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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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Research works printed but not linked: 1
- B.-G. Englert, 1996, Acta Phys. Slov., 46, 249
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Unambiguous path discrimination in a two-path interferometer
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Self-citationResidual and Destroyed Accessible Information after Measurements
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Complete complementarity relations and their Lorentz invariance
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MethodStern-Gerlach Interferometry with the Atom Chip
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Path distinguishability in double scattering of light by atoms
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“The general property of the trace distance is presented in [33], and the meaning of the distinguishability mentioned above was discussed in [32].”
BackgroundQuantum coherence versus interferometric visibility in a biased Mach–Zehnder interferometer
Chrysosthemos, D. S. S., Basso, M. L. W., & Maziero, J. (2023) · Quantum Information Processing · doi:10.1007/s11128-022-03800-6 · Cited by 11
“(7) By using the usual definition of interferometric visibility as the contrast of the interference pattern found in textbooks [28] or in the works of Greenberger and Yasin [4] and Englert [5], a closer examination of Eqs.”
MethodInfluentialViolation of a Leggett-Garg Inequality Using Ideal Negative Measurements in Neutron Interferometry
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“The measurement of the qubit’s state, denoted as qi = ±1, therefore results in a “which-way” measurement [23] in the particular region of interest.”
BackgroundAngular performance measure for tighter uncertainty relations
Hradil, Z., Řeháček, J., Klimov, A. B., Rigas, I., & Sánchez-Soto, L. L. (2010) · Physical Review A · doi:10.1103/physreva.81.014103 · Cited by 11
Experimental test of Bohr's complementarity principle with single neutral atoms
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Busch, P., & Shilladay, C. R. (2003) · Physical Review A · doi:10.1103/physreva.68.034102 · Cited by 11
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Lal, N., Mishra, S., & Singh, R. P. (2022) · AVS Quantum Science · doi:10.1116/5.0083968 · Cited by 11
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“From the mathematical perspective the duality relation cannot be ignored, and has been derived from quantum theory by JSV [52], with a particularly rigorous derivation given by Englert [53].”
BackgroundMethodInfluentialQuantum twist to complementarity: A duality relation
Qureshi, T. (2013) · Progress of Theoretical and Experimental Physics · doi:10.1093/ptep/ptt022 · Cited by 10
Influence of losses on the wave-particle duality
Jia, A., Huang, J., Zhang, T., & Zhu, S. (2014) · Physical Review A · doi:10.1103/physreva.89.042103 · Cited by 10
Experimental tests of the density matrix's property-based complementarity relations
Pozzobom, M. B., Basso, M. L. W., & Maziero, J. (2021) · Physical Review A · doi:10.1103/physreva.103.022212 · Cited by 10
Visibility is not a good measure of a well-defined relative phase
Bjork, G. G., Inoue, S., & Söderholm, J. (2000) · Physical Review A · doi:10.1103/physreva.62.023817 · Cited by 10
Quantum phase resolution and phase distribution
Bjork, G. G., Trifonov, A. S., Tsegaye, T., & Söderholm, J. (1998) · Quantum and Semiclassical Optics Journal of the European Optical Society Part B · doi:10.1088/1355-5111/10/6/004 · Cited by 10
Coherence and entropy complementarity relations of generalized wave-particle duality
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Trapped-ion realization of Einstein’s recoiling-slit experiment
Utter, R. S., & Feagin, J. M. (2007) · Physical Review A · doi:10.1103/physreva.75.062105 · Cited by 9
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“As a next step, we have subjected the obtained data to the Englert-Greenberger-Yasin inequality test [5, 9] expressed in the form of”
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“Increases in particle properties concomitantly increase and “which way information” in double slit experiment according to Bohr's complementarity principle [18].”
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“In the next section we describe these steps in terms of classical and quantum information theory that results in an information-theoretic equality that mirrors (and is completely analogous to) the trade-off between distinguishability and visibility of Greenberger and Yasin [7] as well as Englert [8].”
BackgroundMethodA possible solution to the which-way problem of quantum interference
Hofmann, H. F., Matsushita, T., Kuroki, S., & Iinuma, M. (2023) · Quantum Studies Mathematics and Foundations · doi:10.1007/s40509-023-00304-5 · Cited by 9
“The main conclusion drawn from this experimental approach to the which-way problem was that which-way information is incompatible with the observation of an interference pattern, resulting in a quantitative trade-off of the visibilities [6].”
BackgroundDistance Between Qubit States with Initial System-Environment Correlation
Ban, M., Kitajima, S., & Shibata, F. (2012) · International Journal of Theoretical Physics · doi:10.1007/s10773-012-1121-y · Cited by 9
Complete complementarity relations in curved spacetimes
Basso, M. L. W., & Maziero, J. (2021) · Physical Review A · doi:10.1103/physreva.103.032210 · Cited by 9
“Attempts have been made to formalize the waveparticle duality in a quantitative way [32–34].”
Einselection and decoherence from an information theory perspective
Zurek, W. H. (2000) · Annalen der Physik · doi:10.1002/1521-3889(200011)9:11/12 · Cited by 282
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Randomization of quantum relative phase in welcher Weg measurements
Luis, A., & Sánchez-Soto, L. L. (1999) · Journal of Optics B Quantum and Semiclassical Optics · doi:10.1088/1464-4266/1/6/309 · Cited by 9
Fundamental mechanisms of noise supported energy transfer in biological systemsPreprint
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“This relation was contained implicitly in the work of Wootters and Zurek ( 1979 ) and was obtained independently by Jaeger et al ( 1995 ) and Englert ( 1996 ) .”
MethodOne-particle and two-particle visibilities in bipartite entangled Gaussian states
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“Based on previous studies [37, 38], complementarity holds if the visibility VA and distinguishability DB satisfy the following inequality: V(2) A +D(2) B ≤ 1.”
BackgroundImproving information/disturbance and estimation/distortion trade-offs with non-universal protocols
Olivares, S., & Paris, M. G. A. (2007) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/40/28/s05 · Cited by 8
Complementarity and uncertainty relations for matter-wave interferometry
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Amplification of Schrödinger-cat state: distinguishability and interference in phase space
Filip, R., & Peřina, J. (2001) · Journal of Optics B Quantum and Semiclassical Optics · doi:10.1088/1464-4266/3/2/301 · Cited by 8
Interference and partial which-way information: A quantitative test of duality in two-atom resonance
Abranyos, Y., Jakob, M., & Bergou, J. A. (1999) · Physical Review A · doi:10.1103/physreva.61.013804 · Cited by 8
Cavity QED based tuneable, delayed-choice quantum eraser
Imran, M., Abbas, T., Rameez‐ul‐Islam, & Ikram, M. (2015) · Annals of Physics · doi:10.1016/j.aop.2015.10.025 · Cited by 8
Multipath wave-particle duality with a path detector in a quantum superposition
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“[6] B.”
BackgroundMethodInfluentialQuantum Optical Tests of the Foundations of Physics
Steinberg, A. M., Kwiat, P. G., & Chiao, R. Y. (2006) · Springer handbooks · doi:10.1007/978-0-387-26308-3_80 · Cited by 9
Experimental test of the first- and second-order duality relations for the two-photon states
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The non-dissipative damping of the Rabi oscillations as a “which-path” information
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Ze, Decoherence, and the Quantum Eraser
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Miranda, M., & Orszag, M. (2020) · Physical Review A · doi:10.1103/physreva.102.033723 · Cited by 7
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“The WPD has been expressed in quantitative terms [7, 8], extended to many-body interference [9], and even tested by interferometric scenarios in which neither particle nor wave models can describe the measurement outcomes [10].”
BackgroundAn updated quantum complementarity principle
Starke, D. S., Basso, M. L. W., & Maziero, J. (2024) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2024.0517 · Cited by 7
“Subsequently, Englert [8] extended the concept of a complementarity relation akin to Eq.”
Recovering full coherence in a qubit by measuring half of its environment
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“(x,y,z), the coherence is √ x2 + y2, also known as visibility in view of the analogy of a qubit with the state of a photon in a Mach-Zehnder interferometer [22].”
BackgroundComplementarity via error-free measurement in a two-path interferometer
Liu, Y., Lu, J., & Zhou, L. (2017) · Laser Physics Letters · doi:10.1088/1612-202x/aa6593 · Cited by 7
Fringe visibility and distinguishability in two-path interferometer with an asymmetric beam splitter
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“With the interaction between the quanton and the WPD system, the combined state is expressed as [3]: |ψ〉qd = 1 √ 2 (|0〉q |d0〉d + |1〉q |d1〉d), (3)”
BackgroundMethodInfluentialNeutron Interferometer Experiments Studying Fundamental Features of Quantum Mechanics
Danner, A., Lemmel, H., Wagner, R., Sponar, S., & Hasegawa, Y. (2023) · Atoms · doi:10.3390/atoms11060098 · Cited by 7
“Since the visibility is complementary to the obtainable which-path information [52], only little information can be gained per event.”
BackgroundRevisiting wave–particle duality in Bohr–Einstein debate
Maleki, Y., & Zubairy, M. S. (2023) · AVS Quantum Science · doi:10.1116/5.0148225 · Cited by 7
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Visibility bound caused by a distinguishable noise particle
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“The relation putting a bound on the two is given by the so-called duality relation [3]”
BackgroundTesting complementarity on a transmon quantum processor
Cruz, P. M. Q., & Fernández-Rossier, J. (2021) · Physical Review A · doi:10.1103/physreva.104.032223 · Cited by 7
“Originally, Englert argued that the duality relation in Eq.”
BackgroundInfluentialQuantitative wave-particle duality
Qureshi, T. (2016) · American Journal of Physics · doi:10.1119/1.4948606 · Cited by 7
Pizzi, R., Wang, R., & Rossetti, D. A. D. A. (2016) · PLoS ONE · doi:10.1371/journal.pone.0147464 · Cited by 8
“The importance of the theoretical concepts of the double-slit one-photonat-a-time experiment to the understanding of wave-particle duality and the measurement problem is well known, and extensive literature is available on the subject [2–20].”
BackgroundQuantum delayed-choice experiment with a single neutral atom
Li, G., Zhang, P., & Zhang, T. (2017) · Optics Letters · doi:10.1364/ol.42.003800 · Cited by 7
The Role of Momentum Transfer in Welcher–Weg Experiments
ludwin, D., & Ben-Aryeh, Y. (2001) · Foundations of Physics Letters · doi:10.1023/a:1015505722165 · Cited by 7
Heisenberg optical near-field microscope
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Complementarity and Entanglement in Quantum Information TheoryPreprint
Tessier, T. E. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0412060 · Cited by 7
“relation, valid for an arbitrary pure state of two qubits, which accounts for (and in some cases, generalizes) many of the main results put forward in [104, 105, 106, 107].”
BackgroundResultSerino, L., Chesi, G., Brecht, B., Maccone, L., Macchiavello, C., & Silberhorn, C. (2025) · Physical Review Research · doi:10.1103/v24q-sl6n · Cited by 6
“Otherwise, the value of that property is undefined, and only probabilistic predictions of measurement outcomes are possible.”
Quantum Behavior as a Consequence of Ze Systems
Tkemaladze, J. (2026) · Longevity Horizon · doi:10.65649/93qfwv21 · Cited by 12
“…consulted by an observer, can destroy interference: the presence of the correlated marker in the environment itself constitutes the diagnostic information that any Ze-like system (including the broader experimental apparatus) could, in principle, use to resolve the ambiguity (Englert, 1996).”
Pure dephasing, quantum decoherence, and wave–particle duality
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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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