It is shown that `bipartite' wave functions can present a mathematical formalism of quantum theory for a single particle, in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This extension of Schrödinger's form establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. In particular, this formalism suggests that collapses of Schrödinger's wave functions can be regarded as the simultaneous transition of the particle from many levels to one. Our results shed considerable light on the basis of quantum mechanics, including quantum measurement.
We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, we prove that the same quantity provides a \emph{lower} bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest vs separating two qubits from the rest.
We introduce a complete set of complementary quantities in bipartite, two-dimensional systems. Complementarity then relates the quantitative entanglement measure concurrence which is a bipartite property to the single-particle quantum properties predictability and visibility, for the most general quantum state of two qubits. Consequently, from an interferometric point of view, the usual wave-particle duality relation must be extended to a ``triality'' relation containing, in addition, the quantitative entanglement measure concurrence, which has no classical counterpart and manifests a genuine …
We experimentally verify the link existing between entanglement and the amount of wave-particle duality in a bipartite quantum system, with superconducting qubits in the IBM Q quantum computer. We consider both pure and mixed states, and study the influence of state purity on the observation of the complementarity "triality" relation of Jakob and Bergou. This work confirms the quantitative completion of local Bohr's complementarity principle by the nonlocal quantum entanglement typical of a truly bipartite quantum system.
It is shown that von Neumann-Landau equation for wave functions can present a mathematical formalism of motion of quantum mechanics. The wave functions of von Neumann-Landau equation for a single particle are `bipartite', in which the associated Schrödinger's wave functions correspond to those `bipartite' wave functions of product forms. This formalism establishes a mathematical expression of wave-particle duality and that von Neumann's entropy is a quantitative measure of complementarity between wave-like and particle-like behaviors. Furthermore, this extension of Schrödinger's form suggests …
We provide a method for witnessing nonseparability of quantum processes on shared systems, which uses the channel-state duality. The method uses a maximally entangled state as a resource. We show that using the resource provides significant advantage over the corresponding protocol for nonseparability detection without the resource.
In any probabilistic theory, we may say a bipartite state on a composite system AB steers its marginal state (on, say, system B) if, for any decomposition of the marginal as a mixture, with probabilities p_i, of states b_i of B, there exists an observable a_i on A such that the states of B conditional on getting outcome a_i on A, are exactly the states b_i, and the probabilities of outcomes a_i are p_i. This is always so for pure bipartite states in quantum mechanics, a fact first observed by Schroedinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but per…
Starting with a given generalized boson algebra U_<q>(h(1)) known as the bosonized version of the quantum super-Hopf U_q[osp(1/2)] algebra, we employ the Hopf duality arguments to provide the dually conjugate function algebra Fun_<q>(H(1)). Both the Hopf algebras being finitely generated, we produce a closed form expression of the universal T matrix that caps the duality and generalizes the familiar exponential map relating a Lie algebra with its corresponding group. Subsequently, using an inverse Mellin transform approach, the coherent states of single-node systems subject to the U_<q>(h(1)) …
An differential equation for wave functions is proposed, which is equivalent to Schrödinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schrödinger's wave equation, this equation is on `bipartite' wave functions. It is shown that those `bipartite' wave functions satisfy all the basic properties of Schrödinger's wave functions. Further, it is argued that `bipartite' wave functions can present a mathematical expression of wave-particle duality. This provides an alternative approach to the mathematical formalism of quantum mechanics.
The complementary wave and particle character of quantum objects (or quantons) was pointed out by Niels Bohr. This wave-particle duality, in the context of the two-slit experiment, is now described not just as two extreme cases of wave and particle characteristics, but in terms of quantitative measures of these natures. These measures of wave and particle aspects are known to follow a duality relation. A very simple and intuitive derivation of a closely related duality relation is presented, which should be understandable to the introductory student.
It is shown that the concept of elementary resonator in the theory of thermal radiation implies the indivisible connection between particles (photons) and electromagnetic waves. This wave-particle duality covers both the Wien and Rayleigh-Jeans regions of spectrum.
The Hamilton-Jacobi equation of relativistic quantum mechanics is revisited. The equation is shown to permit solutions in the form of breathers (oscillating/spinning solitons), displaying simultaneous particle-like and wave-like behaviour.
In a recent article Jan Sperling, Syamsundar De, Thomas Nitsche, Johannes Tiedau, Sonja Barkhofen, Benjamin Brecht, and Christine Silberhorn discuss the wave-particle duality using an experiment to demonstrate that "neither the wave nor the particle description is sufficient to predict the utcomes of quantum-optical experiments." I would like to draw your attention to two previous papers that discuss feasible experiments in which a photon does not seem to behave either as particle or as wave and make some brief considerations on the topic.
We experimentally observe that quantum duality of a single photon is controlled by its self-entanglement through a three-way quantum coherence identity $V^2+D^2+C^2=1$. Here V, D, C represent waveness, particleness, and self-entanglement respectively.
We calculate a tunneling time distribution by means of Nelson's quantum mechanics and investigate its statistical properties. The relationship between the average and deviation of tunneling time suggests the exsistence of ``wave-particle duality'' in the tunneling phenomena.
Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory, however, both notions are in some sense identical: outcome probabilities are given by the overlap between two state vectors - quantum theory is self-dual. In this paper, we show that this notion of self-duality can be understood from a dynamical point of view. We prove that self-duality follows from a computational primitive called bit symmetry: every logica…
It is well known that in classical optics, the visibility of interference, in a two-beam light interference, is related to the optical coherence of the two beams. A wave-particle duality relation can be derived using this mutual coherence. The issue of wave-particle duality in classical optics is analyzed here, in the more general context of multipath interference. New definitions of interference visibility and path distinguishability have been introduced, which lead to a duality relation for multipath interference. The visibility is shown to be related to a new multi-point optical coherence f…
Wave-particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave-like properties, in particular, its ability to generate an interference pattern. Recently, duality relations in which the wave-like properties are quantified by using measures of quantum coherence have been proposed. Quantum coherence can be generalized to a property called group asymmetry. Here we derive a generalized duality relation involving group asymmetry, which is closely related to the success probability of discriminating between the actions of the eleme…
Duality games are a way of looking at wave-particle duality. In these games. Alice and Bob together are playing against the House. The House specifies, at random, which of two sub-games Alice and Bob will play. One game, Ways, requires that they obtain path information about a particle going through an $N$-path interferometer and the other, Phases, requires that they obtain phase information. In general, because of wave-particle duality, Alice and Bob cannot always win the overall game. However, if the required amount of path and phase information is not too great, for example specifying a set…
We investigate duality in entanglement of a bipartite multi-photon system generated from a coherent state of light. The system can exhibit polarization entanglement if the two parts are distinguished by their parity, or parity entanglement if the parts are distinguished by polarization. It was shown in [PRL 110, 140404 (2013)] that this phenomenon can be exploited as a method to test indistinguishability of two particles and it was conjectured that one can also test indistinguishability of macroscopic systems. We propose a setup to test this conjecture. Contrary to the previous studies using t…
A polarized photon with well-defined orbital angular momentum that emerges from a Mach-Zehnder interferometer (MZI) is shown to circumvent wave-particle duality. Its polarization-resolved detection probability forms a non-sinusoidal interferometric pattern. For certain phase differences between the MZI arms, this pattern yields both reliable which-path information and high phase-sensitivity.
We approach wave particle duality, the role of the observer and implications on Retrocausality, by starting with the results of a well verified quantum experiment. We analyze how some current theoretical approaches interpret these results. We then provide an alternative theoretical framework that is consistent with the observations and in many ways simpler than usual attempts to account for retrocausality, involving a non-local conscious Observer.
The complementarity of single-photon's particle-like and wave-like behaviors can be described by the inequality $D^2+V^2 \leq 1$, with $D$ being the path distinguishability and $V$ being the fringe visibility. In this paper, we generalize this duality relation to multi-photon case, where two new concepts, higher order distinguishability and higher order fringe visibility, are introduced to quantify the higher order particle-like and wave-like behaviors of multi-photons.
We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the n…
Wave-particle duality relations are fundamental for quantum physics. Previous experimental studies of duality relations mainly focus on the quadratic relation $D^2+V^2\leq1$, based on symmetric beam interference, while a linear form of the duality relation, predicated earlier theoretically, has never been experimentally tested. In addition, the difference between the quadratic form and the linear form has not been explored yet. In this work, with a designed asymmetric beam interference and by utilizing the polarization degree of freedom of the photon as a which-way detector, we experimentally …
The Hamilton-Jacobi equation of relativistic quantum mechanics is revisited. The equation is shown to permit solutions in the form of breathers (oscillating/spinning solitons), displaying simultaneous particle-like and wave-like behavior. The de Broglie wave thus acquires a clear deterministic meaning of a wave-like excitation of the classical action function. The problem of quantization in terms of the breathing action function and the double-slit experiment are discussed.
We predict that if internal and momentum states of an interfering object are correlated (entangled), then by measuring its internal state we may infer both path (corpuscular) and phase (wavelike) information with much higher precision than for objects lacking such entanglement. We thereby partly circumvent the standard complementarity constraints of which-path detection.
In this paper, we show that the quantum bound state problems are mapped to $N$ point vortices with the identical circulation or strength using the Steiljes electrostatic model with imaginary charges. We also show that these $N$ point charges or vortices, become imaginary, in a constant background field will admit a wave solution under paraxial wave approximation. Therefore, in quantum mechanics as long as paraxial approximation is valid it behaves like a wave.
We investigate wave-particle duality in a symmetric two-way interferometer with a which-way detector. We find that it is important to state wether the interfering object or the which-way detector is read out first. In case that the interfering object is read out first, we discover that it is possible to use the information about its state to increase the distinguishability to a value which violates the widely accepted bounds set by an inequality introduced by Jaeger et al. (PRA 51, 54) and Englert (PRL 77, 2154).
Wave-particle duality is a typical example of Bohr's complementarity principle that plays a significant role in quantum mechanics. Previous studies used the visibility of an interference pattern to quantify the wave property and used path information to quantify the particle property. However, coherence is the core and basis of the interference phenomenon. If we could use coherence to characterize the wave property, the understanding of wave-particle duality would be strengthened. A recent theoretical work [Phys. Rev. Lett. 116, 160406 (2016)] found two relations between quantum coherence and …
We formulate a theory for entangled imaging, which includes also the case of a large number of photons in the two entangled beams. We show that the results for imaging and for the wave-particle duality features, which have been demonstrated in the microscopic case, persist in the macroscopic domain. We show that the quantum character of the imaging phenomena is guaranteed by the simultaneous spatial entanglement in the near and in the far field.
The notion of wave-particle duality may be quantified by the inequality V^2+K^2 <=1, relating interference fringe visibility V and path knowledge K. With a single-photon interferometer in which polarization is used to label the paths, we have investigated the relation for various situations, including pure, mixed, and partially-mixed input states. A quantum eraser scheme has been realized that recovers interference fringes even when no which-way information is available to erase.
In a recent paper a mathematical model for quantum measurement was presented. The phenomenon of wave particle duality, which is introduced in every beginning course of quantum theory, can be explained using this model. Although it is a special case of the general theory it has historic interest and involves the use of a continuous spectrum which was treated only casually in the original article. The treatment here is self contained except for the proof of a mathematical theorem appearing in the first paper.
Wave-particle duality is a bizarre feature at the heart of quantum mechanics which refers to the mutually exclusive dual attributes of quantum objects as the wave and the particle. Quantum eraser presents a counterintuitive aspect of the wave-particle duality. In this work, we show that quantum eraser can be quantitatively understood in terms of the recently developed duality--entanglement relation. In other words, we show that wave-particle-entanglement triality captures all the physics of the quantum erasure. We find that a controllable partial erasure of the which-path information is attain…
Wave-particle duality is an essential character of quantum systems. In the last few years, much progress has being made towards formally quantifying these quantum features. The properties of the quantum density matrix were shown to lead to duality inequalities for single-quanton mixed states and to triality identities, also known as complete complementarity relations (CCRs), for two-quanton pure states. In this article, we establish connections between CCRs and EPR (ir)realism, as formalized by Bilobran and Angelo in [Europhys. Lett. 112, 40005 (2015)]. Besides, we show that, for tripartite pu…
On the basis of an alternative approach to micro-cat states (Found. of Phys., 41, No. 9, p.1502 (2011)) we develop a new model of the two-slit experiment. It explains both this particular experiment and how the wave properties of any ensemble of single quantum particles emerge from their corpuscular ones. The key role in this explanation is played by the indistinguishability of identical quantum particles.
A phase space formulation of the filtering process upon an incident quantum state is developed. This formulation can explain the results of both quantum interference and delayed-choice experiments without making use of the controversial wave-particle duality. Quantum particles are seen as localized and indivisible concentrations of energy and/or mass, their probability amplitude in phase space being described by the Wigner distribution function. The wave or particle nature appears in experiments in which the interference term of the Wigner distribution function is present or absent, respective…
Wave-particle duality of photons with losses in the Mach-Zehnder interferometer (MZI) is investigated experimentally and theoretically. The experiment is done with the standard MZI with the beam splitter or the beam merger being continuously varied. The losses are deliberately introduced either inside the MZI (the two arms between the beam splitter and beam mergers) or outside the MZI (after the beam merger). It is proved that the unbalanced losses have great influence on the predictability $P$ (particle nature) and visibility $V$ (wave nature). For the former case the duality inequality holds…
It is well known that in a two-slit interference experiment, acquiring which-path information about the particle, leads to a degrading of the interference. It is argued that path-information has a meaning only when one can umabiguously tell which slit the particle went through. Using this idea, two duality relations are derived for the general case where the two paths may not be equally probable, and the two slits may be of unequal widths. These duality relations, which are inequalities in general, saturate for all pure states. Earlier known results are recovered in suitable limit.
I argue that quantum optical experiments that purport to refute Bohr's principle of complementarity (BPC) fail in their aim. Some of these experiments try to refute complementarity by refuting the so called particle-wave duality relations, which evolved from the Wootters-Zureck reformulation of BPC (WZPC). I therefore consider it important for my forgoing arguments to first recall the essential tenets of BPC, and to clearly separate BPC from WZPC, which I will argue is a direct contradiction of BPC. This leads to a need to consider the meaning of particle-wave duality relations and to question…
Wave-particle duality in multi-path interference is fraught with issues despite substantial progress in recent years. It was experimentally shown that in certain specific conditions, getting path information in a multi-path experiment can actually increase the visibility of interference. As a result, it was argued that in multi-path interference experiments, visibility of interference and 'which-path' information are not always complementary observables. In the present work, a new wave-particle duality relation is presented, based on a sum of visibilities of interference from individual pairs …
The simplest single-photon entanglement is the entanglement of the vacuum state and the single-photon state between two path modes. The verification of the existence of single-photon entanglement has attracted extensive research interests. Here, based on the construction of Bell's inequality in wave space, we propose a new method to verify single photon entanglement. Meanwhile, we define the wave state in two-dimensional space relative to the photon number state, and propose a method to measure it. Strong violation of Bell inequality based on joint measurement of wave states indicates the exis…
We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an n-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they attempt to determine the path of a particle in the interferometer. In another, called phases, they attempt to determine which set of known phases have been applied to the different paths. The House determines which game is to be played by flipping a coin…
We present an architecture to investigate wave-particle duality in $N$-path interferometers on a universal quantum computer involving as low as $2\log N$ qubits and develop a measurement scheme which allows the efficient extraction of quantifiers of interference visibility and which-path information. We implement our algorithms for interferometers with up to $N=16$ paths in proof-of-principle experiments on a noisy intermediate-scale quantum (NISQ) device using down to $\mathcal{O}(\log N)$ gates and despite increasing noise consistently observe a complementary behavior between interference vi…
Bohr's principle of complementarity, in the context of a two-slit interference experiment, is understood as the quantitative measures of wave and particle natures following a duality relation ${\mathcal D}^2+{\mathcal V}^2 \le 1$. Here ${\mathcal D}$ is a measure of distinguishability of the two paths, and ${\mathcal V}$ is the visibility of interference. It is shown that such a relation can be formulated for $N-$slit or $N-$path interference too, with the proviso that the wave nature is characterized by a measure of {\em coherence} (${\mathcal C}$). This new relation, ${\mathcal D}^2+{\mathca…
We formulate a general theory of wave-particle duality for many-body quantum states, which quantifies how wave- and particle-like properties balance each other. Much as in the well-understood single-particle case, which-way information -- here on the level of many-particle paths -- lends particle-character, while interference -- here due to coherent superpositions of many-particle amplitudes -- indicates wave-like properties. We analyze how many-particle which-way information, continuously tunable by the level of distinguishability of fermionic or bosonic, identical and possibly interacting pa…
The development of quantum technologies depends on investigating of the behavior of quantum systems in noisy environments, since complete isolation from its environment is impossible to achieve. In this paper we show that a wave-particle duality experiment performed in a system with an arbitrarily white noise level cannot be explained in classical terms, using hidden-variables models. In the light of our results, we analyze recent optical and NMR experiments and show that a loophole on non-locality is not fundamental.
Quantum Wave-Particle Duality in Free-Electron--Free-Electron Entanglement
Yangtze Normal University（长江师范学院）；Chongqing Municipal Education Commission, Yangtze Normal University（重庆市委教育委员会，长江师范学院）；Tel Aviv University（特拉维夫大学）；Shenkar College of Engineering and Design（申卡尔工程与设计学院）；Hubei University of Arts and Science（湖北文理学院）
We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measurement strategy to obtain the path information. In particular, we present results for two strategies: zero-error identification of the path-detector states, which never produces an error but sometimes fails to return a conclusive answer, and a mixed stra…
We propose and analyze a modified ghost-interference experiment, and show that revealing the particle-nature of a particle passing through a double-slit hides the wave-nature of a spatially separated particle which it is entangled with. We derive a nonlocal duality relation, ${\mathcal D}_1^2 + {\mathcal V}_2^2 \le 1$, which connects the path distinguishability of one particle to the interference visibility of the other. It extends Bohr's principle of complementarity to a nonlocal scenario. We also propose a ghost quantum eraser in which, erasing the which-path information of one particle brin…
We propose three core ideas: 1. the wave-particle duality of the qudit quantum space; 2. the classification of all elementary quantum gates by ordered pairs of qudit functionals; 3. a new type of quantum gates called the "quantum wave gates". We first study the quantum functionals whose relation to the quantum states is analogous to that between the momentum and position wavefunctions in fundamental quantum physics: a Fourier transform and an entropic uncertainty principle can be defined between the dual representations. The quantum functionals are not just mathematical constructs but have cle…
The weak value, introduced by Aharonov et al. to extend the conventional scope of physical observables in quantum mechanics, is an intriguing concept which sheds new light on quantum foundations and is also useful for precision measurement, but it poses serious questions on its physical meaning due to the unconventional features including the complexity of its value. In this paper we point out that the weak value has a direct connection with the wave-particle duality, in the sense that the wave nature manifests itself in the imaginary part while the particle nature in the real part. This is il…
The wave-particle duality is one of the most mysterious phenomena of the quantum theory, in this paper first it's studied the rise of the wave properties of matter from the theory of stochastic electrodynamics (SED), in which de Broglie's idea of a guiding wave that follows the particle and determine its trajectory is taken back. In the frame of the theory (SED) this wave is of electromagnetic character and is a traveling wave that propagates within the zero point field (ZPF) which according the SED is a classic electromagnetic field that permeates all universe. The wave rise from the interact…
