Direct measurement of general quantum states using strong measurement

The direct state measurement (DSM) based on the weak measurement has the advantage of simplicity, versatility, and directness. However, the weak measurement will introduce an unavoidable error in the reconstructed quantum state. We modify the DSM by replacing the weak coupling between the system and the pointer by a strong one, and present two procedures for measuring quantum states, one of which can give the wave function or the density matrix directly. We can also measure the Dirac distribution of a discrete system directly. Furthermore, we propose quantum circuits for realizing these procedures, and the main body of the circuits consists of Toffoli gates. By numerical simulation, we find that our scheme can eliminate the biased error effectively.

Figure 1

Implementation of the coupling between the system initialized in the state $\rho$ (or the pure state $\left|\psi \right.〉$) and the pointer initialized in the state $\left|0\right.〉$, followed by measurements on system and then on pointer. We choose $\left|c_{\alpha }\right.〉$ the usual computational basis; its complementary basis $\left|a_k\right.〉$ is a product state, which can be transformed to $\left|c_{N-1}\right.〉=\left|11\cdots 1\right.〉$ by local rotations. So the unitary coupling $U_k$ consists of local rotations $R_m$ ($R_m^{†}$) and an ($n+1$)-qubit Toffoli gate. A $z$ direction projective measurement is performed on each qubit of the system. For pure states, only $\left|c_0\right.〉$ is postselected; others will be discarded. For general $\rho$, postselection is not needed; all the results provide data that is employed in the reconstruction. On pointer qubit, we need to measure the expectations of ${\sigma }_x,{\sigma }_y,{\sigma }_z$, respectively.

Figure 2

Comparison between the generalized DSM (gDSM) and the standard tomography (QST) for both pure and mixed states of one qubit. The trace distance is plotted as a function of the total number of copies $N_c$ of the system employed in the reconstruction. The pure state is $\rho =\frac{I+1/\sqrt{3}{\sigma }_x+1/\sqrt{3}{\sigma }_y+1/\sqrt{3}{\sigma }_z}{2}$ and the mixed state is $\rho =\frac{I+0.9/\sqrt{3}{\sigma }_x+0.9/\sqrt{3}{\sigma }_y+0.9/\sqrt{3}{\sigma }_z}{2}$. The details of the gDSM are described in the text. The QST is just to measure the expectation of ${\sigma }_x,{\sigma }_y,{\sigma }_z$ of the unknown state, respectively. Because of the removing of the bias, the trace distance of the gDSM continues to decrease when $N_c$ is increasing, as of the QST. For smaller $Nc$ the trace distance has large statistical fluctuation.

Figure 3

Circuit for implementing a direct measurement of the Dirac distribution and the density matrix. The top qubit is the ancillary qubit initialized in state $\frac{|0\rangle +\left|1\right.〉}{\sqrt{2}}$, the bottom lines denote the system with an unknown state $\rho$. One to three controlled-$U$ gates may be required depending on different purposes. If the expectations of ${\sigma }_x$ and ${\sigma }_y$ of the pointer qubit are measured after the first controlled-$U$ gate, the population of the state $\rho$ will be obtained. If the same is performed after the second controlled-$U$ gate, the Dirac distribution of the state $\rho$ will be obtained. By measuring the expectation of ${\sigma }_x$ and ${\sigma }_y$ after the third controlled-$U$ gate, we can obtain the elements of the density matrix $\rho$.