Quantifying the imperfection for non-trace-preserving quantum operations

In quantum information and quantum computing theory, quantum operations are usually assumed to be completely positive trace-preserving linear maps, but this is not the case for realistic quantum systems. Given the measurements and post-selection, non-preserving trace operations occur. We propose a generic metric to compare two non-trace-preserving quantum operations. This metric provides a worst-case error bound for the operation and can be efficiently computed. In addition, we discuss the occurrences of non-trace-preserving operations, classifying them into three main categories: leakage, feed-forward control, and intrinsically probabilistic gates. We also discuss how the metric can be used to study error propagation and estimate thresholds in fault-tolerant quantum computing.