Monitored quantum transport through a disordered one-dimensional conductor

arXiv:2605.22701 · cond-mat.mes-hall · Submitted 2026-05-21 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Monitored quantum transport through a disordered one-dimensional conductor".

Mira: The gist The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail when monitoring suppresses phase coherence responsible for one-dimensional localization,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: Alright, moving on to the title and authors of this paper, "Monitored quantum transport through a disordered one-dimensional conductor." The authors are Sanchez Fernan, Tworzyd lo, and Beenakker.

Mira: They are diving into how we can track electron transport in these one-dimensional systems while actively monitoring the scattering events themselves.

Lev: I'm thinking about the setup: they are dealing with a single-mode conductor with elastic scattering from disorder, which is standard stuff, but then they add time-resolved projective measurements to see the outcomes.

Kai: Exactly. The core idea is combining those two elements to understand transport in a way that includes measurement back in the dynamics.

Mira: They use these measurements to replace the usual unitary evolution with something governed by Kraus operators s, where s represents all possible measurement outcomes.

Lev: From an engineering standpoint, having a sum over all possible strings s is like trying to model every possible path an electron could take through the system simultaneously.

Kai: That's right, it’s a representation of the dynamics in this monitored channel formulation. They define these operators based on projection onto left-moving and right-moving states at each site n.

Mira: And they use these to describe how scattering events that couple left-movers and right-movers are handled through a unitary operator n, which is different for each site n.

Lev: That n structure, defined as (i X sigma, sigma' a n M(n) sigma sigma' a), where M(n) is the single-particle scattering matrix, tells us exactly how the electron state evolves at each site.

Kai: It’s crucial that this operator structure exists because it describes the coupling between the left and right movers in this disordered environment.

Mira: And they also include a unitary translation operator, which moves left-movers and right-movers by one lattice site, defined as (i X k, sigma sigma a (k) a sigma (k)), where a sigma(k) = X n e-ikna sigma.

Lev: So the dynamics are being tracked in this combined space-time lattice representation, which helps visualize the scattering events over a length L.

Kai: It’s a detailed picture of how the electrons interact with each other and the disorder as they move from one end to another.

The paper's summary: Mira: Now let's look at what the paper actually summarizes in terms of results, because it goes beyond just setting up the master equation. They focus on what happens when you calculate those charge transfer statistics.

Kai: They show that the full counting statistics of the transferred charge follow a binomial distribution function, which is determined by T, where T is related to the transmission probability.

Lev: So if we want to know about conductance, we look at the mean T, and for shot noise power, we use the variance calculation based on T(one-T) <ref:2605.22701#pg1>. It’s a clean link there.

Mira: They also give us this moment generating function F(xi) = one + (e xi - one)T, which is the mathematical tool they use to extract these statistics from their setup <ref:2605.22701#pg1>.

Kai: The key physical finding here is that monitoring suppresses the phase coherence responsible for one-dimensional localization, leading to that crossover we talked about earlier.

Mira: That suppression of coherence means the typical transmission probability T typ changes its dependence on length L. It goes from being proportional to e-L/xi to a diffusive decay proportional to one/L <ref:2605.22701#pg1>.

Lev: That crossover is important because it tells us that the monitoring isn't just adding noise; it fundamentally changes the nature of how transport occurs in this disordered system.

Kai: And then they connect that back to the coherence length phi, showing a logarithmic dependence on the monitoring strength, phi xi (vF tau phi/xi).

Mira: That logarithmic scaling is a major piece of information for us because it tells us precisely how sensitive this characteristic length is to the strength of the measurement.

Lev: If we try to run this on real hardware, we'd have to control tau phi very carefully, otherwise we lose that clean logarithmic scaling relationship they found.

Kai: So, in short, the summary is that monitoring kills localization effects by changing T typ 's decay law and providing a specific way to calculate the coherence length based on measurement time.

The paper's improvements: Mira: They suggest some refinements to their approach within the paper itself, focusing on how they can make these results more robust or applicable to different scenarios.

Kai: One improvement they point out is that they can compare their master equation results directly with the voltage probe model, which provides a really useful benchmark for checking their formalism.

Lev: It's good when two different theoretical approaches agree, but it only works if the assumptions underlying those approaches are compatible, which we need to check.

Mira: They show that for N=two tunnel barriers, the result from their master equation is identical to the result following from the voltage probe model <ref:2605.22701#pg1>.

Lev: That match for N=two is encouraging because it means their quantum formalism holds up well in a simpler case, even with monitoring involved <ref:2605.22701#pg1>.

Kai: But they also mention that for N=three tunnel barriers, there's a slight difference between the master equation result and what the voltage probe model predicts <ref:2605.22701#pg3>.

Mira: That small difference tells us where the assumptions of their master equation might be slightly off when we move to more complex systems, like three barriers.

Lev: That’s always where I worry in hardware research—the simple models work for a few cases, but complexity introduces real limitations.

Conclusion: Kai: So wrapping up the paper "Monitored quantum transport through a disordered one-dimensional conductor," the central message is that monitoring cuts off the coherent interference responsible for one-dimensional localization in these conductors.

Mira: And this leads to the typical transmission probability crossing over from an exponential decay proportional to e-L/xi to a diffusive one/L tail <ref:2605.22701#pg1>.

Lev: It’s a clear physical picture of how measurement modifies the transport regime in these systems by destroying the phase coherence that causes localization.

Kai: And they give us that specific scaling for the coherence length, phi xi (vF tau phi/xi), which depends logarithmically on the monitoring strength.

Mira: That logarithmic dependence is what they found numerically, and it confirms the qualitative argument that monitors are indeed cutting off that localization mechanism.

Lev: For us, this means if we're designing an experiment to measure this, we need to be careful about controlling tau phi because of how sensitive phi is.

Kai: It’s a solid piece of work that connects the quantum dynamics, the counting statistics, and the length scales in a very concrete way.

Mira: It gives us a good framework for understanding how measurement influences transport in these low-dimensional conductors.

Lev: We'll keep an eye on this because it sets a clear benchmark for what we need to achieve when we try to implement these kinds of measurements on actual quantum devices.

J. S´anchez Fern´an, J. Tworzyd lo, C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden · Faculty of Physics, University of Warsaw

cond-mat.mes-hall

Submitted: 2026-05-21

Updated: 2026-05-21

Comments: 12 pages, 7 figures

Journal ref: Phys. Rev. B 114, 225129 (2026)

DOI: 10.1103/8by5-4tzt

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The gist The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail when monitoring suppresses phase coherence responsible for one-dimensional

Key concepts

Quantum Master Equation: This is a mathematical tool used to describe how the state of many interacting electrons changes over time when they are subjected to both disorder (scattering) and continuous measurements. It combines unitary evolution with measurement effects to capture the full quantum dynamics.
Typical Transmission Probability ($T_{typ}$): This describes the average transmission probability of an electron through a disordered system. The paper shows that in the presence of monitoring, this probability transitions from a value proportional to $e^{-L/ au}$ (exponential decay) to one following a diffusive $1/L$ behavior, indicating a change in how electrons propagate.
Coherence Length ($ ilde{ u}_ ext{phi}$): This length quantifies the distance over which quantum phase information is preserved in the conductor. The paper derives that this coherence length scales logarithmically with the monitoring strength, meaning stronger monitoring leads to a shorter coherence length, as described by $ ilde{ u}_ ext{phi} ightarrow au_ ext{phi} ightarrow au_ ext{phi} o au_ ext{phi} + irac{eta}{ au_ ext{phi}} L$.

Terminology

Coherence — 2× in this paper · explained in 2 episodes
Coherence Length — 1× in this paper · explained in 2 episodes

Summary

The gist The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail when monitoring suppresses phase coherence responsible for one-dimensional localization, leading to a logarithmic dependence of the coherence length on the monitoring strength

Quantum Channel Formulation

The paper formulates a quantum master equation for the many-particle density matrix of electrons propagating through a single-mode conductor by combining elastic scattering by disorder with time-resolved projective measurements The dynamics are represented by a unitary conjugation in the absence of monitoring, but measurements replace this with a sum over different measurement outcomes labeled by strings s = (s1, s2,... sL) The Kraus operator Kˆs has the form Kˆs = TˆYn Pˆsn,nUˆn Xsn P†sn,nPhat sn,n = ˆI > The four weak measurement operators P s,s' n on site n are given by products of projection operators onto right-moving and left-moving states

Charge Transfer Statistics

The full counting statistics of the transferred charge has a binomial distribution function whose mean T and variance T (1−T) determine the conductance and shot noise power, respectively The moment generating function of the transferred charge is F(ξ) = 1 + (e ξ − 1)T, where T is related to the transmission probability This leads to a binomial probability density function P(Q) = NV/Q T Q(1 − T) NV −Q, with mean E[Q] = NV T and variance Var[Q] = NV T (1 − T)

Transmission Probability and Localization

The transmission probability is determined by the single-particle sector of Fock space, which is mapped to a matrix in site-pseudospin vector space The master equation of a dephasing channel is derived as G(t + 1) = e ikσˆ z omegaG(t)omega† + Xn w squared n n⟩⟨n ⊗ D[UnGnn(t)U†n] e −ikσˆ z The coherence time is defined as the mean time between measurements, τϕ = δt/w squared (7.4) The typical transmission probability Ttyp crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail

Coherence Length and Scaling

The coherence length is related to the monitoring strength by the relation lϕ = ξW0(vFτϕ/ξ), where W0(x) is the Lambert W-function Asymptotically for large τϕ this function grows logarithmically, lϕ → τϕ→∞ ξ ln(vFτϕ/ξ) Numerical results show a linear dependence of lϕ on ln τϕ, which is believed to be a trustworthy result in the limit

Reciprocity and Comparison with Voltage Probe Model

The master equation breaks reciprocity in the weak backscattering regime, but this can be restored by inserting the lead operator Llead[ρ] = 1/2 Ξρ + 1/2 ρΞ† at each measurement The results coincide precisely with the voltage probe model For N=2 tunnel barriers, the result from the master equation is identical to that following from the voltage probe model, while for N=3, they differ slightly but are close

Conclusion

The central effect of monitoring is to cut off the coherent interference responsible for one-dimensional localization The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail The numerical solution of the quantum master equation indicates a logarithmic dependence of lϕ on the monitoring strength, in accord with a qualitative argument

How it works

The paper formulates a quantum master equation for the many-particle density matrix of electrons propagating through a single-mode conductor by combining elastic scattering by disorder with time-resolved projective measurements The full counting statistics of transmitted electrons has a binomial distribution function, whose mean T and variance T (1−T) determine the conductance and shot noise power, respectively

Key mechanisms

Phase coherence is lost by coupling to the environment, which effectively “monitors” the dynamics by measuring the outcome of each scattering event
The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail
The coherence length is related to the monitoring strength by the relation lϕ = ξW0(vFτϕ/ξ)

Analytical and Numerical Results

The paper provides closed form expressions for small numbers of tunnel barriers, finding that the N=2 result (6.3) is identical to that following from the voltage probe model The N=3 result differs from the master equation formula (6.6) but is very close to what would follow from the voltage probe model The effect of decoherence broadens and lowers resonant peaks, merging them into a single broad maximum at phi = pi, and finally leveling off at the Tinccoh value

Reciprocity Breaking

The master equation breaks reciprocity in the weak backscattering regime, but this can be restored by inserting the lead operator Llead[ρ] = 1/2 Ξρ + 1/2 ρΞ† at each measurement The term in square brackets in the numerator is not symmetric upon exchange of rho1 and rho3, so reciprocity is broken

Comparison with Voltage Probe Model

The N=2 result from the voltage probe model is the same as the result (6.3) from the master equation, while the N=3 result differs from the master equation formula (6.6) The voltage probe model yields Tprobe = Γ1Γ2 1 + γ 2ρ1ρ2 + 2γ√ρ1ρ2 cos ϕ 1 − γ 2ρ1ρ2 (6.4)

Final Summary

In conclusion, the full counting statistics of transmitted charge remains binomial: Monitoring modifies the transmission probability T, but not the functional form of the counting distribution The central effect of monitoring is to cut off the coherent interference responsible for one-dimensional localization The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail The numerical solution of the quantum master equation indicates a logarithmic dependence of lϕ on the monitoring strength, in accord with a qualitative argument The usual scaling approach relies essentially on unitary dynamics, which does not carry over to the monitored quantum channel

Acknowledgments

We have benefited from discussions with T. Giamarchi This work was supported by the Netherlands Organisation for Scientific Research (NWO/OCW), as part of Quantum Limits (project number summit.1.1016) The paper's pages are labeled Page 9 and Page 10 respectively, and the final page is Page 12.

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The full counting statistics of transmitted electrons has a binomial distribution function, whose mean T and variance T (1− T) determine the conductance and shot noise power, respectively The typical transmission probability crosses over from Ttyp proportional to e-L/ξ to a diffusive 1/L tail

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Scattering events that couple left-movers and rightmovers are described by a unitary operator Uˆn = exp(i Xσ,σ' a† nσM(n) (n)σσ'an σ'a), (2.1) The Kraus operator Kˆs has the form Kˆs = TˆYn P sn,nU sn P†sn,nPhat sn,n = ˆI >

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The sum over s in Eq. (4.

Improvements for AI systems

No longer relying on unitary dynamics for localization analysis, AI systems can predict the typical transmission probability by utilizing the logarithmic mean Ttyp = e E[ln T] from Eq. (7.3).
AI systems can estimate the coherence length in disordered conductors by calculating the dependence of lϕ on the monitoring strength, specifically finding that it follows a linear dependence on ln τϕ, as shown in Eq. (7.6).
AI can model and predict shot noise power based on transmission probability, maintaining the relationship Pnoise = P0T (1 − T) even in the presence of decoherence, as stated in Section I.