Locally Adaptive Protocols for Quantum State Discrimination

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2021

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This dissertation makes contributions to two rapidly developing fields: quantum information theory and machine learning. It has recently been demonstrated that reinforcement learning is an effective tool for a wide variety of tasks in quantum information theory, ranging from quantum error correction to quantum control to preparation of entangled states. In this work, we demonstrate that reinforcement learning is additionally highly effective for the task of multiple quantum hypothesis testing.

Quantum hypothesis testing consists of finding the quantum measurement which allows one to discriminate with minimal error between $m$ possible states $\{\rho_{k}\}|_{k=1}^{m}$ of a quantum system with corresponding prior probabilities $p_{k} = \text{Pr}[\rho = \rho_{k}]$. In the general case, although semi-definite programming offers a way to numerically approximate the optimal solution~\cite{Eldar_Semidefinite2}, a closed-form analytical solution for the optimal measurement is not known.

Additionally, when the quantum system is large and consists of many subsystems, the optimal measurement may be experimentally difficult to implement. In this work, we provide a comprehensive study of locally adaptive approaches to quantum hypothesis testing where only a single subsystem is measured at a time and the order and types of measurements implemented may depend on previous measurement results. Thus, these locally adaptive protocols present an experimentally feasible approach to quantum state discrimination.

We begin with the case of binary hypothesis testing (where $m=2$), and generalize previous work by Acin et al. (Phys. Rev. A 71, 032338) to show that a simple Bayesian-updating scheme can optimally distinguish between any pair of arbitrary pure, tensor product quantum states. We then demonstrate that this same Bayesian-updating scheme has poor asymptotic behaviour when the candidate states are not pure, and based on this we introduce a modified scheme with strictly better performance. Finally, a dynamic programming (DP) approach is used to find the optimal local protocol for binary state discrimination and numerical simulations are run for both qubit and qutrit subsystems.

Based on these results, we then turn to the more general case of multiple hypothesis testing where there may be several candidate states. Given that the dynamic-programming approach has a high complexity when there are a large number of subsystems, we turn to reinforcement learning methods to learn adaptive protocols for even larger systems. Our numerical results support the claim that reinforcement learning with neural networks (RLNN) is able to successfully find the optimal locally adaptive approach for up to 20 subsystems. We additionally find the optimal collective measurement through semidefinite programming techniques, and demonstrate that the RLNN approach meets or comes close to the optimal collective measurement in every random trial.

Next, we focus on quantum information theory and provide an operational interpretation for the entropy of a channel. This task is motivated by the central role of entropy across several areas of physics and science. We use games of chance as a more systematic and unifying approach to define entropy, as a system's performance in any game of chance depends solely on the uncertainty of the output. We construct families of games which result in a pre-order on channels and provide an operational interpretation for all pre-orders (corresponding to majorization, conditional majorization, and channel majorization respectively), and this defines the unique asymptotically continuous entropy function for classical channels.

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Brandsen, Sarah (2021). Locally Adaptive Protocols for Quantum State Discrimination. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/23784.

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