We formulate a quantum master equation for the many-particle density matrix of electrons propagating through a single-mode conductor, combining elastic scattering by disorder with time-resolved projective measurements that monitor the outcome of scattering events. For noninteracting electrons the charge transfer statistics is governed entirely by the single-particle sector of Fock space, which allows us to resum the exponentially many measurement outcomes into a master equation of dephasing-channel form. The full counting statistics of transmitted electrons has a binomial distribution function, whose mean $\mathcal{T}$ and variance $\mathcal{T}(1-\mathcal{T})$ determine the conductance and shot noise power, respectively. Monitoring suppresses the phase coherence responsible for one-dimensional localization: The decay with conductor length $L$ of the typical transmission probability crosses over at $L\simeq {\ell }_{\varphi }$ from the exponential $e^{-L/\xi }$ (with localization length $\xi$) to the Ohmic $1/L$ decay. Numerical solution of the master equation gives, for weak monitoring, a logarithmic dependence ${\ell }_{\varphi }\simeq \xi ln(v_{\mathrm{F}}{\tau }_{\varphi }/\xi )$ of the coherence length ${\ell }_{\varphi }$ on the mean time ${\tau }_{\varphi }$ between measurements.

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• Anderson localization
• Open quantum systems & decoherence
• Quantum transport
• Quantum-to-classical transition
• Weak values & weak measurements
• Quantum wires
• Quantum master equation