Quantum measurements designed to reduce risk in decision making
Risk-Averse Decision Making via Quantum Measurement Design
Information Theory
Summary
When quantum measurements guide decisions, simply aiming for the best average result can sometimes lead to risky or bad outcomes. The authors study ways to design these measurements to be more careful about avoiding worst-case scenarios, using a concept called the optimized certainty equivalent (OCE). They show that for certain types of risk preferences, the problem can be solved by breaking it into smaller optimization tasks. For a simple case involving two quantum states, they even find an exact solution. Their approach helps improve how often very bad outcomes happen, even if the average result slightly decreases.
quantum measurementrisk aversionoptimized certainty equivalentconditional value at risksemidefinite programmingHelstrom measurementquantum state discriminationutility functiondecision makingoptimization
Authors
Meiyi Zhu, Osvaldo Simeone
Abstract
Quantum measurements are conventionally optimized to maximize the average of a utility that depends on the true state and on the measurement outcome. However, when the outcome of the measurement is used as an action within a larger decision-making system, the average utility does not capture the risk of poor outcomes. This letter addresses the design of quantum measurements that maximize a risk-averse objective given by the optimized certainty equivalent (OCE), a family of criteria that includes the average utility and the conditional value at risk (CVaR) as special cases. For a piecewise linear gain function, defining the OCE, thus including the CVaR, the problem is shown to reduce to a finite number of semidefinite programs, for which a dual formulation is derived. For the discrimination of two states, a closed-form solution is obtained that takes the form of a Helstrom measurement. Numerical results show that the optimized measurement improves the lower tail of the utility distribution at a moderate cost in average utility.