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Fringe Visibility and Which-Way Information: An Inequality
Abstract
An inequality is derived according to which the fringe visibility in a two-way interferometer sets an absolute upper bound on the amount of which-way information that is potentially stored in a which-way detector. In some sense, this inequality can be regarded as quantifying the notion of wave-particle duality. The derivation of the inequality does not make use of Heisenberg's uncertainty relation in any form.
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“Remarkably, monogamy is manifested in the constraint of the energy distribution for the considered spin pair through a complementarity relation[46, 47]”
From Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems (2011) · cites this paper as [46]
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“For weak value experiments aiming to distinguish which-path information, we can link this to the VisibilityDistinguishability Inequality [44, 45],”
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“Though a number of complementarity relations have been cast in quantitative forms [6,7], perhaps complementarity is most frequently observed in terms of quantum uncertainty relations.”
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“…is also known as quantum prob-21 ability), a new set of probability rules were defined, which crucially did not include the axiom of22 1See also Englert (1996) and Greenberger and Yasin (1988) for the expression of the gradual relation between interference visibility and position…”
BackgroundInterference in the resonance fluorescence of two incoherently coupled transitions
Kiffner, M., Evers, J., & Keitel, C. H. (2006) · Physical Review A · doi:10.1103/physreva.73.063814 · Cited by 22
Quantum Nondemolition Circuit for Testing Bipartite Complementarity
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Quantifying the presence of a neutron in the paths of an interferometer
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Minimum-disturbance measurement without postselection
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Experimental investigation of dynamical invariants in bipartite entanglement
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Kaszlikowski, D., Kwek, L. C., Żukowski, M., & Englert, B. (2003) · Physical Review Letters · doi:10.1103/physrevlett.91.037901 · Cited by 21
Self-citationClassical and nonclassical interference
Wódkiewicz, K., & Herling, G. H. (1998) · Physical Review A · doi:10.1103/physreva.57.815 · Cited by 21
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Experimental demonstration of a quantum shutter closing two slits simultaneously
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Wave-particle duality in asymmetric beam interference
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“For symmetric beams and equal slit widths, wave-particle duality can be captured by thewell-known inequality (1), whichwas derived using the ideas of minimum error discrimination of states [5].”
BackgroundMethodInfluentialQuantum complementarity of clocks in the context of general relativity
Zhou, Z., Margalit, Y., Rohrlich, D., Japha, Y., & Folman, R. (2018) · Classical and Quantum Gravity · doi:10.1088/1361-6382/aad56b · Cited by 21
“Quantum complementarity [9] plays a special role at this QM–GR interface, as we show below.”
BackgroundMethodQuantitative wave-particle duality as quantum state discrimination
Lü, X. (2020) · Physical Review A · doi:10.1103/physreva.102.022201 · Cited by 20
Wave-particle duality and polarization properties of light in single-photon interference experiments
Lahiri, M. (2011) · Physical Review A · doi:10.1103/physreva.83.045803 · Cited by 20
Electronic Mach-Zehnder quantum eraser
Kang, K. (2007) · Physical Review B · doi:10.1103/physrevb.75.125326 · Cited by 20
Entanglement monotones from complementarity relations
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Operational geometric phase for mixed quantum states
Andersson, O., & Heydari, H. (2013) · New Journal of Physics · doi:10.1088/1367-2630/15/5/053006 · Cited by 18
“(On the other hand, if ρ is open and the initial and final states ρ(0) and ρ(τ) are distinguishable [30, 31], then Ψ0Π[ρ]Ψ0 = 0n why ζJ(ρ) = 0 for every sequence J , c.”
Generalized coherence concurrence and path distinguishability
Chin, S. (2017) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aa908d · Cited by 19
Independent re-analysis of alleged mind-matter interaction in double-slit experimental data
Tremblay, N. (2019) · PLoS ONE · doi:10.1371/journal.pone.0211511 · Cited by 19
“It has also been verified that the 28 strength of the observed interference pattern is inversely proportional to the amount of 29 which-way information one gathers [11, 12].”
Ou, Z. Y. (2008) · Physical Review A · doi:10.1103/physreva.77.043829 · Cited by 19
Afshar's Experiment does not show a Violation of Complementarity
Steuernagel, O. (2005) · CERN Document Server (European Organization for Nuclear Research) · Cited by 19
Continuous-variable quantum erasing
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Experimental test of the relation between coherence and path information
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Observation of Geometric Phases in Quantum Erasers
Kobayashi, H., Tamate, S., Nakanishi, T., Sugiyama, K., & Kitano, M. (2011) · Journal of the Physical Society of Japan · doi:10.1143/jpsj.80.034401 · Cited by 17
Evidence of the entanglement constraint on wave-particle duality using the IBM Q quantum computer
Schwaller, N., Dupertuis, M. ‐., & Javerzac‐Galy, C. (2021) · Physical Review A · doi:10.1103/physreva.103.022409 · Cited by 18
“Englert [8] also obtained a similar result”
Shatokhin, V., Müller, C. A., & Buchleitner, A. (2006) · Physical Review A · doi:10.1103/physreva.73.063813 · Cited by 18
2003 An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemblePreprint
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No-signaling-in-time as a condition for macrorealism: the case of neutrino oscillations
Massimo Blasone, A., Illuminati, F., Petruzziello, L., Simonov, K., & Smaldone, L. A. (2023) · The European Physical Journal C · doi:10.1140/epjc/s10052-023-11870-8 · Cited by 17
Experimental demonstration of the equivalence of entropic uncertainty with wave-particle duality
Spegel-Lexne, D., Gómez, S., Argillander, J., Pawłowski, M., Dieguez, P. R., Alarcón, A., & XAVIER, G. B. (2024) · Science Advances · doi:10.1126/sciadv.adr2007 · Cited by 17
“…hand, Bohr’s complementarity principle manifested its key role in modern physics by submitting matter and radiation to a unifying framework, where it is expected to exclusively behave either in a wave-or in a particle-like manner, depending on the peculiarities of an interferometric setup [1].”
Complete complementarity relations in tree level QED processes
Blasone, M., Siena, S. D., Lambiase, G., Matrella, C., & Micciola, B. (2025) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.111.016007 · Cited by 26
“The first quantitative version of Bohr’s principle, originally formulated in the context of the two-slit experiment, is due to Wootters and Zurek [74], who studied the effect of introducing a path-detecting device in a two-slit interference experiment, as originally proposed by Einstein.”
Fair sampling perspective on an apparent violation of duality
Bolduc, E., Leach, J., Miatto, F. M., Leuchs, G., & Boyd, R. W. (2014) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1400106111 · Cited by 17
“The presence of decoherence can be modeled very effectively by considering an auxiliary system (5), often called the environment (6), in addition to the two-alternative system.”
BackgroundWave-particle duality employing quantum coherence in superposition with non-orthogonal pointers
Das, S., Mukhopadhyay, C., Roy, S. S., Bhattacharya, S., Sen(De), A., & Sen, U. (2020) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ab741f · Cited by 18
COMPLEMENTARITY RELATIONS FOR MULTI-QUBIT SYSTEMS
Tessier, T. E. (2004) · Cited by 17
Basso, M. L. W., & Maziero, J. (2021) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/ac1bc8 · Cited by 17
“Englert [5] took a different approach by considering a path quantifier which was based on a posteriori path information acquired using a path detector, and derived a stronger inequality between the distinguishability and the visibility.”
MethodComplementarity in multiple beam interference
Luis, A. (2001) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/34/41/314 · Cited by 17
Einstein's Recoiling Slit Experiment, Complementarity and Uncertainty
Qureshi, T., & Vathsan, R. (2013) · Quanta · doi:10.12743/quanta.v2i1.11 · Cited by 17
“The argument here is on the lines of the treatment of a particle going through the double-slit interacting with a 1-bit detector by Scully, Englert and Walther [8].”
BackgroundMethodResultInfluentialObserving the average trajectories of single photons in a two-slit interferometer
Kocsis, S., Braverman, B., Ravets, S., Stevens, M. J., Mirin, R. P., Shalm, L. K., & Steinberg, A. M. (2011) · doi:10.1109/iqec-cleo.2011.6193919 · Cited by 17
Quantification of quantum correlation of ensembles of states
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Visibility for anharmonic fringes
Luis, A. (2002) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/35/41/312 · Cited by 17
Stereographic geometry of coherence and which-path information
Maleki, Y. (2019) · Optics Letters · doi:10.1364/ol.44.005513 · Cited by 16
From wave-particle duality to wave-particle-mixedness triality: an uncertainty approach
Fu, S., & Luo, S. (2022) · Communications in Theoretical Physics · doi:10.1088/1572-9494/ac53a2 · Cited by 16
“…by the fi rst beam splitter, where r r r , , x y r y 2 + + and σ x , σ y , σ z are the Pauli matrices, Englert derived the following inequality [ 6 ] where Π = { | 0 〉〈 0 | , | 1 〉〈 1 | } is the von Neumann measurement induced by the computational basis { | 0 〉 , | 1 〉 } ( two paths ) , while P…”
Self-gravitating interferometry and intrinsic decoherence
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“Conceptually, we are testing the idea that when one forms superpositions of geometries in the interferometer, the nature of time in GR leads at the quantum level to an imprint of which-way information, which is accompanied by a loss of fringe visibility [30].”
BackgroundUncertainty Relations for Coarse-Grained Measurements: An Overview
Toscano, F., Tasca, D., Rudnicki, Ł., & Walborn, S. (2018) · MPG.PuRe (Max Planck Society) · Cited by 16
Complementarity for Generalized Observables
Luis, A. (2002) · Physical Review Letters · doi:10.1103/physrevlett.88.230401 · Cited by 16
On mechanisms that enforce complementarity
Englert, B., Scully, M. O., & WALTHER, H. (2000) · Journal of Modern Optics · doi:10.1080/09500340008235143 · Cited by 16
Self-citationInterferometric information gain versus interaction-free measurement
Krenn, G., Summhammer, J., & Svozil, K. (2000) · Physical Review A · doi:10.1103/physreva.61.052102 · Cited by 16
Fidelity and Coherence Measures from Interference
Oi, D. K. L., & Åberg, J. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.220404 · Cited by 15
Entropic measures of joint uncertainty: Effects of lack of majorization
Luis, A., Bosyk, G. M., & Portesi, M. (2015) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2015.10.097 · Cited by 15
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Incompatible measurements on quantum causal networks
Sedlák, M., Reitzner, D., Chiribella, G., & Ziman, M. (2016) · Physical Review A · doi:10.1103/physreva.93.052323 · Cited by 18
“The existence of incompatible measurements is at the root of the wave-particle duality [3, 4] and, more generally, of many striking distinctions between quantum and classical physics.”
BackgroundDuality relation between coherence and path information in the presence of quantum memory
Bu, K., Li, L., Wu, J., & Fei, S.-M. (2017) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aa9b4f · Cited by 15
Symmetry Allows for Distinguishability in Totally Destructive Many-Particle Interference
Münzberg, J., Dittel, C., Lebugle, M., Buchleitner, A., Szameit, A., Weihs, G., & Keil, R. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.020326 · Cited by 15
Entanglement monotones connect distinguishability and predictability
Basso, M. L. W., & Maziero, J. (2021) · Physics Letters A · doi:10.1016/j.physleta.2021.127875 · Cited by 15
“Summing up, in this letter we pushed foward the conjecture of Englert and Bergou.”
BackgroundInfluentialQuantum nondemolition measurement saturates fidelity trade-off
Mišta, L., & Filip, R. (2005) · Physical Review A · doi:10.1103/physreva.72.034307 · Cited by 14
Uncertainty rescued: Bohr's complementarity for composite systems
Kim, I., & Mahler, G. (2000) · Physics Letters A · doi:10.1016/s0375-9601(00)00280-2 · Cited by 15
“lected to simulate a two-level system (pseudo-spin) [19].”
MethodBohr's complementarity relation and the violation of CP symmetry in the neutral kaon system
Hiesmayr, B. C., & Huber, M. (2008) · Physics Letters A · doi:10.1016/j.physleta.2008.02.055 · Cited by 15
Quantum interference with molecules: The role of internal states
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Coherence in multipath interference via quantum Fisher information
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Quantum Correlation Enhanced Optical Imaging
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From quantum feedback to probabilistic error correction: manipulation of quantum beats in cavity QED
Barberis-Blostein, P., Norris, D. G., Orozco, L. A., & Carmichael, H. J. (2010) · New Journal of Physics · doi:10.1088/1367-2630/12/2/023002 · Cited by 14
“2 − |hb1|ψi|2) and the visibility V, we have numerically checked for the parameters used in this paper that P 2+V ≈ 1. (14) This is an analog of the inequality characterizing a two-way interferometer [18], and it allows us to determine the relative probability to occupy the qubit states from the fringe visibility. The…”
MethodLocal measurement uncertainties impose a limit on nonlocal quantum correlations
Hofmann, H. F. (2019) · Physical Review A · doi:10.1103/physreva.100.012123 · Cited by 14
“For the class of measurements discussed here, the measurement uncertainties described by the visibilities are known to satisfy an uncertainty relation given by [5, 23]”
BackgroundCoherence and complementarity based on modified generalized skew information
Wu, Z., Zhang, L., Fei, S.-M., & Li-Jost, X. (2020) · Quantum Information Processing · doi:10.1007/s11128-020-02651-3 · Cited by 14
Experimental investigation of wave-particle duality relations in asymmetric beam interference
Chen, D., Zhang, Y., Zhao, J., Wu, Q., Fang, Y.-L., Yang, C., & Nori, F. (2022) · npj Quantum Information · doi:10.1038/s41534-022-00610-7 · Cited by 14
“(17) quantifies how much information is obtained by Bob through the measurement (See Methods).”
BackgroundThree-beam interference and which-way information in neutron interferometry
Zawisky, M., Baron, M., & Loidl, R. (2002) · Physical Review A · doi:10.1103/physreva.66.063608 · Cited by 14
Complementarity in variable strength quantum non-demolition measurements
Barbieri, M., Goggin, M. E., de Almeida, M. P., Lanyon, B. P., & White, A. G. (2009) · New Journal of Physics · doi:10.1088/1367-2630/11/9/093012 · Cited by 14
Ultrafast transient interference in pump-probe spectroscopy of band and Mott insulators
Shinjo, K., & Tohyama, T. (2018) · Physical review. B./Physical review. B · doi:10.1103/physrevb.98.165103 · Cited by 14
On Visibility in the Afshar Two-Slit Experiment
Kastner, R. E. (2009) · Foundations of Physics · doi:10.1007/s10701-009-9329-2 · Cited by 14
“A much more general approach to this problem is presented by Englert (1996) who considers an interferometer augmented with a which-way detection device upstream from the region of recombination of the two beams.”
BackgroundInformation-disturbance tradeoff for spin coherent state estimation
Sacchi, M. F. (2007) · Physical Review A · doi:10.1103/physreva.75.012306 · Cited by 15
Bertlmann, R. A., & Hiesmayr, B. C. (2006) · Quantum Information Processing · doi:10.1007/s11128-006-0026-1 · Cited by 14
Universal conservation laws of the wave-particle-entanglement triad: theory and experiment
Ding, Z., Deng, Y., Fei, S.-M., Zhou, S.-Q., Chen, X., Rui, Z., Ma, Z., Xiao, Y., & Wang, J. (2025) · Light Science & Applications · doi:10.1038/s41377-025-01759-4 · Cited by 13
Uncertainty relations: An operational approach to the error-disturbance tradeoff
Renes, J. M., Scholz, V. B., & Huber, S. (2017) · Quantum · doi:10.22331/q-2017-07-25-20 · Cited by 15
“Interestingly, we also find that Englert’s wave-particle duality relation [42] can be viewed as an error-disturbance relation.”
BackgroundHuang, X.-J., Han, P., Ning, W., Yang, S., Xin, Z., Lu, J., Zheng, R., Li, H., Yang, Z., Xu, K., et al. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00714-8 · Cited by 13
“The amount of the which-path information stored in the WPD, is quantified by the distinguishability, defined as [2]”
MethodPath–phase duality with intraparticle translational–internal entanglement
Kolář, M., Opatrný, T., Bar‐Gill, N., Noam Erez, A., & Kurizki, G. (2007) · New Journal of Physics · doi:10.1088/1367-2630/9/5/129 · Cited by 13
Experimental Test of the State Estimation-Reversal Tradeoff Relation in General Quantum Measurements
Chen, G., Zou, Y., Xu, X., Tang, J., Li, Y., Xu, J., Han, Y., Li, C., Guo, G., Ni, H., et al. (2014) · Physical Review X · doi:10.1103/physrevx.4.021043 · Cited by 13
GENERALIZED COMPLEMENTARITY RELATIONS IN COMPOSITE QUANTUM SYSTEMS OF ARBITRARY DIMENSIONS
Jakob, M., & Bergou, J. A. (2006) · International Journal of Modern Physics B · doi:10.1142/s0217979206033851 · Cited by 13
Entanglement dynamics of many-body systems: Analytical results
de Paula, A. L., de Oliveira, J. G. G., de Faria, J. G. P., Freitas, D. S., & Nemes, M. C. (2014) · Physical Review A · doi:10.1103/physreva.89.022303 · Cited by 13
Simulation of wave-particle duality in multipath interferometers on a quantum computer
Amico, M., & Dittel, C. (2020) · Physical Review A · doi:10.1103/physreva.102.032605 · Cited by 13
Biasing a coin after the toss: asymmetric delayed choice quantum eraser via Bragg regime cavity QED
Rameez‐ul‐Islam, Abbas, T., & Ikram, M. (2014) · Laser Physics Letters · doi:10.1088/1612-2011/12/1/015203 · Cited by 13
Schmid, D. (2024) · Quantum · doi:10.22331/q-2024-01-03-1217 · Cited by 19
“But this contradicts the predictions of quantum theory [40].”
ResultYoung's double-slit interference demonstration with single photons
Luo, B., Francis, L., Rodríguez-Fajardo, V., Galvez, E. J., & Khoshnoud, F. (2024) · American Journal of Physics · doi:10.1119/5.0179131 · Cited by 12
“31 This marker can be, for example, the polarization of the light.”
BackgroundNecessary and sufficient condition for joint measurability
Jae, J., Baek, K., Ryu, J., & Lee, J. (2019) · Physical Review A · doi:10.1103/physreva.100.032113 · Cited by 12
“[4] Berthold-Georg Englert, “Fringe visibility and which-way information: An inequality,” Phys.”
BackgroundMethodWave-particle-duality relations based on entropic bounds for which-way information
Bagán, E., Bergou, J. A., & Hillery, M. S. (2020) · Physical Review A · doi:10.1103/physreva.102.022224 · Cited by 12
Bosyk, G. M., Portesi, M. A., Holik, F., & Plastino, Á. L. (2013) · El Servicio de Difusión de la Creación Intelectual (National University of La Plata) · Cited by 12
Recent Progress on the Sum over Paths Approach in Quantum Mechanics Education
Malgieri, M., & Onorato, P. (2022) · Physics · doi:10.3390/physics4030067 · Cited by 13
Hidaka, Y., Iso, S., & Shimada, K. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.106.076018 · Cited by 12
Complementarity Paradox Solved: Surprising Consequences
Flores, E. V., & Tata, J. M. D. (2010) · Foundations of Physics · doi:10.1007/s10701-010-9477-4 · Cited by 12
“However, it is possible to obtain partial which-way information and visibility as long as the results fall within the complementarity inequality [1,2],”
BackgroundInfluentialOperational approach to complementarity and duality relations
Luis, A. (2004) · Physical Review A · doi:10.1103/physreva.70.062107 · Cited by 13
Metadata
- Authors
- Englert, B.-G.
- Affiliations
- Max Planck Institute of Quantum Optics (DE)
- Corresponding author
- Englert, Berthold-Georg
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1996-09-09
- Volume
- 77
- Issue
- 11
- Pages
- 2154-2157
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2156583743
- Cited by
- 821
- References
- 12
- Citations by year
- 2026: 38 · 2025: 41 · 2024: 34 · 2023: 33 · 2022: 40 · 2021: 51 · 2020: 35 · 2019: 43 · 2018: 38 · 2017: 40 · 2016: 36 · 2015: 42 · 2014: 36 · 2013: 18 · 2012: 29
- FWCI
- 5.51
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Keywords
- fringe visibility, which-way information, wave-particle duality, information-theoretic inequalities, quantum interference, complementarity
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