Stronger uncertainty relations with improvable upper and lower bounds

Abstract

The quantum superposition principle is used to establish improved upper and lower bounds for the Maccone–Pati uncertainty inequality, which is based on a “weighted-like” sum of the variances of observables. Our bounds include free parameters that not only guarantee nontrivial bounds but also effectively control the bounds’ tightness. Significantly, these free parameters depend on neither the state nor the observables. A feature of our method is that any nontrivial bound can always be improved. In addition, we generalize both bounds to uncertainty relations with multiple (three or more) observables, maintaining the uncertainty relations’ tightness. Examples are given to illustrate our improved bounds.

Appendix

Lower bound of Eq. (3) in qubit system. First, we emphasize that the nontrivial bound does not only mean that a vanishing bound is given once the sum of the variances does not vanish. It will also be trivial if the lower bound is equal to the sum of variance, because in this case we does not need to look for bound. Of course, this should be distinguished from the case when the state happens to be the eigenstate of one observable. With this in mind, let us study $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ in Eq. (3).

Since the qubit system is considered for Eq. (4), there exists a unique $\vert \psi ^{\perp }\rangle$ orthogonal to $\left| \psi \right\rangle$ neglecting a global phase. So, we have $\vert \psi ^{\perp }\rangle \langle \psi ^{\perp }\vert =1-\left| \psi \right\rangle \left\langle \psi \right|$. Substituting this relation into Eq. (4), one will have

$$\begin{aligned} \mathcal {L}_{1}= & {} \pm i\langle [X,Y]\rangle +|\langle \psi |X\pm iY|\psi ^{\perp }\rangle |^{2} \nonumber \\= & {} \pm i\langle [X,Y]\rangle +\langle \psi |\left( X\pm iY\right) \left( X\mp iY\right) |\psi \rangle \nonumber \\&-\langle \psi |\left( X\pm iY\right) \left| \psi \right\rangle \left\langle \psi \right| \left( X\mp iY\right) |\psi \rangle \end{aligned}$$

(63)

$$\begin{aligned}= & {} \pm i\langle [X,Y]\rangle +\langle X^{2}\rangle +\langle Y^{2}\rangle \mp i\left\langle [X,Y]\right\rangle -\left\langle X\right\rangle ^{2}-\left\langle Y\right\rangle ^{2} \nonumber \\= & {} \Delta X^{2} +\Delta Y^{2}. \end{aligned}$$

(64)

In this sense, the maximum operation in Eq. (3) directly ignores $\mathcal {L}_{2}$ and becomes a trivial “bound”. In this case, if one could give up $\mathcal {L}_{1}$ and turn to $\mathcal {L}_{2}$, this is also of no help, because $\mathcal {L}_{2}$ is trivial once $\left| \psi \right\rangle$ happens to be the eigenstate of $X+Y$, but it is not the eigenstate of either X or Y. So we think the bound given in Eq. (3) is trivial in qubit system. One could argue that the equality could not be a bad thing because it provides a unified form of Eq. (4) for any qudit state. However, this is only a trivial substitution for qubit system. Considering our Eqs. (63) and (64), one will find that they form an equality for any qudit state $\left| \psi \right\rangle$. In particular, all the terms in the left-hand side of Eq. (63) are the average values of observables such as $i\langle [X,Y]\rangle$, $\langle \psi |\left( X\pm iY\right) \left( X\mp iY\right) |\psi \rangle$, $\left\langle \psi \right| X\left| \psi \right\rangle$ and $\left\langle \psi \right| Y\left| \psi \right\rangle$. However, this is obviously an rewriting of Eq. (64) and deviates the original purpose of uncertainty relation.