# Typical bipartite steerability and generalized local quantum measurements

Maximilian Schumacher, Gernot Alber  Institut für Angewandte Physik, Technical
University of Darmstadt, D-64289 Darmstadt, Germany<EMAIL_ADDRESS>darmstadt.de<EMAIL_ADDRESS>
###### Abstract

Recently proposed correlation-matrix based sufficient conditions for bipartite
steerability from Alice to Bob are applied to local informationally complete
positive operator valued measures (POVMs) of the $(N,M)$-type. These POVMs
allow for a unified description of a large class of local generalized
measurements of current interest. It is shown that this sufficient condition
exhibits a peculiar scaling property. It implies that all types of
informationally complete $(N,M)$-POVMs are equally powerful in detecting
bipartite steerability from Alice to Bob and, in addition, they are as
powerful as local orthonormal hermitian operator bases (LOOs). In order to
explore the typicality of steering numerical calculations of lower bounds on
Euclidean volume ratios between steerable bipartite quantum states from Alice
to Bob and all quantum states are determined with the help of a hit-and-run
Monte-Carlo algorithm. These results demonstrate that with the single
exception of two qubits this correlation-matrix based sufficient condition
significantly underestimates these volume ratios. These results are also
compared with a recently proposed method which reduces the determination of
bipartite steerability from Alice’s qubit to Bob’s arbitrary dimensional
quantum system to the determination of bipartite entanglement. It is
demonstrated that in general this method is significantly more effective in
detecting typical steerability provided entanglement detection methods are
used which transcend local measurements.

††: Phys. Scr.

Keywords: EPR steering, (N,M)-POVMs, quantum correlations, quantum information
science

##  1 Introduction

Basic assumptions underlying classical physics in general and special
relativity in particular require consistency of all measurable statistical
correlations with local realism [1]. As a consequence each measurement
scenario imposes particular restrictions on measurable multipartite
correlations. In general these classical correlations can be described within
the framework of a local realistic theory (LRT) or local hidden variable (LHV)
model [2]. These restrictions of possible classical correlations in local
theories manifest themselves in Bell inequalities associated with classical
probability polytopes [3, 4]. A striking phenomenon of quantum systems is
their capability of violating these restrictions imposed by local realism in
special measurement scenarios [5, 6]. By now this phenomenon of Bell-
nonlocality has been demonstrated in a series of impressive experiments with
increasing degrees of sophistication [7, 8, 9].

The possible existence of nonlocal quantum correlations has been pointed out
already as early as 1935 by Einstein, Podolsky and Rosen (EPR) [10] in their
discussion of nonlocal properties of a certain class of pure quantum states.
By now the peculiar properties of these quantum states have inspired numerous
applications [11]. In his response to EPR Schrödinger termed these quantum
states entangled. In addition, he coined the notion of steering for the
characteristic quantum phenomenon underlying the physical discussion of EPR,
namely the ability of one local observer, say Alice, to influence
simultaneously the measurements of another local observer, say Bob, by her
measurements [12, 13]. But it was not until 2007 when Schrödinger’s
qualitative notion of steering, frequently also termed EPR steering, was
clarified by Wiseman, Jones, and Doherty [14, 15]. They based their notion of
EPR steering on the existence of a local hidden state (LHS) model. These LHS
models generalize LHV models by transcending the LHV-specific theoretical
state concept with the help of quantum states thus allowing for a quantum
theoretic description of the locally measurable probability distributions.
This concept of EPR steerability captures some nonlocal aspects of quantum
correlations and differs from Bell nonlocality and from entanglement [16].
Although separability of quantum states is sufficient for EPR unsteerability
or stated equivalently, EPR steerability is sufficient for entanglement, the
precise relation between these different concepts of nonlocal quantum
correlations is intricate, in particular for mixed quantum states. Contrary to
entanglement, for example, EPR steerability is an asymmetric property with
respect to the local observers involved because there are multipartite quantum
states which allow one local observer to steer another observer but not vice
versa. Furthermore, contrary to entanglement EPR steerability is a measurement
dependent notion. Whether a quantum state is EPR steerable or not depends on
the local measurements which can be performed by the observers. Thus,
increasing the measurement capabilities of a local observer should also
increase the number of EPR steerable quantum states.

Apart from their significance for the foundations of quantum theory EPR
steerable quantum states have also potentially promising applications in
quantum information processing, in particular in the areas of quantum key
distribution [17], quantum teleportation [18] and quantum secret sharing [19].
In view of this fundamental and practical significance of EPR steering it is
of interest to develop tests for EPR steerability from Alice to Bob. Although
general necessary and sufficient conditions for bipartite EPR steerability of
quantum states are known [16, 20, 21], these criteria involve intricate
optimization procedures which even in the simplest case of two qubits require
considerable numerical efforts for determining typical statistical features of
bipartite EPR steerability, such as volume ratios of EPR steerable versus all
quantum states, for example [20]. Up to now even in the simple case of two
qubits numerically feasible necessary and sufficient conditions for EPR
steerability from Alice to Bob are only known for cases in which Bob’s local
measurements are restricted to projective measurements. Although there is
numerical evidence that these results also apply to general positive operator
valued measurements (POVMs) [21] a general proof of this conjecture is still
missing.

In view of these difficulties numerically less demanding conditions of EPR
steerability, which are only sufficient and no longer also necessary, offer a
promising alternative for investigating typical statistical properties of EPR
steerable bipartite quantum states from Alice to Bob. In this context
sufficient conditions are of particular interest which are based on measurable
local correlations of Alice and Bob and which are valid for arbitrary
dimensional bipartite quantum states. Recently Lai and Luo [22] have developed
such an approach based on the violation of an inequality which involves the
trace norm of the correlation matrix of local measurements. As a violation of
this inequality can be checked numerically in a rather straight forward way
this approach may offer interesting perspectives for the exploration of
statistical properties of bipartite EPR steerability from Alice to Bob. So
far, these authors have specialized their sufficient condition to cases
involving local orthogonal observables, mutually unbiased measurements (MUMs)
[23] and general symmetric informationally complete positive operator-valued
measures (GSIC-POVMs) [24] for Alice and Bob. For these measurements they have
explored bipartite EPR steerability for special families of qubit-qubit,
qubit-qutrit and qutrit-qutrit states and in addition for a special one-
parameter family of arbitrary dimensional bipartite isotropic quantum states.

Motivated by these developments in this paper we further explore basic
properties of this recently proposed correlation-matrix based sufficient
condition for EPR steerability from Alice to Bob of Lai and Luo [22]. In
particular, we focus on two main questions. Firstly, we want to explore how
the capability of detecting EPR steerability from Alice to Bob in arbitrary
dimensional bipartite quantum systems depends on the nature of the local
quantum measurements for a broader class of measurements. In this respect
informationally complete local measurements are of particular interest, by
which, at least in principle, the reduced density operators of the local
observers can be reconstructed perfectly. For this purpose the recently
introduced informationally complete $(N,M)$-POVMs [25] are well suited as
local measurements. These families of POVMs allow for a unified description of
a large class of generalized measurements of current interest, such as
projective measurements involving mutually unbiased bases [26], mutually
unbiased measurements [23], symmetric informationally complete generalized
measurements (SIC-POVMs) [27, 28] and their generalized analogues (GSIC-POVMs)
[24, 29]. As a main result it will be demonstrated that the recently proposed
correlation-matrix based sufficient condition for bipartite EPR steerability
from Alice to Bob of Lai and Luo [22] exhibits a peculiar scaling property
with respect to local $(N,M)$-POVMs. This scaling property implies that this
sufficient condition becomes independent of the local informationally complete
measurements performed within the class of $(N,M)$-POVMs. Furthermore, it will
be shown that local orthonormal hermitian operator bases (LOOs) are equally
powerful to determine EPR steerability from Alice to Bob as local
informationally complete $(N,M)$-POVMs. Secondly, we want to explore how many
EPR steerable bipartite quantum states from Alice to Bob can be detected by
this correlation-matrix based sufficient condition. For this purpose we sample
random bipartite quantum states by a recently developed hit-and-run Monte-
Carlo procedure [30] and determine the resulting relative Euclidean volume
ratios between EPR steerable states from Alice to Bob and all possible
bipartite quantum states for different dimensions of Alice’s and Bob’s quantum
systems. This way we are capable of exploring typical statistical features of
EPR steerability from Alice to Bob.

This paper is organized as follows. For the sake of completeness in Sec. 2 we
summarize basic facts about the basic concepts of local realism and of
bipartite EPR steerability from Alice to Bob. Basic features of the recently
introduced $(N,M)$-POVMs [25], which are important for our subsequent
investigation, are summarized in Sec. 3. In Sec. 4 the correlation-matrix
based sufficient condition for bipartite EPR steerability from Alice to Bob of
Lai and Luo [22] is introduced and applied to local measurements based on
$(N,M)$-POVMs performed on arbitrary dimensional bipartite quantum systems. In
Sec. 5 it is demonstrated that within the class of local informationally
complete $(N,M)$-POVMs this sufficient EPR steerability condition exhibits a
characteristic scaling property. It implies that this sufficient condition
becomes independent of the particular informationally complete local
measurements testing for EPR steerability and, furthermore, it becomes
identical with the corresponding sufficient condition for LOOs. In Sec. 6
numerical results of Monte-Carlo simulations are presented exploring Euclidean
volume ratios of steerable bipartite quantum states from Alice to Bob and all
bipartite quantum states for different dimensions of Alice’s and Bob’s quantum
systems. Furthermore, these results are compared with the corresponding volume
ratios based on a method recently proposed by Das et al. [31], which reduces
the determination of bipartite steerability from Alice to Bob to the
determination of bipartite entanglement

##  2 Local realism and bipartite steerability

Classical physical theories which are consistent with the physical laws of
special relativity and its fundamental distinction between time-like and
space-like events are governed by local realism [1]. As a consequence
classical correlations between two space-like separated experimenters, say
Alice and Bob, obey locality constraints. They can be expressed in terms of
generalized Bell inequalities which describe probability polytopes [4]. Basic
concepts involved can be described by considering a simple bipartite scenario
in which two space-like separated experimenters, Alice and Bob, perform random
measurements of local observables, say $\alpha\in{\cal O}_{A}$ and
$\beta\in{\cal O}_{B}$ with ${\cal O}_{A}$ and ${\cal O}_{B}$ describing sets
of local observables of Alice and Bob. Each of these local observables has
different possible measurement results, say $a\in{\cal M}_{A}$ and $b\in{\cal
M}_{B}$. In a local realistic classical theory (LRT) the bipartite probability
distribution $P(a,b\mid\alpha,\beta)$ of a possible joint local measurement,
in which Alice and Bob randomly select observables $\alpha$ and $\beta$ and
obtain measurement results $a$ and $b$, has the characteristic structure

$\displaystyle P(a,b\mid\alpha,\beta)$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\Lambda}p(\lambda)P^{A}(a\mid\alpha,\lambda)P^{B}(b\mid\beta,\lambda),$
$\displaystyle\sum_{\lambda\in\Lambda}p(\lambda)$ $\displaystyle=$
$\displaystyle 1$ (1)

with the characteristic normalization $\sum_{a\in{\cal
M}_{A}}P^{A}(a\mid\alpha,\lambda)=\sum_{b\in{\cal
M}_{B}}P^{B}(b\mid\beta,\lambda)=1$. Here and in our subsequent discussion
upper indices, such as $A$ and $B$, refer to local observers, such as Alice
$(A)$ and Bob $(B)$, small Greek letters symbolize measurements and small
Latin letters symbolize measurement results. According to (1), for each random
selection of local observables the joint probability distribution is a convex
sum of local (conditional) probability distributions of Alice and Bob, i.e.
$P^{A}(a\mid\alpha,\lambda)\geq 0$ and $P^{B}(b\mid\beta,\lambda)\geq 0$,
which depend on a random variable $\lambda\in\Lambda$ with probability
distribution $p(\lambda)\geq 0$. In this description this ’hidden variable’
$\lambda\in\Lambda$ characterizes uniquely the classical state of the
bipartite physical system in a classical state space $\Lambda$. According to
Bell’s theorem [1], the constraints imposed on physical theories by local
realism as expressed by Eq. (1) in a bipartite scenario, for example, can be
violated by quantum correlations. In particular, the quantum correlations
originating from entangled quantum state can violate local realism for
particular choices of measurements. By now violations of local realism by
quantum systems have been demonstrated in a series of impressive experiments
with increasing degrees of sophistication [7, 8, 9].

Another concept which also reveals characteristic features of quantum theory
is the concept of steering [16]. It has been introduced originally by
Schrödinger in 1935 [12, 13], in the same year in which also Einstein,
Podolsky and Rosen (EPR) published their work questioning the completeness of
quantum mechanical description and describing the well-known so called EPR
paradoxon [10]. Steering, sometimes also called EPR steering, may be viewed as
a generalization of the experimental scenario which is described by (1) within
a local realistic theory and which also forms the basic scenario discussed in
the EPR paradoxon.

Bipartite scenarios are among the simplest ones by which the concept of EPR
steering can be formulated. Accordingly, let us again consider two
experimenters, Alice and Bob, performing general local measurements which can
be described by local positive operator valued measures (POVMs) [32, 33], say
$\\{\Pi^{A}_{\alpha,a}\geq 0\\}$ for Alice and $\\{\Pi^{B}_{\beta,b}\geq 0\\}$
for Bob with $\alpha$ and $\beta$ denoting the different measurements and $a$
and $b$ their corresponding measurement results. These POVMs obey the
characteristic completeness relations

$\displaystyle\sum_{a\in{\cal M}_{A}}\Pi^{A}_{\alpha,a}$ $\displaystyle=$
$\displaystyle{1\\!\\!1}_{d_{A}},~{}~{}\sum_{b\in{\cal
M}_{B}}\Pi^{B}_{\beta,b}={1\\!\\!1}_{d_{B}}$ (2)

with the unit operators ${1\\!\\!1}_{d_{A}}$ and ${1\\!\\!1}_{d_{B}}$ of the
$d_{A}$ and $d_{B}$ dimensional local Hilbert spaces ${\cal H}^{A}$ and ${\cal
H}^{B}$ of Alice’s and Bob’s quantum systems. The Hilbert space of the
complete bipartite quantum system is given by ${\cal H}={\cal
H}^{A}\otimes{\cal H}^{B}$.

In a typical bipartite EPR steering from Alice to Bob, Alice prepares a
bipartite quantum state, say $\rho\geq 0$, with corresponding reduced local
density operators $\rho^{A}=\Tr_{B}\\{\rho\\}$ and
$\rho^{B}=\Tr_{A}\\{\rho\\}$ for Alice and Bob. It is assumed that Alice can
perform local measurements on the combined quantum state only with the help of
her POVM $\\{\Pi^{A}_{\alpha,a}\\}$ and Bob can perform local measurements on
this quantum state only with his POVM $\\{\Pi^{B}_{\beta,b}\\}$. According to
quantum theory the bipartite probability distribution of a possible joint
local measurement on this quantum state, in which Alice and Bob measure
observables $\alpha$ and $\beta$ with measurement results $a$ and $b$, is
given by

$\displaystyle P(a,b\mid\alpha,\beta,\rho)$ $\displaystyle=$
$\displaystyle\Tr_{AB}\\{\Pi^{A}_{\alpha,a}\otimes\Pi^{B}_{\beta,b}\rho\\}.$
(3)

A quantum state $\rho$ is called EPR unsteerable from Alice to Bob with
respect to measurements $\alpha$ and $\beta$ if this joint probability
distribution can be described within the framework of a local hidden state
(LHS) model [14, 15]. Otherwise this quantum state is called EPR steerable
from Alice to Bob with respect to these measurements. More concretely, EPR
unsteerability from Alice to Bob with respect to local measurements described
by the POVMs $\\{\Pi^{A}_{\alpha,a}\\}$ and $\\{\Pi^{B}_{\beta,b}\\}$ means
that there exists a statistical ensemble of reduced quantum states of Bob, say
$\\{(\lambda,\rho^{B}_{\lambda})\mid\lambda\in\Lambda\\}$ with the random
variable $\\{\lambda\in\Lambda\\}$ being characterized by a probability
distribution $\\{p(\lambda)>0|\lambda\in\Lambda\\}$, which is consistent with
the conditional joint probability (3), i.e.

$\displaystyle P(a,b\mid\alpha,\beta,\rho)$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\Lambda}p(\lambda)P^{A}(a\mid\alpha,\lambda)P^{B}(b\mid\beta,\lambda),$
$\displaystyle P^{B}(b\mid\beta,\lambda)$ $\displaystyle:=$
$\displaystyle\Tr_{B}\\{\Pi^{B}_{\beta,b}\rho^{B}_{\lambda}\\}.$ (4)

It is apparent that this restriction on the joint probability distribution of
Alice $(A)$ and Bob $(B)$ is weaker than the restriction (1) imposed by local
realism as it involves a quantum mechanical evaluation of Bob’s local
conditional probability distribution $P^{B}(b\mid\beta,\lambda)$. Using (2)
and the characteristic normalization for conditional probabilities this
condition (4) implies the relations

$\displaystyle\sum_{a\in{\cal
M}_{A}}P(a,b\mid\alpha,\beta,\rho)=\Tr_{B}\\{\Pi^{B}_{\beta,b}\rho^{B}\\}=$
(5)
$\displaystyle\Tr_{B}\\{\Pi^{B}_{\beta,b}\sum_{\lambda\in\Lambda}p(\lambda)\rho^{B}_{\lambda}\\},$
$\displaystyle\sum_{b\in{\cal
M}_{B}}P(a,b\mid\alpha,\beta,\rho)=\Tr_{A}\\{\Pi^{A}_{\alpha,a}\rho^{A}\\}=$
$\displaystyle\sum_{\lambda\in\Lambda}p(\lambda)P^{A}(a\mid\alpha,\lambda).$

Thus with respect to Bob’s measurement of the POVM $\\{\Pi^{B}_{\beta,b}\\}$
Bob’s reduced quantum state $\rho^{B}$ and the LHS state
$\sum_{\lambda\in\Lambda}p(\lambda)\rho^{B}_{\lambda}$ are indistinguishable.

EPR steering from Bob to Alice is defined in an analogous way. It is worth
mentioning that the two concepts of EPR steerability or EPR unsteerability,
namely from Alice to Bob on the one hand and from Bob to Alice on the other
hand, are asymmetric. In particular, EPR steerability (or EPR unsteerability)
from Alice to Bob does not necessarily imply EPR steerability (or EPR
unsteerability) from Bob to Alice.

##  3 Generalized measurements by positive operator valued $(N,M)$\- measures

In this section we summarize basic results on informationally complete
generalized measurements which can be described by $(N,M)$-POVMs [25]. They
allow for a unified description of various more specialized measurement
schemes, such as projective measurements based on mutually unbiased bases
[26], mutually unbiased measurements (MUMs) [23], informationally complete
generalized measurements (SIC-POVMs) [27, 28] and their generalized analogues
(GSIC-POVMs) [24, 29].

In a $d$-dimensional Hilbert space a $(N,M)$-POVM, say $\Pi$, is a set of $N$
POVMs so that each of them involves $M$ positive (semidefinite) operators
describing the different possible measurement results, i.e.
$\Pi=\\{\Pi_{\alpha,a}\geq 0|~{}\alpha=1,\cdots,N;~{}a=1,\cdots,M\\}$. The
parameter $\alpha$ identifies a particular measurement scheme associated with
an experimental setup and the parameter $a$ identifies the corresponding
different possible measurement results. In a $(N,M)$-POVM the operators
$\Pi_{\alpha,a}$ are constrained by the additional relations [25]

$\displaystyle\Tr\\{\Pi_{\alpha,a}\\}$ $\displaystyle=$
$\displaystyle\frac{d}{M},$ (6)
$\displaystyle\Tr\\{\Pi_{\alpha,a}~{}\Pi_{\alpha,a^{\prime}}\\}$
$\displaystyle=$ $\displaystyle
x~{}\delta_{aa{{}^{\prime}}}+(1-\delta_{aa{{}^{\prime}}})\frac{d-Mx}{M(M-1)},$
(7)
$\displaystyle\Tr\\{\Pi_{\alpha,a}~{}\Pi_{\alpha{{}^{\prime}},a{{}^{\prime}}}\\}$
$\displaystyle=$ $\displaystyle\frac{d}{M^{2}}$ (8)

for all $\alpha\neq\alpha{{}^{\prime}}$. Different real values of $x$ yield
different $(N,M)$-POVMs, and the possible values of $x$ are constrained by the
relation $d/M^{2}<x\leq{\rm min}(d^{2}/M^{2},d/M)$.

A $(N,M)$-POVM $\Pi$ is informationally complete if and only it contains
$d^{2}$ linearly independent positive (semidefinite) operators. As each of the
$N$ POVMs fulfills the completeness relation (2) informational completeness
constrains the parameters $N,M$ and $d$ by the relation [25]

$\displaystyle(M-1)N+1$ $\displaystyle=$ $\displaystyle d^{2}.$ (9)

Therefore, each possible solution of this equation yields a possible
informationally complete $(N,M)$-POVM in a given $d$-dimensional Hilbert
space. As a result at least four possible classes of informationally complete
$(N,M)$-POVMs can always be constructed. They correspond to the possible
solutions $(N,M)\in\\{(1,d^{2}),~{}(d+1,d),~{}(d^{2}-1,2),~{}(d-1,d+2)\\}$. In
particular, the solution $(N,M)=(1,d^{2})$ characterizes a one-parameter
family of generalized symmetric informationally complete positive operator
valued measures (GSIC-POVM) [24, 29]. The solution $(N,M)=(d+1,d)$ describes
mutually unbiased measurements (MUMs) [23], which in the special case
$x=d^{2}/M^{2}=d/M=1$ reduce to projective measurements with maximal sets of
$d+1$ mutually unbiased bases. For a qubit, i.e. $d=2$, these four possible
solutions of (9) collapse to two cases, namely GSIC-POVMs ($(N,M)=(1,4)$) and
MUMs ($(N,M)=3,2)$).

In a $d$-dimensional Hilbert space informationally complete $(N,M)$-POVMs can
be constructed with the help of a basis of $d^{2}$ linearly independent
hermitian operators. For this purpose it is advantageous to choose an
orthonormal basis of hermitian operators, i.e. $\\{G_{i}\mid
G_{i}=G_{i}^{\dagger},i=1,\cdots,d^{2}\\}$, with respect to the Hilbert-
Schmidt (HS) scalar product $\langle
G_{i}|G_{i^{\prime}}\rangle:=\Tr\\{G_{i}^{\dagger}G_{i^{\prime}}\\}$. Such a
basis spans the $d^{2}$-dimensional HS-Hilbert space ${\cal H}_{d^{2}}$ of
hermitian linear operators over the field of real numbers and each linear
operators of ${\cal H}_{d^{2}}$ acts on elements of ${\cal H}_{d}$.

Starting from an arbitrary orthonormal basis ${\cal
B}=\\{|k\rangle;k=1,\cdots,d\\}$ in the $d$-dimensional Hilbert space ${\cal
H}_{d}=({\rm Span}({\cal B}),\langle\cdot|\cdot\rangle)$ of a $d$-dimensional
quantum system, an example of such a hermitian orthonormal operator basis is
given by

$\displaystyle\tilde{G}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{k=1}^{d}|k\rangle\langle k|,$ (10)
$\displaystyle\tilde{G}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{i(i-1)}}\left(\sum_{k=1}^{i-1}|k\rangle\langle
k|-(i-1)|i\rangle\langle i|\right),$ $\displaystyle\tilde{G}_{md+n}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(|m\rangle\langle
n|+|n\rangle\langle m|\right),~{}~{}1\leq m<n\leq d,$
$\displaystyle\tilde{G}_{(m-1)d+n}$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\left(|m\rangle\langle n|-|n\rangle\langle
m|\right),~{}~{}1\leq n<m\leq d$

for $i=2,\cdots,d$. This basis fulfills the additional convenient requirements
$\tilde{G}_{1}\sqrt{d}=1\\!\\!1_{d}$ so that orthonormality implies
$\Tr\\{\tilde{G}_{i}\\}=0$. Any other orthonormal hermitian operator basis can
be constructed from this one by applying an arbitrary real-valued orthogonal
transformation, say $\tilde{O}$ with
$\tilde{O}\tilde{O}^{T}=1\\!\\!1_{d^{2}}$, onto this operator basis. For our
subsequent discussion it is advantageous to arrange the basis operators of
such an arbitrary hermitian operator basis in a $d^{2}$-dimensional column
vector, say $G=(G_{1},\cdots,G_{d^{2}})^{T}=\tilde{O}\tilde{G}$. The hermitian
basis operators of any orthonormal basis fulfill the relation

$\displaystyle\frac{1}{d}\sum_{i=1}^{d^{2}}G_{i}^{2}$ $\displaystyle=$
$\displaystyle 1\\!\\!1_{d}.$ (11)

Analogously, an arbitrary $(N,M)$-POVM can be combined to a $N\times
M$-dimensional row vector of positive (semidefinite) operators, i.e.
$\Pi=(\Pi_{1},\cdots,\Pi_{NM})$. Its expansion in terms of this basis is then
given by $\Pi=G^{T}S$ with the real-valued $d^{2}\times NM$-matrix $S$, whose
matrix elements are constrained by the defining properties (6), (7) and (8) of
$(N,M)$-POVMs.

##  4 Correlation-matrix based sufficient conditions for bipartite EPR
steerability from Alice to Bob

In this section the general form of the recently proposed sufficient condition
for bipartite EPR steerability from Alice to Bob of Lai and Luo [22] is
discussed and applied to local measurements of Alice and Bob which can be
described by arbitrary $(N,M)$-POVMs or by LOOs. It is based on a violation of
an inequality involving the trace- or $1$-norm of the correlation matrix of
these local measurements of Alice and Bob and is valid for arbitrary
dimensional quantum systems.

In order to investigate EPR steerability of bipartite quantum states with
respect to two arbitrary local generalized measurements of the $(N,M)$-type,
say $\Pi^{A}$ and $\Pi^{B}$, performed by Alice and Bob let us consider the
associated correlation matrix $C(\Pi^{A},\Pi^{B}\mid\rho)$ with matrix
elements

$\displaystyle\left(C(\Pi^{A},\Pi^{B}\mid\rho)\right)_{ij}$ $\displaystyle=$
$\displaystyle\Tr_{AB}\\{\Pi^{A}(i)\otimes\Pi^{B}(j)\left(\rho-\rho^{A}\otimes\rho^{B}\right)\\}.$
(12)

Thereby, $\Pi^{A}(i)$ denotes the $i$-th component of Alice’s row vector
$\Pi^{A}$ with the indexing
$i(\alpha,a):=(\alpha-1)M_{A}+a\in\\{1,\cdots,N_{A}M_{A}\\}$ and analogously
for Bob.

If the bipartite quantum state $\rho$ is EPR unsteerable from Alice to Bob
with respect to these local measurements this correlation matrix can be
rewritten in the form (cf. (4))

$\displaystyle\left(C(\Pi^{A},\Pi^{B}\mid\rho)\right)_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\lambda,\lambda^{\prime}\in\Lambda}p(\lambda)p(\lambda^{\prime})V_{i}(\lambda,\lambda^{\prime})W_{j}(\lambda,\lambda^{\prime}),$
$\displaystyle V_{i(\alpha,a)}(\lambda,\lambda^{\prime})$ $\displaystyle=$
$\displaystyle P(a\mid\alpha,\lambda)-P(a\mid\alpha,\lambda^{\prime}),$
$\displaystyle W_{j(\beta,b)}(\lambda,\lambda^{\prime})$ $\displaystyle=$
$\displaystyle\Tr_{B}\\{\Pi^{B}_{\beta,b}\rho^{B}_{\lambda}\\}-\Tr_{B}\\{\Pi^{B}_{\beta,b}\rho^{B}_{\lambda^{\prime}}\\}$
(13)

with $i\in\\{1,\cdots,N_{A}M_{A}\\},j\in\\{1,\cdots,N_{B}M_{B}\\}$. The trace-
or $1$-norm of this $N_{A}M_{A}\times N_{B}M_{B}$ rectangular correlation
matrix can be upper bounded with the help of triangular and Cauchy-Schwarz
inequalities yielding the result

$\displaystyle||C(\Pi^{A},\Pi^{B}\mid\rho)||_{1}\leq$ (14)
$\displaystyle\frac{1}{2}\sum_{\lambda,\lambda^{\prime}\in\Lambda}p(\lambda)p(\lambda^{\prime})||V(\lambda,\lambda^{\prime})||_{2}||W(\lambda,\lambda^{\prime})||_{2}\leq$
$\displaystyle\frac{1}{2}\sqrt{\sum_{\lambda,\lambda^{\prime}\in\Lambda}p(\lambda)p(\lambda^{\prime})||V(\lambda,\lambda^{\prime})||^{2}_{2}}\sqrt{\sum_{\lambda,\lambda^{\prime}\in\Lambda}p(\lambda)p(\lambda^{\prime})||W(\lambda,\lambda^{\prime})||^{2}_{2}}.$

Using the positivity of variances this upper bound can be further upper
bounded with the help of the inequalities

$\displaystyle\sum_{\lambda,\lambda^{\prime}\in\Lambda}\sum_{i=1}^{N_{A}M_{A}}p(\lambda)p(\lambda^{\prime})\left(V_{i}(\lambda,\lambda^{\prime})\right)^{2}\leq$
$\displaystyle
2\sum_{i=1}^{N_{A}M_{A}}\left(\Tr_{A}\\{\left(\Pi^{A}\right)^{2}(i)\rho^{A}\\}-\left(\Tr_{A}\\{\Pi^{A}(i)\rho^{A}\\}\right)^{2}\right):=2V_{>},$
$\displaystyle\sum_{\lambda,\lambda^{\prime}\in\Lambda}\sum_{j=1}^{N_{B}M_{B}}p(\lambda)p(\lambda^{\prime})\left(W_{j}(\lambda,\lambda^{\prime})\right)^{2}\leq$
$\displaystyle
2\sum_{\lambda\in\Lambda}\sum_{j=1}^{N_{B}M_{B}}\left(\Tr_{B}\\{\Pi^{B}(j)\rho^{B}_{\lambda}\\}\right)^{2}-2\sum_{j=1}^{N_{B}M_{B}}\left(\Tr_{B}\\{\Pi^{B}(j)\rho^{B}\\}\right)^{2}$
$\displaystyle\leq
2\left(\max_{\sigma^{B}}\sum_{j=1}^{N_{B}M_{B}}\left(\Tr_{B}\\{\Pi^{B}(j)\sigma^{B}\\}\right)^{2}-\right.$
$\displaystyle\left.\sum_{j=1}^{N_{B}M_{B}}\left(\Tr_{B}\\{\Pi^{B}(j)\rho^{B}\\}\right)^{2}\right):=2W_{>}$
(15)

which involve a maximization over all possible reduced quantum states of Bob
$\sigma^{B}$. Combining these upper bound we finally obtain the inequality

$\displaystyle||C(\Pi^{A},\Pi^{B}\mid\rho)||_{1}$ $\displaystyle\leq$
$\displaystyle\sqrt{V_{>}W_{>}}.$ (16)

Therefore, provided a bipartite quantum state $\rho$ is EPR unsteerable from
Alice to Bob with respect to the arbitrary $(N,M)$-POVMs $\Pi^{A}$ and
$\Pi^{B}$, the $1$-norm of the associated correlation matrix, i.e.
$||C(\Pi^{A},\Pi^{B}\mid\rho)||_{1}$, fulfills inequality (16). Stated
differently, a violation of inequality (16) is a sufficient condition for
bipartite EPR steerability from Alice to Bob with respect to these local
generalized measurements. Consequently inequality (16) can be used to obtain
upper bounds on measures of EPR unsteerable bipartite states from Alice to Bob
and lower bounds on measures of bipartite EPR steerable bipartite states from
Alice to Bob.

It is straight forward to derive an analogous inequality for an arbitrary set
of local hermitian operators, say $\alpha=(\alpha_{1},\cdots,\alpha_{m})$ for
Alice and $\beta=(\beta_{1},\cdots,\beta_{n})$ for Bob describing local
measurements. The resulting inequality is given by [22]

$\displaystyle||C(\alpha,\beta\mid\rho)||_{1}$ $\displaystyle\leq$
$\displaystyle\sqrt{V^{A}_{>}W^{B}_{>}}$ (17)

with

$\displaystyle V^{A}_{>}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\left(\Tr_{A}\\{\left(\alpha_{i}\right)^{2}\rho^{A}\\}-\left(\Tr_{A}\\{\alpha_{i}\rho^{A}\\}\right)^{2}\right),$
(18) $\displaystyle W^{B}_{>}$ $\displaystyle=$
$\displaystyle\left(\max_{\sigma^{B}}\sum_{j=1}^{n}\left(\Tr_{B}\\{\beta_{j}\sigma^{B}\\}\right)^{2}-\sum_{j=1}^{n}\left(\Tr_{B}\\{\beta_{j}\rho^{B}\\}\right)^{2}\right)$

which also involves a maximization over all possible reduced quantum states of
Bob $\sigma^{B}$.

##  5 Local detection of bipartite EPR steerability from Alice to Bob by
informationally complete $(N,M)$-POVMs

The natural question arises how the sufficient condition on bipartite EPR
steerability from Alice to Bob based on a violation of (16) depends on the
type of local measurements performed by Alice and Bob. In this section it is
demonstrated that inequality (16) exhibits a characteristic scaling property
originating from a permutation symmetry inherent in the definition of
$(N,M)$-POVMs. It implies that all informationally complete local measurements
involving $(N,M)$-POVMS and all LOOs lead to one and the same inequality whose
violation yields a sufficient condition for bipartite EPR steerability from
Alice to Bob. The derivation of this scaling property is based on general
relations between informationally complete $(N,M)$-POVMs and LOOs which have
been obtained recently in an investigation on the detection of typical
bipartite entanglement with the help of local measurements [34] and are
summarized in the appendix.

In order to explore this scaling property let us first of all investigate
inequality (17) for the special case of LOOs for Alice and Bob as described by
(10). Accordingly, Alice measures the hermitian basis operators
$\tilde{G}^{A}=(\tilde{G}^{A}_{1},\cdots,\tilde{G}^{A}_{d_{A}^{2}})^{T}$ and
Bob measures the hermitian basis operators
$\tilde{G}^{B}=(\tilde{G}^{B}_{1},\cdots,\tilde{G}^{B}_{d_{B}^{2}})^{T}$.
According to (18) the relevant upper bounds yield the results

$\displaystyle V^{A}_{>}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d_{A}^{2}}\left(\Tr_{A}\\{(\tilde{G}^{A}_{i})^{2}\rho^{A}\\}-\left(\Tr_{A}\\{\tilde{G}^{A}_{i}\rho^{A}\\}\right)^{2}\right)=$
$\displaystyle d_{A}-\Tr_{A}\\{(\rho^{A})^{2}\\},$ $\displaystyle W^{B}_{>}$
$\displaystyle=$
$\displaystyle\max_{\sigma^{B}}\sum_{j=1}^{d_{B}^{2}}\left(\Tr_{B}\\{\tilde{G}^{B}_{j}\sigma^{B}\\}\right)^{2}-\sum_{j=1}^{d_{B}^{2}}\left(\Tr_{B}\\{\tilde{G}^{B}_{j}\rho^{B}\\}\right)^{2}=$
(19) $\displaystyle 1-\Tr_{B}\\{(\rho^{B})^{2}\\}.$

Thus, inequality (17) reduces to the result

$\displaystyle||C(\tilde{G}^{A},\tilde{G}^{B}\mid\rho)||^{2}_{1}$
$\displaystyle\leq$
$\displaystyle\left(d_{A}-\Tr_{A}\\{(\rho^{A})^{2}\\}\right)\left(1-\Tr_{B}\\{(\rho^{B})^{2}\\}\right).$

As the $1$-norm of a matrix is invariant under arbitrary local orthogonal
transformations performed by Alice and Bob, say $\tilde{O}^{A}$ and
$\tilde{O}^{B}$, inequality (LABEL:LOO1) also applies to any other LOOs for
Alice and Bob, say $G^{A}=\tilde{O}^{A}\tilde{G}^{A}$ and
$G^{B}=\tilde{O}^{B}\tilde{G}^{B}$, i.e.

$\displaystyle||C(G^{A},G^{B}\mid\rho)||^{2}_{1}$ $\displaystyle\leq$
$\displaystyle\left(d_{A}-\Tr_{A}\\{(\rho^{A})^{2}\\}\right)\left(1-\Tr_{B}\\{(\rho^{B})^{2}\\}\right).$

Starting from measurements involving LOOs on Alice’s and Bob’s sides, let us
now work out inequality (16) for local informationally complete $(N,M)$-POVMs
for Alice and Bob by representing these POVMs in these orthonormal hermitian
operator bases. Changing the LOOs from $G^{A}$ and $G^{B}$ to two
informationally complete local $(N,M)$-POVMs on Alice’s and Bob’s sides will
affect both sides of inequality (LABEL:LOO). So, let Alice and Bob use local
informationally complete $(N,M)$-POVMs, say $\Pi^{A}$ and $\Pi^{B}$ with basis
expansions $\Pi^{A}=(G^{A})^{T}S^{A}$ and $\Pi^{B}=(G^{B})^{T}S^{B}$ in
arbitrary local orthonormal bases $G^{A}$ and $G^{B}$. As outlined in the
appendix the defining properties of informationally complete $(N,M)$-POVMs
(2), (7), (8) and (9) imply that the $1$-norms of the correlation matrices
exhibit the simple scaling property

$\displaystyle||C(\Pi^{A},\Pi^{B}\mid\rho)||_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{A}\Gamma_{B}}||C(G^{A},G^{B}\mid\rho)||_{1}$ (22)

with

$\displaystyle\Gamma_{A}$ $\displaystyle=$
$\displaystyle\frac{x_{A}M_{A}^{2}-d_{A}}{M_{A}(M_{A}-1)},~{}\Gamma_{B}=\frac{x_{B}M_{B}^{2}-d_{B}}{M_{B}(M_{B}-1)}.$
(23)

Let us now investigate the scaling properties of the right hand sides of (16)
and (17) involving the general upper bounds (15) and (18) which both involve a
maximization over all possible reduced quantum states of Bob $\sigma^{B}$.
Expanding also the relevant reduced density operators of Alice and Bob, i.e.
$\rho^{A}=(G^{A})^{T}r^{A}$, $\rho^{B}=(G^{B})^{T}r^{B}$, and
$\sigma^{B}=(G^{B})^{T}s^{B}$, we find with the help of (11), (39) and (40)
the relations

$\displaystyle\sum_{i=1}^{N_{A}M_{A}}\Tr_{A}\\{(\Pi^{A})^{2}(i)\rho^{A}\\}=\Tr_{A}\\{(G^{A})^{T}S^{A}(S^{A})^{T}G^{A}\rho^{A}\\}=$
$\displaystyle d_{A}\Gamma_{A}+\frac{N_{A}d_{A}/M_{A}-\Gamma_{A}}{d_{A}},$
$\displaystyle\sum_{i=1}^{N_{A}M_{A}}(\Tr_{A}\\{\Pi^{A}(i)\rho^{A}\\})^{2}=(r^{A})^{T}S^{A}(S^{A})^{T}r^{A}=$
$\displaystyle\Gamma_{A}\Tr_{A}\\{(\rho^{A})^{2}\\}+\frac{N_{A}d_{A}/M_{A}-\Gamma_{A}}{d_{A}},$
$\displaystyle\sum_{j=1}^{N_{B}M_{B}}(\Tr_{B}\\{\Pi^{B}(j)\rho^{B}\\})^{2}=(r^{B})^{T}S^{B}(S^{B})^{T}r^{B}=$
$\displaystyle\Gamma_{B}\Tr_{B}\\{(\rho^{B})^{2}\\}+\frac{N_{B}d_{B}/M_{B}-\Gamma_{B}}{d_{B}},$
$\displaystyle\sum_{j=1}^{N_{B}M_{B}}\max_{\sigma^{B}}(\Tr_{B}\\{\Pi^{B}(j)\sigma^{B}\\})^{2}=\max_{s^{B}}(s^{B})^{T}S^{B}(S^{B})^{T}s^{B}=$
$\displaystyle\Gamma_{B}+\frac{N_{B}d_{B}/M_{B}-\Gamma_{B}}{d_{B}}.$ (24)

As a result inequality (16) for local informationally complete $(N,M)$-POVMs
of Alice and Bob becomes

$\displaystyle||C(\Pi^{A},\Pi^{B}\mid\rho)||_{1}=\sqrt{\Gamma_{A}\Gamma_{B}}||C(G^{A},G^{B}\mid\rho)||_{1}\leq$
$\displaystyle\sqrt{\Gamma_{A}\Gamma_{B}}\sqrt{\left(d_{A}-\Tr_{A}\\{(\rho^{A})^{2}\\}\right)\left(1-\Tr_{B}\\{(\rho^{B})^{2}\\}\right)}$
(25)

demonstrating that a violation of inequality (LABEL:LOO) yields the sufficient
condition for bipartite EPR steerability from Alice to Bob not only for
arbitrary LOOs but also for arbitrary informationally complete local
$(N,M)$-POVMS.

For the simple case of Bell diagonal states of the form

$\displaystyle\rho_{Bell}$ $\displaystyle=$
$\displaystyle\frac{1}{4}1\\!\\!1_{2}\otimes
1\\!\\!1_{2}+\sum_{i=1}^{3}\frac{t_{i}}{2}\sigma_{i}^{A}\otimes\sigma_{i}^{B}$
(26)

with the Pauli spin operators $\sigma_{i}^{A}$ and $\sigma_{i}^{B}$ for Alice
and Bob it is straightforward to determine the quantum states which are
steerable from Alice to Bob and which can be detected by a violation of
inequality (25) or equivalently (LABEL:LOO) with the help of LOOs or of local
informationally complete $(N,M)$-POVMs. In Fig. 1 these states are represented
by the yellow regions within the tetrahedron of all Bell diagonal quantum
states. The yellow and blue regions in this figure indicate the quantum states
which are steerable from Alice to Bob with respect to projective measurements
according to the necessary and sufficient condition of Uola et al. [16]. The
central uncolored region is formed by the states which are unsteerable from
Alice to Bob with respect to projective measurements according to this latter
criterion.

Figure 1: Bell diagonal quantum states which are steerable from Alice to Bob
according to a violation of inequality (25) or equivalently (LABEL:LOO)
(yellow region) and according to the criterion of Uola et al. [16] with
respect to projective measurements (yellow and blue regions): The central
uncolored region represents the convex set of states which are unsteerable
from Alice to Bob according to this latter criterion.

The sufficient condition based on a violation of inequality (25) may be
improved further by changing the LOOs of Alice and Bob $G^{A}$ and $G^{B}$ to
another set of local hermitian operators, say $\tilde{\alpha}^{A}$ and
$\tilde{\alpha}^{B}$, by relaxing the orthonormality constraints. This may be
achieved by a simple rescaling of Alice’s local observables, for example, i.e.

$\displaystyle\tilde{\alpha}^{A}_{i}=h_{i}G^{A}_{i}$ (27)

with the real-valued parameters $h_{i}$ [22]. With these local new
measurements inequality (17) assumes the form

$\displaystyle||C(\tilde{\alpha}^{A},G^{B}\mid\rho)||_{1}\leq$
$\displaystyle\sqrt{\tilde{V}^{A}_{>}\left(1-\Tr_{B}\\{(\rho^{B})^{2}\\}\right)}$
(28)

with

$\displaystyle\tilde{V}^{A}_{>}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d_{A}^{2}}h_{i}^{2}\left(\Tr_{A}\\{(G^{A}_{i})^{2}\rho^{A}\\}-\left(\Tr_{A}\\{G^{A}_{i}\rho^{A}\\}\right)^{2}\right).$

For a given state $\rho$ it can be tested whether the parameters $h_{i}$ can
be chosen in such a way that inequality (28) can be violated. Thus, it may be
possible to find for a given state $\rho$ an optimal set of parameters $h_{i}$
so that inequality (28) is violated even if inequality (LABEL:LOO) is still
fulfilled. Such an optimization of the sufficient condition for EPR
steerability from Alice to Bob by a variation over the parameters $h_{i}$ of
Alice’s LOO or local informationally complete $(N,M)$-POVM breaks the scaling
property. Thus, a violation of inequality (28) may be capable of detecting
more EPR steerable states from Alice to Bob than a violation of inequality
(LABEL:LOO).

##  6 Numerical results

Numerous investigations have already concentrated on criteria and sufficient
conditions of steerability and have applied them to restricted classes of
quantum states which typically form zero-measure sets within the convex set of
all quantum states [16]. However, so far questions concerning the statistical
typicality of steerability and its detectability by local measurements are
largely unexplored. In this section we concentrate on this latter issues and
explore Euclidean volume ratios between bipartite steerable quantum states and
all bipartite quantum states. In particular, we concentrate on the detection
of EPR steerability from Alice to Bob by local quantum measurements on the
basis of violations of inequality (28). The required volume ratios between EPR
steerable quantum states from Alice to Bob and all quantum states for
different dimensions of Alice’s and Bob’s quantum systems will be determined
with the help of a recently developed hit-and-run Monte-Carlo method [30, 35].
This way we want to shed light onto the efficiency with which typical
bipartite steerability can be detected with the help of local measurements.

Starting from a $d_{A}\times d_{B}$-dimensional Hilbert space ${\cal
H}_{d_{A}\times d_{B}}$ describing a bipartite quantum system the
corresponding $(d_{A}\times d_{B})^{2}$-dimensional Hilbert space ${\cal
H}_{(d_{A}\times d_{B})^{2}}$ of hermitian linear operators acting on elements
of ${\cal H}_{d_{A}\times d_{B}}$ can be constructed. With respect to the
Hilbert-Schmidt (HS) scalar product $\langle
A|B\rangle_{HS}:=\Tr_{AB}\\{A^{\dagger}B\\}$ for $A,B\in{\cal H}_{(d_{A}\times
d_{B})^{2}}$ this HS-Hilbert space ${\cal H}_{(d_{A}\times d_{B})^{2}}$ is a
Euclidean vector space on which volumes of sets of hermitian linear operators
and of convex sets of quantum states $\rho\geq 0$ can be defined in a natural
way. Numerically the Euclidean volumes of quantum states can be determined in
an efficient way with the help of a recently developed hit-and-run Monte-Carlo
algorithm [30] which has already been applied successfully to the
determination of Euclidean volumes of bipartite quantum states. This efficient
Monte-Carlo method, which has been introduced originally by Smith [36], relies
on the realization of a random walk inside a convex set that converges
efficiently to a uniform distribution over this convex set and, moreover, is
independent of the starting point inside this convex set [37].

With this hit-and-run Monte-Carlo method we have sampled $N=10^{8}$ bipartite
quantum states randomly for different values of $d_{A}$ and $d_{B}$. For each
of these randomly selected states it has been tested whether it violates the
sufficient condition of EPR steerability from Alice to Bob (28) for LOOs or
informationally complete $(N,M)$-POVMs as local measurements and for optimized
parameters $h_{i}$. If inequality (28) is violated this quantum state is EPR
steerable from Alice to Bob and is kept otherwise this quantum state is
dismissed. This way Euclidean volume ratios $R_{S:A\to B}$ between the volumes
of EPR steerable quantum states from Alice to Bob and all bipartite quantum
states have been determined numerically. In view of the optimization over the
parameters $h_{i}$ these Euclidean volume ratios $R_{S:A\to B}$ are always
larger than the corresponding ratios which are obtainable directly from a
violation of inequality (LABEL:LOO).

Our numerical results are summarized in Table 1.

Case | $d_{B}=2$ | $d_{B}=3$
---|---|---
$d_{A}=2$ | $5,011\times 10^{-2}$ | $1,92\times 10^{-5}$
| $\pm 1,5\times 10^{-4}$ | $\pm 4,1\times 10^{-6}$
$d_{A}=3$ | $5,72\times 10^{-5}$ | $0$
| $\pm 6,4\times 10^{-6}$ |
Table 1:  Numerical estimates of lower bounds of the Euclidean volume ratios
$R_{S:A\to B}$ between EPR steerable quantum states from Alice to Bob and all
bipartite quantum states for different dimensions $d_{A}$ and $d_{B}$ of
Alice’s and Bob’s quantum systems: These estimates are based on a violation of
inequality (28) with Alice’ local measurement being optimized by rescaling.
The numerical errors have been estimated with the procedure described in [30].

We have investigated numerically bipartite quantum states with $2\leq
d_{A},d_{B}\leq 4$. The cases not shown in Table 1 yield negligible volume
ratios below our numerical accuracy. These results suggest that detection of
typical bipartite EPR steerability from Alice to Bob on the basis of the
sufficient condition involving a violation of inequality (28) with LOO or
informationally complete $(N,M)$-POVMs as local measurements significantly
underestimates the volume ratios of EPR steerable states in higher dimensional
scenarios beyond two-qubit cases. In view of the peculiar scaling properties
of $(N,M)$-POVMs discussed in Sec. 5 the volume ratios of Table 1 cannot be
increased by changes to other LOOs or informationally complete local
$(N,M)$-POVMs.

In order to quantify this possible underestimation of bipartite steerability
it is of interest to compare the results of Table 1 with the corresponding
results of another sufficient condition for bipartite EPR steerability from
Alice to Bob which has been proposed recently by Das et al. [31]. However,
this method is only applicable whenever Alice’s quantum system is a qubit.
Beyond this requirement it does not impose any restrictions on the
dimensionality of Bob’s quantum system. These authors have proven that given
such a bipartite quantum state $\rho$, entanglement of the mixed quantum state

$\displaystyle\tau_{A\to
B}:=\mu\rho+\frac{(1-\mu)}{2}1\\!\\!1_{2}\otimes\Tr_{A}(\rho)$ (30)

for a value of $\mu\in\left[0,1/\sqrt{3}\right]$ is sufficient for EPR
steerability from Alice to Bob. As there are powerful methods for determining
bipartite entanglement, this approach may possibly yield better lower bounds
on the volume ratios $R_{S:A\to B}$ in its regime of validity. In particular,
according to Peres [38] and Horodecki [39] one may use the existence of a
negative partial transpose (NPT) of $\tau_{A\to B}$ as a sufficient condition
for bipartite entanglement for cases with $d_{B}>3$. In cases with $d_{B}\leq
3$ this latter condition is even also necessary for bipartite entanglement.
However, a possible disadvantage of this approach is that it is not based on
local measurements of Alice and Bob.

$d_{B}$ | $2$ | $3$ | $4$
---|---|---|---
| $0,05167$ | $0,10936$ | $0,17278$
| $\pm 1,5\times 10^{-4}$ | $\pm 3,4\times 10^{-4}$ | $\pm 5,6\times 10^{-4}$
$d_{B}$ | $5$ | $6$ | $7$
| $0,24009$ | $0,3119$ | $0,3842$
| $\pm 8,3\times 10^{-4}$ | $\pm 1,3\times 10^{-3}$ | $\pm 1,5\times 10^{-3}$
Table 2:  Numerical lower bounds on the Euclidean volume ratios $R_{S:A\to B}$
between EPR steerable quantum states from Alice to Bob and all bipartite
quantum states for $d_{A}=2$ and different dimensions $2\leq d_{B}\leq 7$ of
Bob’s quantum system: These estimates are based on the approach of Das et al.
[31]. The Peres-Horodecki condition has been used as a sufficient condition
for bipartite entanglement of $\tau_{A\to B}$ for $d_{B}>3$. The numerical
errors have been estimated with the procedure described in [30].

Table 2 depicts our numerically obtained lower bounds on Euclidean volume
ratios $R_{S:A\to B}$ based on the approach by Das et al. [31] combined with
NPT tests of the states $\tau_{A\to B}$ of (30) for different dimensions
$d_{B}$ of Bob’s quantum system. A comparison with the results of Table 1
demonstrates that for two qubits the detection of EPR steerability by
optimized local measurements leads to a result in agreement with the one of
Table 2. However, in all other cases the local measurement based sufficient
condition for EPR steerability from Alice to Bob significantly underestimates
the volume ratios $R_{S:A\to B}$ of Table 2. Furthermore, the volume ratios of
Table 2 also agree with the intuition suggested by the concept of EPR
steerability from Alice to Bob that increasing the dimensionality of Bob’s
local quantum system increases his abilities to detect EPR steerability by
Alice.

In view of the differences between the results of Tables 1 and 2 one may ask
whether the underestimated volume ratios of Table 1 may still be improved by
using the approach of Das et al [31] but detecting entanglement of the quantum
state $\tau_{A\to B}$ of (30) by local measurements. Recently it has been
demonstrated that local informationally complete $(N,M)$-POVMs are as powerful
in detecting entanglement of bipartite quantum states as LOOs [34]. This is a
consequence of the peculiar scaling properties characterizing local
informationally complete $(N,M)$-POVMs and of their relation to LOOs. It has
already been shown by Gittsovich and Gühne [40] that a sufficient condition
for bipartite entanglement detection by LOOs is given by a violation of the
inequality

$\displaystyle||C(G^{A},G^{B}|\rho)||_{1}\leq\sqrt{\left(1-\Tr_{A}\\{(\rho^{A})^{2}\\}\right)\left(1-\Tr_{B}\\{(\rho^{B})^{2}\\}\right)}$
(31)

with $G^{A}$ and GB denoting the LOOs of Alice and Bob and $\rho^{A}$ and
$\rho^{B}$ denoting their reduced quantum states. Using LOOs of the form (10)
for Alice and Bob and the relations $\tau_{A\to
B}^{A}=\mu\rho^{A}+\left((1-\mu)/2\right)1\\!\\!1_{2}$ and $\tau_{A\to
B}^{B}=\rho^{B}$ for the reduced quantum states of (30), inequality (31)
assumes the form

$\displaystyle\frac{1}{\mu}||C(\tilde{G}^{A},\tilde{G}^{B}\mid\tau_{A\to
B})||_{1}=||C(\tilde{G}^{A},\tilde{G}^{B}\mid\rho)||_{1}\leq$
$\displaystyle\sqrt{\left(\frac{1+\mu^{2}}{2\mu^{2}}-\Tr_{A}\\{(\rho^{A})^{2}\\}\right)\left(1-\Tr_{B}\\{(\rho^{B})^{2}\\}\right)}.$
(32)

For $\mu=1/\sqrt{3}$ inequality (32) reduces to inequality (LABEL:LOO) with
$d_{A}=2$. Thus, for cases with $d_{A}=2$ a violation of inequality
(LABEL:LOO) characterizes once again the sufficient condition for EPR
steerability from Alice to Bob through entanglement detection via correlation
matrices of local measurements involving LOOs or informationally complete
$(N,M)$-POVMs. This demonstrates that the approach of Das et al. [31] combined
with local measurements involving LOOs or informationally complete
$(N,M)$-POVMs as tests for bipartite entanglement cannot improve the results
of Table 1.

##  7 Conclusions

We have applied the correlation-matrix based sufficient condition for
bipartite EPR steerability from Alice to Bob of Lai and Luo [22] to local
measurements based on $(N,M)$-POVMs performed on arbitrary dimensional
bipartite quantum systems. It has been shown that within the class of local
informationally complete $(N,M)$-POVMs this sufficient EPR steerability
condition, which is based on a violation of inequality (16), exhibits a
peculiar scaling property. It implies that a violation of one and the same
inequality characterizes this sufficient condition for measurements involving
LOOs and for all informationally complete local $(N,M)$-POVMs. Thus, local
informationally complete $(N,M)$-POVMs are as powerful as LOOs for detecting
bipartite EPR steerability from Alice to Bob on the basis of a violation of
inequality (LABEL:LOO).

With the help of a hit-and-run Monte-Carlo algorithm we have determined lower
bounds on the Euclidean volume ratios of EPR steerable bipartite quantum
states from Alice to Bob and all bipartite quantum states for low dimensions
of Alice’s and Bob’s quantum systems. These numerical results explore the
statistical typicality of locally detectable bipartite EPR steerability from
Alice to Bob based on a violation of inequality (28). They demonstrate that,
except for the case of two qubits, the sufficient condition for bipartite EPR
steerability from Alice to Bob resulting from a violation of inequality (28)
tends to underestimate the Euclidean volume ratios between EPR steerable
bipartite quantum states from Alice to Bob and all bipartite quantum states
significantly. Our numerical investigations also demonstrate that the recently
introduced approach of Das et al. [31], which relates bipartite EPR
steerability from Alice’s qubit to Bob’s arbitrary dimensional qudit to
bipartite entanglement, can be more efficient provided methods for detecting
bipartite entanglement are used which transcend local measurements, such as
checking for the NPT property of bipartite quantum states. However, besides
not being based on local measurements a further disadvantage of this latter
approach is that so far its validity is restricted to cases in which Alice’s
quantum system is a qubit. Therefore, further research is required for the
development of efficient measurement-based methods for the detection of EPR
steerability and for the exploration of its intricate relation to
entanglement.

It is a pleasure to dedicate this work to Igor Jex on the occasion of his
sixtieth birthday. G.A. is grateful to his friend and regular collaborator
Igor Jex for numerous inspiring discussions on the intricacies of quantum
physics. This research is supported by the Deutsche Forschungsgemeinschaft
(DFG) – SFB 1119 – 236615297.

## Appendix A

In this appendix the derivation of the general scaling relation (22) of Sec.5
between the $1$-norms of the correlation matrices of arbitrary LOOs and local
informationally complete $(N,M)$-POVMs is outlined. The general relations
between informationally complete $(N,M)$-POVMs and orthonormal hermitian
operators bases presented in this appendix, which this derivation is based on
(cf. (39)), have been obtained recently in an investigation on the detection
of typical bipartite entanglement with the help of local measurements [34].

We consider a $d$ dimensional Hilbert space ${\cal H}_{d}=({\rm Span}({\cal
B}),\langle\cdot|\cdot\rangle)$ with orthonormal basis ${\cal
B}=\\{|1\rangle,\cdots,|d\rangle\\}$ and an associated arbitrary basis of
hermitian linear operators $G=(G_{1},\cdots,G_{d^{2}})^{T}$ with
$G_{\mu}=G_{\mu}^{\dagger},\mu\in\\{1,\cdots,d^{2}\\}$ acting on this Hilbert
space. Let us also assume that this operator basis is orthonormal with respect
to the Hilbert-Schmidt (HS) scalar product, i.e. $\langle
G_{\mu}|G_{\nu}\rangle_{HS}:=\Tr_{AB}\\{G_{\mu}G_{\nu}\\}=\delta_{\mu,\nu}$ so
that it spans a HS-Hilbert space ${\cal H}_{d^{2}}=({\rm
Span}(G),\langle\cdot|\cdot\rangle_{HS})$.

An arbitrary $(N,M)$-POVM, say $\Pi=\\{\Pi_{1},\cdots,\Pi_{NM}\\}$ with
$\Pi(i)\geq 0$,
$i(\alpha,a):=(\alpha-1)M+a,~{}\alpha\in\\{1,\cdots,N\\},~{}a\in\\{1,\cdots,M\\}$,
$i\in\\{1,\cdots,NM\\}$, can be expanded in this orthonormal hermitian
operator basis, i.e. $\Pi=G^{T}S$ with the $d^{2}\times(NM)$ matrix $S$ of
real valued coefficients. Using the orthonormality of the basis $G$, for
$N\geq 2$ conditions (7) and (8) characterizing $(N,M)$-POVMs can be rewritten
in the form

$\displaystyle\left(S^{T}S\right)_{i(\alpha,a),j(\alpha^{\prime},a^{\prime})}=\Gamma\delta_{i(\alpha,a),j(\alpha^{\prime},a^{\prime})}-$
(33)
$\displaystyle\frac{\Gamma}{M}\left(\bigoplus_{\alpha=1}^{N}J_{\alpha}\right)_{i(\alpha,a),j(\alpha^{\prime},a^{\prime})}+\frac{d}{M^{2}}J_{i(\alpha,a),j(\alpha^{\prime},a^{\prime})}$

with $\Gamma=(xM^{2}-d)/(M(M-1))$ (cf. (23)), the $(NM)\times(NM)$ matrix $J$
of all ones, i.e. $J_{i(\alpha,a),j(\alpha^{\prime},a^{\prime})}=1$, and with
the $M\times M$ block matrices $J_{\alpha}$ of all ones, i.e.
$\left(J_{\alpha}\right)_{i(\alpha,a),j(\alpha^{\prime},a^{\prime})}=\delta_{\alpha,\alpha^{\prime}}$.
The spectrum of the positive semidefinite symmetric $(NM)\times(NM)$ matrix
(33) is given by

$\displaystyle{\rm Sp}(S^{T}S)$ $\displaystyle=$
$\displaystyle\\{\Gamma^{(N(M-1))},\frac{dN}{M}^{(1)},0^{(N-1)}\\}$ (34)

with the exponents indicating the multiplicities of the eigenvalues. For $N=1$
the zero eigenvalue no longer appears in the spectrum of $S^{T}S$. Therefore,
according to (9) for informationally complete $(N,M)$-POVMs the dimension $D$
of the eigenspace of the non-zero eigenvalues is given by

$\displaystyle D$ $\displaystyle=$ $\displaystyle N(M-1)+1=d^{2}.$ (35)

The spectral representation of this symmetric matrix is given by

$\displaystyle\left(S^{T}S\right)_{i,j}$ $\displaystyle=$
$\displaystyle\sum_{\mu=1}^{N(M-1)+1}X_{i,\mu}\Lambda_{\mu}X^{T}_{\mu,j}$ (36)

with

$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{d}{M^{2}}NM,~{}~{}X_{i,1}=\frac{1}{\sqrt{NM}},$
$\displaystyle\Lambda_{\nu}$ $\displaystyle=$
$\displaystyle\Gamma,~{}~{}\sum_{a=1}^{M}X_{i(\alpha,a),\nu}=0$ (37)

for $i\in\\{1,\cdots,NM\\}$, $\nu\in\\{2,\cdots,N(M-1)+1\\}$. The
$(NM)\times(N(M-1)+1)$ matrix $X_{i,\mu}$ fulfills the orthogonality condition

$\displaystyle\sum_{i=1}^{NM}\left(X^{T}\right)_{\mu,i}X_{i,\nu}$
$\displaystyle=$ $\displaystyle\delta_{\mu,\nu}.$ (38)

As a consequence of (9), for an informationally complete $(N,M)$-POVM the most
general form of the $d^{2}\times(NM)$ matrix $S$ which is consistent with (7)
and (8) is given by

$\displaystyle S_{\mu,i}$ $\displaystyle=$
$\displaystyle\sum_{\mu^{\prime}=1}^{d^{2}}O^{T}_{\mu,\mu^{\prime}}\sqrt{\Lambda_{\mu^{\prime}}}X^{T}_{\mu^{\prime},i}$
(39)

with the arbitrary real-valued orthogonal $d^{2}\times d^{2}$ matrix $O$, i.e.
$OO^{T}=O^{T}O=P_{d^{2}}$. Thereby, $P_{d^{2}}$ denotes the projection
operator onto the $(N(M-1)+1)$-dimensional eigenspace of non-zero eigenvalues
of the linear operator $S^{T}S$ acting in the HS-Hilbert space ${\cal
H}_{NM}$. Note that in the case of an informationally complete $(N,M)$-POVM
this subspace is isomorphic to the HS-Hilbert space ${\cal H}_{d^{2}}$. Let us
finally also add the constraint (2) which yields the relation

$\displaystyle 1\\!\\!1_{d}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{M}\Pi(i(\alpha,a))=\sqrt{d}\sum_{\mu=1}^{d^{2}}G_{\mu}O^{T}_{\mu,1}$
(40)

where we have taken into account the constraints (37) on the eigenvectors of
$S^{T}S$. This relation together with the constraints (37) also implies
condition (6) so that all requirements defining an informationally complete
$(N,M)$-POVM $\Pi=G^{T}S$, namely (2), (6), (7), (8) and (9), are fulfilled.

With the help of (39) it is straight forward to relate the correlation matrix
of two local informationally complete $(N,M)$-POVMs for Alice and Bob, say
$\Pi^{A}$ and $\Pi^{B}$, to the correlation matrix of two LOOs, say $G^{A}$
and $G^{B}$. Using relation (39) for these local bases we find

$\displaystyle
C(\Pi^{A},\Pi^{B}|\rho)=\left(S^{A}\right)^{T}C(G^{A},G^{B}|\rho)S^{B}=$ (41)
$\displaystyle
X^{A}\sqrt{\Lambda^{A}}O^{A}C(G^{A},G^{B}|\rho)\left(O^{B}\right)^{T}\sqrt{\Lambda^{B}}\left(X^{B}\right)^{T}.$

For the corresponding $1$-norm we obtain the result

$\displaystyle||C(\Pi^{A},\Pi^{B}|\rho)||_{1}$ $\displaystyle=$
$\displaystyle||\sqrt{\Lambda^{A}}O^{A}C(G^{A},G^{B}|\rho)\left(O^{B}\right)^{T}\sqrt{\Lambda^{B}}||_{1}$

with $\Lambda^{A}$ and $\Lambda^{B}$ denoting the diagonal matrix of nonzero
eigenvalues of Alice and Bob. Using (40), the degeneracy of all eigenvalues
for $\mu\neq 1$, i.e. $\Lambda_{\mu\neq 1}=\Gamma$, and the invariance of the
$1$-norm under orthogonal transformations this expression simplifies to

$\displaystyle||C(\Pi^{A},\Pi^{B}|\rho)||_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma^{A}}\sqrt{\Gamma_{B}}||C(G^{A},G^{B}|\rho)||_{1}.$

The above arguments demonstrate that this scaling relation is valid for
arbitrary informationally complete local $(N,M)$-POVMs $\Pi^{A}$ and $\Pi^{B}$
and LOOs $G^{A}$ and $G^{B}$.

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