Ability of entanglement and purity to help to detect systematic experimental errors

Abstract

Measurements are central in all quantitative sciences, and a fundamental challenge is to make observations without systematic measurement errors. This holds in particular for quantum information processing, where other error sources, such as noise and decoherence, are unavoidable. Consequently, methods for detecting systematic errors have been developed, but the required quantum state properties are yet unexplored. We theoretically develop a direct and efficient method to detect systematic errors in quantum experiments and demonstrate it experimentally using quantum state tomography of photon pairs emitted from a semiconductor quantum dot. Our method can be scaled to multi-qubit systems, and we find that entanglement and quantum states with high purity can help identify systematic errors.

Figures (10)

• Figure 1: Illustration of the detection scheme for systematic errors in a simple setting. a) The Bloch sphere is the physical state space of a single qubit. The blue dot denotes a quantum state $\rho$ to be determined from Pauli measurements. Severe systematic or statistical error can lead to a least-squares estimate ${\rho_{\text{LS}}}$ (orange dot) outside the Bloch sphere, and thus the distance $D$ (purple arrow) between a physical (but biased) estimate ${\rho_{\text{B}}}$ of Eq. (\ref{['def:PhysicalEstimator']}) (green dot) is finite. If this distance $D$ persists for large statistical samples, this is a signature of a systematic error. The distance $D$ decreases with lower purity $p={\operatorname{Tr}}(\rho^2)$ of the state $\rho$, as suggested by the purple and yellow arrows. b) Single-qubit Pauli measurements for polarized photons: in the brown area the $k$-th Pauli measurement setting for obtaining the two frequencies $f_i$ and $f_j$, corresponding to the eigenvalues, is adjusted by the rotations $\chi_{k}$ and $\psi_{k}$ of the half-wave plate ($Z$ gate) and the quarter-wave plate ($S^{\dagger}$ gate), respectively. The polarizing beam splitter (PBS) and single-photon detectors (D1 and D2) conduct the ${\sigma_{\text{Z}}}$ measurement, shown in the green area. c) The calculated distance $D$ depends on the adjustment angle $\psi_{Z}$ of the Pauli $Z$ basis, and its magnitude decreases with the purity of the true, underlying state $\rho=\ket{\mathrm{i}}\bra{\mathrm{i}}$ (${\sigma_{\text{Y}}}$ eigenstate). The maximum $D$ at $\psi_{\text{Z}}=45°$ occurs when ${\sigma_{\text{Y}}}$ is measured instead of ${\sigma_{\text{Z}}}$, and for a purity $p=0.75$ the error cannot be detected anymore.
• Figure 2: a) Capability of single-qubit states (blue) and two-qubit states (entangled in orange, separable in green) to detect the error of exchanging ${\sigma_{\text{Y}}}$ and ${\sigma_{\text{Z}}}$ on qubit one. We show the maximum positivity constraint values depending on the purity $p$ of the underlying state $\rho$, and find that only entangled states yield values $x_3$ above the dashed line, meaning that they detect this error. b) The two entangled photons from the quantum dot are prepared and separated in the green area, and individual single-qubit Pauli measurements are performed in the red area, see Refs. juliaNeuwirthPRBrota2024source for details. c) We show our experimental results for the distance $D$ depending on $\Delta$, the offset on qubit one that we apply for each measurement setting $k$, i.e. $\psi_{k} = \psi_{k}^{\mathrm{theo}}+\Delta$. The red, blue and green dots are in accordance with the lines from the simulated predictions for all three quantum states, and correspond to purities from $0.56$ to $0.92$. The confidence level to detect a systematic error begins with $90\%$ for a distance $D \geq 0.25$ (grey area), which allows detection of almost all errors for highly entangled states (red points).
• Figure 3: We consider an error of measuring ${\sigma_{\text{Y}}}$ instead of ${\sigma_{\text{Z}}}$ on the first qubit of the three-qubit state probe state $\psi_{\text{probe}}$ at which we apply white noise Eq. (\ref{['eq:whitenoise']}) to change the purity $p$. The orange and blue dots represent the smallest eigenvalue of the erroneous and probe states, respectively, as a function of $p$. The minimal necessary purity for our error detection method is $p^{\text{appr}}_{\text{min}}=0.18$, where the erroneous estimate has a negative eigenvalue as marked by the dashed black line. In comparison, the probe state does not have any negative eigenvalues.
• Figure 4: The maximal reachable positivity constraints $x_l$, as stated in the main text, of the erroneous state are presented as a function of the single- and two-qubit quantum state purity $p$. We assume the systematic error of measuring ${\sigma_{\text{Y}}}$ instead of ${\sigma_{\text{Z}}}$ on one qubit. The black dashed line is located at a value of zero, and if a condition is $x_l$ is above that line, it means that there exists a quantum state with the respective purity which can detect the error. The minimal necessary state purity is $0.75$ and $0.3$ for single- and two-qubit states, respectively.
• Figure 5: Numerical results for the maximal values of $x_1$ in a), $x_2$ in b) and $x_3$ in c) depending on the purity of the true underlying two-qubit state $\rho$ for the error of interpreting the $\operatorname{Y}$ and $\operatorname{Z}$ as $\operatorname{Z}$ and $\operatorname{Y}$ Pauli measurement, respectively. The blue and orange dots represent the $x_l$ values for entangled and separable states as discussed in the main text. If a point is above the black line at $x_l=0$, it means that one of the positivity constraints is violated, and thus, there exists at least one state that can detect the error. Only for entangled states, the results of $x_2$ and $x_3$ are above the dashed line at a purity of $0.45$ and $0.35$, respectively. This error relates to the partial transposition on a subsystem, which explains why only entangled states are sensitive to this error.