Improved quantum entropic uncertainty relations

We study entropic uncertainty relations by using stepwise linear functions and quadratic functions. Two kinds of improved uncertainty lower bounds are constructed: the state-independent one based on the lower bound of Shannon entropy and the tighter state-dependent one based on the majorization techniques. The analytical results for qubit and qutrit systems with two or three measurement settings are explicitly derived, with detailed examples showing that they outperform the existing bounds. The case with the presence of quantum memory is also investigated.

Figure 1

The solid line is for our bound by the quadratic function; the medium size dotted line for our bound by the stepwise linear function; the small size dotted line for the bound (8), the dot-dashed line for the bound given in Ref. [26]; the big size dotted line for (3), the dashed line for the bound (7), and the thick solid line for $H\left({\mathcal{A}}_1\right)+H\left({\mathcal{A}}_2\right)$ using the Mathematica program.

Figure 2

Our bound (15) is represented by the solid line, and our bound (16) is plotted by the dotted line with $n=32$. The bound (6) from Ref. [26] is shown by the dot-dashed line, and the lower bound in (7) is depicted by the dashed line. The thick solid line is the value of $H\left({\mathcal{A}}_1\right)+H\left({\mathcal{A}}_2\right)+H\left({\mathcal{A}}_3\right)$.

Figure 3

Our bound based on quadratic function (stepwise function) is represented by the solid (dotted line with $n=16$). The bound (7) in Ref, [29] is represented by the dashed line, and the bound (6) in Ref. [26] is plotted by the dot-dashed line. The thick solid line stands for the value of ${\sum }_{k=1}^{3}H\left({\mathcal{A}}_k\right)$.

Figure 4

Our bound (21) by majorization is represented by the solid line, and our bound (24) by the stepwise linear function, with $n=32$, is represented by the big size dotted line. The bound (8) in Ref. [32] is represented by the small size doted line, the bound (5) in Ref. [26] is represented by the thick dot-dashed line, the bound (9) in Ref. [34] is represented by the thick dashed line, and the thick solid line is for ${\sum }_{k=1}^{2}H\left({\mathcal{A}}_k\right)$ with $p=0.2$.

Figure 5

Our bound (21) by majorization techniques is represented by the solid line, and our bound (24) with $n=8$ by the stepwise linear function is represented by big size dotted line. The bound (8) in Ref. [32] is represented by the small size dotted line, the bound (5) in Ref. [26] is represented by the thick dot-dashed line, the bound (9) in Ref. [34] is represented by the thick dashed line, and the thick solid line is for ${\sum }_{k=1}^{2}H\left({\mathcal{A}}_k\right)$ with $p=0.055$.

Figure 6

Our bound (22) by majorization is represented by the solid line, and our bound (25) by the stepwise linear function with $n=16$ is represented by the dotted line. The bound (7) in Ref. [29] is represented by the dashed line, and the thick solid line is for ${\sum }_{k=1}^{3}H\left({\mathcal{A}}_k\right)$ with $p=0.2$.