Weak-valued correlation functions: Insights and precise readout strategies

The correlation function in quantum systems plays a vital role in decoding their properties and gaining insights into physical phenomena. Its interpretation corresponds to the propagation of particle excitations between space-time, similar in spirit to the idea of quantum weak measurement in terms of recording the system information by interaction. By defining weak-valued correlation function, we propose the basic insights and the universal methods for recording them on the apparatus through weak measurement. To demonstrate the feasibility of our approach, we perform numerical experiments of perturbed quantum harmonic oscillators, addressing the intricate interplay between the coupling strength and the number of ensemble copies. Additionally, we extend our protocol to the domain of quantum field theory, where joint weak values encode crucial information about the correlation function. Hopefully, this comprehensive investigation can advance our understanding of the fundamental nature of the correlation function and weak measurement in quantum theories.

Figure 1

Weak-valued correlation function determined from Eq. (10) for different measurement strengths $g$ during the time $t\in [0,30)$ plotted as a function of (a) the real part and (b) the imaginary part. The true values of the correlation function are represented by a dark solid line, while the weak values are depicted as a series of colored dotted lines.

Figure 2

Gaussian kernel density distribution of the relative radical vector $z_{\text{RV}}=(z_{\text{R}}-z_{\text{TV}})/\left|z_{\text{TV}}\right|$ on the complex plane, for various numbers of copies $2N$ and coupling strengths $g$. The marginal density distributions are shown at the top and on the right-hand sides sides, providing additional information about the distribution characteristics. In the top right corner of each panel we indicate the optimal value of the coupling strength $g$ for each corresponding $N$.

Figure 3

Detailed trade-off data of optimal $g\in G=\{0.1,0.2,...,1\}$ for each $N\in [1,10000]$.

Figure 4

Numerical simulations of a ${\varphi }^4$ lattice field theory. The weak-valued effective Hermitian function $R_w$ of a ${\varphi }^4$ lattice field theory determined from Eq. (12) for different measurement strengths $g$ during the time $t\in \{-2,-1,...,2\}$ is plotted in (a) the real part, (b) the imaginary part, and (c) the module. The corresponding weak-valued correlation function obtained from $G(t,0)=R_w\langle \mathrm{\Theta }|\mathrm{\Omega }\rangle /\langle \mathrm{\Omega }|\mathrm{\Omega }\rangle$ is plotted in (d) the real part, (e) the imaginary part, and (f) the module. We plot the true values of the correlation function as dark dots, while weak values are plotted as colored crosses.

Figure 5

Correlation function simulated for different cutoffs of the energy level $N_{\text{cutoff}}$. The colored curves are used to show numerical simulation of solving the eigenequations of the Hamiltonians for the $N_{\text{cutoff}}$ at the bottom of (a) and (b). The black curve shows the result calculated from Eq. (A3) for a certain large enough cutoff energy level $N_{\text{cutoff}}=1000$, which is almost covered by the colored curves at large $N_{\text{cutoff}}$. Here we utilize complementary colors to explicitly enhance the distinction between $N_{\text{cutoff}}=5$ and $N_{\text{cutoff}}=6$.

Figure 6

Absolute mean values $|\overline{z}|$ are shown for $M=10000$ sampling points of (a) complex value $z_{\text{RV}}=(z_{\text{R}}-z_{\text{TV}})/\left|z_{\text{TV}}\right|$, (b) module $|z_{\text{RV}}|=|z_{\text{R}}-z_{\text{TV}}|/|z_{\text{TV}}|$, (c) real part $\text{Re}\left(z_{\text{RV}}\right)=\text{Re}(z_{\text{R}}-z_{\text{TV}})/\left|z_{\text{TV}}\right|$, and (d) imaginary part $\text{Im}\left(z_{\text{RV}}\right)=\text{Im}(z_{\text{R}}-z_{\text{TV}}\})/|z_{\text{TV}}|$. Error bars denote confidence intervals of 0.95. Uncertainty is plotted versus the number of copies $1/{\delta }^2-N$ for $g\in G=\{0.1,0.2,...,1\}$ over $M=10000$ sampling points of (e) complex value $z_{\text{RV}}=(z_{\text{R}}-z_{\text{TV}})/\left|z_{\text{TV}}\right|$, (f) module $|z_{\text{RV}}|=|z_{\text{R}}-z_{\text{TV}}|/|z_{\text{TV}}|$, (g) real part $\text{Re}\left(z_{\text{RV}}\right)=\text{Re}(z_{\text{R}}-z_{\text{TV}})/\left|z_{\text{TV}}\right|$, and (h) imaginary part $\text{Im}\left(z_{\text{RV}}\right)=\text{Im}(z_{\text{R}}-z_{\text{TV}})/\left|z_{\text{TV}}\right|$. The relation $1/{\delta }^2-N$ proves to be linear in all four cases.

Figure 7

Comparison of the one-lattice ${\varphi }^4$ theory with the PQHO weak-valued correlation function for different measurement strengths $g$ during the continuous time $t\in [0,30)$. Simulations of weak measurements in the QFT are shown for (a) the real part, (b) the imaginary part, and (c) the module and in QM for (d) the real part, (e) the imaginary part, and (f) the module.