One-way Einstein–Podolsky–Rosen steering beyond qubits

With the asymmetric states being targeted, we start to consider the way to characterize the steerability of the state in both directions, respectively. Indeed, one finds that characterizing steerability for infinite measurement settings is tricky, since the variable $\lambda$ in Eq. (2) could take infinitely many values [34]. However, if we limit the number of measurements and outputs to finite values, the problem would become computationally feasible [35, 36]. In particular, if we further adopt the ${d+1}$ mutually unbiased bases (MUB) as the measurements on Alice’s side—let’s consider Alice to steer Bob first, Alice performs ${d+1}$ measurements and obtains $d$ outcomes.

For each of the conditional state $\sigma_{a|x}$ on Bob’s side, where $a=\{0,...,{d-1}\}$ and $x=\{0,...,d\}$, we are interested in whether the following equation holds

where $\Lambda=d^{({d+1})}-1$, indicating the number of local states that required to construct the LHS model. Here $D(a|x,\lambda)$ is the deterministic single-party conditional probability distribution, which gives a fixed outcome $a$ for each measurement $x$.

It is developed and has been widely acknowledged that the above problem can be fully characterized using semidefinite programming (SDP). Indeed, the SDP approach defines the measure of steering called steering weight (SW) [35], which can be seen as the linear proportion of the steerable part of the given assemblage, i.e.,

where $1-\text{max}(\mu)$ is defined as the steering weight, and the maximization of $\mu$ can be computed via the SDP defined in Ref. [35].

Take the two-qutrit partially entangled states for example, in the following we show in detail how one-way steering manifests in the high-dimensional system. Specifically, the two-qutrit partially entangled states forms as

where $|\psi^{+}_{\theta,\phi}\rangle=\text{cos}(\theta)\text{sin}(\phi)|00\rangle+\text{sin}(\theta)\text{sin}(\phi)|11\rangle+\text{cos}(\phi)|22\rangle$, $p\in[0,1]$ $\theta\in[0,\pi/4]$ and $\phi\in[0,\pi/2]$. In Fig. 2, we present the distribution of the three parameters of two-qutrit partially entangled states demonstrating one-way steering. Here we use $p^{*}$ to work as the measure of steering, which gives the critical value upon which steering weight vanishes. The upper surface shows the critical value $p^{*}$ given certain $\theta$ and $\phi$, when considering Alice to steer Bob. Below the surface, the steering weight is non-positive, which indicates non-steerability. Only if the parameter $p$ is greater than $p^{*}$, which corresponds to the space above the surface, steering from Alice to Bob is certified.

On the other hand, the critical value for Bob to steer Alice turns out to form a plane, which means Bob can always steer Alice, as long as the proportion of the maximally entangled state is greater than 0.4818 (within numerical precision), and is irrelevant to the other two parameters. Therefore we conclude that the partially entangled states with parameters that are in the space above the plane and below the surface demonstrate one-way steering.

The above results are based on four-setting MUB (see results and discussions for two- and three-setting cases in Appendix A). Indeed, it is known that to conclusively characterize one-way steering, one needs to consider general measurements, since to certify non-steerability, it is required to verify if all the assemblages generated from the state admit a LHS model. If, for example, increasing the number of measurement settings, it is shown both theoretically [4, 37, 38] and experimentally [16] that one can turn non-steerability into steerability. The intuitive reason for this is that the extra measurements brings more terms of constraints to the LHS model in Eq. (5), thus limits the LHS model of $\{\sigma_{a|x}^{\text{LHS}}\}$ in Eq. (6) in reproducing the assemblage $\{\sigma_{a|x}\}$, which leads to the increased steering weight value.

For $d$-dimensional partially entangled state, it has been proven analytically [7] that the strict lower bound of the critical value considering Bob to steer Alice is $p^{*}_{B\rightarrow A}(d)=(H_{d}-1)/(d-1)$, where $H_{d}=1+1/2+\cdots+1/d$. This follows the idea that one can always transform the partially entangled state to the $d$-dimensional isotropic state via some filtering operation on Alice [5], without affecting the steerability from Bob to Alice [23].

Certifying the non-steerability in the opposite direction is rather difficult. Indeed, the authors in Refs. [39, 40] proposed the general methods for constructing the LHS model for given states in an asymptotically way. Considering $p^{*}_{B\rightarrow A}$ is independent of $\theta$ and $\phi$, using the aforementioned methods, one may expect that $d$-dimensional partially entangled states can always demonstrate one-way steering for certain combination of parameters, even when it comes to general measurements.

The main insight in finding the lower bound of $p^{*}_{A\rightarrow B}$ with respect to general measurements is to note that applying noisy measurements on a state (which could even be assumed non-physical), is equivalent, at the level of state preparation for Bob, to applying noise-free measurements on a noisy version of the state. Thus if the state admits a LHS model for the noisy measurements, then the noisy version of the state must admit a LHS model for the noise-free measurements. In qubit case specifically, one could start with the noisy measurements $M^{\eta}_{\pm|\hat{v}}=\left[\mathbbm{1}\pm\eta(\hat{v}\cdot\vec{\sigma})\right]/2$ with $\eta=0.79$, where $\vec{\sigma}$ are Pauli matrices, $\hat{v}$ gives the direction of Bloch vectors and 0.79 corresponds to the radius of the inscribed sphere of the icosahedron fitting inside the Bloch sphere. It is known that Werner state with visibility $V\lesssim 0.54$ admits a LHS model for the noisy measurements $M^{\eta=0.79}_{\pm|\hat{v}}$ [40], and thus its noisy version with $V\lesssim 0.54*0.79=0.43$ admits a LHS model for all projective measurements.

Following the terminology in Refs. [39, 40], we call $\eta$ the shrinking factor, which indicates to what extent the given noisy measurements could represent the general measurements in characterizing non-steerability. Then we solve the following SDP problem, which is the relaxation of the feasibility SDP presented in Refs. [39, 40], to detect the lower bound of $p^{*}_{A\rightarrow B}$ given general measurements:

where $O_{AB}$ is any hermitian operator, $\{M_{a|x}\}$ is the $d+1$ MUB, $\{D(a|x)\}$ is the deterministic distribution of the $d+1$ MUB and $\eta$ corresponds to the ratio between the radius of the inscribed sphere of the polytope formed by the $d+1$ MUB to the radius of the generalized unit sphere in $d^{2}-1$ dimensional space, which according to Ref. [46] is $1/\sqrt{(d^{2}-1)(d-1)}$. Note that this result applies only for prime-power dimension, and finding the optimal shrinking factor is always a tricky problem, even for 2-dimensional system—increasing the shrinking factor does not necessarily give a tighter bound of non-steerability [40].

Similar to finding the critical value $p^{*}$ in the previous case, instead of using the vanishing of steering weight, here we find $p^{*}$ upon $Q=1$, and $\rho_{AB}$ is sufficiently unsteerable to general measurements if $p\leq p^{*}$.

Again take two-qutrit case for example, we compute the distribution of $p^{*}_{A\rightarrow B}$, and the results are shown in Fig. 3. The curved surface ranging from approximately 0.1 to 1 corresponds to the computed $p^{*}_{A\rightarrow B}$, while the plane corresponds to the strict lower bound $p^{*}_{B\rightarrow A}$.

It is worth noting that the computed $p^{*}_{A\rightarrow B}$ only gives a sufficient bound for the non-steerability, and could be significantly improved if considering more refined measurement settings and the corresponding shrinking factors. Despite this, one still finds that two-qutrit partially entangled states are of distinct one-way steering zone, which indicates that it can always demonstrate one-way steering for certain combination of parameters, even when the steering party implementing general measurements.

For the higher-dimensional cases (of prime-power number, if adopting the MUB shrinking factor formula), we note that the SDP could become dramatically resource-demanding. Nevertheless, the method provides an elementary solution to estimating a reliable lower bound of non-steerability.

To conclude the non-steerability for a given state, one in principle is required to test infinite-setting measurements, which (without provoking ancillary dimension [28, 29]) is not feasible in the experiment. On the other hand, there is another prominent factor that affects one-way steering demonstration, which is the inevitable experimental losses. In photonic experiment for example, loss happens naturally, which manifests in no-click event being registered at the measurement devices. This should not be a problem if the device is trusted.

However, if the steering party, say Alice, is holding a cracked device, her measurement outcomes could be deliberately discarded, whenever the measurements she performs do not correspond to Bob’s announced measurements, thus faking the perfect correlations and passing the steering test. This opens the so-called detection loophole [41] which is also significant in Bell-nonlocality test [42].

From the view of LHS model, the discarded results grant more flexibility to the combination of local model in Eq. (5), which leads to the decreased steering weight value, thus indicating that steering is demolished—or equivalently, the criterion of demonstrating steering is pushed up. Indeed, the tradeoff relation between the losses (which is commonly characterized by the heralding efficiency $\epsilon$, describing the probability that the steering party heralds the result by declaring a non-null prediction) and the number of measurement settings has been investigated in two-qubit system [37, 16].

Here, we aim to develop a quantitative method to characterize the relation between the heralding efficiency and the number of measurement settings in the steering test for any finite-dimensional system. Then, we apply our method to the one-way steering characterization. The influence of losses in the steering test can be interpreted as the extra outcome for each measurement setting [43, 44]. Take the $d+1$ MUB measurement for example, Alice performs $d+1$ measurements, and in the loss-counted scenario, she obtains $d+1$ outcomes for each measurement setting.

We denote the extra outcome which corresponds to no-event being registered by $a=d$, as for now $a=\{0,...,d\}$. The assemblage in loss-counted scenario is called priori assemblage [42], which we denote as $\{\sigma^{\text{pri}}_{a|x}\}$.

With the priori assemblage, we naturally have

for the probability conservation, where $\rho^{B}$ is the reduced state of Bob. With this decomposition of $\{\sigma^{\text{pri}}_{a|x}\}$, one straightforward way to understand the loss-counted scenario is to consider Alice as the distributor, and she with some probability sends Bob a mixed state that will not be discerned by Bob. In the meantime, she claims that those outcomes from mixed state are lost (discarded), then she can convince Bob that she is indeed able to steer his state—we assume that $\{\sigma_{a|x}\}$ is a steerable assemblage—at a relatively low cost (with a discount of $\epsilon$), since preparing a steerable assemblage $\{\sigma_{a|x}\}$ is obviously harder than preparing a local mixed state $\rho^{B}$.

With this in mind, a decent way for Bob to deal with the losses is to assume that all the losses result from the trick of mixed state, but still believe that the rest of the experimental results are conclusive. Thus the LHS model in loss-counted scenario can be reformed as

where $\Lambda=({d+1})^{({d+1})}-1$.

It is worth noting that the normalization condition is automatically satisfied with $\text{Tr}\sum_{\lambda}\sigma_{\lambda}=1$, should the LHS model $\{\sigma^{\text{LHS}}_{a|x}\}$ can fully reproduce the given assemblage $\{\sigma_{a|x}\}$. Otherwise the LHS model contributes partially to the reconstruction of $\{\sigma_{a|x}\}$ as shown in Eq. (6). Thus we can straightforwardly write down the SDP to solve the steering weight with losses counted [45], namely, to maximize $\text{Tr}\sum_{\lambda}\sigma_{\lambda}$ over $\{\sigma_{\lambda}\}$, subject to the constraints in Eq. (9).

Note here we assume that the probability of loss, or equivalently the heralding efficiency $\epsilon$ is independent of the measurement $x$ that Alice performs, which means the action of discarding is completely random and not specified to particular measurement. However, if the efficiency $\epsilon$ is dependent on the choice of measurement, one can simply replace $\epsilon$ with $\epsilon(x)$ in the above expressions.

Indeed, there are several analytical works have discussed the steering test with losses are taken into account in two- and higher-dimensional systems [37, 43]. In Fig. 4, we present the comparison between our loss-counted model and the known analytical bounds on the cutoff value of visibility of the $d$-dimensional isotropic state versus loss. The circles denote our numerical results for the specific dimension, measurement setting and heralding efficiency. The authors in Ref. [43] derived bounds for dimension up to five given $d+1$ MUB, and the bounds are denoted in blue, orange, red and magenta respectively.

We can see clearly that our results indicate stronger loss-tolerance compared to the known bounds when loss is low, and are approaching to the theoretical limit along with the increasing of loss. This is reasonable since the bounds in Ref. [43] consider the asymptotic behavior of the cutoff value given $d$ and $\epsilon$, which leads to the inexact $p^{*}$ when loss is low. However, we note that a set of refined loss-tolerant steering inequalities from Ref. [43] can probably reproduce the tighter bound.

On the other hand, our results fully reproduced the optimal bounds (denoted in gray) derived in Ref. [38] for two-qubit case with two- and three-setting measurements, as indicated in the figure. It is worth noting that though the examples are tested with $d+1$ MUB in our work, the model we developed is applicable to any finite measurement settings, such as Platonic solid measurements adopted in Ref. [38]. In fact, the two- and three-setting MUB coincide with the square and octahedron measurements respectively.

It is important to note that the analytical bounds were derived specifically for symmetric states, and our method is applicable to any input state, which is shown as follows. In Fig. 5 we present the loss-counted one-way steering demonstration of two-qutrit partially entangled states, using two-, three- and four-setting MUB. For each setting of MUB, we test $\epsilon=1,0.85,0.65$ respectively. Note here we only select the cross section with $\theta=\pi/4$ and we only consider the loss on Alice’s side, and Bob’s device is assumed to be well-characterized—still, Bob cannot steer Alice if the state is unsteerable from his side.

One can see clearly the tradeoff relation between the detection efficiencies and the measurement settings on the steering threshold from Alice to Bob for the partially entangled states. Specifically, we see the state is two-way steerable at the point of $\phi=0.9553$ (within numerical precision [49]) with four measurement settings, and becomes completely one-way steerable when losses are counted. On the other hand, given certain amount of losses, increasing the number of measurement settings can effectively improve the loss-tolerance of Alice to steer Bob, thus giving a larger area of two-way steering.