Weak and strong measurement of a qubit using a switching-based detector

We analyze the operation of a switching-based detector that probes a qubit’s observable that does not commute with the qubit’s Hamiltonian, leading to a nontrivial interplay between the measurement and free-qubit dynamics. In order to obtain analytical results and develop intuitive understanding of the different possible regimes of operation, we use a theoretical model where the detector is a quantum two-level system that is constantly monitored by a macroscopic system. We analyze how to interpret the outcome of the measurement and how the state of the qubit evolves while it is measured. We find that the answers to the above questions depend on the relation between the different parameters in the problem. In addition to the traditional strong-measurement regime, we identify a number of regimes associated with weak qubit-detector coupling. An incoherent detector whose switching time is measurable with high accuracy can provide high-fidelity information, but the measurement basis is determined only upon switching of the detector. An incoherent detector whose switching time can be known only with low accuracy provides a measurement in the qubit’s energy eigenbasis with reduced measurement fidelity. A coherent detector measures the qubit in its energy eigenbasis and, under certain conditions, can provide high-fidelity information.

Figure 1

Schematic diagram of the theoretical model that we use to analyze the operation of a switching-based detector. The qubit’s energy eigenstates are denoted by $|0⟩$ and $|1⟩$ and they have energy separation $E$. The detector is a two-state system (TSS) (with a nonswitched state $|N⟩$ and a switched state $|S⟩$) that is constantly monitored by a macroscopic system. The detector probes the operator ${\widehat{\sigma }}_{\mathbf{n}}$ of the qubit. $\chi$ represents the detector’s operator through which the detector is coupled to the qubit. The macroscopic system monitors whether the TSS remains in the initially prepared state $|N⟩$ or switches to the state $|S⟩$.

Figure 2

The measurement fidelity $F_s\left(t,\Delta t\right)$ as a function of the switching time $t$ for ${\gamma }_R/{\gamma }_L=10$ and $\beta =0$. The accuracy to which the switching time is known, $\Delta t$, is assumed to be much smaller than $1/E$. Therefore, the fidelity is independent of $\Delta t$ and is a function of $t$ only. Recall that ${\tau }_0\equiv \left(\text{log}{\gamma }_R-\text{log}{\gamma }_L\right)/\left({\gamma }_R-{\gamma }_L\right)$.

Figure 3

Overall fidelity as a function of the ratio of switching rates ${\gamma }_R/{\gamma }_L$ when the qubit Hamiltonian commutes with the probed qubit operator, i.e., in the special case $\beta =0$.

Figure 4

Maximum fidelity as a function of the angle between the qubit Hamiltonian and measurement axes, assuming that the exact switching time is not known (up to the measurement duration, which is taken to be much larger than $1/E$).

Figure 5

The $S$ curves (i.e., switching probabilities as functions of readout bias current with fixed pulse duration) for three different types of detectors: (a) a strongly coupled detector (see Sec. 4B), (b) a weakly coupled fast-decohering detector (see Sec. 4C), and (c) a weakly coupled slow-decohering detector (see Sec. 5). The dashed lines are the $S$ curves for the qubit’s clockwise- and counterclockwise-current states, i.e., the ones that are obtained when the qubit is biased far from the degeneracy point. In the example plotted in this figure, the probabilities for these $S$ curves (i.e., the dashed lines) are given by $P=1-e^{-{\gamma }_{L/R}t}$ with ${\gamma }_{L}t=e^{5\left(x-2\right)}$ and ${\gamma }_{R}t=e^{5x}$ ($x$ is a dimensionless parameter that represents the bias current used in the readout pulse and ${\gamma }_{L/R}$ represent the switching rates for the two different persistent-current states). The solid lines are the $S$ curves for the qubit’s energy eigenstates when these states have the form $|0⟩=\sqrt{0.7}|L⟩+\sqrt{0.3}|R⟩$ and $|1⟩=\sqrt{0.3}|L⟩-\sqrt{0.7}|R⟩$. In (a) the switching probabilities for the states $|L⟩$ and $|R⟩$ are averaged (in a weighted manner) in order to obtain the switching probabilities for the states $|0⟩$ and $|1⟩$. This reasoning is the one that would be consistent with the picture of a strong measurement. In (b) the switching rates are averaged and the switching probabilities are calculated from the averaged switching rates (see Sec. 4C). In (c) the parameter $x$ is averaged, and from the two weighted averages of $x$ the switching rates and probabilities are calculated (see Sec. 5). Here we are treating $x$ as the physical observable that is modified by the qubit at the Hamiltonian level.

Figure 6

The maximum measurement fidelity (i.e., at the optimal value of $x$) as a function of the angle $\beta$ (between the qubit Hamiltonian’s axis and the probed operator axis) for the three different types of detectors whose $S$ curves are plotted in Fig. 5. The fidelity is derived by finding the maximum vertical separation between the two $S$ curves for a given value of $\beta$. In (a) the fidelity is given by $\text{cos}\beta$, as expected for the textbooklike projective measurement. In (b) the fidelity is given by Eq. (51). In (c) the fidelity is calculated numerically. For further demonstration that the coherent detector’s fidelity can remain close to 100% even for $\beta \ne 0$, we plot in (c) (dashed line) the fidelity when the coefficient 5 in the switching-rate formula of Fig. 5 is replaced with the coefficient 10 (i.e., in going from the solid to the dashed line, the sensitivity of the detector is increased).