un interféromètre à fentes d'Young. Nous abordons au chapitre 3 les implications de l'annulation d'une valeur faible au regard de l'apparition de trajectoires faibles dans des interféromètres de Mach-Zender imbriqués. Enfin, le chapitre 4 traite des critiques théoriques et expérimentales présentes dans la littérature concernant l'effet du Chat du Cheshire quantique qui est défini dans le cadre de mesures faibles.
... With the usage of weak interaction, we are able to measure qubit without destroying it [15], directly measure the quantum wave function [16], observe trajectory in quantum systems in the semiclassical regime [17], study three-box paradox [18], determine the past of photons passing through an interferometer [14] and even amplify the nonlinear effect of the single photon [19]. Typically, weak value measurement requires an infinitesimally weak interaction between the measured system and a pointer state [20][21][22][23]. Such interaction, however, might not always be available or practical. ...
Weak values are traditionally obtained using a weak interaction between the measured system and a pointer state. In this paper, we show that weak values can also be measured using strong interaction accompanied by either a suitably prepared pointer state or quantum erasure. Presented theoretical derivations prove analytical equivalence of these approaches. Moreover, we have performed an experimental verification of our model on a linear-optical controlled phase gate. Our results open new ways of performing non-invasive quantum measurements without collapsing the measured system.
... A deeper understanding of the dynamics of these weak values is offered by "mirage particles", momentary particles springing from the initial particle during the above "between-measurements" interval. This concept was already alluded in earlier works of ours [14,[19][20][21][22] and colleagues [23,24]. Among these mirage particles, there are some of which the presence has a minus sign, implying that, upon a weak enough interaction, their properties, including mass and charge, reverse their sign [15,25,26]. ...
While quantum reality can be probed through measurements, the Two-State Vector Formalism (TSVF) reveals a subtler reality prevailing between measurements. Under special pre- and post-selections, odd physical values emerge. This unusual picture calls for a deeper study. Instead of the common, wave-based picture of quantum mechanics, we suggest a new, particle-based perspective: Each particle possesses a definite location throughout its evolution, while some of its physical variables (characterized by deterministic operators, some of which obey nonlocal equations of motion) are carried by “mirage particles” accounting for its unique behavior. Within the time interval between pre- and post-selection, the particle gives rise to a horde of such mirage particles, of which some can be negative. What appears to be “no-particle”, known to give rise to interaction-free measurement, is in fact a self-canceling pair of positive and negative mirage particles, which can be momentarily split and cancel out again. Feasible experiments can give empirical evidence for these fleeting phenomena. In this respect, the Heisenberg ontology is shown to be conceptually advantageous compared to the Schrödinger picture. We review several recent advances, discuss their foundational significance and point out possible directions for future research.
... This interesting relationship resonates with past works which studied nondemolition and continuous quantum measurements [1][2][3][4], while connecting them with path integration [5]. Recently, the above relationship was further analyzed and strengthened by different researchers [6][7][8][9][10][11], but here we focus on the notion of sequential weak values as a pivotal issue, which has not been mentioned before in the above literature. In particular, we show that sequential weak values are able to probe directly the quantum probability amplitudes along individual virtual Feynman histories thereby possibly supporting their physical meaningfulness. ...
Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories, thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multitime projection operators for a complete set of orthogonal quantum histories is unity, complete sets of weak values could be interpreted in agreement with the standard quantum mechanical picture. We also elucidate the relationship between sequential weak values of quantum histories with different coarse graining in time and establish the incompatibility of weak values for nonorthogonal quantum histories in history Hilbert space. Bridging theory and experiment, the presented results may enhance our understanding of both weak values and quantum histories.
Quantum mechanics, in its orthodox version, imposes severe limits on what can be known, or even said, about the condition of a quantum system between two observations. A relatively new approach, based on so‐called “weak measurements”, suggests that such forbidden knowledge can be gained by studying the system's response to an inaccurate weakly perturbing measuring device. It goes further to propose revising the whole concept of physics variables, and offers various examples of counterintuitive quantum behavior. Both views go to the very heart of quantum theory, and yet are rarely compared directly. A new technique must either transcend the orthodox limits, or just prove that these limits are indeed necessary. Both possibilities are studied and orthodoxy is vindicated.
The interference pattern produced by a quantum particle in Young's double-slit setup is attributed to the particle's wave function having gone through both slits. In the path integral formulation, this interference involves a superposition of paths, going through either slit, linking the source to the detection point. We show how these paths superpositions can in principle be observed by implementing a series of minimally perturbing weak measurements between the slits and the detection plane. We further propose a simplified protocol in order to observe these “weak trajectories” with single photons.
The past of a Quantum particle and the investigations of the particle’s unitary evolution in between two strong measurements have generated a heated debate recently in quantum foundational context mainly studied earlier through photonics techniques. We, however, here propose an atom interferometric exploration based on the time tested tools of Atomic Bragg Diffraction to study the past of an atom, more akin to a real, concrete particle as compared with its counterpart i.e. photon. Two-State Vector Formalism (TSVF) along with the methodology of weak measurement have been invoked for the analysis of the past of an atom traversing a Mach–Zehnder Bragg atom interferometer and the results obtained are quite interesting. It is also being demonstrated explicitly that the suggested schematics are inherently deterministic and the proposal can be experimentally executed with high enough fidelity under the prevailing atom-field interaction based laboratory tools.
We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in quantum-chaos-related studies (an illustration involving quantum scars is given), and also in solving current weak-value-related controversies (we discuss the existence of discontinuous trajectories in interferometers and the issue of anomalous weak values in the classical limit).
The quantum Cheshire cat is an effect demonstrated within the framework of weak measurement aided with post-selection scenario where the property (say, grin) of a quantum particle (the cat) can be found in a spatially separated location from its position. In this work, we first propose interesting variants of quantum Cheshire cat where disembodiment of two different properties of a "cat" is demonstrated, so that, the grin and the meowing of it can be found in two different paths on an interferometer while the cat is in another path. We further extend this proposal for three-qutrit and d-qudit systems of a single particle. We provide sketches of the experimental proposals for testing our scheme by using the path, spin and energy degrees of freedoms of a single neutron and the path, polarization and orbital angular momentum of a single photon.
Recent experiments with identically tuned nested Mach-Zehnder interferometers which attempted to observe the location of particles inside these interferometers are analyzed. In spite of claims to the contrary, it is argued that all experiments support the same surprising picture according to which the location of the particles inside the interferometers is not described by continuous trajectories.
We analyse a proposition which considers quantum theory as a mere tool for calculating probabilities for sequences of outcomes of observations made by an Observer, who him/herself remains outside the scope of the theory. Predictions are possible, provided a sequence includes at least two such observations. Complex valued probability amplitudes, each defined for an entire sequence of outcomes, are attributed to Observer's reasoning, and the problem of wave function's collapse is dismissed as a purely semantic one. Our examples include quantum “weak values”, and a simplified version of the “delayed quantum eraser”.
The recent criticism of Vaidman's proposal for the analysis of the past of a particle in the nested interferometer is refuted. It is shown that the definition of the past of the particle adopted by Englert et al. [B. G. Englert et al., Phys. Rev. A 96, 022126 (2017)] is applicable only to a tiny fraction of photons in the interferometer which indeed exhibit different behavior. Their proof that all pre- and postselected particles behave this way, i.e., follow a continuous trajectory, does not hold, because it relies on the assumption that it is intended to prove.
The history of photons in a nested Mach–Zehnder interferometer with an inserted Dove prism is analyzed. It is argued that the Dove prism does not change the past of the photon. Alonso and Jordan correctly point out that an experiment by Danan et al. demonstrating the past of the photon in a nested interferometer will show different results when the Dove prism is inserted. The reason, however, is not that the past is changed, but that the experimental demonstration becomes incorrect. The explanation of a signal from the place in which the photon was (almost) not present is given. Bohmian trajectory of the photon is specified.
An experiment with nested Mach-Zehnder interferometer [Phys. Rev. Lett. 111, 240402 (2013)] has been recently implemented with neutrons [Phys. Rev. A 97, 052111 (2018)]. Which-path information has been extracted from faint traces the neutrons left, providing operational meaning to "the particle's path". The authors of the neutron interference experiment criticised the conclusions obtained by the authors of the optical experiment. I refute the criticism and argue that the results of the neutron interference experiment actually support the surprising picture of the past of the particle in the nested Mach-Zehnder interferometer.
The past of the photon in a nested Mach-Zehnder interferometer with inserted Dove prism is analyzed. It is argued that the Dove prism does not change the past of the photon. Alonso and Jordan correctly point out that an experiment by Danan et al. demonstrating the past of the photon in nested interferometer will show different results when the Dove prism is inserted. The reason, however, is not that the past is changed, but that the experimental demonstration becomes incorrect. The explanation of a signal from the place in which the photon was (almost) not present is given. Bohmian trajectory of the photon is specified.
The recent criticism of Vaidman's propsal for the analysis of the past of a particle in the nested interferometer is refuted. It is shown that the definition of the past of the particle adopted by Englert et al. [Phys. Rev. A 96, 022126 (2017)] is applicable only to a tiny fraction of photons in the interferometer which indeed exhibit different behaviour. Their proof that all pre- and postselected particles behave this way, i.e. follow a continuous trajectory, does not hold, because it relies on the assumption that it is intended to prove.
In a recent paper [Phys. Rev. A 95, 032110 (2017)2469-992610.1103/PhysRevA.95.032110], Duprey and Matzkin investigated the meaning of vanishing weak values and their role in the retrodiction of the past of a preselected and postselected quantum system in the presence of interference. Here we argue that any proposition regarding the weak values should be understood as a statement about the probability amplitudes. With this in mind, we revisit some of the conclusions reached in Duprey and Matzkin's work.
We discuss the preceding Comment and conclude that the arguments given there against the relevance of null weak values as representing the absence of a system property are not compelling. We give an example in which the transition matrix elements that make the projector weak values vanish are the same ones that suppress detector clicks in strong measurements. Whether weak values are taken to account for the past of a quantum system or not depend on general interpretional commitments of the quantum formalism itself rather than on peculiarities of the weak measurements framework.
Hashmi et al. [J. Phys. A 49, 345302 (2016)] claimed that the approach to the past of a quantum particle introduced by Vaidman [Phys. Rev. A 87, 052104 (2013)] has difficulties in certain examples and that it even can be refuted. Here I reply to their criticism showing that the approach provides a good explanation of all examples they considered. It is fully consistent with standard quantum mechanics and provides a useful tool for analyzing interference experiments.
In the year 2013, Danan et al. published a paper [Phys. Rev. Lett. 111, 240402 (2013)] demonstrating a counterintuitive behavior of photons in nested Mach-Zehnder interferometers. The authors then proposed an explanation based on the two-state vector formalism. This experiment and the authors' explanation raised a vivid debate within the scientific community. In this paper, we contribute to the ongoing debate by presenting an alternative experimental implementation of the Danan et al. scheme. We show that no counterintuitive behavior is observed when performing direct spectrally-resolved detection.
Bartkiewicz et al. [Phys. Rev. A 91, 012103 (2015)] provided an alternative
analysis of experiment performed by Danan et al. [Phys. Rev. Lett. 111, 240402
(2013)] which presented surprising evidence regarding the past of photons
passing through an interferometer. They argued that the quantity used by Danan
et al. is not a suitable which-path witness, and proposed an alternative. It is
argued that the quantum and classical analyses of Bartkiewicz et al. are
inconsistent and both are inappropriate for describing the past of photons in a
properly working interferometer.
The question: "Where was a quantum particle between pre- and postselection
measurements?" is analyzed in view of a recent proposal that it was in the
overlap of the forward and backward evolving wave functions. It is argued that
this proposal corresponds not only to the criterion of where the particle
leaves a weak trace, but also where the local interactions can affect the
probability of postselection and where finding the particle in a strong
nondemolition measurement is possible. The concept of a "secondary presence" of
a pre- and postselected particle where local interactions affect the weak trace
in the overlap region is introduced.
Linear-optical interferometers play a key role in designing circuits for quantum information processing and quantum communications. Even though nested Mach-Zehnder interferometers appear easy to describe, there are occasions when they provide unintuitive results. This paper explains the results of a highly discussed experiment performed by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013).] using a standard approach. We provide a simple and intuitive one-state vector formalism capable of interpreting their experiment. Additionally, we cross-checked our model with a classical-physics-based approach and found that both models are in complete agreement. We argue that the quantity used in the mentioned experiment is not a suitable which-path witness, producing seemingly contraintuitive results. To circumvent this issue, we establish a more reliable which-path witness and show that it yields well-expected outcomes of the experiment.
We reexamine the thought experiment and real experiment of Vaidman et al., by
placing Dove prisms in the nested Mach-Zehnder interferometer arms. In those
previous works, the criterion of whether a single photon was present, or not,
was the presence of a "weak trace", indicating the presence of a nonzero weak
value. This was verified by slightly varying the mirror angle at a given
frequency, which was then detected on a position sensitive detector at the
oscillation frequency. We show the presence of the Dove prisms gives identical
weak values everywhere to the previous configuration because the prisms change
neither the path difference, nor the mode profile in the aligned case.
Nevertheless, the same slight variations of the interferometer mirrors now give
a signal at the first mirror of the nested interferometer. We can interpret
this result as a misaligned optical interferometer, whose detailed response
depends on the stability of the elements, or as the detector coupling to a
nonzero effective weak value.
Misinterpretation in the preceding Comment of my recent analysis of the
past of a photon is corrected. There is nothing in this analysis which
is “contrary to the usual quantum expectations” but,
nevertheless, it does provide “further understanding and
interpretation of the system considered.” In particular, it
indicates that the common sense argument used in the Comment should be
abandoned.
We present surprising experimental evidence regarding the past of photons passing through an interferometer. The information about the positions through which the photons pass in the interferometer is retrieved from modulations of the detected signal at the vibration frequencies of mirrors the photons bounce off. From the analysis we conclude that the past of the photons is not represented by continuous trajectories, although a "common sense" analysis adopted in various welcher weg measurements, delayed-choice which-path experiments, and counterfactual communication demonstrations yields a single trajectory. The experimental results have a simple explanation in the framework of the two-state vector formalism of quantum theory.
Since its introduction 25 years ago, the quantum weak value has gradually
transitioned from a theoretical curiosity to a practical laboratory tool. While
its utility is apparent in the recent explosion of weak value experiments, its
interpretation has historically been a subject of confusion. Here, we present a
pragmatic introduction to the weak value in terms of measurable quantities and
explain how it can be determined in the laboratory. Further, we review its
application to three distinct experimental techniques. First, as a large
interaction parameter it can amplify small signals above technical background
noise. Second, as a measurable complex value it enables novel techniques for
quantum state tomography and geometric phase determination. Third, as a
conditioned average of generalised observable eigenvalues it provides a
measurable window into nonclassical features of quantum mechanics. In this
selective review, we use a single experimental configuration to discuss and
clarify each of these applications.
We propose in this work a definite theoretical implementation of the
three-box paradox - a scheme in which a single quantum particle appears to be
present with certainty in two separate boxes - with spin-1 atoms. We further
show how our setup can give rise to a "Cheshire cat grin" type of situation, in
which an atom can apparently be found with certainty in one of the boxes while
one of its properties (the angular momentum projection along a specifically
chosen axis) appears to be in a different box. The significance of our findings
are discussed relative to the status of the properties of a system obtained
from weak measurements.
Although there is no consensus regarding the "reality" of the past of a
quantum particle, in situations where there is only one trajectory with
nonvanishing quantum wave of the particle between its emission and detection
points, it seems "safe" to associate the past of the particle with this
trajectory. A method for analyzing the past of a quantum particle according to
the weak trace it leaves is proposed. Such a trace can be observed via
measurements performed on an ensemble of identically pre- and post-selected
particles. Examples, in which this method contradicts the above common sense
description of the past of the particle are presented. It is argued that it is
possible to describe the past of a quantum particle, but the naive approach has
to be replaced by both forward and backward evolving quantum states.
We consider weak measurements (WM) of a spin observable in quantum mechanics
beyond the usual asymptotic regime. This is done by obtaining the exact
time-dependent wave functions of the measuring apparatus for general non-ideal
measurements. Ideal strong measurements and the usual WM regime are obtained as
the two extreme limiting cases of our exact treatment. We show that in the
intermediate regime non-ideal measurements lead to "semiweak" eccentric values
that differ from the usual weak values. We further show that even in the WM
regime the exact treatment leads to a meter behavior that can be markedly
distinct from the one predicted by the usual WM formalism. We give several
illustrations and discuss an application to the distinction of different
realizations of the same density matrix.
The authors of Phys.Rev.Lett. \textbf{111}, 240402 (2013) conclude that "the past of the photons is not represented by continuous trajectories". A simple analysis by standard quantum mechanics shows that this claim is false.
In our recent paper [Phys. Rev. A 91, 012103 (2015)10.1103/PhysRevA.91.012103] we have provided an alternative description of a highly debated experiment by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)10.1103/PhysRevLett.111.240402]. The former paper has been recently commented on by Vaidman [Phys. Rev. A 93, 036103 (2016)10.1103/PhysRevA.93.036103]. In this Reply we demonstrate that the comments are either invalid or, where valid, do not affect the results of our original paper.
Possible paths of a photon passing through a nested Mach-Zehnder interferometer on its way to a detector are analyzed using the consistent histories formulation of quantum mechanics, and confirmed using a set of weak measurements (but not weak values). The results disagree with an analysis by Vaidman [ Phys.\ Rev.\ A 87 (2013) 052104 ], and agree with a conclusion reached by Li et al.\ [ Phys.\ Rev.\ A 88 (2013) 046102 ]. However, the analysis casts serious doubt on the claim of Salih et al.\ (whose authorship includes Li et al.) [ Phys.\ Rev.\ Lett.\ 110 (2013) 170502 ] to have constructed a protocol for counterfactual communication: a channel which can transmit information even though it contains a negligible number of photons.
Conventional quantum mechanics describes a pre- and post-selected system in terms of virtual (Feynman) paths via which the final state can be reached. In the absence of probabilities, a weak measurement (WM) determines the probability amplitudes for the paths involved. The weak values (WV) can be identified with these amplitudes, or their linear combinations. This allows us to explain the "unusual" properties of the WV, and avoid the "paradoxes" often associated with the WM. (C) 2016 Published by Elsevier B.V.
Recently, A. Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.240402] performed a much-discussed experiment in which which-way information was obtained from the light in a nested Mach-Zehnder interferometer by weak measurement. The presented analysis using the two-state vector formalism drew the conclusions that the photons followed disconnected paths. We analyze this experiment using standard quantum optical methods and arrive at analytical expressions that match the experimental results without the need for such disconnected photon paths. We also propose a simple amendment to the experiment capable of displaying new phenomena, highlighting the advantages of our description. © 2015 American Physical Society.
In an recent work with the title "Asking Photons Where They Have Been," Danan et al. experimentally demonstrate an intriguing behavior of photons in an interferometer [Phys. Rev. Lett. 111, 240402 (2013), 10.1103/PhysRevLett.111.240402]. In their words: "The photons tell us that they have been in the parts of the interferometer through which they could not pass." They interpret the results using the two-state vector formalism of quantum theory and say that, although an explanation of the experimental results in terms of classical electromagnetic waves in the interferometer is possible (and they provide a partial description), it is not so intuitive. Here we present a more detailed classical description of their experimental results, showing that it is actually intuitive. The same description is valid for the quantum wave function of the photons propagating in the interferometer. In particular, we show that it is essential that the wave propagates through all parts of the interferometer to describe the experimental results. We hope that our work helps to give a deeper understanding of these interesting experimental results.
A double atom interferometer based upon the Stern-Gerlach method in
which the first interferometer acts as a ``nonordinary'' source for the
second one is described. Experiments have been carried out with a
wide-velocity-distribution beam of metastable hydrogen atoms, using two
different operating modes: (i) continuous mode in which the magnitude of
the magnetic field in the first interferometer is fixed whereas that of
the second one is scanned; (ii) time-of-flight mode in which both
magnetic fields are fixed. In the former case lateral interference
patterns appear, in the latter one an overmodulation of the
time-of-flight distribution is observed. It is predicted that this
overmodulation becomes more and more marked as the number of
interferometers is increased, leading to a limiting behavior
characteristic of a quantum Zeno effect.
We present an analysis of a nested Mach-Zehnder interferometer in which an ensemble of identical pre- and postselected particles leaves a weak trace. A knowledge of the weak value partially destroys the quantum interference. The results, contrary to some recent claims [Vaidman, Phys. Rev. A 87, 052104 (2013)], are in accordance with the usual quantum-mechanical expectations.
We propose a scheme allowing us to observe the evolution of a quantum system in the semiclassical regime along the paths generated by the propagator. The scheme relies on performing consecutive weak measurements of the position. We show how "weak trajectories" can be extracted from the pointers of a series of devices having weakly interacted with the system. The properties of these weak trajectories are investigated and illustrated in the case of a time-dependent model system.
Realism - the idea that the concepts in physical theories refer
to `things' existing in the real world - is introduced as a tool
to analyse the status of the wavefunction. Although the physical
entities are recognized by the existence of invariant quantities,
examples from classical and quantum physics suggest that not all
the theoretical terms refer to the entities: some terms refer to
properties of the entities, and some terms have only an epistemic
function. In particular, it is argued that the wavefunction may
be written in terms of classical non-referring and epistemic
terms. The implications for realist interpretations of quantum
mechanics and on the teaching of quantum physics are examined.
When Humpty-Dumpty had his great fall nobody could put him together again. A vastly more moderate challenge is to reunite the two partial beams of a Stern-Gerlach apparatus with such precision that the original spin state is recovered. Nevertheless, as we demonstrate, a substantial loss of spin coherence always occurs, unless the experimenter is able to control the magnetic field's inhomogeneity with an accuracy of at least one part in 105.
We have found that the usual measuring procedure for preselected and postselected ensembles of quantum systems gives unusual results. Under some natural conditions of weakness of the measurement, its result consistently defines a new kind of value for a quantum variable, which we call the weak value. A description of the measurement of the weak value of a component of a spin for an ensemble of preselected and postselected spin-(1/2 particles is presented.
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