Fundamental Bounds in Measurements for Estimating Quantum States

Quantum measurement unavoidably disturbs the state of a quantum system if any information about the system is extracted. Recently, the concept of reversing quantum measurement has been introduced and has attracted much attention. Numerous efforts have thus been devoted to understanding the fundamental relation of the amount of information obtained by measurement to either state disturbance or reversibility. Here, we experimentally prove the trade-off relations in quantum measurement with respect to both state disturbance and reversibility. By demonstrating the quantitative bound of the trade-off relations, we realize an optimal measurement for estimating quantum systems with minimum disturbance and maximum reversibility. Our results offer fundamental insights on quantum measurement and practical guidelines for implementing various quantum information protocols.

Figure 1

(a) Measurement $\{{\widehat{M}}_r\}$ is performed for estimating an unknown initial state $|\psi ⟩$. Information about $|\psi ⟩$ can be extracted by making a guess according to the measurement outcome $r$. However, $|\psi ⟩$ is inevitably disturbed by the measurement. (b) The post-measurement state $|{\psi }_r⟩$ can be probabilistically reversed to the initial state $|\psi ⟩$ by applying the reversing measurement $\{{\widehat{R}}_{r,l}\}$.

Figure 2

An arbitrary single-photon qutrit state is prepared by using half-wave plates (HWPs), quarter-wave plates (QWPs), and a beam displacer. Quantum measurement and measurement reversal are performed with additional partially polarizing beam splitters (PPBSs), which fully transmit the horizontally polarized photons but partially transmit the vertically polarized photons. The operations of PPBS1, PPBS2, and PPBS3 can be described as ${\widehat{X}}_{\mathrm{PPBS}}=|H⟩⟨H|+\sqrt{t}|V⟩⟨V|$ with, respectively, $t=p$, $(1-p)/2$, and $(1-p)/2p$. By combining HWPs and PPBSs, a general qutrit measurement operator $\widehat{Y}=\sqrt{t_0}|0⟩⟨0|+\sqrt{t_1}|1⟩⟨1|+\sqrt{t_2}|2⟩⟨2|$ can be implemented. Here, (a) and (b) correspond to Figs. 1 and 1, respectively. $G$ and $F$ are evaluated in (a), while $P_{\text{rev}}$ is obtained in (b). This figure shows the experimental setup for implementing ${\widehat{M}}_1^{(1)}$ and ${\widehat{R}}_{1,0}^{(1)}$ described in Eqs. (3) and (4). For other measurement operators, see [27].

Figure 3

An optimal guessing strategy is adopted for (a), and a nonoptimal guessing strategy is adopted for (b). In (b), the state guessing strategy is as follows. For ${\widehat{M}}_r^{(1)}$, we guess the initial state to be ${\rho }_r^{(\varphi )}=p|r⟩⟨r|+(1-p)/2(\mathbb{1}-|r⟩⟨r|)$. For ${\widehat{M}}_r^{(2)}$, we guess the initial state to be ${\rho }_0^{(\varphi )}=p|0⟩⟨0|+(1-p)|1⟩⟨1|$ for $r=0$ and ${\rho }_1^{(\varphi )}=|1⟩⟨1|$ for $r=1$. The solid lines represent the ideal trade-off relations for each operator set. Since the initial states used in experiment are not perfectly pure, the experimental data points lie slightly below the ideal trade-off relations. The dashed lines represent the theoretical trade-off relations assuming a nonideal input state $\rho (r)=r|\psi ⟩⟨\psi |+(1-r)\mathbb{1}/3$. $r=0.958$ and $r=0.969$ for ${\widehat{M}}_r^{(1)}$ and ${\widehat{M}}_r^{(2)}$, respectively. The error bars represent the statistical error of $\pm 1$ standard deviation.

Figure 4

(a) When the optimal guessing strategy is adopted, the $G-P_{\text{rev}}$ trade-off relation is linear. The red line is the ideal trade-off relation, $6G+P_{\text{rev}}=3$, for ${\widehat{M}}_r^{(1)}$. The blue line is the ideal trade-off relation, $12G+P_{\text{rev}}=5$, for ${\widehat{M}}_r^{(2)}$. (b) When the same nonoptimal guessing strategy adopted in Fig. 3 is used, the result shows that $P_{\text{rev}}$ cannot reach the bound even with ${\widehat{M}}_r^{(1)}$. The error bars represent the statistical error of $\pm 1$ standard deviation.