Optimal uncertainty relations in a modified Heisenberg algebra

Various theories that aim at unifying gravity with quantum mechanics suggest modifications of the Heisenberg algebra for position and momentum. From the perspective of quantum mechanics, such modifications lead to new uncertainty relations that are thought (but not proven) to imply the existence of a minimal observable length. Here we prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows us to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, we make use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min and Shannon entropies. We compute the corresponding entropic minimal lengths and find that the minimal length in terms of min entropy is exactly 1 bit.

Figure 1

Probability distributions of a position measurement on quantum states suspected to a UV cutoff [see Eq. (36)]. Which one is the “sharpest”? All three distributions attain a minimal length in their own sense: the blue, dotted one attains minimal length in terms of variances as typically considered in the literature. The black, solid distribution has the highest peak and the red, dashed distribution has the smallest entropy. The latter two distributions have infinite variance, although they appear to be localized to some extent.

Figure 2

Theorem 1 gives sufficient conditions for representing the operator $\mathbf{p}$ as a function $p(\mathbf{k})$, where $\mathbf{k}$ denotes the unmodified momentum operator [here shown for the modification $f(p)=1+\beta p^2$]. We show that the support of $\mathbf{k}$ must be restricted to an interval $[-k_{\max },k_{\max }]$; i.e., modifying the algebra directly leads to UV cutoff and, hence, minimal length.

Figure 3

For an unmodified algebra, minimal uncertainty is obtained for states $\varphi$ from a state space $\mathcal{H}$. In the case of the usual Heisenberg algebra the corresponding minimal length becomes trivial; that is, all states in $\mathcal{H}$ are physical. If a modification of the algebra directly leads to a restricted state space $\mathcal{P}$ that does not contain such states, one obtains a nontrivial minimal length as a direct consequence of modifying the algebra.

Figure 4

The boundary of $\mathcal{U}$ for the modified [with $f(p)=1+\beta p^2$] and standard (long dashed black curve) Heisenberg algebra. We show the point corresponding to the maximally localized state (26), for which $\mathrm{\Delta }\mathbf{x}=1/\mathrm{\Delta }\mathbf{p}=\beta$. The tradeoff curve (solid red line) ${\gamma }_{\mathcal{U}}(\lambda )$ branch below this point corresponds to ground states of (deformed) harmonic oscillators (36) with, from right to left, increasing frequency $\omega \in ]0,\infty [$. The upper part of the curve (short dashed red line) is obtained by considering states $\psi (k)\propto \cos (\sqrt{\beta }k{)}^{{\gamma }_{\lambda }}$ with ${\gamma }_{\lambda }<1$, which are not the ground state of a harmonic oscillator.

Figure 5

Uncertainty regions $\mathcal{U}$ for the modifications $f_1=\cosh (\sqrt{\beta }p)$ (left panel) and $f_2=1+\beta p^2+{\beta }^2{p}^4/4$ (right panel) bounded by the red lines. The black dashed lines show the standard Heisenberg bound for the position and unmodified momentum operator. The blue lines show the not optimal but state-independent bounds from (19).

Figure 6

Different binnings of measurement data lead to different uncertainty measures. Here we choose a binning such that all bins have the same probability in our definition of the Shannon entropy.

Figure 7

Entropic uncertainty region in the $h_{\mathbf{x}}\text{−}{h}_{\mathbf{p}}$ plane for the modification $f(\mathbf{p})=1+\beta {\mathbf{p}}^2$. The dashed line is the Bialynicki-Birula bound, which is still valid in our setting but not longer optimal. The horizontal dotted line shows the upper bound on the entropy $h_{\mathbf{p}}$. The marked points correspond to the $h_{\mathbf{x}}\text{−}{h}_{\mathbf{p}}$ pairs for the maximally localized states, the state of minimal position entropy, and the state with maximal momentum entropy, respectively. Physical states are conjectured to lie on the right of the solid line and below the dotted horizontal line.

Figure 8

Minimal length states (36) for $\beta =1$ and ${\gamma }_{\lambda }=1$ (red, dashed, variance), ${\gamma }_{\lambda }=1/2$ (blue, dotted, Shannon entropy) and ${\gamma }_{\lambda }\approx 0$ (black, solid, min entropy) as in Fig. 1 but in the $k$ representation. These minimal length states differ from the states with maximal momentum uncertainty in the case of variances and Shannon entropy. For min entropies, however, the minimal length state also yields maximal possible momentum uncertainty.

Figure 9

Entropic uncertainty region for min entropies $h_{\infty }$. The minimal length ${\mathrm{\Gamma }}_{\min }^{\infty }$ is analytically computed in (72). As for Shannon entropies, the min entropy of momentum attains an upper bound (dashed line), $\max h_{\infty }^p=\log (2k_{\max })$, for the uniform distribution on the interval $I$. Numerical evaluation indicates that any entropy pair satisfying these two constraints can be attained. Hence the uncertainty region ${\mathcal{U}}_{h_{\infty }}$ and the corresponding trade-off curve are given by the shaded region and the solid line, respectively.

Figure 10

The lower convex hull ${\mathcal{U}}_{lc}$ (blue region) of some set $\mathcal{U}$ (grey region) is obtained by considering hyperplanes (dashed blue lines) for $\lambda \in (0,1)$. As in (20), the trade-off curve $\mathrm{\Gamma }({\mathcal{U}}_{lc})$ (solid blue line) of the convex hull can be parametrized by values $u(\lambda )$. The trade-off curve $\mathrm{\Gamma }({\mathcal{U}}_{lc})$ is never “worse” than the trade-off curve $\mathrm{\Gamma }(\mathcal{U})$ (solid red line); both curves coincide if $\mathcal{U}$ is convex itself.