# Bimodal behavior of post-measured entropy and one-way quantum deficit for two-qubit X states

## TL;DR

This paper develops a method to analyze the bimodal behavior of post-measured entropy in two-qubit X states, revealing phase transitions in the one-way quantum deficit and identifying subregions with variable optimal measurement angles.

## Contribution

It introduces a new approach for calculating the one-way quantum deficit and uncovers bimodal entropy behavior leading to phase boundary identification in X states.

## Key findings

- Bimodal entropy behavior with two interior extrema in certain X states.
- Phase boundaries characterized by finite jumps in optimal measurement angles.
- Subregions with variable optimal angles occupy about 1% of the state space, with high fidelity.

## Abstract

A method for calculating the one-way quantum deficit is developed. It involves a careful study of post-measured entropy shapes. We discovered that in some regions of X-state space the post-measured entropy $\tilde S$ as a function of measurement angle $\theta\in[0,\pi/2]$ exhibits a bimodal behavior inside the open interval $(0,\pi/2)$, i.e., it has two interior extrema: one minimum and one maximum. Furthermore, cases are found when the interior minimum of such a bimodal function $\tilde S(\theta)$ is less than that one at the endpoint $\theta=0$ or $\pi/2$. This leads to the formation of a boundary between the phases of one-way quantum deficit via {\em finite} jumps of optimal measured angle from the endpoint to the interior minimum. Phase diagram is built up for a two-parameter family of X states. The subregions with variable optimal measured angle are around 1$\%$ of the total region, with their relative linear sizes achieving $17.5\%$, and the fidelity between the states of those subregions can be reduced to $F=0.968$. In addition, a correction to the one-way deficit due to the interior minimum can achieve $2.3\%$. Such conditions are favorable to detect the subregions with variable optimal measured angle of one-way quantum deficit in an experiment.