Simultaneous measurement of two noncommuting quantum variables: Solution of a dynamical model

The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin–$\frac{1}{2}$ system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be nonideal, as (i) the joint statistics do not coincide with the one obtained by separately measuring each spin component, and (ii) the density matrix of the spin does not collapse in either of the measured observables. However, we give an operational interpretation of the process as a generalized quantum measurement, and show that it is fully informative: The expected value of the measured spin components can be found with arbitrary precision for sufficiently many runs of the experiment.

Figure 1

Schematic figure of the measurement, containing the measured spin system (S), the magnet (M), and the bath (B). The main idea of the measurement is that the SM interaction brakes the metastability of the initial state of M, triggering a phase transition from the initial paramagnetic state to a ferromagnetic state, which is driven by B. The two ferromagnetic states correspond to the pointer states of the measurement.

Figure 2

Free energy functions of M: (i) in the absence of S ($F_{\mathrm{eq}}$), (ii) in the presence of $|\uparrow \rangle$ ($F_{\Uparrow }$), and (iii) in the presence of $|\downarrow \rangle$ ($F_{\Downarrow }$). The minimum of $F_{\Uparrow }$ defines the value of $m_F\approx 1$. The parameters are chosen as $J_2=0, J_4=1, \beta =3.3, g=0.08$.

Figure 3

Schematic figure of the simultaneous measurement. The system S simultaneously interacts with M and M' through the $z$ and $x$ spin components, respectively.

Figure 4

Free energy functions of MM': (i) in the absence of $\mathrm{S},F_{\mathrm{eq}}+F_{\mathrm{eq}}^{\prime }$, (ii) in the presence of ${|\uparrow }_{\mathbf{u}}\rangle , F_{\Uparrow }^{\left(\mathbf{u}\right)}$, and (iii) in the presence of $|{\downarrow }_{\mathbf{u}}\rangle , F_{\Downarrow }^{\left(\mathbf{u}\right)}$.

Figure 5

Plot of $P(m,m^{\prime },t)$ for three different times: $t=0,6\tau ,12\tau$ (from top to bottom) with $\tau =1/\gamma J$. We take $N=161,J_4=J,g=0.1J,1/\beta =0.2,J_2=0$, and the initial condition $\langle s^z\left(0\right)\rangle =1$. The results are obtained by solving numerically the equations of motion (B36) derived in Appendix pp2. Notice that at the end of the measurement only one of the two apparatuses registers a result, to be expected as the interaction satisfies $h_c<g<h_d$.

Figure 6

Ratio $h_d/h_c$ as a function of $\beta$. In the figure we take $J_2=0$ and $J_4=1$.

Figure 7

Plot of $P(m,m^{\prime },t)$ for three different times: $t=0,6\tau ,8\tau$ (from top to bottom) with $\tau =1/\gamma J$. We take exactly the same conditions as in Fig. 5, except for the interaction, which is increased to $g=0.4J$. In this case $g\ge h_d$, and hence both magnets can register results from the measurement.

Figure 8

Numerical estimates of $\lambda$ using the equations of motion (B36), in the case of two identical apparatuses. The parameters are $J_4=J,J_2=0,1/\beta =0.2J$, and $\lambda$ is evaluated at $t_f=10\tau$ with $\tau =1/\lambda$, a time for which the measurement is finalized.

Figure 9

Plot of the field velocity of the Fokker-Plank equation associated with $P_{\uparrow \uparrow }^{\left(\mathbf{u}\right)}$ (left) and $P_{\downarrow \downarrow }^{\left(\mathbf{u}\right)}$ (right). The parameters are chosen as $T=0.2J,g=T$, and we take the two apparatuses to be identical. The plots clearly show how $P_{\uparrow \uparrow }^{\left(\mathbf{u}\right)}$ tends towards the corners whereas $P_{\downarrow \downarrow }^{\left(\mathbf{u}\right)}$ tends towards the center.