Heisenberg uncertainty relation for three canonical observables

Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.

Figure 1

Phase-space contour lines of the Wigner functions associated with the states $|{\Xi }_0\rangle$ (full line) and a standard coherent state $|0\rangle$ (dashed), respectively; both lines enclose the same area.

Figure 2

Dimensionless pair and triple uncertainties for squeezed states with $\gamma =ln\sqrt[4]{3}$, rotated away from the position axis by an angle $\phi \in [0,\pi ]$ . The pair uncertainty $\Delta p\Delta q$ starts out at its minimum value of $1/2$ which is achieved again for $\phi =\pi /2$ and $\phi =\pi$ (dashed line). The triple uncertainty has period $\pi$, reaching its minimum for $\phi =3\pi /4$ for the state $|{\Xi }_0\rangle$ (full line). The dotted lines (top to bottom) represent the bounds (2), (5), and (10), with values $1/2$, ${(\tau /2)}^{3/2}$, and ${(1/2)}^{3/2}$.