Coherence and contextuality in a Mach-Zehnder interferometer

Rafael Wagner1,2, Anita Camillini1,2, and Ernesto F. Galvão1,3

1International Iberian Nanotechnology Laboratory (INL), Av. Mestre José Veiga, 4715-330 Braga, Portugal
2Centro de Física, Universidade do Minho, Braga 4710-057, Portugal
3Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, Niterói, RJ, 24210-340, Brazil

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We analyse nonclassical resources in interference phenomena using generalized noncontextuality inequalities and basis-independent coherence witnesses. We use recently proposed inequalities that witness both resources within the same framework. We also propose, in view of previous contextual advantage results, a systematic way of applying these tools to characterize advantage provided by coherence and contextuality in quantum information protocols. We instantiate this methodology for the task of quantum interrogation, famously introduced by the paradigmatic bomb-testing interferometric experiment, showing contextual quantum advantage for such a task.

In this paper, we explore nonclassical resources in interference phenomena by analyzing generalized noncontextuality inequalities and basis-independent coherence witnesses. We apply recently proposed inequalities to characterize coherence and contextuality in quantum information protocols, focusing on Mach-Zehnder interferometers (MZIs). Our study reveals that basis-independent quantum coherence within MZIs can be witnessed and quantified using coherence-free inequalities, providing experimentally accessible methods for assessing coherence. Using novel techniques, we show a quantifiable advantage provided by quantum contextuality to the task of quantum interrogation. Our contributions range from novel inequalities, analytical results, and proposed experimental protocols, shedding light on the relationship between coherence and contextuality in MZIs and offering a general approach for proving quantum advantages in interferometric experiments.

► BibTeX data

► References

[1] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM review, 41(2):303–332, (1999).

[2] S. Parker and Martin B. Plenio. Efficient factorization with a single pure qubit and $\log N$ mixed qubits. Physical Review Letters, 85 (14):3049, Oct (2000).

[3] Felix Ahnefeld, Thomas Theurer, Dario Egloff, Juan Mauricio Matera, and Martin B. Plenio. Coherence as a Resource for Shor's Algorithm. Physical Review Letters, 129 (12):120501, Sep (2022).

[4] Olaf Nairz, Markus Arndt, and Anton Zeilinger. Quantum interference experiments with large molecules. American Journal of Physics, 71 (4): 319–325, Apr (2003).

[5] Eric Chitambar and Gilad Gour. Quantum resource theories. Reviews of Modern Physics, 91 (2), Apr (2019).

[6] Niels Bohr. The Quantum Postulate and the Recent Development of Atomic Theory, Nature. 121: 580–590 Apr (1928).

[7] William K. Wootters and Wojciech H. Zurek. Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle. Physical Review D, 19 (2): 473, Jan (1979).

[8] Berthold-Georg Englert. Fringe visibility and which-way information: An inequality. Physical Review Letters, 77 (11): 2154, May (1996).

[9] Shuming Cheng and Michael J.W. Hall. Complementarity relations for quantum coherence. Physical Review A, 92 (4): 042101, Aug (2015).

[10] Marcos L.W. Basso and Jonas Maziero. Complete complementarity relations: Connections with Einstein-Podolsky-Rosen realism and decoherence, and extension to mixed quantum states. EPL (Europhysics Letters), 135 (6): 60002, Nov (2021).

[11] Avshalom C. Elitzur and Lev Vaidman. Quantum mechanical interaction-free measurements. Foundations of Physics, 23(7):987–997, Jul (1993).

[12] Lucien Hardy. On the existence of empty waves in quantum theory. Physics Letters A, 167 (1): 11–16, Jul (1992).

[13] Tillmann Baumgratz, Marcus Cramer, and Martin B. Plenio. Quantifying coherence. Physical Review Letters, 113 (14): 140401, Feb (2014).

[14] Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio. Colloquium: Quantum coherence as a resource. Reviews of Modern Physics, 89: 041003, Oct (2017).

[15] Diego SS Chrysosthemos, Marcos LW Basso, and Jonas Maziero. Quantum coherence versus interferometric visibility in a biased Mach–Zehnder interferometer. Quantum Information Processing 22 (68), Jan (2023).

[16] Sandeep Mishra, Anu Venugopalan, and Tabish Qureshi. Decoherence and visibility enhancement in multipath interference. Physical Review A, 100 (4): 042122, Jul (2019).

[17] Tabish Qureshi. Coherence, interference and visibility. Quanta, 8 (1): 24–35, Jun (2019).

[18] Tanmoy Biswas, María García Díaz, and Andreas Winter. Interferometric visibility and coherence. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2203): 20170170, Jul (2017).

[19] Tania Paul and Tabish Qureshi. Measuring quantum coherence in multislit interference. Physical Review A, 95(4):042110, Feb (2017).

[20] Kang-Da Wu, Alexander Streltsov, Bartosz Regula, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo. Experimental progress on quantum coherence: detection, quantification, and manipulation. Advanced Quantum Technologies, 4(9):2100040, Jul (2021).

[21] Alexander Streltsov, Uttam Singh, Himadri Shekhar Dhar, Manabendra Nath Bera, and Gerardo Adesso. Measuring quantum coherence with entanglement. Physical Review Letters, 115 (2): 020403, Mar (2015).

[22] Alexander Streltsov, Eric Chitambar, Swapan Rana, Manabendra N. Bera, Andreas Winter, and Maciej Lewenstein. Entanglement and coherence in quantum state merging. Physical Review Letters, 116 (24): 240405, Jun (2016).

[23] Lu-Feng Qiao, Alexander Streltsov, Jun Gao, Swapan Rana, Ruo-Jing Ren, Zhi-Qiang Jiao, Cheng-Qiu Hu, Xiao-Yun Xu, Ci-Yu Wang, Hao Tang, et al. Entanglement activation from quantum coherence and superposition. Physical Review A, 98 (5): 052351, Nov (2018).

[24] Michele Masini, Thomas Theurer, and Martin B. Plenio. Coherence of operations and interferometry. Physical Review A, 103(4):042426, Apr (2021).

[25] Laura Ares and Alfredo Luis. Beam splitter as quantum coherence-maker. Physica Scripta, 98: 015101, Dec (2022).

[26] Artur K. Ekert, Carolina Moura Alves, Daniel K.L. Oi, Michał Horodecki, Paweł Horodecki, and Leong Chuan Kwek. Direct estimations of linear and nonlinear functionals of a quantum state. Physical Review Letters, 88 (21): 217901, May (2002).

[27] Paweł Horodecki and Artur Ekert. Method for direct detection of quantum entanglement. Physical Review Letters, 89 (12): 127902, Aug (2002).

[28] Michał Oszmaniec, Daniel J. Brod, and Ernesto F. Galvão. Measuring relational information between quantum states, and applications. New Journal of Physics, (in press) Jan (2024).

[29] Sébastien Designolle, Roope Uola, Kimmo Luoma, and Nicolas Brunner. Set coherence: basis-independent quantification of quantum coherence. Physical Review Letters, 126 (22): 220404, Jun (2021).

[30] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Physical Review A, 40 (8): 4277, Oct (1989).

[31] Robert W. Spekkens. Evidence for the epistemic view of quantum states: A toy theory. Physical Review A, 75 (3): 032110, Mar (2007).

[32] Lucien Hardy. Disentangling nonlocality and teleportation. arXiv preprint quant-ph/​9906123, Jun (1999).

[33] Lorenzo Catani, Matthew Leifer, David Schmid, and Robert W. Spekkens. Why interference phenomena do not capture the essence of quantum theory. Quantum, 7: 1119, (2023).

[34] Ernesto F. Galvão and Daniel J. Brod. Quantum and classical bounds for two-state overlaps. Physical Review A, 101: 062110, Jun (2020).

[35] Rafael Wagner, Rui Soares Barbosa, and Ernesto F. Galvão. Inequalities witnessing coherence, nonlocality, and contextuality. arXiv preprint arXiv:2209.02670, Sep (2022).

[36] Matteo Lostaglio and Gabriel Senno. Contextual advantage for state-dependent cloning. Quantum, 4: 258, Apr (2020).

[37] Lev Vaidman. Interaction-free measurements. arXiv preprint quant-ph/​9610033, Oct (1996).

[38] Paul Kwiat, Harald Weinfurter, Thomas Herzog, Anton Zeilinger, and Mark A. Kasevich. Interaction-free measurement. Physical Review Letters, 74: 4763, Jun (1995).

[39] Paul G Kwiat, AG White, JR Mitchell, O Nairz, G Weihs, H Weinfurter, and A Zeilinger. High-efficiency quantum interrogation measurements via the quantum Zeno effect. Physical Review Letters, 83 (23): 4725, Dec (1999).

[40] T. Rudolph. Better schemes for quantum interrogation in lossy experiments. Physical Review Letters, 85 (14): 2925, Oct (2000).

[41] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson. Kochen-Specker contextuality. Review of Modern Physics, 94: 045007, Dec (2021).

[42] Simon Kochen and Ernst Specker. The problem of hidden variables in quantum mechanics. J. Math. and Mech., 17: 59–87, (1967).

[43] John S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1: 195–200, Nov (1964).

[44] John S. Bell. On the problem of hidden variables in quantum mechanics. Reviews of Modern Physics, 38: 447–452, Jul (1966).

[45] Ehtibar N Dzhafarov and Janne V Kujala. Contextuality-by-default 2.0: Systems with binary random variables. In International Symposium on Quantum Interaction, pages 16–32. Springer, Jan (2017).

[46] Janne V. Kujala and Ehtibar N. Dzhafarov. Contextuality and dichotomizations of random variables. Foundations of Physics, 52 (1): 1–25, Dec (2022).

[47] Janne V. Kujala and Ehtibar N. Dzhafarov. Measures of contextuality and non-contextuality. Philosophical Transactions of the Royal Society A, 377 (2157): 20190149, Sep (2019).

[48] Víctor H. Cervantes and Ehtibar N. Dzhafarov. Snow queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices. \hrefhttps:/​/​​10.1037/​dec0000095 Decision, 5 (3): 193, (2018).

[49] Robert W. Spekkens. Contextuality for preparations, transformations, and unsharp measurements. Physical Review A, 71: 052108, May (2005).

[50] David Schmid, Robert W. Spekkens, and Elie Wolfe. All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences. Physical Review A, 97 (6): 062103, Jun (2018).

[51] Anubhav Chaturvedi, Máté Farkas, and Victoria J. Wright. Characterising and bounding the set of quantum behaviours in contextuality scenarios. Quantum, 5: 484, Jun (2021).

[52] Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Roope Uola, and Alastair A Abbott. Bounding and simulating contextual correlations in quantum theory. PRX Quantum, 2 (2): 020334, Jun (2021).

[53] David Schmid and Robert W. Spekkens. Contextual advantage for state discrimination. Physical Review X, 8: 011015, Feb (2018).

[54] Ravi Kunjwal, Matteo Lostaglio, and Matthew F Pusey. Anomalous weak values and contextuality: robustness, tightness, and imaginary parts. Physical Review A, 100 (4): 042116, Oct (2019).

[55] David Schmid, John H. Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens. Characterization of noncontextuality in the framework of generalized probabilistic theories. PRX Quantum, 2 (1): 010331, Feb (2021).

[56] Farid Shahandeh. Contextuality of general probabilistic theories. PRX Quantum, 2 (1): 010330, Feb (2021).

[57] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality. Physical Review A, 107: 062203 Jun (2023).

[58] John H. Selby, Elie Wolfe, David Schmid, and Ana Belén Sainz. An open-source linear program for testing nonclassicality. arXiv preprint arXiv:2204.11905, Oct (2022).

[59] Matthew S. Leifer. Is the quantum state real? An extended review of $\psi$ ontology theorems. Quanta, 3 (1): 67–155, (2014).

[60] Yeong-Cherng Liang, Robert W. Spekkens, and Howard M. Wiseman. Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Physics Reports, 506 (1-2): 1–39, Sep (2011).

[61] Matteo Lostaglio. Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response. Physical Review Letters, 125 (23): 230603, Dec (2020).

[62] Ravi Kunjwal. Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements. Quantum, 3: 184, Sep (2019).

[63] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens. A structure theorem for generalized-noncontextual ontological models. arXiv preprint arXiv:2005.07161, May (2020).

[64] Roberto D. Baldijão, Rafael Wagner, Cristhiano Duarte, Bárbara Amaral, and Marcelo Terra Cunha. Emergence of Noncontextuality under Quantum Darwinism. PRX Quantum, 2(3):030351, Sep (2021).

[65] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of reality be considered complete? Physical Review, 47 (10): 777–780, May (1935).

[66] M. Pusey, J. Barrett, and T. Rudolph. On the reality of the quantum state Nature Physics, 8(6):475–478, May (2012).

[67] Robert W. Spekkens. The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of Einstein. arXiv preprint arXiv:1909.04628, Aug (2019).

[68] Michael D. Mazurek, Matthew F. Pusey, Ravi Kunjwal, Kevin J. Resch, and Robert W. Spekkens. An experimental test of noncontextuality without unphysical idealizations. Nature communications, 7 (1): 1–7, Jun (2016).

[69] Michael D. Mazurek, Matthew F. Pusey, Kevin J. Resch, and Robert W. Spekkens. Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories. PRX Quantum, 2: 020302, Apr (2021).

[70] Ravi Kunjwal. Contextuality beyond the Kochen-Specker theorem. arXiv preprint arXiv:1612.07250, Dec (2016).

[71] M. S. Leifer and O. J. E. Maroney. Maximally epistemic interpretations of the quantum state and contextuality. Physical Review Letters, 110: 120401, Mar (2013).

[72] Manik Banik, Some Sankar Bhattacharya, Sujit K. Choudhary, Amit Mukherjee, and Arup Roy. Ontological models, preparation contextuality and nonlocality. Foundations of Physics, 44 (11): 1230–1244, Oct (2014).

[73] Piers Lillystone, Joel J. Wallman, and Joseph Emerson. Contextuality and the single-qubit stabilizer subtheory. Physical Review Letters, 122 (14): 140405, Apr (2019).

[74] Cristhiano Duarte and Bárbara Amaral. Resource theory of contextuality for arbitrary prepare-and-measure experiments. Journal of Mathematical Physics, 59(6):062202, Jun (2018).

[75] Rafael Wagner, Roberto D. Baldijão, Alisson Tezzin, and Bárbara Amaral. Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios. Journal of Physics A: Mathematical and Theoretical, 56: 505303, Nov (2023).

[76] Miguel Navascués, Stefano Pironio, and Antonio Acín. Bounding the set of quantum correlations. Physical Review Letters, 98(1):010401, Jul (2007).

[77] George Boole. An Investigation on The Laws of Thought. Cambridge University Press, Nov (2009).

[78] Mateus Araújo, Marco Túlio Quintino, Costantino Budroni, Marcelo Terra Cunha, and Adán Cabello. All noncontextuality inequalities for the $n$-cycle scenario. Physical Review A, 88: 022118, Aug (2013).

[79] Bárbara Amaral and Marcelo Terra Cunha. On graph approaches to contextuality and their role in quantum theory. Springer, Aug (2018).

[80] Adán Cabello, Simone Severini, and Andreas Winter. Graph-theoretic approach to quantum correlations. Physical Review Letters, 112 (4): 040401, Jan (2014).

[81] Taira Giordani, Chiara Esposito, Francesco Hoch, Gonzalo Carvacho, Daniel J. Brod, Ernesto F. Galvão, Nicolò Spagnolo, and Fabio Sciarrino. Witnesses of coherence and dimension from multiphoton indistinguishability tests. Physical Review Research, 3: 023031, Apr (2021).

[82] Taira Giordani, Daniel J Brod, Chiara Esposito, Niko Viggianiello, Marco Romano, Fulvio Flamini, Gonzalo Carvacho, Nicolò Spagnolo, Ernesto F Galvão, and Fabio Sciarrino. Experimental quantification of four-photon indistinguishability. New Journal of Physics, 22 (4): 043001, Apr (2020).

[83] Samuraí Gomes de Aguiar Brito, Bárbara Amaral, and Rafael Chaves. Quantifying Bell nonlocality with the trace distance. Physical Review A, 97 (2): 022111, Feb (2018).

[84] Rodney Loudon. The quantum theory of light. OUP Oxford, (2000).

[85] KP Zetie, SF Adams, and RM Tocknell. How does a Mach-Zehnder interferometer work? Physics Education, 35 (1): 46, Jan (2000).

[86] Markus Rambach, Mahdi Qaryan, Michael Kewming, Christopher Ferrie, Andrew G. White, and Jacquiline Romero. Robust and efficient high-dimensional quantum state tomography. Physical Review Letters, 126 (10): 100402, Mar (2021).

[87] Sitan Chen, Brice Huang, Jerry Li, Allen Liu, and Mark Sellke. Tight bounds for state tomography with incoherent measurements. arXiv preprint arXiv:2206.05265, May (2022).

[88] Da-Jian Zhang, C.L. Liu, Xiao-Dong Yu, and D.M. Tong. Estimating coherence measures from limited experimental data available. Physical Review Letters, 120 (17): 170501, Apr (2018).

[89] Carmine Napoli, Thomas R Bromley, Marco Cianciaruso, Marco Piani, Nathaniel Johnston, and Gerardo Adesso. Robustness of coherence: an operational and observable measure of quantum coherence. Physical Review Letters, 116 (15): 150502, Apr (2016).

[90] Yi-Tao Wang, Jian-Shun Tang, Zhi-Yuan Wei, Shang Yu, Zhi-Jin Ke, Xiao-Ye Xu, Chuan-Feng Li, and Guang-Can Guo. Directly measuring the degree of quantum coherence using interference fringes. Physical Review Letters, 118 (2): 020403, Jan (2017).

[91] Wenqiang Zheng, Zhihao Ma, Hengyan Wang, Shao-Ming Fei, and Xinhua Peng. Experimental demonstration of observability and operability of robustness of coherence. Physical Review Letters, 120 (23): 230504, Jun (2018).

[92] Caterina Taballione, Reinier van der Meer, Henk J Snijders, Peter Hooijschuur, Jörn P Epping, Michiel de Goede, Ben Kassenberg, Pim Venderbosch, Chris Toebes, Hans van den Vlekkert, Pepijn W H Pinkse and Jelmer J Renema A universal fully reconfigurable 12-mode quantum photonic processor. Materials for Quantum Technology, I 035002, Aug (2021).

[93] Peter Janotta and Raymond Lal. Generalized probabilistic theories without the no-restriction hypothesis. Physical Review A, 87 (5): 052131, May (2013).

[94] Markus P. Müller and Cozmin Ududec. Structure of reversible computation determines the self-duality of quantum theory. Physical Review Letters, 108 (13): 130401, Mar (2012).

[95] Kieran Flatt, Hanwool Lee, Carles Roch I Carceller, Jonatan Bohr Brask, and Joonwoo Bae. Contextual advantages and certification for maximum-confidence discrimination. PRX Quantum, 3: 030337, Sep (2022).

[96] Gilberto Borges, Marcos Carvalho, Pierre-Louis de Assis, José Ferraz, Mateus Araújo, Adán Cabello, Marcelo Terra Cunha, and Sebastião Pádua. Quantum contextuality in a Young-type interference experiment. Physical Review A, 89 (5): 052106, May (2014).

[97] B. H. Liu, Y. F. Huang, Y. X. Gong, F. W. Sun, Y. S. Zhang, C. F. Li, and G. C. Guo. Experimental demonstration of quantum contextuality with nonentangled photons. Physical Review A, 80 (4): 044101, Oct (2009).

[98] Carles Roch i Carceller, Kieran Flatt, Hanwool Lee, Joonwoo Bae, and Jonatan Bohr Brask. Quantum vs noncontextual semi-device-independent randomness certification. Physical Review Letters, 129 (5): 050501, Jul (2022).

[99] Sumit Mukherjee, Shivam Naonit, and A. K. Pan. Discriminating three mirror-symmetric states with a restricted contextual advantage. Physical Review A, 106: 012216, Jul (2022).

Cited by

[1] Rafael Wagner, Rui Soares Barbosa, and Ernesto F. Galvão, "Inequalities witnessing coherence, nonlocality, and contextuality", Physical Review A 109 3, 032220 (2024).

[2] Massy Khoshbin, Lorenzo Catani, and Matthew Leifer, "Alternative robust ways of witnessing nonclassicality in the simplest scenario", Physical Review A 109 3, 032212 (2024).

[3] Jingyan Lin, Kunkun Wang, Lei Xiao, Dengke Qu, Gaoyan Zhu, Yong Zhang, and Peng Xue, "Experimental investigation of contextual robustness and coherence in state discrimination", Physical Review A 109 5, 052208 (2024).

[4] Lorenzo Catani, Matthew Leifer, David Schmid, and Robert W. Spekkens, "Why interference phenomena do not capture the essence of quantum theory", Quantum 7, 1119 (2023).

[5] Vinicius P. Rossi, David Schmid, John H. Selby, and Ana Belén Sainz, "Contextuality with vanishing coherence and maximal robustness to dephasing", Physical Review A 108 3, 032213 (2023).

[6] Rafael Wagner, Zohar Schwartzman-Nowik, Ismael L. Paiva, Amit Te'eni, Antonio Ruiz-Molero, Rui Soares Barbosa, Eliahu Cohen, and Ernesto F. Galvão, "Quantum circuits for measuring weak values, Kirkwood-Dirac quasiprobability distributions, and state spectra", Quantum Science and Technology 9 1, 015030 (2024).

[7] Lorenzo Catani, Matthew Leifer, Giovanni Scala, David Schmid, and Robert W. Spekkens, "Aspects of the phenomenology of interference that are genuinely nonclassical", Physical Review A 108 2, 022207 (2023).

[8] Rafael Wagner, Roberto D. Baldijão, Alisson Tezzin, and Bárbara Amaral, "Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios", Journal of Physics A Mathematical General 56 50, 505303 (2023).

[9] Taira Giordani, Rafael Wagner, Chiara Esposito, Anita Camillini, Francesco Hoch, Gonzalo Carvacho, Ciro Pentangelo, Francesco Ceccarelli, Simone Piacentini, Andrea Crespi, Nicolò Spagnolo, Roberto Osellame, Ernesto F. Galvão, and Fabio Sciarrino, "Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor", Science Advances 9 44, eadj4249 (2023).

[10] Rafael Wagner and Ernesto F. Galvão, "Simple proof that anomalous weak values require coherence", Physical Review A 108 4, L040202 (2023).

[11] Holger F. Hofmann, "Sequential propagation of a single photon through five measurement contexts in a three-path interferometer", arXiv:2308.02086, (2023).

[12] Marcos L. W. Basso, Ismael L. Paiva, and Pedro R. Dieguez, "Unveiling quantum complementarity trade-offs in relativistic scenarios", arXiv:2306.08136, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 07:48:27) and SAO/NASA ADS (last updated successfully 2024-05-21 07:48:28). The list may be incomplete as not all publishers provide suitable and complete citation data.