Detecting high-dimensional entanglement by randomized product projections

Abstract

The characterization of high-dimensional entanglement plays a crucial role in the field of quantum information science. Conventional entanglement criteria measuring coherent superpositions of multiple basis states face experimental bottlenecks on most physical platforms due to limited multi-channel control. Here, we introduce a practically efficient detection strategy based on randomized product projections. We show that the first-order moments of such projections can be used to estimate entanglement fidelity, thereby enabling practical and efficient certification of the Schmidt number in high-dimensional bipartite systems. By constructing optimal observables, it is sufficient to merely measure a single basis state, substantially reducing experimental overhead. Moreover, we present an algorithm to obtain a lower bound of the Schmidt number with a high confidence level from a limited number of experimental data. Our results open up resource-efficient experimental avenues to detect high-dimensional entanglement and test its implementations in modern information technologies.

Figures (6)

• Figure 1: Illustration of randomized product projections for estimating the Schmidt number of a bipartite state $\varrho$. The first stage implements local operations $U_l^{\otimes2}$ and $O_l^{\otimes2}$ and then measures on the well-constructed observable $\mathcal{M}^{\otimes2}$. The random unitaries $U_l$ and random orthogonal matrices $O_l$ are sampled independently from the Haar measure on unitary and orthogonal groups goodman2000representationscollins2006integrationgross2007evenlydankert2009exact. From the obtained expectation values $E_{U}^{l}$ and $E_{O}^{(l)}$, the algorithm outputs the Schmidt number estimation $\mu_{\mathrm{est}}$.
• Figure 1: Test for algorithm based on Bootstrapping resampling technique. Numerical results for the states $\varrho_{\mathrm{iso}}^v$ with dimension (a) $d=20$ and (b) $d=30$ by fixing $N=15$ and the CL $99.9\%$. We set the number of resampling $B=2000$.
• Figure 2: (a-d) shows the distribution of estimated $\mathrm{SN}$ lower bound $\mu_{\mathrm{est}}$ of the state $\varrho_{\mathrm{iso}}^v$ with $d=20$ as a function of the number of $N$, where (a) $v=0.95$, (b) $v=0.77$, (c) $v=0.52$, and (d) $v=0.3$. For each $N$, the expectation values of product observables are exactly calculated, and the CL is $99.9\%$. Running the $\mathrm{SN}$ estimation algorithm $N_{\mathrm{iter}}=500$ times, we collect $N_{\mathrm{iter}}$ estimators, $\{\mu_{\mathrm{est}}^{(l)}\}_{l=1}^{N_{\mathrm{iter}}}$, in which the minimal and maximal values are denoted as $\mu_{\mathrm{est}}^{\min}$ and $\mu_{\mathrm{est}}^{\max}$. $\mu_{\mathrm{fid}}$ denotes the $\mathrm{SN}$ estimated by the fidelity-based witness. (e,f) shows the error bar of the fidelity estimation for the state $\varrho_{\mathrm{iso}}^{v}$ with CL $99.9\%$ and $N=30$ random operations. (e) $d=20$ and (f) $d=30$. $f_{\mathrm{est}}$ and $f_{\mathrm{exa}}$ denote the estimated and exact fidelities. The expectation values are calculated exactly.
• Figure 2: Numerical results for the states $\varrho_{\mathrm{iso}}^v$ with dimension (a) $d=10$, (b) $d=15$, (c) $d=40$, (d) $d=50$, (e) $d=60$, (f) $d=80$. For (a,b), the number of measurements for each product observable is $K=1000$. For (c-f), we directly calculate the expectation values of observables. In (a,b,d,e,f), the CL is $99\%$.
• Figure 3: Results for the states $\varrho_{\mathrm{iso}}^v$ with (a) $d=20$, (b) $d=30$, (c) $d=40$, and $d=50$ by fixing $N=30$ and the CL $99.9\%$. After performing the $\mathrm{SN}$ estimation algorithm $N_{\mathrm{iter}}=500$ times, we collect $N_{\mathrm{iter}}$ estimators, $\{\mu_{\mathrm{est}}^{(l)}\}_{l=1}^{N_{\mathrm{iter}}}$, in which the minimal and maximal values are denoted as $\mu_{\mathrm{est}}^{\min}$ and $\mu_{\mathrm{est}}^{\max}$. $\mu_{\mathrm{fid}}$ denotes the $\mathrm{SN}$ estimated by the fidelity-based witness. $3$-MUBs denote the results of the criterion by using $3$ MUBs morelli2023resource. $\mu_{\mathrm{fid}}$ is the $\mathrm{SN}$ estimated by the fidelity-based witness. The expectation values are calculated exactly.

Theorems & Definitions (3)

• Corollary 1
• proof
• proof