[Paper Review] Scrutinizing single-qubit quantum channels: Theory and experiment with trapped ions

Thilo Hannemann, Christof Wunderlich+3 more|ArXiv.org|Apr 6, 2009

Quantum Information and Cryptography2 references3 citations

TL;DR

This paper presents a comprehensive experimental and theoretical study of single-qubit quantum channels using trapped ions, demonstrating robust quantum process tomography under realistic imperfections such as imperfect state preparation and biased measurements. It shows that even with mixed test states and finite statistics, maximum likelihood and constrained estimation methods achieve high-fidelity channel reconstruction, with process fidelity reaching 0.97 for phase-damping channels.

ABSTRACT

We report experimental implementation of various types of qubit channels using an individual trapped ion. We analyzed experimental data and we performed tomographic reconstruction of quantum channels based on these data. Specifically, we studied phase damping channels, where the damping acts either in the xy-plane of the Bloch sphere or in an arbitrary plane that includes the origin of the Bloch sphere. We also experimentally realized and consequently analyzed quantum channels that in addition to phase damping affect also a polarization rotation. We used three reconstruction schemes for estimation of quantum channels from experimental data: (1) a linear inverse method, (2) a maximum likelihood estimation, and (3) a constrained maximum likelihood estimation. We took into account realistic experimental conditions where imperfect test-state preparations and biased measurements are incorporated into the estimation schemes. As a result we found that imperfections present in the process of preparation of test states and as well as in measurements of the considered ion trap system do not limit the control of the implementation of the desired channel. Even imperfect preparation of test state and subsequent measurements still provide sufficient resources for the complete quantum-channel tomography.

Motivation & Objective

• To experimentally implement and characterize various types of single-qubit quantum channels, including phase damping and rotation-damping channels, in a trapped-ion system.
• To evaluate the performance of different quantum process estimation techniques—linear inverse, maximum likelihood, and constrained maximum likelihood—under realistic experimental noise and imperfections.
• To assess whether imperfect initial state preparation and biased measurements limit the accuracy of quantum channel reconstruction.
• To validate theoretical models of quantum channels by comparing estimated channels with expected physical behavior, particularly for phase-damping dynamics.
• To quantify the robustness of quantum process tomography when test states are mixed and detection is imperfect, using process fidelity as a metric.

Proposed method

• Engineered quantum channels were implemented on a single trapped 40Ca+ ion using laser-driven Raman transitions to simulate phase damping and polarization rotation effects.
• Quantum process tomography was performed using three reconstruction schemes: linear inverse method, standard maximum likelihood estimation, and constrained maximum likelihood estimation assuming a phase-damping model.
• Theoretical modeling used the Bloch-sphere representation, where channels are described by affine transformations of the form $\vec{r} \to M\vec{r} + \vec{v}$, with $M$ and $\vec{v}$ derived from trace and Pauli expectation values.
• The process fidelity $F({\cal E}^{\rm est}, {\cal E}_{\lambda_{\rm est}})$ was computed using the standard formula involving the maximally entangled state $\Psi_+$ and the channel's Choi matrix.
• Imperfections such as mixed test states (fidelity ~0.91) and biased detection were explicitly modeled and incorporated into the estimation frameworks.
• The constrained maximum likelihood method assumed a single-parameter phase-damping model $D = \mathrm{diag}(\lambda, \lambda, 1)$, reducing measurement requirements while maintaining high accuracy.

Experimental results

Research questions

• RQ1 Can quantum process tomography accurately reconstruct engineered quantum channels in the presence of imperfect state preparation and biased measurements?
• RQ2 How do different reconstruction methods—linear inverse, maximum likelihood, and constrained maximum likelihood—compare in terms of fidelity and robustness under experimental noise?
• RQ3 To what extent do state preparation infidelity and detection bias limit the accuracy of quantum channel estimation?
• RQ4 What is the achievable process fidelity when reconstructing phase-damping channels with mixed test states and finite statistics?
• RQ5 Does the constrained maximum likelihood method significantly reduce the number of required measurements while preserving accuracy for known channel classes?

Key findings

• The maximum likelihood estimation method consistently produced physically valid quantum channels, even when the linear inverse method yielded unphysical results due to statistical fluctuations.
• The process fidelity between the estimated channel and the true phase-damping channel reached $F \approx 0.97$, indicating high reconstruction accuracy despite imperfect test states with ~91% fidelity to ideal pure states.
• Imperfect state preparation and biased detection did not limit the ability to perform complete quantum channel tomography, as all reconstruction methods yielded reliable results.
• The constrained maximum likelihood method, assuming a phase-damping model, achieved high accuracy with significantly fewer measurements, demonstrating its efficiency for known channel types.
• The estimated phase-damping rate $\lambda$ showed a consistent dependence on the control parameter $s$, allowing the extraction of $S_v^0(0) = 0.38\,\mathrm{ms}^{-1}$ from experimental data.
• The agreement between theoretical expectations and experimental estimates validated the underlying physical model, confirming that the engineered channels behaved as intended despite experimental imperfections.