Quantum interference of two independent particles in pure quantum states is fully described by the particles' distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability, taking on a surprisingly rich character. Here, we study many-particle interference using three photons. We show that the distinguishability between pairs of photons is not sufficient to fully describe the photons' behaviour in a scattering process, but that a collective phase, the triad phase, plays a role. We are able to explore the full parameter space of three-photon interference by generating heralded single photons and interfering them in a fibre tritter. Using multiple degrees of freedom - temporal delays and polarisation - we isolate three-photon interference from two-photon interference. Our experiment disproves the view that pairwise two-photon distinguishability uniquely determines the degree of non-classical many-particle interference.
A number of recent interference experiments involving multiple photons are reviewed. These experiments include generalized photon bunching effects, generalized Hong-Ou-Mandel interference effects and multi-photon interferometry for demonstrations of multi-photon de Broglie wavelength. The multi-photon states used in these experiments are from two pairs of photons in parametric down-conversion. We find that the size of the interference effect in these experiments, characterized by the visibility of interference pattern, is governed by the degree of distinguishability among different pairs of ph…
The complementary principle of quantum mechanics relates qualitatively the visibility of quantum interference with path indistinguishability. Here we propose a scheme of constructive quantum interference involving superposition between an $N$-photon state and a single-photon state to characterize quantitatively the degree of temporal distinguishability of the $N$-photon state. This scheme is based on a generalized photon bunching effect. Such a scheme can be extended to other more general cases.
It is known that quantum interference can disappear with the mere possibility of distinguishability without actually performing the act. We create such distinguishability in an unbalanced SU(1,1) interferometer and indeed observe no interference in the direct photodetection of the outputs. On the other hand, such distinguishability can be erased with a projective measurement. Here, we report a method of homodyne detection that can also recover interference effect. We find that it is the indistinguishability in amplitude measurement that leads to the recovery of interference, and the quantum na…
Multimode two-particle systems show interference effects in one-particle detections when both particles have common modes. We explore the possibility of extending the usual concepts of distinguishability and visibility to these types of systems. Distinguishability will refer now to the balance between common and different modes of a two-particle system, instead of the standard definition concerning available alternatives for a one-particle system. On the other hand, the usual concept of visibility is not suitable for our problem and must be replaced with that of contrast, measuring the ratio o…
We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a …
We present a multi-mode model to describe an arbitrary N-photon state with a wide spectral range and some arbitrary temporal distribution. In general, some of the $N$ photons are spread out in time while other may overlap and become indistinguishable. From this model, we find that the temporal (in)distinguishability of photons is related to the exchange symmetry of the multi-photon wave function. We find that simple multi-photon detection scheme gives rise to a more general photon bunching effect with the famous two-photon effect as a special case. We then send this N-photon state into a recen…
We demonstrate the conspiration of many-particle interferences of different degree to determine the transmission of four photons of tunable indistinguishability through a four-port beam splitter array. The probability of certain output events depends non-monotonically on the degree of distinguishability, due to distinct multi-particle interference contributions to the transmission signal.
An experiment is performed to demonstrate the temporal distinguishability of a four-photon state and a six-photon state, both from parametric down-conversion. The experiment is based on a multi-photon interference scheme in a recent discovered NOON-state projection measurement. By measuring the visibility of the interference dip, we can distinguish the various scenarios in the temporal distribution of the pairs and thus quantitatively determine the degree of temporal (in)distinguishability of a multi-photon state.
Typically, quantum superpositions, and thus measurement projections of quantum states involving interference, decrease (or increase) monotonically as a function of increased distinguishability. Distinguishability, in turn, can be a consequence of decoherence, for example caused by the (simultaneous) loss of excitation or due to inadequate mode matching (either deliberate or indeliberate). It is known that for some cases of multi-photon interference, non-monotonic decay of projection probabilities occurs, which has so far been attributed to interference between four or more two photons. We show…
Quantitative measures are introduced for the indistinguishability $U$ of two quantum states in a given measurement and the amount of interference $I$ observable in this measurement. It is shown that these measures obey an inequality $U\geq I$ which can be seen as an exact formulation of Bohr's claim that one cannot distinguish between two possible paths of a particle while maintaining an interference phenomenon. This formulation is applied to a neutron interferometer experiment of Badurek e.a. It is shown that the formulation is stronger than an argument based on an uncertainty relation for ph…
The complementarity relation between the visibility and the spectral distinguishability of frequency-displaced photonic wave-packets in a Hong-Ou-Mandel interferometer is studied. An experimental definition of $K$, the distinguishability parameter, is proposed and tested for the $K^2+\mathcal{V}^2\leq 1$ complementarity inequality when a consistent visibility parameter is defined. The results show that the spectral distinguishability is, indeed, complementary to the visibility and that the quantum aspect of the two-photon interference phenomenon can be examined by employing weak-coherent state…
High-dimensional quantum entanglement is currently one of the most prolific fields in quantum information processing due to its high information capacity and error resilience. A versatile method for harnessing high-dimensional entanglement has long been hailed as an absolute necessity in the exploration of quantum science and technologies. Here we exploit Hong-Ou-Mandel interference to manipulate discrete frequency entanglement in arbitrary-dimensional Hilbert space. The generation and characterization of two-, four- and six-dimensional frequency entangled qudits are theoretically and experime…
Recent times have seen a spurt of research activity focused on "completing" certain wave-particle duality relations using entanglement or polarization. These studies use a duality relation involving path-predictability, and not path-distinguishability. Quantum origins of these results are explored here, in the more general framework of multipath quantum interference. Multipath interference with a path-detector is theoretically analyzed to find the connection between predictability and distinguishability. It is shown that entanglement is what quantitatively connects distinguishability with pred…
We investigate, in a four photon interference experiment in a laser-written waveguide structure, how symmetries control the suppression of many-body output events of a $J_x$ unitary. We show that totally destructive interference does not require mutual indistinguishability between all, but only between symmetrically paired particles, in agreement with recent theoretical predictions. The outcome of the experiment is well described by a quantitative simulation which accounts for higher order emission of the photon source, imbalances in the scattering network, partial distinguishability, and phot…
Multi-photon interference is at the heart of the recently proposed linear optical quantum computing scheme and plays an essential role in many protocols in quantum information. Indistinguishability is what leads to the effect of quantum interference. Optical interferometers such as Michaelson interferometer provide a measure for second-order coherence at one-photon level and Hong-Ou-Mandel interferometer was widely employed to describe two-photon entanglement and indistinguishability. However, there is not an effective way for a system of more than two photons. Recently, a new interferometric …
Bohr's Complementarity Principle is quantitatively formulated in terms of the distinguishability of various paths a quanton can take, and the measure of the interference it produces. This phenomenon results from the interference of single-quanton amplitudes for various paths. The distinguishability of paths puts a bound on the sharpness of the interference the quanton can produce. However there exist other kinds of quantum phenomena where interference of two-particle amplitudes results in a two-particle interference, if the particles are indistinguishable. The Hong-Ou-Mandel (HOM) effect and t…
An interference experiment with entangled particles is theoretically analyzed, where one of the entangled pair (particle 1) goes through a multi-slit before being detected at a fixed detector. In addition, one introduces a mechanism for finding out which of the n slits did particle 1 go through. The other particle of the entangled pair (particle 2) goes in a different direction, and is detected at a variable, spatially separated location. In coincident counting, particle 2 shows n-slit interference. It is shown that the normalized quantum coherence of particle 2, $\mathcal{C}_2$, and the path-…
Wave-particle duality finds a natural application for electrons or light propagating in disordered media where coherent corrections to transport are given by two-wave interference. For scatterers with internal degrees of freedom, these corrections are observed to be much smaller than would be expected for structureless scatterers. By examining the basic example of the scattering of one photon by two spin-1/2 atoms--a case-study for coherent backscattering--we demonstrate that the loss of interference strength is associated with which-path information stored by the scattering atoms.
The issue of interference and which-way information is addressed in the context of 3-slit interference experiments. A new path distinguishability ${\mathcal D_Q}$ is introduced, based on Unambiguous Quantum State Discrimination (UQSD). An inequality connecting the interference visibility and path distinguishability, ${\mathcal V} + {2{\mathcal D_Q}\over 3- {\mathcal D_Q}} \le 1$, is derived which puts a bound on how much fringe visibility and which-way information can be simultaneously obtained. It is argued that this bound is tight. For 2-slit interference, we derive a new duality relation wh…
Erasing quantum-mechanical distinguishability is of fundamental interest and also of practical importance, particularly in subject areas related to quantum information processing. We demonstrate a method applicable to optical systems in which single-mode filtering is used with only linear optical instruments to achieve quantum indistinguishability. Through "heralded" Hong-Ou-Mandel interference experiments we measure and quantify the improvement of indistinguishability between single photons generated via spontaneous four-wave mixing in optical fibers. The experimental results are in excellent…
A three-slit ghost interference experiment with entangled photons is theoretically analyzed using wave-packet dynamics. A non-local duality relation is derived which connects the path distinguishability of one photon to the interference visibility of the other.
The frequency distinguishability of two single photons was successfully erased using single photon frequency up-conversion. A frequency non-degenerate photon pair generated via spontaneous four-wave mixing in a dispersion shifted fiber was used to emulate two telecom-band single photons that were in the same temporal mode but in different frequency modes. The frequencies of these photons were converted to the same frequency by using the sum frequency generation process in periodically poled lithium niobate waveguides, while maintaining their temporal indistinguishability. As a result, the two …
Non-classical interference of photons lies at the heart of optical quantum information processing. This effect is exploited in universal quantum gates as well as in purpose-built quantum computers that solve the BosonSampling problem. Although non-classical interference is often associated with perfectly indistinguishable photons this only represents the degenerate case, hard to achieve under realistic experimental conditions. Here we exploit tunable distinguishability to reveal the full spectrum of multi-photon non-classical interference. This we investigate in theory and experiment by contro…
Many recent experiments employ several parametric down conversion (PDC) sources to get multiphoton interference. Such interference has applications in quantum information. We study here how effects due to photon statistics, misalignment, and partial distinguishability of the PDC pairs originating from different sources may lower the interference contrast in the multiphoton experiments.
Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of $n$ pairs of photons, whose normalization $q_{2n}$ is the probability that $2n$ photons interfere as indistinguishable. For two pairs of photons $q_{4}=(1+ 2\mathbb{P} )/3$, where $\mathbb{P} $ is the purity of the squeezed states ($K=1/\mathbb{P} $ is the Schmidt numb…
We study distinguishability of photons in multiphoton interference on a multiport when fast detectors, capable of precise time resolution, are employed. Such a setup was previously suggested for experimental realization of boson sampling with single photons. We investigate if fast photodetection allows to circumvent distinguishability of realistic single photons in mixed states. To this goal we compare distinguishability of photons in two setups: (a) with photons in the same average (temporal) profile on a spatial interferometer and photodetection incapable of (or with strongly imprecise) time…
We study the hardness of classically simulating Gaussian boson sampling at nonzero photon distinguishability. We find that similar to regular boson sampling, distinguishability causes exponential attenuation of the many-photon interference terms in Gaussian boson sampling. Barring an open problem in the theory of matrix permanents, this leads to an efficient classical algorithm to simulate Gaussian boson sampling in the presence of distinguishability. We also study a new form of boson sampling based on photon number superposition states, for which we also show noise sensivity. The fact that su…
Particle distinguishability is a significant challenge for quantum technologies, in particular photonics where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantisation, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum information inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes clear that the HOM experiment is an instance of a (mix…
We experimentally demonstrate the non-monotonic dependence of genuine many-particle interference signals on the particles' mutual distinguishability. Our theoretical analysis shows that such non-monotonicity is a generic feature of the quantum to classical transition in multiparticle correlation functions of more than two particles.
In this article we show that, in a two-arm interferometer, pure quantum states of perfect path distinguishability (particles) are geometrically equidistant from all states with constant path distinguishability D. This property is not shared by other states, such as perfect fringe-visibility (waves) or maximally entangled quantum states (entanglon). Indeed, the Bures distance between a particle and any other state depends only the distinguishability of the latter. On the contrary, the Bures distance between a wave or an entanglon, and any other single photon state depends on other set of parame…
We study the fringe visibility and the distinguishability of a general Mach-Zehnder interferometer with an asymmetric beam splitter. Both the fringe visibility V and the distinguishability D are affected by the input state of the particle characterized by the Bloch vector S=(Sx,Sy,Sz) and the second asymmetric beam splitter characterized by paramter /beta. For the total system is initially in a pure state, it is found that the fringe visibility reaches the upper bound and the distinguishability reaches the lower bound when cos(/beta) = -Sx. The fringe visibility obtain the maximum only if Sx =…
Quantum mechanical wave-particle duality is quantified in terms of a trade-off relation between the fringe visibility and the which-way distinguishability in an interference experiment. This relation is recently generalized by Banaszek et. al., (Nature Communications 4, 2594, 2013) when the particle is equipped with an internal degree of freedom such as spin. Here, we extend the visibility-distinguishability trade-off relation to quantum interference of a particle possessing an internal degree of freedom, when the which-way detector state is entangled with an ancillary system. We introduce an …
The aim of this paper is to revisit the implications of complementarity when we inject into a Mach Zehnder interferometer particles with internal structure, prepared in special translational-internal entangled (TIE) states. This correlation causes the path distinguishability to be phase dependent in contrast to the standard case. We show that such a TIE state permits us to detect small phase shifts along with almost perfect path distinguishability, beyond the constraints imposed by complementarity on simultaneous which-way and which-phase measurements for standard cases (when distinguishabilit…
We demonstrate two-photon interference using two remote single molecules as bright solid-state sources of indistinguishable photons. By varying the transition frequency and spectral width of one molecule, we tune and explore the effect of photon distinguishability. We discuss future improvements on the brightness of single-photon beams, their integration by large numbers on chips, and the extension of our experimental scheme to coupling and entanglement of distant molecules.
Measuring an entangled state of two particles is crucial to many quantum communication protocols. Yet Bell state distinguishability using a finite apparatus obeying linear evolution and local measurement is theoretically limited. We extend known bounds for Bell-state distinguishability in one and two variables to the general case of entanglement in $n$ two-state variables. We show that at most $2^{n+1}-1$ classes out of $4^n$ hyper-Bell states can be distinguished with one copy of the input state. With two copies, complete distinguishability is possible. We present optimal schemes in each case…
The suggestion that particles of the same kind may be indistinguishable in a fundamental sense, even so that challenges to traditional notions of individuality and identity may arise, has first come up in the context of classical statistical mechanics. In particular, the Gibbs paradox has sometimes been interpreted as a sign of the untenability of the classical concept of a particle and as a premonition that quantum theory is needed. This idea of a quantum connection stubbornly persists in the literature, even though it has also been criticized frequently. Here we shall argue that although thi…
Partial polarization is the manifestation of the correlation between two mutually orthogonal transverse field components associated with a light beam. We show both theoretically and experimentally that the origin of this correlation can be purely quantum mechanical. We perform a two-path first-order (single photon) interference experiment and demonstrate that the degree of polarization of the light emerging from the output of the interferometer depends on path distinguishability. We use two independent methods to control the distinguishability of the photon paths. While the distinguishability …
Numerous quantum information protocols make use of maximally entangled two-particle states, or Bell states, in which information is stored in the correlations between the two particles rather than their individual properties. Retrieving information stored in this way means distinguishing between different Bell states, yet the well known no-go theorem establishes that projective linear evolution and local measurement (LELM) detection schemes can only reliably distinguish three of the four qubit Bell states. We establish maximum distinguishability of the qutrit Bell states of bosons via projecti…
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 develop a framework to systematically investigate the influence of many-particle interference on the dynamics of generic $-$ possibly interacting $-$ bosonic systems. We consider mixtures of bosons which belong to several distinguishable species, allowing us to tune the level of many-particle interference, and identify the corresponding signatures in the time-dependent expectation values of observables. Interference contributions to these expectation values can be classified based on the number of interfering particles. Interactions are shown to generate a series of additional, higher-order…
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…
A recent work (2014 New J. Phys. 16 013006) claims that nonmonotonic structures found in the many-particle quantum-to-classical transition (2013 Proc. Natl Acad. Sci. USA 110 1227-1231; 2011 Phys. Rev. A 83 062111) are not exclusive to the many-body domain, but they also appear for single-photon as well as for semi-classical systems. We show that these situations, however, do not incorporate any quantum-to-classical transition, which makes the claims unsustainable.
We demonstrate how boson sampling with photons of partial distinguishability can be expressed in terms of interference of fewer photons. We use this observation to propose a classical algorithm to simulate the output of a boson sampler fed with photons of partial distinguishability. We find conditions for which this algorithm is efficient, which gives a lower limit on the required indistinguishability to demonstrate a quantum advantage. Under these conditions, adding more photons only polynomially increases the computational cost to simulate a boson sampling experiment.
Multi-photon interference in large multi-port interferometers is key to linear optical quantum computing and in particular to boson sampling. Silicon photonics enables complex interferometric circuits with many components in a small footprint and has the potential to extend these experiments to larger numbers of interfering modes. However, loss has generally limited the implementation of multi-photon experiments in this platform. Here, we make use of high-efficiency grating couplers to combine bright and pure quantum light sources based on ppKTP waveguides with silicon circuits. We interfere u…
A scheme for generating an entangled state in a two spin-1/2 system by means of a spin-dependent potential scattering of another qubit is presented and analyzed in three dimensions. The entanglement is evaluated in terms of the concurrence both at the lowest and in full order in perturbation with an appropriate renormalization for the latter, and its characteristics are discussed in the context of (in)distinguishability of alternative paths for a quantum particle.
We show that the possibility of distinguishing between single and two photon detection events, usually not met in the actual experiments, is not a necessary requirement for the proof that the experiments of Alley and Shih [Phys. Rev. Lett. 61, 2921 (1988)], and Ou and Mandel [Phys. Rev. Lett. 61, 50 (1988)], are modulo fair sampling assumption, valid tests of local realism. We also give the critical parameters for the experiments to be unconditional tests of local realism, and show that some other interesting phenomena (involving bosonic type particle indistinguishability) can be observed duri…
We present a quantum fingerprinting protocol relying on two-photon interference which does not require a shared phase reference between the parties preparing optical signals carrying data fingerprints. We show that the scaling of the protocol, in terms of transmittable classical information, is analogous to the recently proposed and demonstrated scheme based on coherent pulses and first-order interference, offering comparable advantage over classical fingerprinting protocols without access to shared prior randomness. We analyze the protocol taking into account non-Poissonian photon statistics …
Classical wisdom of wave-particle duality says that it is impossible to observe simultaneously the wave and particle nature of microscopic object. Mathematically the principle requests that the interference visibility V and which-path distinguishability D satisfy an orthodox limit of square(V)+square(D)<=1. This work presents a new wave-particle duality test experiment with single photon in a modified Mach-Zehnder interferometer and convincingly show the possibility of breaking the limit. The key element of the interferometer is a weakly-scattering total-internal reflection prism surface, whic…
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 …
