Unified Framework for Direct Characterization of Kraus Operators, Observables, Density Matrices, and Weak Values Without Weak Interaction

TL;DR

The paper introduces a unified framework for direct characterization of Kraus operators, POVMs, density matrices, and observables without relying on weak interactions. It leverages a probe–system–environment setup with a single fixed unitary $U_{PSE}$ and two fixed probe measurements to extract matrix elements of Kraus operators, density matrices, and unitary/observable information, and extends naturally to weak and modular values. The authors provide exact derivations (e.g., Eq. $FW-3$) and demonstrate resource-efficient protocols—such as using $d_S/2+1$ Pauli $X$-gates for density matrices and a minimal gate set for Kraus reconstruction—while offering precise error analyses and comparisons to prior methods (e.g., Xu-2021, Vallone). The framework is applicable across optical, superconducting, and trapped-ion platforms and holds promise for scalable, high-precision tracking of open quantum system dynamics and multi-partite measurements. By removing the weak-coupling constraint and unifying various direct-measurement strategies, the work provides a versatile tool for direct, efficient quantum tomography and value estimation with broad practical impact.

Abstract

Generalized quantum measurements, described by positive operator-valued measures (POVMs), are essential for modeling realistic processes in open quantum systems. While quantum process tomography can fully characterize a POVM, it is resource-intensive and impractical when only specific POVM elements or matrix elements of a particular POVM element are of interest. Direct quantum measurement tomography offers a more efficient alternative but typically relies on weak interactions and complex structures of the system, environment, and probe as the dimension of the system increases, limiting its precision and scalability. Furthermore, characterizing a POVM element alone is insufficient to determine the underlying physical mechanism, as multiple Kraus operators can yield the same measurement statistics. In this work, we present a unified framework for the direct characterization of individual matrix elements of Kraus operators associated with specific POVM elements and arbitrary input states without requiring weak interaction, complex structures of the system-environment-probe or full process and state tomography. This framework naturally extends to projective measurements, enabling direct observable tomography, and to the characterization of unitary operations. Our method also captures modular and weak values of observables and Kraus operators, without invoking weak interaction approximations. We demonstrate potential implementations in optical systems, highlighting the experimental feasibility of our approach.

Figures (5)

• Figure 1: Quantum circuit for implementation of the unitary operator $U_{\text{PSE}}$.
• Figure 2: Schematic illustration of a modified Mach–Zehnder interferometer (MZI) used for the direct characterization of an unknown Kraus operator.
• Figure 3: Comparison of our results in Eqs. \ref{['S3.1-7']} and \ref{['S3.1-8']} with those of Ref. Xu-2021-DirectMeasurements for the error in estimating an unknown POVM element $E_k$ as well as the entire POVM, $E = \{E_k\}_{k=0}^{d_{\text{E}}-1}$. In (1.a), we plot $\delta E_k$ [our result from Eq. \ref{['S3.1-7']}] versus $\widetilde{\delta}E_k$ [the result of Ref. Xu-2021-DirectMeasurements, recalculated in Eq. \ref{['S3.2-11']}]. For $d_{\text{S}} = 2$, if the POVM element $E_k$ has a low trace value, e.g.,$\text{Tr}(E_k) = 0.5$, both methods perform comparably. However, for high trace values, e.g.,$\text{Tr}(E_k) = 1.8$, our method significantly outperforms that of Ref. Xu-2021-DirectMeasurements for any value of the unitary interaction parameter $\theta$ introduced therein. A similar performance trend is observed in (1.b) for $d_{\text{S}} = 20$, indicating that the performance gain of our method increases with system dimension. In (2.a), we compare $\delta E$ [our result from Eq. \ref{['S3.1-8']}] with $\widetilde{\delta}E$ [from Ref. Xu-2021-DirectMeasurements, recalculated in Eq. \ref{['S3.2-12']}]. For $d_{\text{S}} = 2$ and $d_{\text{E}} = 2$, our method performs slightly better. However, when $d_{\text{E}} \geq d_{\text{S}}$, e.g.,$d_{\text{E}} = 5$, the method of Ref. Xu-2021-DirectMeasurements performs better. In (2.b), for $d_{\text{S}} = 20$, a similar trend holds, but when the difference $d_{\text{S}} - d_{\text{E}} \gg 1$ (e.g.,$d_{\text{S}} - d_{\text{E}} = 15$), our method again outperforms the method of Ref. Xu-2021-DirectMeasurements.
• Figure 4: Schematic illustration of a modified Mach–Zehnder interferometer (MZI) used for the direct characterization of an unknown density matrix.
• Figure 5: Description of this figure is given in the "Example: $d_\text{S}=10$".