Repeatability of measurements: Non-Hermitian observables and quantum Coriolis force

A noncommuting measurement transfers, via the apparatus, information encoded in a system's state to the external “observer.” Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-Hermitian operators should volens nolens be excluded as their eigenstates are not orthogonal. Here, we show that non-Hermitian operators with real spectra can be treated within the standard framework of quantum mechanics. Furthermore, we propose a quantum canonical transformation that maps Hermitian systems onto non-Hermitian ones. Similar to classical inertial forces this map is accompanied by an energetic cost, pinning the system on the unitary path.

Figure 1

Schematic representation of the unitary transfer of information from a quantum system $\mathcal{S}$ to the measuring device—apparatus $\mathcal{A}$. Information encoded in the quantum state $\left|\psi \right.〉$ is being recorded by $\mathcal{A}$ and is written down into its state $\left|A_{\psi }\right.〉$. The recording process $\left|A_0\right.〉\to \left|A_{\psi }\right.〉$ is unitary and does not influence the information carried by $\left|\psi \right.〉$.

Figure 2

The average position $\langle x\rangle$ of a quantum particle as a function of time $t$. The blue solid curve is the exact solution to the Schrödinger equation obtained in the Hermitian representation. Red points represent a numerical solution to the same problem computed in the non-Hermitian (time-dependent) representation. The two paths coincide only if the inertial force is accounted for [see Eq. (18)]. Finally, the green dashed line depicts a “naive” solution obtained without taking into account this contribution. Parameters are $\hslash =m=1.0,\omega =0.5$, and ${\eta }_t=t/\tau$, where $\tau =10$. The initial state is $\left|{\psi }_0\right.〉=(\left|0\right.〉+\left|1\right.〉)/\sqrt{2}$, where $\left|0\right.〉$ is the ground state of $K=H_0$ and $\left|1\right.〉=a^{†}\left|0\right.〉$.