Contradictions between different measures of quantum uncertainty

We show that variance and Shannon entropy provide contradictory conclusions for the uncertainty associated with the number operator for some families of states of harmonic oscillator systems with fixed mean number, and for the uncertainty of a spin component for states with and without fixed mean. We analyze this behavior in terms of the properties of these uncertainty measures. We explore their impact on quantum metrology, examining the limits to resolution caused by number fluctuations in diverse scenarios of phase-shift detection.

Figure 1

Numerical evaluation of the Shannon entropy for the pure coherent state (2.12) (dashed line) and the classical mixed states (2.15) (solid line) as functions of $\nu$ for the same $\overline{n}=10$. Throughout this work, plotted quantities are dimensionless.

Figure 2

Shannon entropy (solid line) and logarithm of the square root of variance (dash-dotted line) for a mixed classical state (2.15) as functions of $\nu$, as well as the Shannon entropy for pure coherent states (2.12) (dashed line) for $\overline{n}=50$.

Figure 3

Plot of the Shannon entropy (solid line) and the logarithm of the square root of the variance (dashed-dotted line) for the mixed state (3.6) as functions of $j$, as well as the Shannon entropy for the pure SU(2) coherent state (dashed line).

Figure 4

Histograms for variance $V$ and entropy $S$ grouped in 100 and 200 uniformly distributed bins, respectively, for an ensemble of $5\times {10}^6$ randomly generated $j_3$ statistics.

Figure 5

Percentages of cases in different regions of a $V$-$S$ plane with joint $V$ and $S$ values above and below the critical values $V_{\max },\min$ and $S_{\max },\min$ in arbitrary units.