Suppressing technical noise in weak measurements by entanglement

Postselected weak measurement has aroused broad interest for its distinctive ability to amplify small physical quantities. However, the low postselection efficiency to obtain a large weak value has been a big obstacle to its application in practice since it may waste resources, and reduce the measurement precision. An improved protocol was proposed in Pang et al., Phys. Rev. Lett. 113, 030401 (2014) to make the postselected weak measurement dramatically more efficient by using entanglement. Such a protocol can increase the Fisher information of the measurement to approximately saturate the well-known Heisenberg limit. In this paper, we review the entanglement-assisted protocol of postselected weak measurement in detail, and study its robustness against technical noises. We focus on readout errors. Readout errors can greatly degrade the performance of postselected weak measurement, especially when the readout error probability is comparable to the postselection probability. We show that entanglement can significantly reduce the two main detrimental effects of readout errors: inaccuracy in the measurement result and the loss of Fisher information. We extend the protocol by introducing a majority vote scheme to postselection to further compensate for readout errors. With a proper threshold, almost no Fisher information will be lost. These results demonstrate the effectiveness of entanglement in protecting postselected weak measurement against readout errors.

Figure 1

Quantum circuit to initialize the $n$ system qubits. The qubits are prepared in the maximally entangled state $|{\mathrm{\Psi }}_i{\rangle =\left(\right|0\rangle }^{\otimes n}+{|1\rangle }^{\otimes n})/\sqrt{2}$ by a sequence of cnot gates.

Figure 2

Quantum circuit equivalent to the weak interaction $H_{\mathrm{int}}=\phi {\widehat{\sigma }}_z\otimes {\widehat{\sigma }}_x,\phi \ll 1,$ between $n$ system qubits and one common pointer qubit.

Figure 3

Quantum circuit to postselect the $n$ system qubits in the entangled state $|{\mathrm{\Psi }}_f\rangle$ [Eq. (66)], which is in essence the reverse process of preparing $|{\mathrm{\Psi }}_f\rangle$ from ${|0\rangle }^{\otimes n}$.

Figure 4

Plot of the relative error probability $p{\left(\mathrm{error}\right||0\rangle }^{\otimes n})$ versus the weak value $A_w$ for different $n$. The lines from left to right in each figure are for $n=2,3,4,5,6,$ respectively. (a) When $q_{1\to 0}=q_{0\to 1}=0.05$, the $p{\left(\mathrm{error}\right||0\rangle }^{\otimes n})$ for $n=4,5,6$ are much lower than $n=2,3$, and they almost overlap since they are very close to each other. (b) Similarly, when $q_{1\to 0}=q_{0\to 1}=0.01$, the $p{\left(\mathrm{error}\right||0\rangle }^{\otimes n})$ for $n=3,4,5,6$ are much lower than $n=2$, and they almost overlap with each other.

Figure 5

This figure plots the relative error probability $p{\left(\mathrm{error}\right||0\rangle }^{\otimes n})$ versus the number of entangled qubits $n$ for different weak values $A_w$. It shows that the relative error rate $p{\left(\mathrm{error}\right||0\rangle }^{\otimes n})$ can decrease very fast with $n$, which implies the advantage of entanglement in suppressing the relative error rate.

Figure 6

Plot of the complementarity relation between the maximum relative error probability $p{\left(\mathrm{error}\right||0\rangle }^{\otimes n})$ and the loss rate $r_{\mathrm{loss}}$ for different weak values $A_w$. The lines from left to right in each figure are for $A_w=20,50,100,150,\infty ,$ respectively. The $r_{\mathrm{loss}}$ axis in (b) is rescaled.

Figure 7

This figure plots the correction factor $\gamma \left(n\right)$ of the measurement result versus the number of entangled qubits $n$ for different weak values $A_w$. It can be seen that the correction factor can increase to 1 very fast with $n$, which shows the effectiveness of entanglement in protecting the weak value amplification against readout errors.

Figure 8

This figure plots the modification factor $f\left(n\right)$ of the Fisher information versus the number of entangled qubits $n$ in the presence of readout errors, but without majority voting, for different weak values. When $n$ is small, the Fisher information can increase with $n$ because the entanglement eliminates some errors in the postselection results and raises the proportion of correct postselected states that have higher Fisher information. However, when $n$ becomes larger, the Fisher information starts to fall since the loss rate of correct postselected states dramatically increases with $n$ in this case.

Figure 9

This figure plots the modification factor $f(n,k)$ of the Fisher information versus the allowed number of readout errors $k$ for different $n$ when the majority voting scheme is used. The number of allowed readout errors $k$ is from 0 to 3. The weak value is $A_w=30$. The figure shows that when $k$ climbs to half of $n,f(n,k)$ can be very close to 1. It implies that with a proper number of readout errors allowed in postselecting the $n$ qubits, the majority voting strategy can recover almost all the Fisher information lost by the readout errors.

Figure 10

This figure plots the correction factor $\gamma (n,k)$ of the measurement result versus the allowed number of readout errors $k$ for different $n$, when the majority voting scheme is employed. The weak value is $A_w=30$. The lines for $n=8,9,10$ almost overlap since they are very close to each other. The figure shows that when $k$ increases, a small drop in the measurement result may occur, due to more errors being induced by the majority voting scheme.