Two-state Vector Formalism Research Articles

Exploring postselection-induced quantum phenomena with time-bidirectional state formalism

Here we present the time-bidirectional state formalism (TBSF) unifying in a general manner the standard quantum mechanical formalism with no postselection and the time-symmetrized two-state (density) vector formalism, which deals with postselected states. In the proposed approach, a quantum particle's state, called a time-bidirectional state, is equivalent to a joined state of two particles propagating in opposite time directions. For a general time-bidirectional state, we derive outcome probabilities of generalized measurements, as well as mean and weak values of Hermitian observables. We also show how the obtained expressions reduce to known ones in the special cases of no postselection and generalized two-state (density) vectors. Then we develop tomography protocols based on mutually unbiased bases and a symmetric informationally complete positive operator-valued measure, allowing experimental reconstruction of an unknown single qubit time-bidirectional state. Finally, we employ the developed techniques for tracking of a qubit's time-reversal journey in a quantum teleportation protocol realized with a cloud-accessible noisy superconducting quantum processor. The obtained results justify an existence of a postselection-induced qubit's proper time-arrow, which is different from the time-arrow of a classical observer, and demonstrate capabilities of the TBSF for exploring quantum phenomena brought forth by a postselection in the presence of noise.

No contextual advantage in nonparadoxical scenarios of the two-state vector formalism

The two-state vector formalism (TSVF) proposed by Aharonov, Bergmann, and Lebowitz [A. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964)] allows a counterfactual assignment of probabilities of outcomes of contemplated (but unperformed) measurements on quantum systems. The probabilities assigned by the Aharonov-Bergmann-Lebowitz (ABL) rule and the associated weak values have been used to provide insights into quantum situations and to unearth underlying quantum contextuality. We apply the principle of exclusivity on ABL probabilities which are assigned to mutually orthogonal projectors to define paradoxical and nonparadoxical scenarios. Given a pre- and a postselected pair of states, we consider the nonparadoxical sector with a view to explore the demonstration of quantum contextuality. For the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenario, we numerically show that the ABL probabilities of the TSVF in the nonparadoxical sector do not offer any contextual advantage. Our approach can be easily generalized to other contextual scenarios as well. We thus argue that several previous proofs of the emergence of contextuality in pre- and postselected scenarios are only possible if the principle of exclusivity is violated and are therefore classified as paradoxical.

Implementation of the quantum three box paradox through atomic internal states

Quantum three box paradox is one of the key dynamical mysteries of the quantum evolution explored through Two-State Vector Formalism (TSVF) of the quantum theory in conjunction with the weak measurements. The paradox further deepens the notation of quantum reality while highlighting the strange and counterintutive traits of the quantum unitary evolution in-between two strong measurements. We present an experimentally feasible protocol for implementing the paradox through internal states of a three-level ladder atomic system. The techniques invoked here are all state of the art cavity QED tools and the suggested schematics can be executed practically with good overall results.

Past of a Quantum particle: An atom interferometric based study

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.

Do weak values capture the complete truth about the past of a quantum particle?

Weak values inferred from weak measurements have been proposed as a tool to investigate trajectories of pre- and post-selected quantum systems. Are the inferences drawn from the weak values about the past of a quantum particle fully true? Can the two-state vector formalism predict everything that the standard formalism of quantum mechanics can? To investigate these questions we present a “which-path” gedanken experiment in which the information revealed by a pre- and post-selected quantum system is surprisingly different from what one would expect from the weak values computed using the two-state vector formalism. In our gedanken experiment, a particle reveals its presence in locations where the weak value of the projection operator onto those locations was vanishingly small. Therefore our predictions turn out to be in contradistinction to those made based on the non-vanishing weak values as the presence indicators of the quantum particle. We propose a six port photon-based interferometer setup as a possible physical realization of our gedanken experiment.

Evidence of Predictive Power and Experimental Relevance of Weak-Values Theory

The concepts of Weak Values (WV) and Two-State Vector Formalism (TSVF) appear to motivate new experiments and to offer novel insights into dynamical processes in various materials of several scientific and technological fields. To support this view, here we consider the dynamics of hydrogen atoms and/or molecules in nanostructured materials like e.g., carbon nanotubes. The experimental method applied is incoherent scattering of thermal (i.e., non-relativistic) neutrons (INS). In short, the main finding consists in the following effect: the measured energy and momentum transfers are shown to contradict even qualitatively the associated expectations of conventional scattering theory. This effect was recently observed in INS experiments, e.g., in H2 adsorbed in carbon nanotubes, where a large momentum transfer deficit was found. Due to the broad abundance of hydrogen, these findings may be also of technological importance, since they indicate a considerably enhanced H mobility in specific structured material environments. A new INS experiment is proposed concerning the H mobility of an ultra-fast proton conductor (H3OSbTeO6) being of technological relevance. Further neutron scattering investigations on other systems (metallic hydrides and H2 encapsulated inside C60) are proposed. As concerns theoretical implications, the analysis of the experimental results strongly supports the view that the wavefunction (or state vector) represents an ontological physical entity of a single quantum system.

Quantum three-box paradox: a proposal for atom optics implementation

Quantum three-box paradox is one of the major mysteries revealed through two-state vector formalism of the quantum theory in conjunction with the weak measurements. We suggest a proposal for the experimental implementation of the paradox using time tested tools of Bragg regime atom optics with a two-level atom diffracted from a cavity field in the momentum space. The proposed schematics can be easily executed within the limits of contemporary laboratory resources using either quantized or classical fields.

Quantum Cheshire cat: a physically realistic interpretation by invoking entangled correlations

The phenomenon of the quantum Cheshire cat (QCC) and its interpretation by Aharanov et al. [New J. Phys. 15, 113015 (2013)NJOPFM1367-263010.1088/1367-2630/15/11/113015], with the conjecture that any quantum entity can be disembodied from its physical attributes, has resulted in a heated debate leading to interpretational controversy as well as practical consequences. Here, we propose an experimentally testable and physically more realistic and logically plausible interpretation. We utilize a specifically engineered Mach–Zehnder-type interferometeric setup that is quite similar to the original QCC setup but with the slight difference that now a single-photon, bipartite entangled state traverses the interferometer such that each path is designated to a photon with different tags. With this specific setup, we demonstrate that the photon’s polarization is never physically separated from the photon itself. Rather, it becomes dormant and hence inaccessible along the designated interferometric path. We also generalize the schematics and show that any precisely oriented photon’s polarization that stands inaccessible or dormant re-emerges along the same spatially separated and isolated arm as we tune the polarization vector away from the selected angle. Thus our proposal persuasively proves that polarization is never stripped off the photon itself and instead becomes inaccessible along the interferometeric arm for a certain particularly selected orientation. The schematics further reveal that this inaccessibility of the photon’s polarization is not permanent, fixed, and universal, but rather it is entirely constrained to a specific orientation in the Hilbert space, governed by the particular pre- and post-selected state under two-state vector formalism.

Weak Values and Quantum Information in Scattering Physics — New Theoretical and Experimental Effects

Weak Values (WV) and Two-State-Vector Formalism (TSVF) provide novel insights in various quantum physical and technological fields. In the first part of the paper we consider a new quantum effect of scattering accompanying an elementary collision of two quantum systems A and B, in which the latter interacts with a quantum environment. In clear contrast to a classical environment, the quantum case can exhibit counter-intuitive effects of momentum- and energy-transfer which contradict conventional expectations. Experimental evidence of a new effect—deficit of momentum transfer (equivalently: reduced effective mass) in a neutron-atom collision—is presented and theoretically interpreted. Here, non-relativistic incoherent inelastic neutron scattering (INS) is applied. INS on single H2 molecules confined in multi-walled carbon nanotube channels has been experimentally investigated. Interpreted within conventional theory, the results reveal a counter-intuitive reduced effective mass of the recoiling H2 molecule, i.e. M = 0.64 a.m.u. (atomic mass units). In contrast, this finding has a simple qualitative interpretation within WV and TSVF theory. In the second part of the paper we report on current experimental and theoretical investigations in the field of X-ray diffraction (XRD), which belongs to coherent scattering. Preliminary XRD results from cubic crystalline materials show a surprising variation of the measured lattice parameter (usually called “lattice constant”) with momentum transfer. A first theoretical model of the effect in the light of the new theory is presented. These findings give further evidence for the broad character and significance of the novel WV and TSVF theory.

Effects of Quantum Confinement of Hydrogen in Nanocavities – Experimental INS Results and New Insights

Current developments of non-relativistic quantum mechanics appear to predict, and reveal, counter-intuitive dynamical effects of hydrogen in nanostructured materials that are of considerable importance for basic research as well as for technological applications. In this review, the experimental focus is on HO and H molecules in carbon nanotubes and other nanocavities that have been experimentally investigated using the well-established technique of incoherent inelastic neutron scattering (INS). For instance, the momentum and energy transfers, as obtained from the commonly used standard data analysis techniques, from a (I) H2 molecule in a C-nanotube resulting in a roto-translational motion along the nanotube axis seems to (1) either violate the standard conservation laws or (2) to attribute to the H molecule undergoing translation the effective mass a.m.u. (atomic mass units) instead of the expected 2 a.m.u. A similar striking anomalous effect has been found in the neutron-H scattering from the (II) H2O molecules in nano-channels of some solid materials, in which O-H stretching vibrations along the channel axis are created. The results of this scattering process seem to once again either violate the standard conservation laws or to attribute to the effective mass of the struck H2 molecule as a.m.u. instead of the expected value of 1 a.m.u. We show that these counterintuitive observations from the INS studies have no conventional interpretation within the standard non-relativistic scattering theory. However, they can be qualitatively interpreted “from first principles” within the framework of modern theories of (III) time-symmetric quantum dynamics, as provided by the weak values (WV) and two-state- vector formalism (TSVF), and/or (IV) quantum correlations, especially quantum discord (QD) and quantum thermodynamics (QTD). The theoretical analysis provides an intuitive understanding of the experimental results, gives strong evidence that the nano-structured cavities do represent quantum systems which participate significantly in the dynamics of the neutron-H scattering, and, surprisingly, shows that new physical information can be derived from the experimental data. This latter point may also have far-reaching consequences for technology and material sciences (e.g., fuel cells, H storage materials, etc.). Moreover, novel insights into the short-lived quantum dynamics and/or quantum information theory can be gained.

Some Notes on Counterfactuals in Quantum Mechanics.

Counterfactuals, i.e., events that could have occurred but eventually did not, play a unique role in quantum mechanics in that they exert causal effects despite their non-occurrence. They are therefore vital for a better understanding of quantum mechanics (QM) and possibly the universe as a whole. In earlier works, we have studied counterfactuals both conceptually and experimentally. A fruitful framework termed quantum oblivion has emerged, referring to situations where one particle seems to "forget" its interaction with other particles despite the latter being visibly affected. This framework proved to have significant explanatory power, which we now extend to tackle additional riddles. The time-symmetric causality employed by the Two State-Vector Formalism (TSVF) reveals a subtle realm ruled by “weak values,” already demonstrated by numerous experiments. They offer a realistic, simple and intuitively appealing explanation to the unique role of quantum non-events, as well as to the foundations of QM. In this spirit, we performed a weak value analysis of quantum oblivion and suggest some new avenues for further research.

Entanglement and the paradox of untying the defining feature from a quantum entity

We analyze the quantum Cheshire cat dilemma under an entangled scenario based on atom-field interferometric setup that can be readily executed experimentally. Once the accuracy and the faithful workability of the two-state vector formalism along with the weak measurement theme is guaranteed, our analysis suggests that the interpretation, i.e., separation of a certain specific feature from the quantum entity, furnished in the pioneering Cheshire cat paper [Y. Aharonov, S. Popescu, D. Rohrlich, and P. Skrzypczyk, New J. Phys. 15, 113015 (2013)], cannot be taken for granted as the sole reason for the phenomenon. Rather, our analysis of the considered entangled interferometric scenario suggests an alternative point of view where the inaccessibility of the defining features of a quantum system appears to be a more plausible reason than the disembodying of the feature from the system itself. Our proposal is based on the time-tested tools of the atomic Bragg diffraction, operating in off-resonant regime, and utilizes the entangled correlations engineered between two identical, two-level atoms in the momentum space.

Weak Values and Two-State Vector Formalism in Elementary Scattering and Reflectivity—A New Effect

The notions of Weak Value (WV) and Two-State Vector Formalism (TSVF), firstly introduced by Aharonov and collaborators, provide a quantum-theoretical formalism of extracting new information from a system in the limit of small disturbances to its state. Here, we explore two applications to the case of non-relativistic two-body scattering with one body weakly interacting with its environment. We present a physically compelling analysis of a new quantum effect: momentum transfer deficit and an accompanying enhanced energy transfer; or, equivalently, an apparent mass-deficit of the struck body. First, incoherent inelastic neutron scattering (INS) from protons of H 2 molecules in C-nanotubes is investigated. The data of the H 2 translational motion along the nanotube shows that the neutron apparently exchanges energy and momentum with a fictitious particle with mass of 0.64 atomic mass units (a.m.u.), which is in blatant contradiction with the expected value of 2 a.m.u. Second, the same theory is applied to neutron reflectivity—which is elastic and coherent—from the interface of (single crystal) Si with H 2 O-D 2 O liquid mixtures. The data shows a striking enhanced reflectivity in a wide range of momentum transfers, which is tantamount to a momentum-transfer deficit with respect to conventional expectations. However, these effects find a natural interpretation within the WV-TSVF theoretical analysis under consideration. In summary, both scattering effects contradict conventional theoretical expectations, thus also supporting the novelty of the theoretical framework of WV and TVSF. Additionally, it should be pointed out that the two dynamical variables in the interaction Hamiltonian of the theoretical model belong to two different physical bodies.

The Weak Reality That Makes Quantum Phenomena More Natural: Novel Insights and Experiments.

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.

Weak measurement and weak values — New insights and effects in reflectivity and scattering processes

Recently, the notions of Weak Measurement (WM), Weak Value (WV) and Two-State-Vector Formalism (TSVF), firstly introduced by Aharonov and collaborators, have extended the theoretical frame of standard quantum mechanics, thus providing a quantum-theoretical formalism for extracting new information from a system in the limit of small disturbance to its state. Here we provide an application to the case of two-body scattering with one body weakly interacting with its environment — e.g. a neutron being scattered from a H2 molecule physisorbed in a carbon nanotube. In particular, we make contact with the field of incoherent inelastic neutron scattering from condensed systems. We provide a physically compelling prediction of a new quantum effect — a momentum transfer deficit; or equivalently, an enhanced energy transfer; or an apparent reduction of the mass of the struck particle. E.g., when a neutron collides with a H2 molecule in a C-nanotube and excites its translational motion along the nanotube, it apparently exchanges energy and momentum with a fictitious particle with mass of 0.64 atomic mass units. Experimental results are shown and discussed in the new theoretical frame. The effect under consideration has no conventional interpretation, thus also supporting the novelty of the quantum theoretical framework of WV and TVSF. Some speculative remarks about possible applications being of technological interest (fuel cells and hydrogen storage; Li+ batteries; information and communication technology) are shortly mentioned.

Implementation Scheme of Two-Photon Post-Quantum Correlations**Supported by National Natural Science Foundation of China (NSFC) under Grant Nos. 11274010, 11374085, and 61370090, Anhui Provincial Natural Science Foundation under Grant Nos. 1408085MA20 and 1408085MA16, the Key Program of Domestic Visiting of Anhui Province under Grant No. gxfxZD2016192

The pre- and post-selection processes of the “two-state vector formalism” lead to a fair sampling loophole in Bell test, so it can be used to simulate post-quantum correlations. In this paper, we propose a physical implementation of such a correlation with the help of quantum non-demolition measurement, which is realized via the cross-Kerr nonlinear interaction between the signal photon and a probe coherent beam. The indirect measurement on the polarization state of photon is realized by the direct measurement on the phase shift of the probe coherent beam, which enhances the detection efficiency greatly and leaves the signal photon unabsorbed. The maximal violation of the CHSH inequality 4 can be achieved by pre- and post-selecting maximally entangled states. The reason why we can get the post-quantum correlation is that the selection of the results after measurement opens fair-sampling loophole. The fair-sampling loophole opened here is different from the one usually used in the currently existing simulation schemes for post-quantum correlations, which are simulated by selecting the states to be measured or enlarging the Hilbert space. So, our results present an alternative way to mimic post-quantum correlations.

Interaction-Free Effects Between Distant Atoms

A Gedanken experiment is presented where an excited and a ground-state atom are positioned such that, within the former’s half-life time, they exchange a photon with 50% probability. A measurement of their energy state will therefore indicate in 50% of the cases that no photon was exchanged. Yet other measurements would reveal that, by the mere possibility of exchange, the two atoms have become entangled. Consequently, the “no exchange” result, apparently precluding entanglement, is non-locally established between the atoms by this very entanglement. This quantum-mechanical version of the ancient Liar Paradox can be realized with already existing transmission schemes, with the addition of Bell’s theorem applied to the no-exchange cases. Under appropriate probabilities, the initially-excited atom, still excited, can be entangled with additional atoms time and again, or alternatively, exert multipartite nonlocal correlations in an interaction free manner. When densely repeated several times, this result also gives rise to the Quantum Zeno effect, again exerted between distant atoms without photon exchange. We discuss these experiments as variants of interaction-free-measurement, now generalized for both spatial and temporal uncertainties. We next employ weak measurements for elucidating the paradox. Interpretational issues are discussed in the conclusion, and a resolution is offered within the Two-State Vector Formalism and its new Heisenberg framework.

A quasi-current representation for information needs inspired by Two-State Vector Formalism

Recently, a number of quantum theory (QT)-based information retrieval (IR) models have been proposed for modeling session search task that users issue queries continuously in order to describe their evolving information needs (IN). However, the standard formalism of QT cannot provide a complete description for users’ current IN in a sense that it does not take the ‘future’ information into consideration. Therefore, to seek a more proper and complete representation for users’ IN, we construct a representation of quasi-current IN inspired by an emerging Two-State Vector Formalism (TSVF). With the enlightenment of the completeness of TSVF, a “two-state vector” derived from the ‘future’ (the current query) and the ‘history’ (the previous query) is employed to describe users’ quasi-current IN in a more complete way. Extensive experiments are conducted on the session tracks of TREC 2013 & 2014, and show that our model outperforms a series of compared IR models.

Experimental observation of anomalous trajectories of single photons

A century after its conception, quantum mechanics still hold surprises that contradict many "common sense" notions. The contradiction is especially sharp in case one consider trajectories of truly quantum objects such as single photons. From a classical point of view, trajectories are well defined for particles, but not for waves. The wave-particle duality forces a breakdown of this dichotomy and quantum mechanics resolves this in a remarkable way: Trajectories can be well defined, but they are utterly different from classical trajectories. Here, we give an operational definition to the trajectory of a single photon by introducing a novel technique to mark its path using its spectral composition. The method demonstrates that the frequency degree of freedom can be used as a bona fide quantum measurement device (meter). The analysis of a number of setups, using our operational definition, leads to anomalous trajectories which are non-continuous and in some cases do not even connect the source of the photon to where it is detected. We carried out an experimental demonstration of these anomalous trajectories using a nested interferometer. We show that the Two-state vector formalism provides a simple explanation for the results.

The Case of the Disappearing (and Re-Appearing) Particle

A novel prediction is derived by the Two-State-Vector-Formalism (TSVF) for a particle superposed over three boxes. Under appropriate pre- and post-selections, and with tunneling enabled between two of the boxes, it is possible to derive not only one, but three predictions for three different times within the intermediate interval. These predictions are moreover contradictory. The particle (when looked for using a projective measurement) seems to disappear from the first box where it would have been previously found with certainty, appearing instead within the third box, to which no tunneling is possible, and later re-appearing within the second. It turns out that local measurement (i.e. opening one of the boxes) fails to indicate the particle’s presence, but subtler measurements performed on the two boxes together reveal the particle’s nonlocal modular momentum spatially separated from its mass. Another advance of this setting is that, unlike other predictions of the TSVF that rely on weak and/or counterfactual measurements, the present one uses actual projective measurements. This outcome is then corroborated by adding weak measurements and the Aharonov-Bohm effect. The results strengthen the recently suggested time-symmetric Heisenberg ontology based on nonlocal deterministic operators. They can be also tested using the newly developed quantum router.