Quantum state tomography (QST) is a central task for quantum information processing, enabling quantum cryptography, computation, and state certification. Traditional QST relies on projective measurements of single- and two-qubit Pauli operators, requiring the system of interest to be isolated from environmental dissipation. In this Letter, we demonstrate that measuring currents and associated transport quantities flowing through a quantum system in an open configuration enables the reconstruction of its quantum state. This result relies on an exact relation between transport quantities and the Krylov subspaces associated with the Lindbladian which encodes the dynamical evolution of an open quantum system. We illustrate this transport approach to QST with the explicit example of a two-qubit system embedded in a two-terminal setup. As a direct consequence of our framework, we are able to provide a transport-based entanglement measure to certify the presence of quantum correlations, expressing the concurrence in terms of current averages and correlation functions only. Our findings are analytical, providing fundamental insights into quantum information processing in open quantum systems. They establish new connections between the fields of mesoscopic physics and quantum information theory.

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• Entanglement measures
• Open quantum systems & decoherence
• Quantum tomography
• Quantum transport