Quantum state exclusion through offset measurement

The quantum state discrimination problem has Alice sending a quantum state to Bob who wins if he correctly identifies the state. The pretty good measurement, also known as the square root measurement, performs pretty well at this task. We study the version of this problem where Bob tries to lose with the greatest probability possible (which is harder than it sounds). We define the pretty bad measurement, which performs pretty well at this task, or in other words, pretty poorly for the original task. We show that both the pretty good measurement and the pretty bad measurement are always no worse than blind guessing at their respective tasks. As an application, we apply the pretty bad measurement to the quantum state anomaly detection problem and show how to avoid pretty bad qubits.