# One-way Einstein–Podolsky–Rosen steering beyond qubits

Qiang Zeng<EMAIL_ADDRESS>Beijing Academy of Quantum Information
Sciences, Beijing 100193, China  Key Laboratory of Advanced Optoelectronic
Quantum Architecture and Measurement of Ministry of Education, School of
Physics, Beijing Institute of Technology, Beijing 100081, China  Centre for
Quantum Computation and Communication Technology (Australian Research
Council), Centre for Quantum Dynamics, Griffith University, Brisbane,
Queensland 4111, Australia

###### Abstract

Quantum steering has been exploited as an important resource in modern quantum
information processing. Owing to its directional nature, some quantum states
that are asymmetric under the exchange of parties have been found to manifest
steering only in specific direction, thus called one-way steering. Existing
works focused on one-way steering in systems of qubits. Here we propose a
family of two-party states that are one-way steerable in systems of
$d$-dimension. In particular, we validate the one-way steerability of the
states for $d=3$, and demonstrate how one-way steering parameter space
manifests in two-qutrit system. A general numerical approach for
characterizing higher-dimensional one-way steering is provided. Moreover, we
develop a method to characterize one-way steering with the experimental loss
taken into account, with which the tradeoff relation between losses and
measurement settings in steering test in higher-dimensional system is
investigated. Our loss-counted model works for finite-dimensional system with
finite measurement settings.

##  I Introduction

Quantum entanglement, which has a clear mathematical structure (that is, the
state of the system is non-factorizable), is seen as a counter-intuitive
effect exemplified by real physical systems. This led to the dispute on the
completeness of quantum mechanics that was raised from the famous
Einstein–Podolsky–Rosen argument [1], and the dispute was later reformed into
the discussion and certification of the violation of Bell inequalities [2].
Indeed, it is not revealed until recently that quantum entanglement, in the
view of correlation, has far richer structure between non-factorizable and
Bell-nonlocal [3, 4, 5].

The intermediate form of entanglement structure is named steering [6], which
describes the effect that one of the entangled parties, by performing local
measurements, can affect the local state of another remote party. Different
from the other two entanglement structures, quantum steering is defined in the
asymmetric form, as the steering and the steered parties are distinct and
cannot be interchanged. To capture the essence of quantum steering, one can
interpret it in the view of theoretical information tasks, where steering is
demonstrated if the steered ensemble cannot be explained by a local hidden
state (LHS) model [7].

As steering has direct correspondence to information tasks, various
applications of steering have been found, such as one-sided device-independent
quantum key distribution [8], randomness certification [9, 10], subchannel
discrimination [11], device-independent quantification of measurement
incompatibility [12], secure quantum teleportation [13] and secret sharing
[14].

In particular, the asymmetry of steering plays a pivotal role in those
applications, which manifests in the different levels of trust to the involved
parties [15]. It has been shown that the detection loophole at the untrusted
side can be effectively closed using loss-tolerant steering criteria, thus
promising information-theoretical security [16]. However, the asymmetry of
steering does not only manifest when imposed different levels of trust to
parties, which should be addressed as the directionality of steering [17], but
rather with the state per se is of an asymmetric form, in which case the
steering is unidirectional—known as one-way steering.

One-way steering, as identified in 2007 [3], was first studied in the Gaussian
regime [18, 19, 20, 21], and was subsequently studied in discrete variable
regime. Specifically, two-qubit states in tailored forms were found to exhibit
one-way steering, by using projective measurements [22] and positive-operator-
valued measures (POVMs) [5] respectively. Further, a family of two-parameter
mixed states that exhibit one-way steering are characterized [23]. Recently,
sufficient conditions for certifying one-way steering of two-qubit system have
also been further investigated [24, 25].

Following the theoretical works, some experimental demonstrations of one-way
steering have been carried out. It was first observed using two-setting
projective measurements [26], and later extended to using multisetting
projective measurements [27]. One-way steering with POVMs has also been
demonstrated [28]. Recently, conclusive one-way steering with infinite number
of measurements was first demonstrated for qubit-qutrit system (or two-qubit
with one of them has a third outcome of vacuum state) [29], and later was
demonstrated reducing to the genuine two-qubit system [30, 31].

However, existing works on one-way steering focused on two-qubit or qubit-
qutrit systems, and the genuine high-dimensional one-way steering has not been
studied to date. Indeed, steering that is beyond qubit system is of special
interests, since high-dimensional steering possesses strong robustness against
noise, which has been theoretically predicted and experimentally demonstrated
[32].

In this work, we extend the exploration of one-way steering to genuine high-
dimensional systems. Specifically, we first introduce the $d$-dimensional
partially entangled states, which is the generalization of its two-qubit form.
Then, taking three-dimensional partially entangled states for example, we
illustrate how one-way steering manifests in high-dimensional systems. Next,
we propose a loss-counted model for characterizing $d$-dimensional EPR
steering in a more practical manner, and accordingly develop an operational
numerical method. Finally, we apply our method to characterizing
quantitatively the relation between the measurement settings and detection
efficiencies of two-qutrit one-way steering states.

This article is organized as follows: In Sec. II we briefly review the LHS
model and propose the $d$-dimensional partially entangled states. In Sec. III
we characterize the one-way steerability of the proposed states for $d=3$
based on numerical approach with respect to mutually unbiased bases
measurements and further the general measurements. In Sec. IV we introduce our
loss-counted model and apply it to high-dimensional two-way steering states
(in comparison with known analytical results) and two-qutrit one-way steering
states respectively. We summarize the main results and discuss potential
applications of the work in Sec. V.

##  II Partially entangled states

Consider the two-party system, where Alice and Bob are sharing a quantum state
$\rho_{AB}$. If Alice performs local measurements on her subsystem, which are
denoted by a set of operators $\\{M_{a|x}\\}_{a,x}$, with $M_{a|x}\geq 0$ and
$\sum_{a}M_{a|x}=\mathbbm{1}$, where $x$ denotes her measurement setting and
$a$ the corresponding outcome, then on Bob’s subsystem, conditioned on the
measurement $x$ and outcome $a$, a collection of subnormalized density
matrices $\\{\sigma_{a|x}\\}_{a,x}$ are generated, with

$\sigma_{a|x}=\text{Tr}_{A}(M_{a|x}\otimes\mathbbm{1}\rho_{AB}).$ (1)

This set of matrices is called assemblage [33], and is proven to be useful in
characterizing the steerability from Alice to Bob (or vice versa if
measurements are performed on Bob’s side). Specifically, the assemblage is
called unsteerable if it can be reproduced by a LHS model [3], namely, it can
be decomposed into

$\sigma^{\text{LHS}}_{a|x}=\int\sigma_{\lambda}q(\lambda)p(a|x,\lambda)d\lambda,~{}\forall
a,x,$ (2)

where $\\{\sigma_{\lambda}\\}$ is a set of positive matrices with some
probability distribution $q(\lambda)p(a|x,\lambda)$. Any assemblage that
refutes a LHS model demonstrates EPR-steering. Interestingly, assemblages
generated from entangled states do not guarantee steering. Indeed, given an
assemblage, certifying whether it admits a LHS model is difficult, not mention
to one has to consider the experimental imperfections in practical
demonstration.

Figure 1:  The correspondence of symmetric states to two-way steering and
asymmetric states to one-way steering. Note here the symmetric state does not
only refer to the state with symmetric form, but rather the ones can
demonstrate steering under the exchange of the steering and steered parties.
The black squares represent the uncharacterized devices, while the white
squares denote the trusted devices.

It can be seen that the steering party and the steered party have already been
designated for an assemblage, given a quantum state and the local
measurements. Thus certifying steering is in itself directional, even if the
given quantum state has a symmetric form. Traditionally, people use symmetric
states to study steering, and the conclusion can be straightforwardly deduced
when the role of two parties are interchanged.

As illustrated in Fig. 1, for the symmetric states, if Alice is able to steer
Bob, then Bob must be able to steer Alice. While if the states are asymmetric
under the exchange of parties, then the conclusions may differ for different
roles of the parties. It is pointed out that one-way steering can only occur
for mixed states, as the pure entangled states can always be transformed to a
symmetric form by local basis change [22].

Recently, a family of mixed states with two independent parameters have been
found, which are shown to be one-way steerable for specific combination of
parameters. This family of states are called two-qubit partially entangled
state [23], with

$\rho(p,\theta)=p|\psi_{\theta}\rangle\langle\psi_{\theta}|+(1-p)\rho_{\theta}^{A}\otimes\mathbbm{1}/2,$
(3)

where
$|\psi_{\theta}\rangle=\text{cos}(\theta)|00\rangle+\text{sin}(\theta)|11\rangle$,
$\rho_{\theta}^{A}=\text{Tr}_{B}|\psi_{\theta}\rangle\langle\psi_{\theta}|$,
$p\in[0,1]$ and $\theta\in[0,\pi/4]$.

Here we generalize the above states to its high-dimensional form, i.e.,

$\rho(p,\bm{a})=p|\psi^{+}_{\bm{a}}\rangle\langle\psi^{+}_{\bm{a}}|+(1-p)\rho_{\bm{a}}^{A}\otimes\mathbbm{1}/d,$
(4)

where $|\psi^{+}_{\bm{a}}\rangle=\sum_{i=1}^{d}a_{i}|ii\rangle$,
$\rho_{\bm{a}}^{A}=\text{Tr}_{B}|\psi^{+}_{\bm{a}}\rangle\langle\psi^{+}_{\bm{a}}|$,
$p\in[0,1]$, $d$ is the dimension of the system and $\bm{a}$ is a
$d$-dimensional unit vector.

##  III One-way steering of two-qutrit state

###  III.1 Mutually unbiased bases measurements

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

$\sigma_{a|x}=\sum_{\lambda=0}^{\Lambda}D(a|x,\lambda)\sigma_{\lambda},$ (5)

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.,

$\sigma_{a|x}=\mu\sigma_{a|x}^{\text{LHS}}+(1-\mu)\sigma_{a|x}^{\text{ST}},~{}\forall
a,x,~{}0\leq\mu\leq 1,$ (6)

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

$\rho(p,\theta,\phi)=p|\psi^{+}_{\theta,\phi}\rangle\langle\psi^{+}_{\theta,\phi}|+(1-p)\rho_{\theta,\phi}^{A}\otimes\mathbbm{1}/3,$
(7)

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.

Figure 2:  One-way steering demonstration of two-qutrit partially entangled
state using four-setting mutually unbiased bases. The upper surface denotes
the critical value of parameter $p$ given various $\theta$ and $\phi$
considering Alice to steer Bob. The lower plane corresponds to the results for
Bob to steer Alice. The strict lower bound of $p^{*}_{B\rightarrow A}$ given
general measurements is also presented at the lowest in gray for reference.

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.

###  III.2 General measurements

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:

$\displaystyle\max~{}~{}$ $\displaystyle
Q=\text{Tr}\sum_{\lambda}\sigma_{\lambda}$ (8) over $\displaystyle
O_{AB},~{}\sigma_{\lambda}$ s. t.
$\displaystyle\text{Tr}_{A}(M_{a|x}\otimes\mathbbm{1}O_{AB})=\sum_{\lambda}D(a|x,\lambda)\sigma_{\lambda},~{}\forall
a,x,\lambda$ $\displaystyle\eta O_{AB}+(1-\eta)\rho^{A}\otimes
O_{B}=\rho_{AB},$

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 $p\leq
p^{*}$.

Figure 3:  Demonstration of one-way steering of two-qutrit state with respect
to general measurements. For those states with parameters are in the space
above the plane and below the curved surface, the one-way steering effect is
conclusively certified, in the sense that the general measurements are
considered in both directions. The curved surface with
$p^{*}<p^{*}_{B\rightarrow A}$ is painted in gray, for the steerability of the
states with parameters in such space is uncertain using current SDP settings.
Note that this uncertain space is not the same as the steerability-reversing
space appearing in the two-setting MUB case in Appendix A.

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.

##  IV Loss-counted one-way steering

###  IV.1 Loss-counted model

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

$\sigma^{\text{pri}}_{a|x}=\left\\{\begin{array}[]{ll}\epsilon\sigma_{a|x},&\forall
a,x~{}\text{with}~{}a\neq d,\\\ (1-\epsilon)\rho^{B},&\forall
a,x~{}\text{with}~{}a=d,\end{array}\right.$

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

$\begin{array}[]{rl}\sigma^{\text{LHS}}_{a|x}=\frac{1}{\epsilon}\sum_{\lambda=0}^{\Lambda}D(a|x,\lambda)\sigma_{\lambda},&\forall
a,x~{}\text{with}~{}a\neq d,\\\
\rho^{B}=\frac{1}{1-\epsilon}\sum_{\lambda=0}^{\Lambda}D(a|x,\lambda)\sigma_{\lambda},&\forall
a,x~{}\text{with}~{}a=d,\end{array}$ (9)

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

Figure 4:  Comparison of loss-counted model and known analytical bounds. The
circles denote our numerical results for specific dimension, measurement
setting and heralding efficiency. The colored curves correspond to bounds
derived in Ref. [43] with ${d+1}$ MUB. Specifically, the six curves from top
to bottom correspond to dimension two to five respectively. Note that the two
gray curves at the top (specified in the figure) correspond to the optimal
bounds derived in Ref. [38], which are for two-qubit system with square- and
octahedron-Platonic solid measurements respectively.  Figure 5:  Tradeoff
relation between the heralding efficiency and the measurement settings for
two-qutrit partially entangled state. The cross section with $\theta=\pi/4$ is
selected as the representative, for there is a two-way steerable point.
Results of two-, three-, and four-setting measurements are denoted in blue,
orange, and red respectively (additional labels on top of each curve are for
grayscale version). Results of $\epsilon=1,0.85,0.65$ are denoted with solid,
dash-dotted and dotted lines respectively. The threshold of Bob steering Alice
is also presented at the lowest forming a straight line.

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.

###  IV.2 Examples

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.

##  V Conclusion

In this work, we have introduced a family of high-dimensional one-way
steerable states, and taking the three-dimensional case for example, we have
fully characterized the parameter regimes of one-way steering, even with
respect to general measurements. It is worth noting that the validation of
one-way steerability of the proposed states for higher-dimensional case is not
presented in this work, but a general numerical method is provided—with the
dimension is of (relatively small) prime-power number, if adopting the MUB
shrinking factor. Beyond merely the analysis on the ideal case, we devised an
alternative interpretation (which is the variation of the non-click-event
interpretation) accounting for the effect of experimental loss and accordingly
developed the loss-counted version of steering weight measure. We tested our
loss-counted model and compared it with known results of symmetric states.

More importantly, our model applies to asymmetric input states. Thus we have
demonstrated quantitatively the tradeoff relation between the experimental
losses and measurement settings for two-qutrit one-way steering. The results
of this work sheds a light on the experimental demonstrations of high-
dimensional one-way steering, and on the corresponding quantum cryptography in
the one-sided device-independent scenario, where the quantification of loss is
crucial.

## Acknowledgments

We are grateful to Travis Baker and Howard Wiseman for useful discussions, and
especially Chau Nguyen for helpful advice and elaborate reviews on the
manuscript. We gratefully thank Paul Skrzypczyk for useful discussions on non-
steerability certification and helpful advice on the steering weight code.
This work was supported by the National Natural Science Foundation of China
through Grant No. 12105010. Q.Z. acknowledges support of The International
Postdoctoral Exchange Fellowship Program by China Postdoctoral Science
Foundation (Grant No. 20190096).

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##  Appendix A One-way steering of two-qutrit partially entangled states with
two- and three-setting MUB

Figure 6:  One-way steering demonstration of two-qutrit partially entangled
state using two-setting and three-setting mutually unbiased bases.

In Fig. 6 we present the one-way steering parameter space for two- and three-
setting MUB. It is clear that the overall steering effect with two- and three-
settings from both directions are underestimated compared to four-setting case
presented in the main text. It is worth noting that the one-way steering
parameter space in three-setting case is “shrunk” , but still keeps the
similar parameter combination compared to the four-setting case for
demonstrating one-way steering. Interestingly, the one-way steering is
reversed given two-setting MUB for certain combination of parameter as shown
in Fig. 6(a), which we see it resulting from the deficiency of two-setting
measurements in characterizing three-dimensional system.