## Abstract

Genuine high-dimensional (HD) quantum steering was recently proposed for certifying HD entanglement in the one-sided device-independent setting, as quantified by the Schmidt number [Phys. Rev. Lett. **126**, 200404 (2021) [CrossRef] ]. As one of the central challenges in the steering test is the tolerated noise threshold, here we develop a more robust method for certifying genuine HD steering in noisy environments. We derive genuine HD steering criteria using multiple mutually unbiased bases, surpassing the previous restriction of only two measurement settings. Although the present multisetting criteria are not tight, we theoretically show that they are more robust to noise than the tight two-setting criteria. To test the practicality of our method under realistic conditions, we report an experimental demonstration using photonic orbital angular momentum entangled states in dimensions $d = 3,4,5$. Our work offers a more robust way to witness the entanglement dimension in practical one-sided device-independent quantum information processing.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. INTRODUCTION

Entanglement [1], quantum steering [2], and Bell nonlocality [3], the three types of quantum correlations, play a pivotal role in the development of quantum communication [4]. The above three concepts are nonequivalent, forming a hierarchy where states exhibiting steering are a strict subset of entangled states, and a strict superset of Bell nonlocal states [5]. Among these quantum correlations, steering is rather unique in its inherent asymmetry [6], providing a way to certify entanglement with the trustworthy requirement of measurement devices on only one side [7–9]. During the past decade, steering has attracted increasing attention for potential applications in hybrid quantum networks with both trusted and untrusted components [10–16].

Complex entangled systems involving multilevel quantum particles open many interesting perspectives [17,18]. Protocols based on higher dimensions possess the advantages of larger channel capacity [19–21], stronger noise resistance [22–24], and enhanced security against attacks [25,26]. Remarkably, high-dimensional (HD) quantum steering has been experimentally generated recently in photonic systems encoding in various degrees of freedom [27–31].

Based on the available prior information about quantum systems, the Schmidt number of HD entanglement can be quantified in different contexts [32]. Quantum steering criteria have been derived via many means, such as entropic uncertainty relations [33–35] and semidefinite programming (SDP) [8,10]. However, both theoretical [7] and experimental [27–30] tests of steering developed so far mostly characterize unsteerable states, thus verifying the states exhibiting steering by violating these descriptions. These methods witness only the presence of steering (certify a Schmidt number $n \gt 1$), but fail to capture its effective dimension in the one-sided device-independent (1SDI) setting. Recently, by developing the notion of genuine HD steering, Designolle *et al.* exploited a way to characterize the dimension of HD steering [31]. They derived a genuine HD steering inequality using pairs of mutually unbiased bases (MUBs) and experimentally demonstrated it with photons entangled in their discretized transverse position momentum. Quantum measurements based on MUBs are generally viewed as being “maximally noncommutative” and “complementary” [36].

One of the most crucial problems in quantum steering is the level of tolerated noise threshold [7,8]. Here we propose a more robust method to certify genuine HD steering in noisy environments. Considering a more complex scenario with multiple MUB measurements, we derive multisetting steering inequalities for certifying the presence of genuine HD steering. Even though the multisetting criteria are not tight, we show that they are more robust against white noise compared with the tight two-setting genuine steering criteria derived in Ref. [31]. Then, to verify this advantage under realistic conditions, we experimentally prepare the simplified isotropic states encoding in the orbital angular momentum (OAM) degree of freedom in dimensions $d = 3,4,5$ and perform projective measurements onto MUBs. The experiment results are in good agreement with theoretical expectations and show that one can demonstrate a higher genuine steering dimension using the multisetting criteria in noisy environments. Hence, this would lead to a more robust protocol for certifying the lower bound on entanglement dimensions in the 1SDI setting.

## 2. THEORETIC FRAMEWORK

Assuming a bipartite state ${\rho _{\textit{AB}}}$ shared by Alice and Bob, two space-like separate observers, Alice implements the measurement ${M_{a|x}}$ on her subsystem (labeled by $x = 1,\ldots, m$) and receives a result $a \in \{{1,\ldots, d} \}$, where ${M_{a|x}}$ denotes positive operators satisfying $\sum\nolimits_a {M_{a|x}} = \unicode{x1D7D9}1$ for each $x$. The unnormalized post-measurement states prepared for Bob are given by

The set ${\{{{\sigma _{a|x}}} \}_{a,x}}$ is called an assemblage. Then, Bob performs the trusted measurements on his subsystem and confirms the members of the assemblage he holds.If the source distributes a separable state, denoted $\rho = \sum\nolimits_i {p_i}\rho _i^A \otimes \rho _i^B$, Alice can generate only the conditional states at Bob’s side that possess the local hidden state (LHS) model [5]

We consider a general ($m,d$) scenario where $d$ is the dimension of the Hilbert space and each of the two observers chooses $m$ sets of measurements, each of which can result in one of $d$ outcomes. Alice may generate assemblages for Bob that could not be obtained via any lower-dimensional entanglement. In Ref. [31], the authors formalized them by the notion of genuine HD steering. An assemblage features genuine $n$-dimensional steering (with $1 \le n \le d$) when it admits the decomposition in Eq. (1) with a shared state having at most $n$-dimensional entanglement ${\rho _{\textit{AB}}} = \sum\nolimits_l {p_l}| {{\phi _l}} \rangle \langle {{\phi _l}} |$, where all ${\phi _l}$ are pure entangled states with the Schmidt number at most $n$. In particular, any one-preparable assemblage can be produced via a state with Schmidt number $n = 1$, i.e., a separable state. As $(n - 1)$-preparable assemblages form a strict subset of $n$-preparable assemblages, a nested structure naturally emerges, shown in Fig. 1. So we can see that standard steering criteria allow for characterization of only one-preparable assemblages, detect steering by violating this description, but fail to capture the effective steering dimension.

A crucial problem that arises when carrying out a steering test is the level of tolerated noise threshold [7,8]. While in Ref. [31], the authors consider only a ($2,d$) scenario, here we consider a general ($m,d$) scenario and derive multisetting criteria for certifying genuine steering, which has better robustness in noisy environments. We first introduce a notion of steering robustness, measuring the general noise threshold of a state assemblage ${\sigma _{a|x}}$ before becoming unsteerable [10], under the form of SDP [8]:

Incompatibility robustness is a similar quantifier to describe incompatibility of quantum measurements, measuring the generelized noise threshold of measurements before becoming jointly measurable [40]:

Then, to evaluate the multisetting criteria, we consider $d$-dimensional isotropic states ${\rho _{{\text{iso}}}} = p| {{\Phi _d}} \rangle \langle {{\Phi _d}} | + (1 - p)\unicode{x1D7D9}/{d^2}$, where $| {{\Phi _d}} \rangle$ is the $d$-dimensional maximally entangled state (MES) and $\unicode{x1D7D9}$ denotes the identity matrix of dimension ${d^2}$. $p \in [0, 1]$ is the mixing parameter, while here we denote ${\text{NF}}$ as the fraction of white noise, that is, ${\text{NF}} = 1 - p$. Alice still performs projective measurements ${M_{a|x}}$ and the assemblage created for Bob is

Thus, the lower bound of steering robustness for the isotropic state is given by Combined with Eq. (7) and Eq. (9), we can obtain the lower bound of witness ${\delta _m}({\sigma _{a|x}})$ as a function of noise fraction NF for $m \ge 3$ and $m = 2$, respectively, certifying at least genuine $n= \lfloor {{\delta }_{m}}({{\sigma }_{a|x}})+1\rfloor$-dimensional steering. We theoretically show the functions of multisetting and two-setting criteria for fixed dimensions $d = 3,4,5$ in Figs. 2(a)–2(c). As we can see, with the same noise fraction NF, the values of the lower bound of witness ${\delta _m}({\sigma _{a|x}})$ obtained by multisetting criteria are larger in the gray colored regions, which gives rise to potentially higher certified genuine steering dimensions $n$. On the other hand, Fig. 2(d) shows the noise thresholds for $d$-dimensional isotropic states exhibiting genuine $n$-dimensional steering as functions of $m$, which are obtained from the intersections of the black dotted lines with the curves in Figs. 2(a)–2(c). The noise thresholds almost increase as $m$ increases, which means that the noise resistance of genuine HD steering enhances. Note that there is one point declining due to the looseness of the present multisetting criteria. Even though $\delta ({\sigma _{a|x}})$ cannot reach $d + 1$ in the loose multisetting criteria, it is still straightforward to show that multisetting criteria offer a more robust method to detect genuine HD steering in noisy environments.Furthermore, during submission of the paper, we noticed that a tighter bound has been derived [44]. The improvement occurs for $m$ not too large with a fixed $n$. We also evaluate this tighter bound in the experiment.

## 3. EXPERIMENTAL DETAILS AND RESULTS

To test the practicability of the multisetting criteria in realistic circumstances with actual noise, we implement an experiment based on photons entangled in their OAM degree of freedom. Our experimental setup is sketched in Fig. 3. A CW 405 nm diode laser is first spatially filtered by a single-mode fiber (SMF). The output beam pumps a 5 mm long periodically poled potassium titanyl phosphate (ppKTP) crystal via a 250 mm lens L1, and generates the original OAM entangled photon pairs with orthogonal polarization at 810 nm through the type-II spontaneous parametric downconversion (SPDC) process. A dichroic mirror (DM) is used to remove the pump. Then the photons are recollimated by a 100 mm lens L2, separated by a polarization beam splitter (PBS) and manipulated by the first spatial light modulator (SLM1), where a designed animation is loaded to construct tunable isotropic states statistically [28]. Each SLM2 is displayed with computer-generated holograms (CGHs) to covert the photons carrying OAM or OAM-MUB modes to the fundamental Gaussian mode. They are effectively coupled to the SMFs by a group of coupling lenses. The SMFs are connected to single-photon avalanche detectors, whose output is subsequently fed into a coincidence circuit with a coincidence time window of 1.6 ns. The steering robustness ${\text{SR}}({\sigma _{a|x}})$ is estimated via the coincidence count numbers for MUB measurements, and then used to evaluate Eq. (7) and Eq. (9).

The original OAM entangled state produced by the SPDC process are not a MES owing to the limited spiral bandwidth [45], expressed as $| \Phi \rangle | = \sum\nolimits_{- \infty}^\infty {c_l}|{l \rangle _A}|{{- l} \rangle _B}$, where the mode amplitudes ${c_l}$ vary for the different index $l$. To tailor a MES, we use the local operation to concentrate entanglement, which is called the Procrustean method. This is done by altering the diffraction efficiencies of blazed phase gratings to equalize the amplitudes for different OAM modes. And to test within $d$-dimensional subspace, for odd $d$, we choose the modes $l$ from ${-}(d - 1)/2$ to $(d - 1)/2$ as the computational basis. For even $d$, we choose $l$ from ${-}d/2$ to $d/2$, omitting the $l = 0$ mode.

Next, we explain how to design an animation loaded on SLM1 to construct an isotropic state with a specific mixing parameter $p$. Since there is no need to implement full-state tomography, here we considered a simplified isotropic noise for the $d + 1$ sets of MUBs [28]:

In the experiment, we consider the general scenarios $(m,d)$, where $d = 3,4,5$ in the OAM subspace and $m = 2,\ldots,d + 1$. We set different noise fractions in different dimensions by altering different mixing parameters $p$ of the animation. Note that choosing $m$ MUBs from the complete set of $d + 1$ MUBs in prime power dimension, there are $C_{d + 1}^m$ possible subsets of them, which may exhibit different certified genuine steering dimensions owing to the different visibility in a practical experiment. In Table 1, we show the experimental values of the maximum lower bounds of $\delta ({\sigma _{a|x}})$ for $m \lt d + 1$, the definite bounds of $\delta ({\sigma _{a|x}})$ for $m = d + 1$, and their corresponding certified Schmidt number $n= \lfloor \delta ({{\sigma }_{a|x}})+1 \rfloor$. As we can see, the lower bounds of $\delta ({\sigma _{a|x}})$ steadily increase as the number of measurement settings $m$ increases in each case, which means potentially higher Schmidt number $n$ can be certified with more measurement settings. For instance, a Schmidt number $n = 3$ can be certified when we use $m = 4$ measurement settings in dimension $d = 3$ and noise fraction ${\text{NF}} = 0.10$. However, using $m = 2$ and 3 measurements in this case, we obtain the maximum bound of $\delta ({\sigma _{a|x}}) \ge 1.8 \pm 0.1$ and $\delta ({\sigma _{a|x}}) \ge 1.9 \pm 0.1$, respectively, which both demonstrate only genuine two-dimensional steering (Schmidt number $n = 2$). We also notice a particular case ${\delta _2} \gt {\delta _3}$ in $d = 5$ and ${\text{NF}} = 0.08$. This is still consistent with the theoretical prediction in Fig. 2(d) because of the looseness of the present multisetting criteria. Furthermore, we also tested the experimental values of the noise thresholds in Fig. 2(d) (see Supplement 1 for more details). In conclusion, it is clear from our results that by increasing measurement settings, one can detect genuine HD steering more robustly in noisy environments.

## 4. CONCLUSION

In summary, we have developed multisetting genuine HD steering criteria that give a dimension certificate in the 1SDI process, going beyond the previous restriction of only two settings. Although the present genuine steering witnesses for more than two measurement settings are not tight, we theoretically analyze their noise resistance via isotropic states and find that they still offer higher robustness compared with the tight two-setting criteria derived in Ref. [31]. We have carried out an experiment to demonstrate the practicality of multisetting criteria by introducing tunable isotropic noise into the OAM entanglement system. The experimental results show that a higher certified genuine steering dimension can be observed using more measurement settings. It is worthy to note that our method is an incremental step towards accurately witnessing genuine HD steering but at a cost of more measurement settings. Furthermore, the multisetting method is not limited to transverse modes, and also readily applies to other quantum platforms in different degrees of freedom. Our work offers a more robust way to witness the entanglement dimension in practical 1SDI quantum information processing.

## Funding

China Postdoctoral Science Foundation (2020M673366); National Natural Science Foundation of China (11534008, 11804271, 12074307, 12174301, 91736104, 92050103); State Key Laboratory of Applied Optics.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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