Measurement-device agnostic quantum tomography

TL;DR

The paper addresses systematic errors in quantum tomography caused by mismatches between actual measurement operators and those assumed during reconstruction, which distort tomographic characterizations of quantum states. It introduces a measurement-device agnostic tomography framework based on self-calibrating probe states distributed quasi-uniformly on the Bloch sphere, using reconstruction artifacts to infer the true measurement operators $\hat{\pi}_j$. By optimizing an assumed error vector $\vec{\delta}'$ to minimize the state-dependent purity modulation $\Delta P$, the method remains robust to noise and scalable to multi-qubit and device tomography, without requiring perfect probe states. Experimental demonstrations on polarization-encoded qubits show substantial fidelity improvements (e.g., $F_{\min}=0.9907(2)$, $F=0.997(3)$) and successful application to on-chip path qubits, highlighting practical utility for photonic quantum technologies. The approach generalizes to other tomography schemes and state preparation calibration, offering a versatile tool for accurate quantum characterization in complex devices.

Abstract

Characterization of quantum states and devices is paramount to quantum science and technology. The characterization consists of individual measurements, which must be precisely known. A mismatch between actual and assumed constituent measurements limits the accuracy of this characterization. We show that such a mismatch introduces reconstruction artifacts in quantum state tomography. We use these artifacts to detect and quantify the mismatch, gaining information about the actual measurement operators. It consequently allows the mitigation of systematic errors in both quantum measurement and state preparation, improving the precision of state control and characterization. The practical utility of our approach is experimentally demonstrated.

Figures (3)

• Figure 1: Purity modulation due to the measurement operator mismatch. (a-c) States are represented as points on the Bloch sphere plotted using Hammer projection, and the purity of their reconstructions is color-coded. Although all probe states were prepared with the same purity, the reconstruction using quantum tomography with mismatched measurement operators yields purity modulation. The purity modulation scales with the magnitude of error parameters. Panel (b) shows the purity modulation for randomly selected error parameters described in the main text. Panels (a) and (c) show how the purity modulation, if we scale the error vector by a factor of $\lambda = 0.25$ and $\lambda = 4$, respectively. The black curve in panel (d) depicts the scaling trend, and red dots on the curve mark the selected values of $\lambda$ plotted in panels (a-c).
• Figure 2: Experimental demonstration for a polarization-encoded photonic qubit. Panel (a) shows the experimental scheme: LD - laser diode, HWP (QWP) - half-(quarter)-wave plate, GT - Glan-Taylor polarizer, BD - calcite beam displacer, PD - photodiode. Panels (b) and (c) show the purity and fidelity of reconstructed test states before (blue) and after (orange) application of measurement-device agnostic calibration. The fidelity of reconstructed states is calculated with respect to theoretical test states. The theoretical test states are depicted as empty black circles on the projected Bloch sphere in panels (d, e). Panel (d) depicts the reference case -- the method was not used, and we assumed perfect waveplates in preparation and tomographic projections. In contrast, panel (e) shows the improved precision of preparation and measurement when we use the method and take into account the retardance deviation. The purity modulation is reduced and the test states match closely the theoretical expectations. In both panels, the reconstructed states are represented on the Bloch sphere using Hammer projection, and their purity is color-coded.
• Figure 3: Application of the method for an on-chip photonic path qubit. (a) Projection apparatus for a path qubit. Applied voltage $V_{1,2}$ on heater elements (orange) causes thermal-based phase shift in the waveguide interferometers. These shifts determine the measurement projector. The photons in the upper output path are detected with a single-photon detector (SPAD). (b) 1D-section of 6D function $\Delta P(c'_1, \dots, c'_6)$. The value of $\Delta P$ at the initial assumption $c'_j$ is $1.8 \times 10^{-2}$ on the ensemble of probe states. Minimum $\Delta P = 4.5 \times 10^{-4}$ lies at the true value $c_6$. (c) Initial mismatch results in a state-dependent purity artifact. We plot test states on the Bloch sphere as circles and color-code their reconstructed purity. (d) When we consider the true retardance, the artifact vanishes.