Retrieving High-Dimensional Quantum Steering from a Noisy Environment with $N$ Measurement Settings

One of the most often implied benefits of high-dimensional (HD) quantum systems is to lead to stronger forms of correlations, featuring increased robustness to noise. Here, we experimentally demonstrate the $n$-setting linear HD quantum steering criterion. We verify the large violation of the steering inequalities without full-state tomography. The lower bound of the violation is $2.24\pm 0.01$ in 11 dimensions, exceeding the bound ($V<2$) of two-setting criteria. Hence, a higher strength of steering has been revealed. Moreover, we demonstrate the method for enhancing the noise robustness without increasing dimension, alternatively, by increasing measurement settings. Using the entanglement in 11 dimensions, we experimentally retrieve steering nonlocality with $63.4\pm 1.4\%$ isotropic noise fraction, surpassing the 50% limitation of two-setting criteria. Our Letter offers the potential for practical one-sided device-independent quantum information processing that tolerates the noisy environment, lossy detection, and transcends the present transmission distance limitation.

Figure 1

Experimental setup. Pumper: diode laser centered at 405 nm; single mode fiber (SMF); periodically poled potassium titanyl phosphate (PPKTP); lens (${\mathrm{f}}_{1,2,3,4}$); dichroic mirror (DM); polarizing beam splitter (PBS); half-wave plate (HWP); spatial light modulator (SLM); high-pass optical filter (Filter); single-photon avalanche detector (SPAD).

Figure 2

Experimental results. (a) Experimental two-photons coincidence counts in computational basis ${|l_i⟩}_A{|l_j⟩}_B$ with $l_i,l_j=-5,\dots ,5$. The colored and transparent bars depict the coincidence counts with and without entanglement concentration. (b) Normalized two-photon coincidence rates in two 11-dimensional MUBs ($x=1$ and 2). When Alice’s projective measurement is $|{\phi }_x^a⟩$, Bob measures on the complex conjugation bases ${|{\phi }_x^b⟩}^{*}$ ($a,b=0,\dots ,10$). All the actual coincidence rates in 11 dimensions are in the Supplemental Material [43]. (c) Experimental results of the steering functional $S$ with dimension up to 11. The error bars are of the order of ${10}^{-2}$, much smaller than the marker size.

Figure 3

Experimental results of the bounds of the violation $V$ in the $n$-setting steering criteria with dimension up to 11. The error bars represent three standard deviations.

Figure 4

Experimental results of the noise threshold. (a)–(d) The steering functional of isotropic state $S_{\mathrm{iso}}$ for different mixing parameters in dimensions of $d=4$, 5, 7, 11. The blue inclined lines are fitted by the blue experimental points, and the blue dashed lines represent the range of error bars. The red lines are the theoretical prediction for the isotropic states $S_{\mathrm{iso}}=n[p+(1-p)/d]$, and the black horizontal lines represent the bounds of the LHS steering functional. The blue vertical lines denote the experimental lower bounds of the mixing parameter and their errors. (e) The theoretical and experimental values of the lower bound on mixing parameter $p_{\min }$ as a function of dimension $d$. The theoretical prediction of the two-setting criterion and its asymptotes are given for comparison.

Figure 5

Experimental results of the noise threshold $1-p_{\min }$ for different numbers of measurement settings $m$ in dimension $d=7$. For each $m$, the minimum and maximum experimental values are provided.