Combined-probability space and certainty or uncertainty relations for a finite-level quantum system

The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, called the combined-probability space, is presented here for a $d$-level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.

Figure 1

Contour plot of $\mathfrak{u}(p,q)$ on $\mathbf{\omega }$ for $d=2$ and $r=\frac{3}{4}$, where a darker shade represents a smaller value of $\mathfrak{u}$. Square-shaped and elliptical regions represent $\mathrm{\Omega }$ and $\mathbf{\omega }$, respectively. Note that $\mathbf{\omega }\subset \mathrm{\Omega }\subset {\mathbb{R}}^4$ and the unseen coordinates are $p_2=1-p$ and $q_2=1-q$ for each point. For every $r\in [0,1]$, $\mathfrak{u}$ hits its global minimum $\mathfrak{c}$ [given in (54)] on $\mathbf{\omega }$ at all four points $E_1,\cdots ,E_4$, which are shown by small black circles. And $\mathfrak{u}$ always achieves its global maximum $2\sqrt{2}$ [stated in (40)] at the center, $p=\frac{1}{2}=q$, indicated by the black star, of $\mathbf{\omega }$.

Figure 2

Contour plot of ${\mathbf{\Delta }}^{\text{sq}}(p,q)$ of (76) on $\mathbf{\omega }$, where a darker shade illustrates a smaller value of ${\mathbf{\Delta }}^{\text{sq}}$. Here $r=\frac{1}{4}$, therefore ${\mathbf{\Delta }}^{\text{sq}}$ reaches its global minimum $2r$ [see the UR, (77) and (78)] at the two points $F_2$ and $F_4$. However, ${\mathbf{\Delta }}^{\text{sq}}$ always gains its global maximum 2 at the center, $p=\frac{1}{2}=q$, denoted by the black star, of $\mathbf{\omega }$. As in Fig. 1, $\mathbf{\omega }$ is the region bounded by the ellipse, (55), but $\theta =\frac{\pi }{3}$ here.