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Bound on the distance between controlled quantum state and target state under decoherence

Kohei Kobayashi

TL;DR

This work addresses the challenge of steering a quantum system to a target state under decoherence by deriving a computable upper bound $D(T) \le \delta$ on the distance between the ideal and noisy evolutions. The bound depends on the time-averaged Frobenius norm of the decoherence operator and whether the operator is Hermitian, and extends to multiple decoherence channels. The main contribution, Theorem 1, provides a practical, equation-free limit $\delta$ (with a corresponding quantum-speed-limit $T_\delta$) and is demonstrated through qubit and two-qubit examples, along with a bound on the target-state probability under decoherence. These results offer a general, actionable benchmark for evaluating and planning quantum control tasks in noisy environments, including implications for algorithmic success probabilities under realistic decoherence levels.

Abstract

To implement quantum information technologies, carefully designed control for preparing a desired state plays a key role. However, in realistic situation, the actual performance of those methodologies is severely limited by decoherence. Therefore, it is important to evaluate how close we can steer the controlled state to a desired target state under decoherence. In this paper, we provide an upper bound of the distance between the two controlled quantum systems in the presence and absence of decoherence. The bound quantifies the degree of achievement of the control for a given target state under decoherence, and can be straightforwardly calculated without solving any equation. Moreover, the upper bound is applied to derive a theoretical limit of the probability for obtaining the target state under decoherence.

Bound on the distance between controlled quantum state and target state under decoherence

TL;DR

This work addresses the challenge of steering a quantum system to a target state under decoherence by deriving a computable upper bound on the distance between the ideal and noisy evolutions. The bound depends on the time-averaged Frobenius norm of the decoherence operator and whether the operator is Hermitian, and extends to multiple decoherence channels. The main contribution, Theorem 1, provides a practical, equation-free limit (with a corresponding quantum-speed-limit ) and is demonstrated through qubit and two-qubit examples, along with a bound on the target-state probability under decoherence. These results offer a general, actionable benchmark for evaluating and planning quantum control tasks in noisy environments, including implications for algorithmic success probabilities under realistic decoherence levels.

Abstract

To implement quantum information technologies, carefully designed control for preparing a desired state plays a key role. However, in realistic situation, the actual performance of those methodologies is severely limited by decoherence. Therefore, it is important to evaluate how close we can steer the controlled state to a desired target state under decoherence. In this paper, we provide an upper bound of the distance between the two controlled quantum systems in the presence and absence of decoherence. The bound quantifies the degree of achievement of the control for a given target state under decoherence, and can be straightforwardly calculated without solving any equation. Moreover, the upper bound is applied to derive a theoretical limit of the probability for obtaining the target state under decoherence.
Paper Structure (10 sections, 31 equations, 2 figures)

This paper contains 10 sections, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Dynamics of controlled quantum system under decoherence
  4. Main result
  5. Example
  6. Qubit
  7. Two qubits
  8. Bound on probability for obtaining the target state
  9. Conclusion
  10. Proof of the theorem 1

Figures (2)

  • Figure 1: Plots of $\Delta$ as a function of $\gamma$ when $\Omega=1$, $3$, and $10$. These plots are computed via numerical simulation. (b) Comparison of the exact distance $D(T)$ and its upper bound $\delta$ as a function of $\gamma T$.
  • Figure 2: Plots of $\Delta$ for (a) $\hat{M}_{\rm AD}$ and (b) $\hat{M}_{\rm PD}$, as a function of $\gamma$ when $\Omega=1$ and $5$.