Direct Characterization of Quantum Measurements Using Weak Values

The time-symmetric formalism endows the weak measurement and its outcome, the weak value, with many unique features. In particular, it allows a direct tomography of quantum states without resorting to complicated reconstruction algorithms and provides an operational meaning to wave functions and density matrices. Here, we propose and experimentally demonstrate the direct tomography of a measurement apparatus by taking the backward direction of weak measurement formalism. Our protocol works rigorously with the arbitrary measurement strength, which offers improved accuracy and precision. The precision can be further improved by taking into account the completeness condition of the measurement operators, which also ensures the feasibility of our protocol for the characterization of the arbitrary quantum measurement. Our work provides new insight on the symmetry between quantum states and measurements, as well as an efficient method to characterize a measurement apparatus.

Figure 1

Experimental setup. (a) The pulsed laser at 830 nm first gets through a $\beta$-barium borate crystal (BBO) for the second harmonic generation and then inputs the potassium di-hydrogen phosphate (KDP) crystal for spontaneous parametric down-conversion (SPDC). Two photons are simultaneously generated in SPDC, one of which is used for heralding, and the other is collected to input the “direct tomography” module for the characterization of the unknown quantum measurement labeled “Measurement (I, II).” The modules “Measurement I” and “Measurement II” illustrate the experimental implementation of projective measurements and symmetric informationally complete positive-operator-valued measure (SIC POVM) in the polarization degree of freedom, respectively.

Figure 2

The experimental results for the characterization of “Measurement I.” (a) We plot the experimentally (points) estimated and theoretical (circles) ${\omega }_{j,s}^{(n)}$ and ${\omega }_{j,s}^{(n),\perp }$ by denoting the real and the imaginary parts of $\omega$ as the horizontal and the vertical axes, respectively. (b) In the Bloch sphere representation, the solid (dashed) arrows refer to the theoretical retrodicted states $|{\xi }_n⟩$ ($|{\xi }_n^{\perp }⟩$), and the points (cubes) represent the experimental retrodicted states $|{\psi }_n⟩$ ($|{\psi }_n^{\perp }⟩$). The red, green, blue, and yellow colors refer to $n=1$, 2, 3, 4, respectively. (c) The schematic error distributions of the measured $|{\psi }_3⟩$ and $|{\psi }_3^{\perp }⟩$ before (pink shaded area) and after (green shaded area) the normalization are shown in the cross section of the Bloch sphere. (d) The precision of the experimentally measured POVM (points) obtained from the Monte Carlo simulation with different $g$ is compared with the theoretical results (lines). The ${\delta }^2({\widehat{\mathrm{\Pi }}}_n)$ and ${\delta }^2({\widehat{\mathrm{\Pi }}}_n^{\perp })$ denote the variances of the POVM elements before the normalization. The variance ${\delta }^2({\widehat{\mathrm{\Pi }}}_n^{\prime })$ for both the POVM elements (${\widehat{\mathrm{\Pi }}}_n$ and ${\widehat{\mathrm{\Pi }}}_n^{\perp }$) after the normalization is the same. (e) The fidelities ($\mathcal{F}=|⟨\psi |\varphi ⟩{|}^2$ for $|\psi ⟩$ and $|\varphi ⟩$) between the experimental retrodicted states $|{\psi }_n⟩$ ($|{\psi }_n^{\perp }⟩$) and the theoretical states $|{\xi }_n⟩$ ($|{\xi }_n^{\perp }⟩$) for $n=1$, 2, 3, 4 are given in the table.

Figure 3

The experimental results for the characterization of “Measurement II.” (a) By regarding “Measurement II” as a rank-1 POVM, we illustrate the experimentally (solid markers) estimated and theoretical (hollow markers) ${\omega }_{j,s}^{(n)}$. (b) The experimentally derived pure retrodicted states (points) are compared with those of the ideal SIC POVM (arrows) in the Bloch sphere with fidelities 0.9781 (red), 0.9912 (green), 0.9882 (blue), 0.9984 (yellow) for $n=1$, 2, 3, 4, respectively. (c) The experimentally estimated (the cylinders) Dirac distributions of “Measurement II” are compared with those inferred from the results of conventional tomography (solid edges). The fidelities ($\mathcal{F}=[\mathrm{Tr}(\sqrt{\sqrt{{\rho }_1}{\rho }_2\sqrt{{\rho }_1}}){]}^2$ for ${\rho }_1$ and ${\rho }_2$) between the retrodicted states of the directly measured POVM with those of CT are 0.9994, 0.9994, 0.9999, 0.9997 for $n=1$, 2, 3, 4, respectively. The precision of the obtained POVM is illustrated in (d) (rank-1 situation) and (e) (rank-2 situation). In (d), we compare the precision of the experimentally measured POVM (points and triangles) obtained from the Monte Carlo simulation with the calculated precision of the ideal SIC POVM (dashed lines) and the estimated POVM (lines). In (e), the calculated precision inferred from the results of the CT (lines) and the ideal SIC POVM (dashed lines) coincide with each other. In both (d) and (e), ${\mathrm{\Delta }}^2$ in $\lg ({\mathrm{\Delta }}^2)$ of the $y$ axis represents the total variance of the POVM. The two insets show the precision of each POVM element for $g=-0.105$ and $g=0.733$, respectively. Here, we denote the precision of the directly measured results as “Primary” (blue) and that of the normalized results as “Normalized” (red).