Dynamical model for positive-operator-valued measures

We tackle the dynamical description of the quantum measurement process by explicitly addressing the interaction between the system under investigation and the measurement apparatus, the latter ultimately considered as a macroscopic quantum object. We consider arbitrary positive-operator-valued measures (POVMs) such that the orthogonality constraint on the measurement operators is relaxed. We show that, as with the well-known von Neumann scheme for projective measurements, it is possible to build up a dynamical model holding a unitary propagator characterized by a single time-independent Hamiltonian. This is achieved by modifying the standard model so as to compensate for the possible lack of orthogonality among the measurement operators of arbitrary POVMs.

Figure 1

Scheme of the Naimark representation for a POVM $\left\{F_{{}_S}^{\left(\gamma \right)}\right\}$ on $S,F_{{}_S}^{\left(\gamma \right)}={\mathrm{Tr}}_A{[{\mathbb{I}}_{{}_S}\otimes |0\rangle }_{{}_A}\langle 0\left|V_{{}_{SA}}^{†}{({\mathbb{I}}_{{}_S}\otimes |\gamma \rangle }_{{}_A}\langle \gamma \left|\right)V_{{}_{SA}}\right]$.

Figure 2

Schematic representation of the von Neumann–Ozawa dynamical scheme for projective measures: an observable $O_{{}_S}={\sum }_{\gamma }{o}_{\gamma }{\mathrm{\Pi }}_{{}_S}^{\left(\gamma \right)}$ is measured on $S$ by letting it interact with the measurement apparatus $\mathrm{\Xi }$, so as to encode the information on the states of the apparatus $|{\mathrm{\Xi }}^{\gamma }\left(t\right){\rangle }_{{}_{\mathrm{\Xi }}}$.

Figure 3

Schematic representation of our dynamical model for POVMs. The principal system $S$ interacts with an ultimately macroscopic ancilla $A$. The $S+A$ coupling is ruled by a time-independent generator $H_{{}_{SA}}$. In fact, the unitary transformation $U_{{}_{SA}}\left(t\right)$ induces the transition from an arbitrary initial state ${\rho }_{{}_S}^{\mathrm{in}}\otimes |{\psi }_0{\rangle }_{{}_A}\langle {\psi }_0|$ to the final state (13). The information about the possible outputs ${\mu }_{\gamma }$ is encoded in the orthonormal states of ancilla $|A^{\gamma }{\left(t\right)\rangle }_{{}_A}$.

Figure 4

Plot of the probability function $P_0\left(t\right):={\left|{\beta }_0\left(t\right)\right|}^2$ entering Eq. (46) obtained by solving Eq. (41) for $n_L=20$ [panel (a)] and $n_L=70$ [panel (b)] for the case in which the frequency parameters ${\omega }_{\ell }$ of Eq. (35) are taken to be uniform and equal to ${\omega }_0$. In both cases, $P_0\left(t\right)$ drops from 1 to almost zero around $t\sim 2/{\omega }_0$. Then small revivals appear quite periodically after a time interval approximately given by $n_L$. Therefore, in the limit of $n_L\to \infty$, and after a given lapse of time (see the insets), the state of the $S+A$ can be safely approximated by (48).

Figure 5

Correspondence between the statistics $\left\{p_{\gamma }\right\}$ yielded by the POVM $\left\{F_{{}_S}^{\left(\gamma \right)}\right\}=\{{}_A\langle {\psi }_0|P_{{}_{SA}}^{\left(\gamma \right)}|{\psi }_0{\rangle }_{{}_A}\}$ and the one resulting from $\left\{F_{{}_S}^{\left(\gamma k\right)}\right\}$.