Monitored quantum transport: full counting statistics of a quantum Hall interferometer

C.W.J. Beenakker; Jin-Fu Chen

doi:10.22331/q-2025-10-02-1874

Monitored quantum transport: full counting statistics of a quantum Hall interferometer

C.W.J. Beenakker¹ and Jin-Fu Chen^1,2

¹Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
²$\langle aQa^L\rangle$ Applied Quantum Algorithms Leiden, The Netherlands

Published:	2025-10-02, volume 9, page 1874
Editor:	Michele Campisi
Eprint:	arXiv:2504.07773v2
Doi:	https://doi.org/10.22331/q-2025-10-02-1874
Citation:	Quantum 9, 1874 (2025).

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Abstract

We generalize the Levitov-Lesovik formula for the probability distribution function of the electron charge transferred through a phase coherent conductor, to include projective measurements that monitor the chiral propagation in quantum Hall edge modes. When applied to an electronic Mach-Zehnder interferometer, the monitoring reduces the visibility of the Aharonov-Bohm conductance oscillations while preserving the binomial form of the counting statistics, thereby removing a fundamental shortcoming of the dephasing-probe model of decoherence.

► BibTeX data

@article{Beenakker2025monitoredquantum,
  doi = {10.22331/q-2025-10-02-1874},
  url = {https://doi.org/10.22331/q-2025-10-02-1874},
  title = {Monitored quantum transport: full counting statistics of a quantum {H}all interferometer},
  author = {Beenakker, C.W.J. and Chen, Jin-Fu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1874},
  month = oct,
  year = {2025}
}

► References

[1] E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, Dephasing in electron interference by a 'which-path' detector, Nature 391, 871 (1998).
https://doi.org/10.1038/36057

[2] D. Sprinzak, E. Buks, M. Heiblum, and H. Shtrikman, Controlled dephasing of electrons via a phase sensitive detector, Phys. Rev. Lett. 84, 5820 (2000).
https://doi.org/10.1103/PhysRevLett.84.5820

[3] Yang Ji, Yunchul Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H. Shtrikman, An electronic Mach-Zehnder interferometer, Nature 422, 415 (2003).
https://doi.org/10.1038/nature01503

[4] L. V. Litvin, H.-P. Tranitz, W. Wegscheider, and C. Strunk, Decoherence and single electron charging in an electronic Mach-Zehnder interferometer, Phys. Rev. B 75, 033315 (2007).
https://doi.org/10.1103/PhysRevB.75.033315

[5] P. Roulleau, F. Portier, D. C. Glattli, P. Roche, A. Cavanna, G. Faini, U. Gennser, and D. Mailly, Finite bias visibility of the electronic Mach-Zehnder interferometer, Phys. Rev. B 76 161309 (2007).
https://doi.org/10.1103/PhysRevB.76.161309

[6] P. Roulleau, F. Portier, P. Roche, A. Cavanna, G. Faini, U. Gennser, and D. Mailly, Noise dephasing in edge states of the integer quantum Hall regime, Phys. Rev. Lett. 101, 186803 (2008).
https://doi.org/10.1103/PhysRevLett.101.186803

[7] E. Weisz, H. K. Choi, M. Heiblum, Y. Gefen, V. Umansky, and D. Mahalu, Controlled dephasing of an electron interferometer with a path detector at equilibrium, Phys. Rev. Lett. 109, 250401 (2012).
https://doi.org/10.1103/PhysRevLett.109.250401

[8] A. Helzel, L. V. Litvin, I. P. Levkivskyi, E. V. Sukhorukov, W. Wegscheider, and C. Strunk, Counting statistics and dephasing transition in an electronic Mach-Zehnder interferometer, Phys. Rev. B 91, 245419 (2015).
https://doi.org/10.1103/PhysRevB.91.245419

[9] I. Gurman, R. Sabo, M. Heiblum, V. Umansky, and D. Mahalu, Dephasing of an electronic two-path interferometer, Phys. Rev. B 93, 121412(R) (2016).
https://doi.org/10.1103/PhysRevB.93.121412

[10] M. Jo, P. Brasseur, A. Assouline, G. Fleury, H.-S. Sim, K. Watanabe, T. Taniguchi, W. Dumnernpanich, P. Roche, D. C. Glattli, N. Kumada, F. D. Parmentier, and P. Roulleau, Quantum Hall valley splitters and a tunable Mach-Zehnder interferometer in graphene, Phys. Rev. Lett. 126, 146803 (2021).
https://doi.org/10.1103/PhysRevLett.126.146803

[11] M. Jo, June-Young M. Lee, A. Assouline, P. Brasseur, K. Watanabe, T. Taniguchi, P. Roche, D. Glattli, N. Kumada, F. Parmentier, H.-S. Sim, and P. Roulleau, Scaling behavior of electron decoherence in a graphene Mach-Zehnder interferometer, Nature Comm. 13, 5473 (2022).
https://doi.org/10.1038/s41467-022-33078-2

[12] G. Seelig and M. Büttiker, Charge-fluctuation-induced dephasing in a gated mesoscopic interferometer, Phys. Rev. B 64, 245313 (2001).
https://doi.org/10.1103/PhysRevB.64.245313

[13] A. A. Clerk and A. D. Stone, Noise and measurement efficiency of a partially coherent mesoscopic detector, Phys. Rev. B 69, 245303 (2004).
https://doi.org/10.1103/PhysRevB.69.245303

[14] F. Marquardt and C. Bruder, Influence of dephasing on shot noise in an electronic Mach-Zehnder interferometer, Phys. Rev. Lett. 92, 056805 (2004); Phys. Rev. B 70, 125305 (2004).
https://doi.org/10.1103/PhysRevB.70.125305

[15] E. V. Sukhorukov and V. V. Cheianov, Resonant dephasing in the electronic Mach-Zehnder interferometer, Phys. Rev. Lett. 99, 156801 (2007).
https://doi.org/10.1103/PhysRevLett.99.156801

[16] I. P. Levkivskyi and E. V. Sukhorukov, Dephasing in the electronic Mach-Zehnder interferometer at filling factor $\nu=2$, Phys. Rev. B 78, 045322 (2008).
https://doi.org/10.1103/PhysRevB.78.045322

[17] C. Neuenhahn and F. Marquardt, Dephasing by electron-electron interactions in a ballistic Mach-Zehnder interferometer, New J. Phys. 10, 115018 (2008).
https://doi.org/10.1088/1367-2630/10/11/115018

[18] Seok-Chan Youn, Hyun-Woo Lee, and H.-S. Sim, Nonequilibrium dephasing in an electronic Mach-Zehnder interferometer, Phys. Rev. Lett. 100, 196807 (2008).
https://doi.org/10.1103/PhysRevLett.100.196807

[19] M. Schneider, D. A. Bagrets, and A. D. Mirlin, Theory of the nonequilibrium electronic Mach-Zehnder interferometer, Phys. Rev. B 84, 075401 (2011).
https://doi.org/10.1103/PhysRevB.84.075401

[20] J. Dressel, Y. Choi, and A. N. Jordan, Measuring which-path information with coupled electronic Mach-Zehnder interferometers, Phys. Rev. B 85, 045320 (2012).
https://doi.org/10.1103/PhysRevB.85.045320

[21] E. G. Idrisov, I. P. Levkivskyi, and E. V. Sukhorukov, Dephasing in a Mach-Zehnder interferometer by an Ohmic contact, Phys. Rev. Lett. 121, 026802 (2018).
https://doi.org/10.1103/PhysRevLett.121.026802

[22] L. Bellentani, A. Beggi, P. Bordone, A. Bertoni, Dynamics and Hall-edge-state mixing of localized electrons in a two-channel Mach-Zehnder interferometer, Phys. Rev. B 97, 205419 (2018).
https://doi.org/10.1103/PhysRevB.97.205419

[23] V. S.-W. Chung, P. Samuelsson, and M. Büttiker, Visibility of current and shot noise in electrical Mach-Zehnder and Hanbury Brown Twiss interferometers, Phys. Rev. B 72, 125320 (2005).
https://doi.org/10.1103/PhysRevB.72.125320

[24] S. Pilgram, P. Samuelsson, H. Förster, and M. Büttiker, Full-counting statistics for voltage and dephasing probes, Phys. Rev. Lett. 97, 066801 (2006).
https://doi.org/10.1103/PhysRevLett.97.066801

[25] H. Förster, P Samuelsson, S. Pilgram, and M. Büttiker, Voltage and dephasing probes in mesoscopic conductors: A study of full-counting statistics, Phys. Rev. B 75, 035340 (2007).
https://doi.org/10.1103/PhysRevB.75.035340

[26] M. Büttiker, Coherent and sequential tunneling in series barriers, IBM J. Res. Dev. 32, 63 (1988). The voltage probe model introduced by Büttiker was adapted to a dephasing-probe model by M. J. M. de Jong and C. W. J. Beenakker, Semiclassical theory of shot noise in mesoscopic conductors, Physica A 230, 219 (1996).
https://doi.org/10.1016/0378-4371(96)00068-4

[27] We do wish to emphasise that this shortcoming of the dephasing-probe model only applies to the current fluctuations, not to the time-averaged current and hence not to the conductance. Most applications of the dephasing-probe model only address the conductance, the reservoir fluctuations play no role.

[28] Ya. M. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Phys. Reports 336, 1 (2000).
https://doi.org/10.1016/S0370-1573(99)00123-4

[29] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).

[30] L. S. Levitov and G. B. Lesovik, Charge distribution in quantum shot noise, JETP Lett. 58, 230 (1993).

[31] L. S. Levitov, H.-W. Lee and G. B. Lesovik, Electron counting statistics and coherent states of electric current, J. Math. Phys. 37, 10 (1996).
https://doi.org/10.1063/1.531672

[32] K. Schönhammer, Full counting statistics for noninteracting fermions: Exact results and the Levitov-Lesovik formula, Phys. Rev. B 75, 205329 (2007).
https://doi.org/10.1103/PhysRevB.75.205329

[33] A. Bednorz and W. Belzig, Formulation of time-resolved counting statistics based on a positive-operator-valued measure, Phys. Rev. Lett. 101, 206803 (2008).
https://doi.org/10.1103/PhysRevLett.101.206803

[34] J. E. Avron, S. Bachmann, G. M. Graf, and I. Klich, Fredholm determinants and the statistics of charge transport, Comm. Math. Phys. 280, 807 (2008).
https://doi.org/10.1007/s00220-008-0449-x

[35] F. Hassler, M. V. Suslov, G. M. Graf, M. V. Lebedev, G. B. Lesovik, and G. Blatter, Wave-packet formalism of full counting statistics, Phys. Rev. B 78, 165330 (2008).
https://doi.org/10.1103/PhysRevB.78.165330

[36] I. Klich, An elementary derivation of Levitov's formula, in: Quantum Noise in Mesoscopic Physics, NATO Science Series II, 97, 397 (2003).
https://doi.org/10.1007/978-94-010-0089-5_19

[37] S. Bravyi, Lagrangian representation for fermionic linear optics, Quantum Inf. Comput. 5, 216 (2005).
https://doi.org/10.26421/QIC5.3-3

[38] F. de Melo, P. Ć wikliński, and B. M. Terhal, The power of noisy fermionic quantum computation, New J. Phys. 15, 013015 (2013).
https://doi.org/10.1088/1367-2630/15/1/013015

[39] The sum (17) over $4^p$ determinants can be reduced to a sum over $2^p$ determinants by avoiding the substitution (14), and directly evaluating combinations of projectors onto filled and empty states, see App. A.3. The resulting expressions, in terms of the Schur complements of the scattering matrices, are more complicated to analyze, so we do not take that approach.

[40] V. B. Bulchandani, S. L. Sondhi, and J. T. Chalker, Random-matrix models of monitored quantum circuits, J. Stat. Phys. 191, 55 (2024).
https://doi.org/10.1007/s10955-024-03273-0

[41] F. Gerbino, P. Le Doussal, G. Giachetti, and A. De Luca, A Dyson brownian motion model for weak measurements in chaotic quantum systems, Quantum Rep. 6, 200 (2024).
https://doi.org/10.3390/quantum6020016

[42] Z. Xiao, T. Ohtsuki, and K. Kawabata, Universal stochastic equations of monitored quantum dynamics, Phys. Rev. Lett. 134, 140401 (2025).
https://doi.org/10.1103/PhysRevLett.134.140401

[43] C. W. J. Beenakker, Entropy and singular-value moments of products of truncated random unitary matrices, Phys. Rev. E 111, 064108 (2025).
https://doi.org/10.1103/PhysRevE.111.064108

[44] Chao-Ming Jian, Bela Bauer, Anna Keselman, and Andreas W. W. Ludwig, Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions, Phys. Rev. B 106, 134206 (2022).
https://doi.org/10.1103/PhysRevB.106.134206

[45] Haining Pan, Hassan Shapourian, and Chao-Ming Jian, Topological modes in monitored quantum dynamics, arXiv:2411.04191.
arXiv:2411.04191

[46] V. Gurarie, Randomly measured quantum particle, arXiv:2504.05479.
https://doi.org/10.48550/arXiv.2504.05479
arXiv:2504.05479

[47] E. Knill, Fermionic linear optics and matchgates, arXiv:quant-ph/0108033.
https://doi.org/10.48550/arXiv.quant-ph/0108033
arXiv:quant-ph/0108033

[48] Eq. (48) was pointed out to us by Fred Hucht.

Cited by

[1] C. W. J. Beenakker, "Pure dephasing increases partition noise in the quantum Hall effect", Physical Review B 113 7, 075416 (2026).

[2] C. W. J. Beenakker, "Shot noise from which-path detection in a chiral Majorana interferometer", Physical Review B 112 8, L081402 (2025).

[3] Joon Hyung Lee, "Free Multiplicative Convolution and Erlang Moments in Monitored Quantum Transport", arXiv:2607.05693, (2026).

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