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# Analysis of Quantum Steering Measures

Lucas Maquedano ID Department of Physics, Federal University of Paraná,
Curitiba, PR, Brazil  Ana C. S. Costa ID Department of Physics, Federal
University of Paraná, Curitiba, PR, Brazil

###### Abstract

The effect of quantum steering describes a possible action at a distance via
local measurements. In the last few years, several criteria have been proposed
in to detect this type of correlation in quantum systems. However, there are a
few approaches presented in order to measure the degree of steerability of a
given system. In this work, we are interested in investigating possible ways
to quantify quantum steering, where we based our analysis on different
criteria presented in the literature.

##  1 Introduction

In 1935, Einstein, Podolsky, and Rosen [1] pointed out contradictions between
local realism and completeness assumptions of quantum mechanics. In response
to this argument, Schrödinger [2, 3] introduced the term steering to present
the idea that the source of these contradictions is based on Alice’s ability,
from her choice of measurement basis, to influence Bob’s state. This debate
gave rise to a new field in Physics, dedicated to exploring these quantum
correlations.

The study of quantum correlations has contributed fundamentally to the area of
quantum information and various aspects of the foundations of quantum theory.
From the point of view of applications, quantum correlations have been used
for the development of several quantum information and quantum computing
protocols. Unlike Bell nonlocality [4] and entanglement [5], steering is
fundamentally asymmetrical. It allows, for example, entangled systems that are
only steerable from one observer to another, known as one-way steering [6, 7,
8, 9]. This property makes quantum steering an interesting resource for
quantum information processing where some of the parties are considered
untrusted, such as quantum key distribution [10, 11], randomness generation
[12, 13, 14], subchannel discrimination [15], and quantum teleportation [16].

Quantum systems may describe different types of correlation, including
nonlocality [4], steering [17, 18, 19, 20], and entanglement [5, 21]. There is
a hierarchy among them, meaning that quantum states that present Bell
nonlocality are also steerable and entangled, but not all entangled states are
steerable and violate a Bell inequality [22]. In a practical scenario, trust
in the measurement apparatus plays the role here, where certifying
entanglement requires trusted devices on both sides of the experiment, Bell
nonlocality is demonstrated with untrusted devices on both sides, and steering
is possible in a one-trusted device scenario.

Quantum steering is a well-defined type of quantum correlation, and there are
several ways to detect it, however, its quantification is still a matter of
debate. Several proposals have been presented in the literature, where we
have, for example, the steering weight [7] and the steering robustness [15].
Another attempt in the direction of quantifying quantum steering was given in
[23], where the authors based their measure on the maximal violation of a
linear steering inequality [18].

In this work, we are interested in analyzing different steering quantifiers.
Our goal is to compare the existing measures of quantum steering with new
proposals that we will introduce here. Note that it is not our intention to
declare that one measure is better than the other, but to show that some are
more efficient in a given context. We analyzed the following criteria: the
linear steering [18, 23], the generalized entropic steering [24, 25], the
rotationally invariant steering [26], and the dimension bounded steering [27].

To do this, we will associate each criterion with a quantifier based on the
method presented in [23], giving a steerability degree for a given state. Once
we have the most advantageous quantifier, we will compare it to a numerical
measure based on semi-definite programming [19]. Finally, we will consider the
approach defined in [28], where the authors considered the volume of
violations of a given Bell inequality in order to quantify the degree of non-
locality in the system. We will extend their analysis to quantum steering and
compare it with our previous method for a specific example, considering a
family of Werner states [29].

##  2 Quantum steering

In a typical quantum steering scenario, we consider two parties, Alice and
Bob, who share a bipartite quantum state. In each round of the experiment,
Alice performs different measurement settings $x$ with outcome $a$, while Bob
remains with an unnormalized conditional state $\sigma^{B}_{a|x}$ for each of
Alice’s choices. The collection of these conditional states is called the
steering assemblage. According to the quantum theory, the members of the
assemblage can be calculated by
$\sigma^{B}_{a|x}:=\Tr_{A}(M^{A}_{a|x}\otimes\mathbbm{1}^{B}\rho)$, where
$\\{M^{A}_{a|x}\\}_{a}$ is a set of positive operator valued measure (POVM),
with $\sum_{a}M^{A}_{a|x}=\mathbbm{1}$ and $M^{A}_{a|x}\geq 0$. Note that the
conditional states must obey the condition
$\sum_{a}\sigma^{B}_{a|x}=\rho^{B}$, which means that Bob’s reduced state
$\rho^{B}=\Tr_{A}(\rho)$ is independent of Alice’s measurement choices.

After characterizing the assemblage, Bob wants to check whether the state he
shares with Alice is entangled. This test can be done by checking if Bob’s
assemblage satisfies a local hidden state (LHS) model [30], which is
constructed by considering that Bob’s particle is in some hidden state
$\rho^{B}_{\lambda}$ with probability $p(\lambda)$, parametrized by a hidden
variable $\lambda$. The outcome of a measurement performed by Alice would only
give him extra information about Bob’s state probability. According to this
scenario, we arrive at [30]

$\displaystyle\sigma^{B}_{a|x}=\int d\lambda
p(\lambda)p(a|x,\lambda)\rho^{B}_{\lambda}.$ (1)

Equation (1) is known as an LHS model, and violating this model would mean
that Bob’s state is steerable. In this case, it is possible for Alice to steer
Bob’s state in some non-local way. It is important to note that, unlike
entanglement and Bell nonlocality, quantum steering is an asymmetric quantum
feature, and there are states where Alice can steer Bob but not the other way
around [6, 7, 8, 9].

##  3 Detection of quantum steering

In order to verify if a given state presents quantum steering, one needs to
check whether it is possible to construct an LHS model, assuming that Alice
performed a set of measurements and Bob ended up with unnormalized conditional
states, which can be determined using full tomography on such states. However,
depending on the complexity of some states, this is not an easy task. One
possibility is to make use of semidefinite programming (SDP) in order to
verify whether the state assemblage satisfies an LHS model. Depending on the
system, this technique might be demanding, given that it needs full tomography
on Bob’s conditional states and an optimization procedure that can become
intrinsically hard to perform. Given this difficulty, quantum steering can be
detected by the violation of certain inequalities. The construction of these
inequalities is based on an LHS model, and its violation would mean that it is
not possible to construct such a model.

Thus, most of these inequalities are built using a combination of mean values
(or probability distributions of certain outcomes) of specific measurements
performed by Alice and Bob. Using the correlations presented by these results,
one can check whether the shared system is steerable without needing to
reconstruct the assemblage completely. In the following, we are going to
present some criteria used to detect steering in bipartite quantum systems.

###  3.1 Linear Steering Criteria

In [18], a linear steering criterion was proposed, based on the uncertainty
relation between the average values of maximally incompatible observables, and
it is given by

$F^{(m)}_{\mathrm{L}}(\rho,\nu)=\quantity|\sum_{i=1}^{m}\langle A_{i}\otimes
B_{i}\rangle|\leqslant\mathcal{B}^{(m)}_{\mathrm{L}}.$ (2)

where $A_{i}=\hat{u}_{i}\cdot\vec{\sigma}$ and
$B_{i}=\hat{v}_{i}\cdot\vec{\sigma}$, with $\vec{\sigma}$ is a vector composed
of the Pauli matrices111For now on, we will use the operators $A_{i}$ and
$B_{i}$ to denote projective measurements performed by Alice and Bob,
respectively.. Here, $\mathcal{B}^{(m)}_{\mathrm{L}}=\sqrt{m}$, where $m$ is
the number of measurements performed in each site. Note that
$\hat{u}_{i}\in\mathbb{R}^{3}$ are unit vectors, and $\hat{v}_{i}\in$
$\mathbb{R}^{3}$ are orthonormal vectors. The mean value is given by $\langle
A_{i}\otimes B_{i}\rangle=\operatorname{Tr}[(A_{i}\otimes B_{i})\rho]$, and
$m\in\\{2,3\\}$. Violation of (2) means that the state is steerable.

Later on, these criteria were extended to a more general form [31],

$\left|\sum_{i=1}^{m}\langle A_{i}\otimes
B_{i}\rangle\right|\leqslant\max_{\\{a_{i}\\}}\left[\lambda_{\max}\left(\sum_{i=1}^{n}a_{i}B_{i}\right)\right].$
(3)

Here, $a_{i}=\pm 1$ and $\lambda_{\max}(X)$ is the highest eigenvalue of $X$.
If we consider Alice and Bob performing $m=\\{2,3\\}$ observables, with Bob’s
measurements being orthogonal, we recover the linear steering criteria
presented in Eq. (2).

It is important to note that Eq. (2) depends on the choice of the measurements
performed by Alice and Bob. So, for a given state, it can happen that the
inequality is violated by one set of Alice and Bob’s measurements, and not by
another one. In order to emphasize the dependence on the measurements, we
inserted the $\nu$ symbol in the definition of $F^{(m)}_{\mathrm{L}}$.

###  3.2 Generalized Entropic Steering

A different type of steering criteria is based on entropic uncertainty
relations. The first proposal [32] considered the Shannon entropy and
continuous variables, and in [33], the authors extended the criteria for
discrete variables. Later, a generalization of the criteria was proposed
considering the Tsallis entropy [24, 25] and the Rényi entropy [34]. In this
work, we will focus on the proposal based on the Tsallis entropy, since
numerical investigations suggest that it is more sensitive to detect steering
than the one based on the Shannon entropy [24] and it is equivalent to the one
based on Rényi entropy [34] for a given choice of parameters [35].

Then, considering the Tsallis entropy [36] for a given probability
distribution, $S_{q}(\mathcal{P})=-\sum_{n}p^{q}_{n}\ln_{q}(p_{n})$, with
$\ln_{q}(p)=(p^{1-q}-1)/(1-q)$, the steering criteria derived in [24] are
given by

$F^{(q,m)}_{\mathrm{E}}(\rho,\nu)=\sum_{i=1}^{m}\left[S_{q}(B_{i}|A_{i})+(1-q)C(A_{i},B_{i})\right]\geqslant\mathcal{B}^{(q,m)}_{\mathrm{E}},$
(4)

where $C(A_{i},B_{i})$ is a correction term, and $S_{q}(B_{i}|A_{i})$ is the
conditional Tsallis entropy [37]. For the case in which $q\in(0,2]$ and the
measurements are mutually unbiased with dimension $d$, the bound in Eq. (4)
can be calculated in the following way [38]

$\mathcal{B}^{(q,m)}_{\mathrm{E}}=m\ln_{q}\quantity(\frac{md}{m-1+d}).$ (5)

It is also possible to rewrite this inequality in terms of the probabilities
of Alice and Bob’s measurements,

$F^{(q,m)}_{\mathrm{E}}(\rho,\nu)=\frac{1}{q-1}\left[\sum_{i=1}^{m}\left(1-\sum_{ab}\frac{(p_{ab}^{(i)})^{q}}{(p_{a}^{(i)})^{q-1}}\right)\right]\geqslant\mathcal{B}^{(q,m)}_{\mathrm{E}}.$
(6)

Here, $p_{ab}^{(i)}$ is the probability of Alice and Bob for outcome $(a,b)$
when measuring $A_{i}\otimes B_{i}$, and $p_{a}^{(i)}$ are the marginal
outcome probabilities of Alice’s measurement $A_{i}$, assuming that in this
protocol, Alice aims to steer Bob’s state. We are considering that Alice and
Bob can perform $m$ measurements. The entropic criteria have been investigated
in different contexts, and recently it was tested experimentally [35].

###  3.3 Rotationally Invariant Steering

The rotationally invariant steering criterion, presented by Wollmann et al.
[26], is motivated by the fact that the steering can be verified, for a larger
class of states, in a rotationally invariant way. Returning to the general
scenario for steering presented at the beginning, after many runs of the
protocol, Bob should be able to estimate the correlation matrix
$\mathcal{M}_{ij}:=\left\langle A_{i}\otimes B_{j}\right\rangle$ and test
whether it is compatible with an LHV model (1), such that

$\mathcal{M}_{ij}=\int d\lambda p(\lambda)\left\langle
A_{i}\right\rangle_{\lambda}\left\langle
B_{j}\right\rangle_{\rho^{B}_{\lambda}}.$ (7)

Restricting to the case where all results are labeled by $\pm 1$ and Bob’s
measurements correspond to a set of orthogonal measurements, it was shown that
any LHV model must satisfy the steering inequality

$F^{(m)}_{\mathrm{RI}}(\rho,\nu)=\|\mathcal{M}\|_{\mathrm{tr}}\leqslant\mathcal{B}^{(m)}_{\mathrm{RI}},$
(8)

with
$\|\mathcal{M}\|_{\mathrm{tr}}:=\Tr\quantity[\sqrt{\mathcal{M}^{\top}\mathcal{M}}]$
and $\mathcal{B}^{(m)}_{\mathrm{RI}}=\sqrt{m}$.

###  3.4 Dimension Bounded Steering

Another criterion proposed in the literature is the dimension bounded steering
[27]. It is based on the fact that an unsteerable assemblage can be reproduced
by measurements in a separable state of the form

$\rho=\sum_{a_{1},\ldots,a_{m}}\ket{a_{1},\ldots,a_{m}}_{A}\bra{a_{1},\ldots,a_{m}}\otimes\omega^{B}_{a_{1}\ldots
a_{m}},$ (9)

being $x=\\{1,\ldots,m\\}$ the number of measurements performed by Alice with
$a_{x}$ outcomes. Here, the set of operators $\\{\omega^{B}_{a_{1}\ldots
a_{m}}\\}$ is described by an LHS model. Since $\rho^{B}=\Tr_{A}(\rho)$ and
$\rho^{B}=\sum_{a}\sigma^{B}_{a|x}$, it follows directly that an ensemble is
unsteerable if and only if there exist unnormalized density operators
$\\{\omega^{B}_{a_{1}\ldots a_{m}}\\}$ such that the conditional states can be
written as

$\rho^{B}_{k|x}=\sum_{a_{1},\ldots,a_{m}}\delta_{a_{x},k}\omega^{B}_{a_{1}\ldots
a_{m}}.$ (10)

The authors follow with a method to remove the discord zero structure from the
state, exchanging $\outerproduct{a_{1}\ldots a_{m}}{a_{1}\ldots a_{m}}$ for
another operator of dimension $d_{A}$ and adding constrictions relations in
such a way that the resulting separable state $\Sigma$ becomes completely
determined.

Considering dichotomic measurements and spin operators, the relevant criterion
is evaluated through the data matrix $\mathcal{D}:=\Tr[(A_{i}\otimes
B_{j})\Sigma]$, with $A_{0}=B_{0}=\mathbbm{1}$. The determinant of the data
matrix can be used to lower bound the trace norm of a correlation matrix by
adapting the CCNR criterion [39, 40], so one has a quantity for which an upper
bound is known for separable states. This leads to the dimension bounded
steering inequality (for more details see [27]):

$F^{(m)}_{\mathrm{DB}}(\rho,\nu)=|\det\mathcal{D}|\leq\mathcal{B}^{(m)}_{\mathrm{DB}},$
(11)

with the bound

$\mathcal{B}^{(m)}_{\mathrm{DB}}=\frac{1}{\sqrt{d_{A}}}\left(\frac{\sqrt{2d_{A}}-1}{m\sqrt{d_{A}}}\right)^{m}.$
(12)

The violation of Eq. (11) implies that the state is steerable.

##  4 Steering Quantifiers

So far, we have presented several steering criteria, given by the functions
$F^{(m)}_{\mathrm{K}}(\rho,\nu)$ and their respective bounds
$\mathcal{B}^{(m)}_{\mathrm{K}}$, with K=L,E,RI,DB. Although all of them can
be used to detect steering in different contexts, and for some specific cases
they all agree, these functions don’t give the information of how steerable a
given state is. Considering this fact, in this work, we are interested in
proposing different ways to quantify the steerability of a given state, based
on all functions presented above. Then, in hands with these steering measures,
we can compare them with themselves and with another well-known steering
measure, the steering robustness [15].

Our strategy is the following. We associate each inequality with a measure
$S^{(m)}_{\mathrm{K}}$, which will serve as a metric of the degree of
steerability of a given state. Thus, $S^{(m)}_{\mathrm{K}}\in[0,1]$, meaning
that this measure is one for maximally steerable states and zero for
unsteerable states according to a given criterion222Note that the non-
violation of a given criterion does not mean that the system is unsteerable.
This answer is given only if it is possible to construct an LHS model for such
a state considering infinite measurements.. In order to construct this measure
we will consider the same approach used in [23], where the authors proposed a
steering quantifier based on the linear steering criteria (2). In their work,
they considered the amount by which a steering inequality is maximally
violated. The reasoning behind this strategy is based on a similar approach
used to quantify Bell nonlocality in bipartite states [4, 41]. Operationally,
one can think that a state that violates more an inequality is said to be more
nonlocal (or quantum correlated) because it is more resilient under the
presence of noise.

Then, using the approach considered in [23], the structure of the quantifiers
associated with each inequality is given by

$S^{(m)}_{\mathrm{K}}(\rho):=\max\left[0,\frac{F^{(m)}_{\mathrm{K}}(\rho)-\mathcal{B}^{(m)}_{\mathrm{K}}}{F^{(m)}_{\mathrm{K}}(\rho_{\max})-\mathcal{B}^{(m)}_{\mathrm{K}}}\right],$
(13)

being $F^{(m)}_{\mathrm{K}}(\rho)=\max_{\nu}F^{(m)}_{\mathrm{K}}(\rho,\nu)$,
meaning that we optimize over all measurements performed by Alice and Bob. In
order to normalize this measure, so it is restricted to the interval $[0,1]$,
we considered the denominator of Eq. (13) such as
$F^{(m)}_{\mathrm{K}}(\rho_{\max})$ to be the state that violates maximally
the inequality. For bipartite systems, the Bell states, which are maximally
entangled, are the ones that fulfill this feature.

Although we have written all $F^{(m)}_{\mathrm{K}}(\rho)$ considering $m$
measurements per site, we restricted our analysis to the case where Alice and
Bob are performing three orthogonal measurements each. Besides that, we will
focus on general two-qubit states, which can be written in Bloch
representation [42] as

$\varrho=\frac{1}{4}\left(\mathbbm{1}\otimes\mathbbm{1}+\vec{a}\cdot\vec{\sigma}\otimes\mathbbm{1}+\mathbbm{1}\otimes\vec{b}\cdot\vec{\sigma}+\sum_{r=1}^{3}c_{r}\sigma_{r}\otimes\sigma_{r}\right),$
(14)

where $\\{\vec{a},\vec{b},\vec{c}\\}\in\mathbb{R}^{3}$ and
$\vec{a}^{2}+\vec{b}^{2}+\vec{c}^{2}\leq 3$.

###  4.1 Linear steering measure

As we already mentioned, the quantifier for the linear steering criteria (2)
was developed in [23]. Here, we will reproduce some of their results because
they will be important in the analysis of other criteria we considered in this
work.

Given that Alice and Bob perform projective measurements $A_{i}$ and $B_{i}$
respectively, we have that

$\left\langle A_{i}\otimes
B_{i}\right\rangle=\sum_{r=1}^{3}u_{ir}c_{r}v_{ir}\equiv\left\langle
u_{i}|C|v_{i}\right\rangle\equiv C_{i},$ (15)

where $C\equiv\sum_{r}e_{r}\outerproduct{e_{r}}{e_{r}}$ is a Hermitian
operator with eigenvalues $e_{r}$. Here,
$\hat{u}_{i}\equiv\left|u_{i}\right\rangle=\sum_{r}u_{ir}\left|e_{r}\right\rangle$
and
$\hat{v}_{i}\equiv\left|v_{i}\right\rangle=\sum_{r}v_{ir}\left|e_{r}\right\rangle$.

In [23], the authors defined the transformation
$\left|\alpha_{i}\right\rangle\equiv C\left|u_{i}\right\rangle$, such that
$C_{i}=\left\langle\alpha_{i}|v_{i}\right\rangle$, to maximize
$\sum_{i}C_{i}$. The upper bound of this sum is obtained from the
orthogonality between the vectors $\hat{v}_{i}$ and, consequently, between
$\ket{\alpha_{i}}$. Thus, the linear function (2) depends exclusively on the
parameter $\vec{c}$ of the state $\varrho$, and

$F^{(2)}_{\textrm{L}}(\varrho)=\sqrt{c^{2}-c_{\min}^{2}}\quad\text{and}\quad
F^{(3)}_{\textrm{L}}(\varrho)=c,$ (16)

where $c=\sqrt{\vec{c}^{\,2}}$ and
$c_{\min}\equiv\min\left\\{\left|c_{1}\right|,\left|c_{2}\right|,\left|c_{3}\right|\right\\}$.

Then, in this case, we have the linear steering measure [23] given by

$S^{(2)}_{\mathrm{L}}(\varrho):=\max\left[0,\frac{\sqrt{c^{2}-c_{\min}^{2}}-1}{\sqrt{2}-1}\right],$
(17)

in the case of two measurement settings, and

$S^{(3)}_{\mathrm{L}}(\varrho):=\max\left[0,\frac{c-1}{\sqrt{3}-1}\right],$
(18)

for three measurement settings.

###  4.2 Generalized entropic steering measure

After analyzing the linear steering criteria, we will consider the entropic
steering criteria presented in Eq. (4). Optimizing the measurements in order
to obtain $F^{(q,m)}_{\textrm{E}}(\varrho)$ is not trivial, even in the simple
case of $q=2$. Although it is possible to perform a numerical optimization, we
are interested in obtaining a closed analytical formula. Given this
difficulty, we focus on the case of fixed orthogonal measurements based on the
Pauli operators. This choice means that
$\hat{u}_{i}=\hat{v}_{i}=\left\\{(1,0,0)^{T},(0,1,0)^{T},(0,0,1)^{T}\right\\}$
for $m=3$, and we also consider $q=2$, based on numerical investigations which
show that this criterion is the strongest among this class. Thus, we obtain
the following entropic steering measure (for $q=2$ and fixed measurements):

$S^{(2,3)}_{\mathrm{E}}(\varrho):=\max\left\\{0,1-\sum_{r=1}^{3}\left[\frac{1-a_{r}^{2}-b_{r}^{2}-c_{r}^{2}+2a_{r}b_{r}c_{r}}{2\left(1-a_{r}^{2}\right)}\right]\right\\}.$
(19)

It remains an open question whether it is possible to obtain analytically a
closed formula to quantify steering considering the criteria based on the
Tsallis entropic uncertainty relations, for all values of $q$.

###  4.3 Rotationally invariant steering measure

The rotationally invariant steering criterion (8) is invariant under
rotations. Then, in this case, it is not necessary to perform an optimization
over all measurements, being enough to fix Alice and Bob’s measurements to the
ones based on the Pauli operators. In this case, we arrive at the following
measure for a system of two qubits

$S^{(3)}_{\mathrm{RI}}(\varrho):=\max\left[0,\frac{(|c_{1}|+|c_{2}|+|c_{3}|)-\sqrt{3}}{3-\sqrt{3}}\right].$
(20)

###  4.4 Dimension bounded steering measure

The dimension bounded steering criterion is very useful in detecting steering,
given that one can simply assume that Bob’s measurements act on a qubit system
instead of trusting his devices. It is also rotationally invariant [35]. In
order to obtain a steering quantifier for the dimension bounded steering, one
has to analyze the elements of the data matrix $\mathcal{D}$ constructed with
mean values $\langle A_{i}\otimes B_{j}\rangle$, $\langle A_{i}\rangle$ and
$\langle B_{i}\rangle$. To maximize $|\det\mathcal{D}|$ in order to obtain
$F^{(m)}_{\textrm{DB}}(\varrho)$ we adopt a similar analysis to the linear
case (2). The determinant is optimal when the $\langle A_{i}\otimes
B_{j}\rangle$, for $i\neq j$, elements are null, a condition for which each of
Alice and Bob’s measurement settings must be orthogonal, as we can verify
directly from the equation (15). Thus, the dimension bounded steering measure
is given by

$S^{(3)}_{\textrm{DB}}(\varrho)=\max\left[0,\frac{3\sqrt{3}|c_{1}c_{2}c_{3}-(a_{1}b_{1}c_{2}c_{3}+a_{2}b_{2}c_{1}c_{3}+a_{3}b_{3}c_{1}c_{2})|-1}{3\sqrt{3}-1}\right].$
(21)

It is interesting to note that we would arrive at the same result if we had
chosen fixed measurements based on the Pauli operators. This result shows that
the rotational invariance, first noted in [35], is confirmed for this
criterion.

##  5 Steering measures comparison

In the last section, we have presented steering quantifiers based on different
criteria. These analytical formulas to quantify steering is one of our main
results. Now we can check if they are all equivalent, and if they are not,
which one can detect more steerable states than the others. For this analysis,
we generated $10^{6}$ random states and constructed parametric plots to
compare the different quantifiers. In Fig. 1, each point represents a random
state while the black points represent the Werner states
$\varrho_{w}=w\outerproduct{\psi_{s}}{\psi_{s}}+\frac{1-w}{4}\mathbbm{1}_{4}$,
where $\ket{\psi_{s}}=(\ket{01}-\ket{10})/\sqrt{2}$ is the singlet state. For
$w=0$, corresponding to maximally mixed states, all measures are null, and for
$w=1$, being the maximally entangled states, all the quantifiers are maximal.
In Fig. 1, one can notice that the Werner states form a lower bound in the
measures comparison.

An important result, extracted from Fig. 1, is the fact that the measure based
on the generalized entropic direction (19) appears to be the most efficient to
quantify steering for two-qubit systems. When analyzing all the figures, it is
possible to notice that some quantifiers disappear for certain states while
the entropic one does not. It is our second important result, an analytical
formula that appears to be more sensitive to detect (and quantify) when
compared to other steering criteria in the literature. It is important to note
that this conclusion is only valid for the case of three measurements per
set333This conclusion can also be made for two measurements per set, but since
taking three measurements provides stronger steering criteria, we restrict the
analysis for the latter case., if we consider the possibility of carrying out
an infinite amount of measurements, better quantifiers can be found in [43,
44].

Figure 1: Diagrams of (a) $S^{(2,3)}_{E}(\varrho)$ with
$S^{(3)}_{L}(\varrho)$, (b) $S^{(3)}_{L}(\varrho)$ with
$S^{(3)}_{RI}(\varrho)$, and (c) $S^{(3)}_{L}(\varrho)$ with
$S^{(3)}_{DB}(\varrho)$, for $10^{6}$ general two-qubit states. The red points
represent when one of the measures is zero while the other one is not. The
black points represent Werner states $\varrho_{w}$ with $w\in[0,1]$. The
values $w=0$ ($w=1$) correspond respectively to the minimum (maximum) for all
quantifiers.

##  6 Steering Robustness

In the section above, we defined different steering measures based on the
maximal violation of a respective steering inequality. However, it is
important to note that there is also a quantifier known as steering
robustness, based on numerical optimization and structured on the framework of
a semidefinite program (SDP) problem. Here, we present an overview of the SDP
structure for steering (more details on how to implement the algorithm, see
[19]).

For a finite number of measurement choices and outcomes, the probability
$p(a|x,\lambda)$ can be decomposed as a convex combination of a finite number
of deterministic probability distributions:

$p(a|x,\lambda)=\sum_{\lambda^{\prime}}p\left(\lambda^{\prime}|\lambda\right)D\left(a|x,\lambda^{\prime}\right),$
(22)

where $p\left(\lambda^{\prime}|\lambda\right)$ is the weight of the
deterministic distribution of the hidden variable $\lambda^{\prime}$ given
$\lambda$. An LHS assemblage can be written by inserting (22) into (1) in
order to obtain the members of the assemblage as

$\sigma_{a|x}=\sum_{\lambda^{\prime}}D\left(a|x,\lambda^{\prime}\right)\sigma_{\lambda^{\prime}},$
(23)

where we have defined
$\sigma_{\lambda^{\prime}}:=\int\mathrm{d}\lambda\mu(\lambda)p(\lambda^{\prime}|\lambda)\rho_{\lambda}$.
Given that the distribution $D\left(a|x,\lambda^{\prime}\right)$ is fixed, in
contrast with $p(a|x,\lambda)$ which was unknown, the model (23) is a
simplification of the LHS assemblages structure.

We are now able to implement an SDP to verify whether an assemblage
$\left\\{\sigma_{a|x}\right\\}_{a,x}$ is a member of the LHS set:

$\begin{array}[]{lll}\text{ given
}&\left\\{\sigma_{a|x}\right\\}_{a,x}\;;\\{D(a|x,\lambda)\\}_{\lambda}&\\\
\text{ find }&\left\\{\sigma_{\lambda}\right\\}_{\lambda}&\\\ \text{ s.t.
}&\sum_{\lambda}D(a|x,\lambda)\sigma_{\lambda}=\sigma_{a|x}\quad\forall
a,x.\\\ &\sigma_{\lambda}\geqslant 0\quad\forall\lambda.\end{array}$ (24)

To construct a steering measure, we start by asking how much noise
($\varepsilon$) one has to add to a given assemblage in order for it to have
an LHS model. For any convex subset of assemblages ($\mathcal{R}$)
characterized by positive semi-definite constraints and linear matrix
inequalities, we can define the steering robustness of an assemblage as

$\displaystyle\operatorname{SRA}\left(\\{\sigma_{a|x}\\}\right)=$
$\displaystyle\min_{\pi_{a|x},\sigma_{\lambda},\varepsilon}\;\varepsilon$ (25)
s.t.
$\displaystyle\;\;\;\frac{\sigma_{a|x}+\varepsilon\,\pi_{a|x}}{1+\varepsilon}=\sigma_{a|x}^{\mathrm{LHS}}\quad\forall
a,x,$
$\displaystyle\;\;\;\sigma_{a|x}^{\mathrm{LHS}}=\sum_{\lambda}D(a|x,\lambda)\sigma_{\lambda}\quad\forall
a,x,$ $\displaystyle\;\;\;\pi_{a|x}\in\mathcal{R},\quad\sigma_{\lambda}\geq
0\quad\forall\lambda.$

The types of noise that can be applied and their respective robustness
quantifiers are given from the subset $\mathcal{R}$. The next steps are to
transform (25) explicitly into an SDP and then into an explicitly feasible
problem. Details can be found at [19].

In the strategy used to derive the steering robustness quantifier, we need to
fix a choice of measurements for Alice. Another approach would be to optimize
over the possible choices of $M^{A}_{a|x}$, a method known as the see-saw
algorithm. Formally, it can be presented as

$\displaystyle\mathrm{SRS}(\rho)=$
$\displaystyle\max_{\left\\{M^{A}_{a|x}\right\\}}\mathrm{SR}(\\{\sigma_{a|x}\\})$
(26) s.t.
$\displaystyle\;\;\;\sigma_{a|x}=\Tr_{\mathrm{A}}\left[\left(M_{a|x}\otimes\mathbbm{1}\right)\rho\right]\quad\forall
a,x.$

Using this construction, we would be able to find the amount of how steerable
a given state is by the calculation of the steering robustness considering $m$
measurements and performing an optimization over all of them.

Since the see-saw algorithm requires a lot of computational effort (it
incorporates a second maximization), we compared the statistics for a set of
generic states $\varrho$ calculating its steerability via the steering
robustness (25) assuming fixed measurements based on the Pauli operators and
the steering robustness supported by the see-saw algorithm (26). As a result,
for $3\times 10^{4}$ random states, for $12\%$ of the states, the latter
present higher values of steerability. Although is a low statistic, the
difference between the values is around $10^{-3}$. This result is not a
surprise since the SDP in (25) performs an optimization over all possible
inequalities to detect the steerability of a given state. Considering this, we
focus on steering robustness with fixed measurements to perform the following
analysis.

Finally, we compared the steering robustness (which is a numerical procedure
to detect the steerability of a given state), with the closed formula provided
by the Tsallis entropy-based steering measure (19), which we showed in the
previous section to be the measure that best detects and quantify the
steerability. For this analysis, we generated $2.4\times 10^{5}$ random states
and calculated both measures. The results can be visualized in Fig. 2. Given
this set of states, around $5.5\%$ is only detected by the steering robustness
and by the analytical formula provided in Eq. (19). This is one of the main
results of this work. Although the closed formula to quantify steerability
given in Eq. (19) is not optimal, we still have a very good agreement with
another quantifier that relies on a computational method.

Figure 2: Comparison of steering robustness
$\operatorname{SRA}\left(\\{\sigma_{a|x}\\}\right)$ and the generalized
entropic steering measure $S^{(2,3)}_{\textrm{E}}$ for $2.4\times 10^{5}$
random states. For the red points ($\thicksim 5.5\%$), SR is non-null while
the entropic measure vanishes. The black points are the Werner states
$\varrho_{w}$ with $w\in[0,1]$.

##  7 Volume of Violations

So far, we have compared the methods by generating a collection of states and
calculating the value of the optimized quantifiers – or assuming orthogonal
and aligned measurements between the parts – for each member of the set.
Another way to compare different approaches to detect steering is by fixing a
quantum state and performing on it a collection of measurements, which are
orthogonal but randomly distributed. In the end, we get a statistic of how
many rounds of measurements violated each quantifier.

This idea was proposed by Fonseca and Parisio [28] in order to quantify Bell
nonlocality. They defined the space of all possible parameters
$\text{Z}=\\{x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}\\}$ that determines the
settings for a Bell-like inequality. For a state $\rho$, the set of points
$\zeta_{\rho}\subset\text{Z}$ that lead to violation of the inequality will
have a corresponding quantity $V(\rho)$ proportional to the volume of
$\zeta_{\rho}$. The authors proposed, if $V(\rho)>V(\rho^{\prime})$, then the
state $\rho$ is more nonlocal than the other $\rho^{\prime}$, with

$V(\rho):=\frac{1}{\mathcal{N}}\int_{\zeta_{\rho}}d^{m}xd^{n}y,$ (27)

where $\mathcal{N}$ is a constant of normalization.

In the following analysis, we investigate the quantification of steering for
all the criteria presented in Sec. 3 using the approach based on the volume of
violations. This would give us another method to quantify the steerability of
a given state, with the difference that here we do not optimize over all
possible measurements but we check the volume of violations of a specific
inequality. Unlike the previous method, here it is not possible to derive
analytical formulas.

Using the approach presented in [28], we checked, for specific Werner states,
the volume of violations for the inequalities of Sec. 3. The results can be
seen in Fig. 3, where we performed numerical simulations for $10^{5}$
orthogonal random measurements. When considering two measurement settings per
site, one can see that there are no significant differences between the
criteria. They all give approximated values, and, most importantly, they are
all null and maximal for the same states, showing some notion of equivalence
between them. However, if we check the case of three measurement settings per
site, the rotationally invariant steering (8) and the dimension bounded
steering (11) give different results when compared to the linear steering (2)
and the generalized entropic steering (19) for $q=2$. This result is not
surprising, given that the former is known to be invariant under rotations so
they would violate for a larger amount of measurements than the latter ones.
Interestingly, according to this quantification method, all Werner states,
with $w>1/\sqrt{3}$, are maximally steerable, according to the volume of
violations of the dimension bounded and the rotationally invariant steering.

To complement our analysis, we also plotted the analytical measures developed
in Sec. 4 in Fig. 3(b). It is very interesting that, except for the dimension
bounded and the rotationally invariant steering with three measurements per
site, both ways of quantification have similar behavior, being the generalized
entropic steering the one that fits better with each other.

Figure 3: Comparison between the different steering measures for (a) $m=2$ and
(b) $m=3$ measurement settings per site. Each point represents a Werner state
whose volume of violations was computed for $10^{5}$ measurements. In (b) we
also plotted the analytical formulas presented in Sec. 4 to compare both ways
of quantification.

##  8 Discussion

In this work, we analyzed two methods to quantify steerability, one based on
the maximization over all measurements and the other based on the volume of
violations of steering inequalities. Using the former method, we obtained
analytical formulas that can be used to quantify steering. We focused on four
known criteria from the literature, the linear steering [18], the generalized
entropic steering [24, 25], the rotationally invariant steering [45], and the
dimension bounded steering [27]. Among the quantifiers, the generalized
entropic measure (19) was more efficient in detecting and quantifying steering
for a larger class of states. In the sequence, we compared this measure to
another well-known quantifier, the steering robustness, which is based on
semidefinite programming. The computational method proved to be more efficient
only for $5.5\%$ of the states.

While considering the volume of violations of these inequalities, the results
showed that this method gives similar results to the one based on the
maximization of the measurements, except for the rotational invariant
criteria. Here, we focus only on the family of Werner states, which already
showed us interesting results.

For future works, it would be interesting to extend this analysis to higher
dimensional states, where the numerical method is more demanding, and
obtaining an analytical formula to quantify steering for this class of states
could be very helpful for different applications. Another aspect to be
investigated is the extension to multipartite systems, where it could be
possible to look at the monogamy of steering.

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