Math. Model. Nat. Phenom.
Vol. 10, No. 5, 2015, pp. 1-5
DOI: 10.1051/mmnp/201510501

Three waves of chemical dynamics

A.N. Gorbana111Corresponding author. E-mail<EMAIL_ADDRESS>G.S. Yablonskyb

aDepartment of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

bParks College of Engineering, Aviation and Technology,

Saint Louis University, Saint Louis, MO63103, USA

Abstract.Three epochs in development of chemical dynamics are presented. We
try to understand the modern research programs in the light of classical
works.

Key words:Chemical kinetics, reaction kinetics, chemical reaction networks,
mass action law

The first Nobel Prize in Chemistry was awarded in 1901 to Jacobus H. van’t
Hoff “in recognition of the extraordinary services he has rendered by the
discovery of the laws of chemical dynamics and osmotic pressure in solutions”.
This award celebrated the end of the first epoch in chemical dynamics,
discovery of the main laws. This epoch begun in 1864 when Waage and Guldberg
published their first paper about Mass Action Law [42]. Van’t Hoff
rediscovered this law independently in 1877. In 1984 the first book about
chemical dynamics was published, van’t Hoff’s “Études de Dynamique chimique
[40]”, in which he proposed the ‘natural’ classification of simple reactions
according to the number of molecules that are simultaneously participate in
the reaction. Despite his announcement, “I have not accepted a concept of mass
action law as a theoretical foundation”, van’t Hoff did the next step to the
development of the same law. He also studied the relations between kinetics
and thermodynamics and found the temperature dependence of equilibrium
constant (the van’t Hoff equation). In particular, he proved the “principle of
mobile equilibrium”: the equilibrium in a system tends to shift in such a
direction as to oppose the temperature change which is imposed upon the
system. This principle was later on generalized by Le Chatelier. The
temperature dependence of the reaction rate constants was analyzed further by
Arrhenius (Nobel prize in chemistry in 1903). The famous Arrhenius equation is
a particular case of the van’t Hoff equation. Chemical dynamics was developed
in line with thermodynamics. In particular, chemical equilibrium was
recognised as a dynamic process, i.e. the balance between the forward and
backward reactions. Their rates must be equal at the chemical equilibrium.
During this period, Gibbs created modern chemical thermodynamics [14]. The
kinetic law of a single reaction and its relation to thermodynamics were
studied in full. Chemical dynamics was prepared to the next epoch, that is to
the analysis of complex reaction networks.

The scientific context of the first decades of chemical dynamics was also very
important. Maxwell [28] and Boltzmann [5] developed physical kinetics of
gases. They used analogue of the Mass Action Law for collisions and discover
the principle of detailed balance: at the equilibrium, each elementary process
should be equilibrated by the reverse process. Maxwell used the principle of
sufficient reason to justify detailed balance. For the same purpose Boltzmann
employed reversibility of collision (for more detail see [16]). Detailed
balance was used by Boltzmann to prove the $H$-theorem (entropy growth) in
kinetics. Lorentz raised an objection against this theory [30]. He stated that
the collision of polyatomic molecules are irreversible and therefore the
principle of detailed balance is wrong. Boltzmann immediately invented a
weaker condition, the principle of cyclic balance (it is known also as
semidetailed balance or complex balance) [6] which is sufficient for the proof
of $H$-theorem. Much later, Stueckelberg proved that the cyclic balance
condition should hold for all systems with Markov micro-description [38] (he
used the $S$-matrix representation of collisions, for more detail see [20]).
Finally, in 1981, it was demonstrated that Lorentz was wrong and detailed
balance holds for collisions of polyatomic molecules [8]. In 1902, equilibrium
statistical mechanics was developed by Gibbs and mechanical backgrounds of
thermodynamics became clear [15].

In 1901, Wegscheider studied reaction networks which consist of several
elementary reactions [43]. He discovered that the equilibrium of such a
network may not be an equilibrium of each reaction from the network if we
define their rate constants independently (“Wegscheider’s paradox”). He found
the conditions on the reaction rate constants which are necessary and
sufficient for coincidence of the equilibrium of the network with the joint
equilibrium of their elementary reactions (Wegscheider’s conditions of
detailed balance). For example, in a simple linear cycle these conditions
state that clockwise and anticlockwise products of reaction rate constants
coincide. (The modern explanation could be found in many textbooks [44].) Of
course, if we define the rate of each backward reaction thermodynamically,
through the thermodynamic equilibrium constant and the reaction rate of the
forward reaction, then Wegscheider’s conditions are satisfied automatically
and Wegscheider’s paradox vanishes: the elementary reactions “do not know”
about other reactions, they all “just know” the same thermodynamic properties
of the reagents. Wegscheider’s conditions were used by Onsager [32] in his
work about reciprocal relations awarded by Nobel Prize in Chemistry in 1968.
The Onsager relations are, in their essence, the detailed balance conditions
linearized near thermodynamic equilibrium.

Wegscheider’s work was a first milestone in analysis of chemical reaction
networks. The great achievement here was the theory of chain reactions and of
the critical effects in chain reactions [37, 22] In 1956, Nobel Prize in
Chemistry was awarded to Semenov and Hinshelwood “for their researches into
the mechanism of chemical reactions”. The great problem approached by the
theory of chain reaction may be formulated as follows: how the structure of
reaction network affects its dynamics? Of course, it is solved only partially
and remains the source of challenges for several generations of researchers.
It appears that the work with this problem requires methods for model
reduction [9]. Besides the obvious reason (it is easier to analyse systems of
smaller dimension) there is a very important issue: in the whole system the
critical effects are hidden. To see the critical effect, we have to simplify
the model and to separate variables into fast and slow ones. The bifurcations
in the fast subsystem correspond to observable critical phenomena like
inflammation, ignition or, inverse, to quenching. This trick allowed Semenov
and Hinshelwood to study critical effects in chain reactions. Several methods
for model reduction were developed. Most famous of them are: quasi-steady
states, quasi-equilibrium states and limiting steps and subsystems. Later on
they were integrated into several more general technologies: methods of
invariant manifolds [18, 19] and computation singular perturbations [29]. The
mathematical backgrounds of all these methods are the theory of singular
perturbations [35] and the theory of invariant manifolds of dynamical systems.
Non-trivial dynamics of chemical systems attracted much attention and various
oscillations, bifurcations and non-linear waves were found. Among them the
famous example gives the Belousov-Zhabotinskii oscillatory reaction [45].

In 1965, the scientific program of new synthesis and rational analysis of
chemical reaction networks was proposed by Aris [3, 4]. His enthusiasm,
creativity and research reputation attracted many scientists and a series of
research project was performed [23, 12, 13, 25, 33, 36]. The Aris program was
driven by the needs of chemical engineering [2]. Nevertheless, the fundamental
results of this program spread wider and are now important to various areas of
science. Horn, Jackson and Feinberg rediscovered Boltzmann’s cyclic balance in
the context of chemical kinetics [23, 12] (they call it the “complex balance”
condition). This complex balance condition is sufficient for positivity of
entropy production (it is the exact analogue of the 1887 version of
Boltzmann’s $H$-theorem [6]). It guarantees decrease of the Helmholtz free
energy of closed systems under isothermal isochoric conditions and of the
Gibbs free energy (the free enthalpy) under isothermal isobaric conditions. It
appears that this condition is important not only in chemical engineering [39]
but also in algebraic geometry and related areas, in particular, in the theory
of toric dynamical systems [10]. There are numerous links between chemical
dynamics and modern algebraic geometry. Classification of the limits of
systems with detailed balance when some of reactions become irreversible [21]
practically coincides with the recently proved classification of binomial
manifolds [11]. The methods of tropical geometry are efficient for model
reduction in large reaction networks [31].

It is worth to mention other types of global Lyapunov functions which exist
for some reaction mechanisms. For example, if we consider the reaction
networks with elementary reactions of the form $mA_{i}\to\ldots$ (with various
coefficients $m$ and right hand sides of the reaction equations, but with a
positive linear conservation law like conservation of mass) then the
$l_{1}$-distance between all kinetic curves decreases in time [17]. For
systems with arbitrary monotone kinetics the Lyapunov functions are
constructed from reaction rate functions [1].

The question about connection between the structure of reaction network and
its dynamics, remains one of the central problem of chemical dynamics. In
parallel with answering this question we need to answer the question what is a
kinetic law of complex chemical reaction. If we separate times and select fast
and slow subsystems then this second question could be understood as an
elimination problem: what are equations of slow dynamics after elimination of
fast variables? This problem stimulated development of a new chapter of
computer algebra [7]. For several classes of catalytic reactions this
exclusion was performed and the steady-state kinetic law of the single overall
reaction was presented as the single polynomial regarding the reaction rate.
Such a polynomial may have several roots which correspond to different steady-
states. At the same time its free term has a rigorous thermodynamic form and
validates thermodynamic correctness of this presentation [26].

Various methods of graph theory were employed for analysis of chemical
reaction networks. The general theory of equations on graphs was developed
[41]. Now, several areas of applied and pure mathematics are used in chemical
dynamics and are developed further due to this applications. “The beginning of
this era was marked by the concerted effort of a few to raise the mathematical
consciousness of the profession to think fundamentally about processes” [34].
After these 50 years of efforts we have to look back, to review the past and
to ask about the future: what should we expect soon? New technologies will
generate new questions. The quantum world will become closer to industry.
Biological engineering will be as usual as is chemical engineering. The
artificial intelligence and rapid computation will change the practice of
mathematical modelling.

Three eras (or waves) of chemical dynamics can be revealed in the flux of
research and publications. These waves may be associated with leaders: the
first is the van’t Hoof wave, the second may be called the Semenov–Hinshelwood
wave and the third is definitely the Aris wave. Of course, the whole building
was impossible without efforts of hundreds of other researchers. Few of of
them are mentioned in our brief review, more are cited in the papers of the
issue.

This issue is opened by the reprint of the classical paper of A.I. Volpert,
where he introduced the differential equations on graphs. The work of Gorban
and Kolokoltsov analyzed appearance of Mass Action Law with complex balance
conditions and their generalizations in the Michaelis–Menten–Stueckelberg
limit of general Markov processes. Joshi and Shiu studied minimal reaction
networks with multiple steady states. In the paper by Bykov and Tsybenova the
classical nonlinear models of catalytic reaction were augmented by the
additional variable, i.e. temperature, and the extended models were
systematically studied. Müller and Hofbauer applied the formalism of chemical
reaction networks to kinetics of genetic recombination and analyze existence,
uniqueness, and global stability of an equilibrium in such networks.
Grigoriev, Samal, Vakulenko and Weber developed algebraic algorithms for
analysis and reduction of larger metabolic reaction networks and studied
several biochemical networks. Constales, Yablonsky and Marin analyzed kinetics
of pulse-response experiments in the Temporal Analysis of Products (TAP) setup
and demonstrated that in these special conditions the activity profile of a
prepared catalytic system depends only on the total amount of admitted
substance. Gorban demonstrated how to use a special geometric procedure,
forward-invariant peeling, to produce forward-invariant subset from a given
set in concentration space space and to prove persistence of a chemical
dynamic system.

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#  Einstein-Podolsky-Rosen uncertainty limits for bipartite multimode states

Paulina Marian<EMAIL_ADDRESS>Tudor A. Marian
<EMAIL_ADDRESS>Centre for Advanced Quantum Physics, Department of
Physics, University of Bucharest, R-077125 Măgurele, Romania

###### Abstract

Certification and quantification of correlations for multipartite states of
quantum systems appear to be a central task in quantum information theory. We
give here a unitary quantum-mechanical perspective of both entanglement and
Einstein-Podolsky-Rosen (EPR) steering of continuous-variable multimode
states. This originates in the Heisenberg uncertainty relations for the
canonical quadrature operators of the modes. Correlations of two-party
$(N\,\text{vs}\,1)$-mode states are examined by using the variances of a pair
of suitable EPR-like observables. It turns out that the uncertainty sum of
these nonlocal variables is bounded from below by local uncertainties and is
strengthened differently for separable states and for each one-way unsteerable
ones. The analysis of the minimal properly normalized sums of these variances
yields necessary conditions of separability and EPR unsteerability of
$(N\,\text{vs}\,1)$-mode states in both possible ways of steering. When the
states and the performed measurements are Gaussian, then these conditions are
precisely the previously-known criteria of separability and one-way
unsteerability.

##  I Introduction

The use of uncertainty arguments to understand quantum properties of composite
systems was initiated in the seminal work of Einstein, Podolsky and Rosen EPR
and deepened by Schrödinger, shortly thereafter Sch . Specifically, in
Refs.EPR ; Sch  the following paradox of quantum mechanics is described: two
distant parties of a given system share an entangled state and one party, by
measuring its subsystem, can remotely change (steer) the state of the other
party’s subsystem, in contrast to the requirements of the local realism. Many
years after, Bell proved that no local realistic theory can give a complete
description of the predictions of quantum mechanics (Bell’s theorem) Bell . A
further contribution in defining quantum correlations between the parties of a
composite system was made by Werner Werner . What was subsequently known as a
separable state, namely, a convex combination of product states, was termed by
Werner as being classically correlated. Werner’s original terminology for all
the states not having such an expansion has been Einstein-Podolsky-Rosen (EPR)
correlated states. Nowadays they are called entangled. Soon after Werner’s
definition, separability conditions were found for finite-dimensional systems
in terms of $2$-entropy inequalities H1 ; H2  or preservation of the non-
negativity of the density matrix under partial transposition Peres ; H3 . Note
that in 1989, a practical procedure to demonstrate the EPR paradox by using
products of inferred variances was first proposed by Reid Reid1  for
continuous-variable bipartite states.

A significant progress in understanding quantum correlations manifested
through entanglement, EPR steering, and Bell non-locality, in the framework of
a unified quantum-information description was given by Wiseman et al. as three
rather different tasks for confirming inseparability W1 ; W2 ; W3 . First, for
certification of two-party entanglement, use is made of the general aspect of
a separable state Werner  and quantum state tomography is performed with all-
trusted measurement devices. Second, quantum steering corresponds to the task
of verifiable entanglement distribution by one untrusted party. Third, in the
case of Bell nonlocality, both parties do not trust their measuring devices.
In the EPR steering scenario, the two parties, Alice and Bob, play a different
role and thus EPR steering is asymmetric when interchanging the subsystems,
unlike entanglement and Bell nonlocality W2 . Being perceived as another type
of quantum correlations, EPR steering has recently attracted considerable
interest especially related to its verification and quantification from a
quantum information perspective RMP ; Pusey ; SNC ; QVB ; UMG ; PW ;
Adesso2015 ; Kiukas ; exp2012 ; Das2019 . Note that for continuous-variable
settings, much attention was given from the very beginning to the conditions
and quantifiers of steering for Gaussian states (GSs) Reid1 ; W2 ; Reid2 ;
Adesso2 ; Adesso3 ; Nha ; Kiukas2 ; He .

In the present paper we take a different approach that treats on an equal
footing entanglement and steering. This generalizes and enlarges the EPR-line
of reasoning initiated by Reid for a two-mode state Reid1 . Essentially, Reid
used the variances of two nonlocal observables linearly built with the one-
mode canonical quadrature operators ${\hat{q}}_{j},{\hat{p}}_{j},(j=1,2)$:

$\displaystyle{\hat{Q}}(\lambda):={\hat{q}}_{1}-\lambda{\hat{q}}_{2},\qquad{\hat{P}}(\mu):={\hat{p}}_{1}+\mu{\hat{p}}_{2},$
(1)

where $\lambda$ and $\mu$ are adjustable positive parameters. The coordinates
and momenta in Eq. (1) are defined in terms of the amplitude operators of the
modes:

$\displaystyle\hat{q}_{j}:=\frac{1}{\sqrt{2}}(\hat{a}_{j}+\hat{a}_{j}^{{\dagger}}),\qquad\hat{p}_{j}:=\frac{1}{\sqrt{2}i}(\hat{a}_{j}-\hat{a}_{j}^{{\dagger}}).$
(2)

Unless $\lambda\mu=1$, the operators (1) are not proper EPR observables since
they do not commute. Consequently, we get the weak (Heisenberg) form of the
uncertainty relation (UR)

$\Delta Q(\lambda)\,\Delta P(\mu)\geqq\frac{1}{2}|1-\lambda\mu|,$ (3)

which has to be fulfilled by any quantum state. In Eq. (3) and in the sequel
as well, $\Delta A$ denotes the standard deviation of the observable $\hat{A}$
in the state ${\hat{\rho}}$, which is the square root of the variance

$\displaystyle\left(\Delta
A\right)^{2}:=\left<\left({\hat{A}}-{\langle{\hat{A}}\rangle}{\hat{I}}\right)^{2}\right>={\langle{\hat{A}}^{2}\rangle}-{\langle{\hat{A}}\rangle}^{2}.$
(4)

In Refs. Reid1 ; Reid2 , a possible experimental observation of the inequality

$\Delta Q(\lambda)\,\Delta P(\mu)<\frac{1}{2}$ (5)

was interpreted as a signature of detecting an EPR paradox. Moreover, the EPR
paradox as invoked by Reid and the concept of EPR steering as analyzed in
Refs. W1 ; W2 ; W3  were proven to be equivalent. The way this equivalence
arises from two different perspectives became a first reason for our interest
in understanding the modification of the URs for unsteerable states. A second
motivation for the present work stems from a recent result on EPR-like URs
valid for separable two-mode states PT2018 . Essentially, in Ref. PT2018  it
was proven that the minimum normalized product and sum of the uncertainties of
appropriate EPR-like observables are separability indicators for two-mode GSs,
due to the Peres-Simon separability theorem. According to this, such a state
is separable if and only if its density matrix has a positive-semidefinite
partial transpose (PPT) Peres ; Simon .

Consequently, the remainder of this paper develops an approach for a parallel
study of the separability and unsteerability of two-party
$(N\,\text{vs}\,1)$-multimode states. This study exploits the sum-form URs for
a pair of appropriate EPR-like observables. In Sec. II we extend the Reid
treatment of two-mode states to the larger class of two-party
$(N\,\text{vs}\,1)$-mode states. We thus define two linear combinations of
canonical quadrature operators of the same kind, each one depending on $N+1$
positive parameters, and then write their URs in an arbitrary bipartite
$(N\,\text{vs}\,1)$-mode state. In Sec. III we apply an important theorem of
Hofmann and Takeuchi HT  to find out the enhancement of the URs written for a
separable $(N\,\text{vs}\,1)$-mode state. Accordingly, a modification of the
lower bound of URs for separable states is imposed by observing the local
uncertainty relations (LURs). By extremization of the normalized uncertainty
sum with respect to the parameters, we then establish a necessary condition of
separability which appears to reduce to the Peres PPT one. Section IV is
devoted to an analysis of the steering process which aims to enable us to put
forward the efficient way of applying the Hofmann-Takeuchi theorem in the case
of one-way unsteerability. In Sec. V we perform the extremization of the
properly normalized uncertainty sums in order to get necessary conditions of
unsteerability for both ways of potential steering. It appears that they
coincide with those found by Wiseman et al. W1 ; W2  in the Gaussian
framework. We revisit in Sec. VI the special case of GSs, for which it is well
known that the necessary conditions of separability and unsteerability we have
derived here are also sufficient ones, as proven in Refs. Simon ; WW2001  and,
respectively, in Refs. W1 ; W2 . Section VII contains a recapitulation and a
discussion of the methods and results of the paper.

##  II EPR-like uncertainty relations for bipartite
${\bm{(}}{\bm{N}}\,\text{vs}\,{\bm{1}}{\bm{)}}$-mode states

Qualification and quantification of quantum correlations for multimode states
continue to be a central task of continuous-variable information theory LF2003
; H2009 ; RMP . For simplicity and feasibility reasons, we consider here a
two-party state ${\hat{\rho}}$ with $(N\,\text{vs}\,1)$ modes shared by Alice
and Bob as follows. Alice performs measurements on the $N$-mode part of the
state with the canonical quadrature operators
${\hat{q}}_{j},{\hat{p}}_{j},(j=1,\dots,N)$, while the $(N+1)$th mode with the
pair of canonical quadrature operators $\hat{q}_{N+1},\hat{p}_{N+1}$ is
controlled by Bob. In order to generalize Eq.(1), we introduce the following
EPR-like observables depending on the positive parameters
${\alpha}_{j}>0,\;{\beta}_{j}>0,\\\ (j=1,\dots,N+1)$:

$\displaystyle{\hat{Q}}({\bm{\alpha}}):=\sum_{j=1}^{N}{\alpha}_{j}{\hat{q}}_{j}-{\alpha}_{N+1}{\hat{q}}_{N+1},$
$\displaystyle{\hat{P}}_{\pm}({\bm{\beta}}):=\sum_{j=1}^{N}{\beta}_{j}{\hat{p}}_{j}\pm{\beta}_{N+1}{\hat{p}}_{N+1},\quad\text{where}$
$\displaystyle{\bm{\alpha}}:=\\{{\alpha}_{1},\dots,{\alpha}_{N+1}\\},\qquad{\bm{\beta}}:=\\{{\beta}_{1},\dots,{\beta}_{N+1}\\}.$
(6)

Their commutation relations,

$\displaystyle[{\hat{Q}}({\bm{\alpha}}),{\hat{P}}_{\pm}({\bm{\beta}})]=i\left(\sum_{j=1}^{N}{\alpha}_{j}\beta_{j}\mp{\alpha}_{N+1}{\beta}_{N+1}\right){\hat{I}},$
(7)

lead to the weak (Heisenberg) form of the URs in product form:

$\displaystyle\Delta Q({\bm{\alpha}})\,\Delta
P_{\pm}({\bm{\beta}})\geqq\frac{1}{2}\left|\sum_{j=1}^{N}{\alpha}_{j}{\beta}_{j}\mp{\alpha}_{N+1}{\beta}_{N+1}\right|.$
(8)

This implies the sum-form inequalities

$\displaystyle\left[\Delta Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{\pm}({\bm{\beta}})\right]^{2}\geqq\left|\sum_{j=1}^{N}{\alpha}_{j}{\beta}_{j}\mp{\alpha}_{N+1}{\beta}_{N+1}\right|.$
(9)

The variances of the EPR-like observables (6),

$\displaystyle\left[\Delta
Q({\bm{\alpha}})\right]^{2}=\sum_{j=1}^{N+1}\sum_{k=1}^{N+1}{\alpha}_{j}{\alpha}_{k}{\sigma}(q_{j},q_{k})$
$\displaystyle-4{\alpha}_{N+1}\sum_{j=1}^{N}{\alpha}_{j}{\sigma}(q_{j},q_{N+1}),$
$\displaystyle\left[{\Delta
P}_{+}({\bm{\beta}})\right]^{2}=\sum_{j=1}^{N+1}\sum_{k=1}^{N+1}{\beta}_{j}{\beta}_{k}{\sigma}(p_{j},p_{k}),$
$\displaystyle\left[{\Delta
P}_{-}({\bm{\beta}})\right]^{2}=\sum_{j=1}^{N+1}\sum_{k=1}^{N+1}{\beta}_{j}{\beta}_{k}{\sigma}(p_{j},p_{k})$
$\displaystyle-4{\beta}_{N+1}\sum_{j=1}^{N}{\beta}_{j}{\sigma}(p_{j},p_{N+1})$
(10)

are quadratic forms in the positive variables ${\bm{\alpha}}$ and
${\bm{\beta}}$, respectively. Their coefficients are covariances of the
canonical quadrature operators of the $(N\,\text{vs}\,1)$-mode state
$\hat{\rho}$:

$\displaystyle\sigma(q_{j},q_{k}):=\langle\hat{q}_{j}\hat{q}_{k}\rangle-\langle\hat{q}_{j}\rangle\langle\hat{q}_{k}\rangle,$
$\displaystyle\sigma(p_{j},p_{k}):=\langle\hat{p}_{j}\hat{p}_{k}\rangle-\langle\hat{p}_{j}\rangle\langle\hat{p}_{k}\rangle,$
$\displaystyle(j,k=1,\dots,N+1).$ (11)

These are the entries of the covariance matrix (CM) of the two-party state
$\hat{\rho}$, which is denoted ${\mathcal{V}}\in M_{2(N+1)}({\mathbb{R}})$, is
symmetric, and has the block structure:

$\displaystyle{\mathcal{V}}=\left(\begin{matrix}{\mathcal{V}}_{A}\;&\;{\mathcal{C}}\\\
\\\ {\mathcal{C}}^{\rm T}\;&\;{\mathcal{V}}_{B}\end{matrix}\right).$ (12)

The partition (12) allows one to vizualize the contribution of each subsystem.
Thus, $\mathcal{V}_{A}$ is the $2N\times 2N$ CM of the reduced $N$-mode state
held by Alice, $\mathcal{V}_{B}$ is the $2\times 2$ CM of the reduced one-mode
state handled by Bob, while ${\mathcal{C}}$ is the $2N\times 2$ matrix
describing the cross-correlations between the parties.

Notice that the covariances $\sigma(q_{j},p_{k})$ of any coordinate-momentum
pair do not appear in the quadratic forms (2.5). Apparently, the EPR-like
inequalities (2.4) hold only for ($N$ vs 1)-mode state having a standard-form
CM with vanishing covariances $\sigma(q_{j},p_{k})$. One could ask whether
this might be a restriction on the class of states specified at the beginning
of this section. This is not the case, because it was shown in Refs. Duan ;
SA2007 ; GK2014  that any multimode CM can be transformed to the standard form
described above with local symplectic matrices such as one-mode rotations and
squeezings. Since two multimode states which are similar via a tensor product
of one-mode unitaries possess the same amount of entanglement or steerability,
they are equivalent as nonlocal resources. Therefore, our discussion applies
to all the multimode states whose CMs are connected by one-mode symplectic
transformations. The particular form acquired by the CM (12) discussed above
is very convenient in many aspects. For instance, due to the identity
$\sigma(q_{j},p_{k})=0,$ it is more productive for subsequent evaluations to
write the CM ${\cal V}$ as the direct sum

$\displaystyle{\cal V}={\cal V}^{(q)}\oplus{\cal V}^{(p)}.$ (13)

The two terms of the above decomposition are the $(N+1)\times(N+1)$ CMs in the
position and momentum spaces, written with the reordered canonical operators
$\\{{\hat{q}}_{1},{\hat{q}}_{2},\dots,{\hat{q}}_{N+1}\\}$ and
$\\{{\hat{p}}_{1},{\hat{p}}_{2},\dots,{\hat{p}}_{N+1}\\}$, respectively. A
direct consequence of Eq. (13) is the factorization rule

$\det({\cal V})=\det\left({\cal V}^{(q)}\right)\det\left({\cal
V}^{(p)}\right).$ (14)

Let us recall the restrictions imposed by the canonical commutation relations
to the CM ${\mathcal{V}}$ of an arbitrary $n$-mode state Simon0 ; Simon .
Essentially, any bona fide CM ${\mathcal{V}}$ fulfills the Robertson-
Schrödinger matrix uncertainty relation, namely, the positive semidefiniteness
condition

$\displaystyle{\mathcal{V}}+\frac{i}{2}J\geq 0,$ (15)

where $J$ is the standard matrix of the symplectic form on
${\mathbb{R}}^{2n}$:

$\displaystyle
J:=\bigoplus_{k=1}^{n}\,J_{k},\;\;J_{k}:=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right),\;\;\;(k=1,\dots,n).$ (18)

We stress that the conditions (8), (9), and (15) stem from the URs for the
canonical observables and hence they have to be satisfied by any physical
multimode state.

The quantumness requirement (15) implies, via Williamson’s theorem Williamson
, the positive definiteness of the CM ${\mathcal{V}}$,

$\displaystyle{\mathcal{V}}>0,$ (19)

as well as the inequality

$\displaystyle\det({\mathcal{V}})\geqq\frac{1}{2^{2n}},\qquad(n=1,2,3,\dots).$
(20)

It is worth mentioning that the inequality (20) can be recovered by selecting
from the whole class of $n$-mode states possessing the same CM ${\mathcal{V}}$
a Gaussian one, denoted ${\hat{\rho}}_{G}$, which is determined up to a
translation in the phase space. Indeed, its purity depends only on the
determinant of the CM ${\mathcal{V}}$, as follows PTH2001 ; PTH2003 :

$\displaystyle{\rm
Tr}\left[\left({\hat{\rho}}_{G}\right)^{2}\right]=\frac{1}{2^{n}\sqrt{\det({\cal
V})}}\leqq 1.$ (21)

This universal property of the purity, when applied to any $n$-mode Gaussian
state, Eq. (21), coincides therefore with the general inequality (20).

##  III EPR-like separability bounds for
bipartite ${\bm{(}}{\bm{N}}\,\text{vs}\,{\bm{1}}{\bm{)}}$-mode states

Use of uncertainty relations between nonlocal operators of the type (1)
appeared to be a fruitful idea in quest of separability conditions for
continuous-variable two-party states and an enforcement of the role of
uncertainty principle in understanding quantum correlations. Besides the
original approach of Duan et al. Duan , some other necessary conditions of
separability which employ pairs of more general nonlocal observables depending
on one or more parameters have been written Simon ; GMVT . Essentially, all
these treatments rely on the important finding that for separable states the
uncertainty relations should modify to become stronger. This was first shown
in Refs. Duan ; Simon  by directly exploiting the expansion of a two-party
separable state shared by Alice and Bob as a convex combination of product
states Werner :

$\displaystyle\hat{\rho}_{sep}:=\sum_{k}w_{k}{\hat{\rho}}_{A}^{(k)}\otimes{\hat{\rho}}_{B}^{(k)},\quad(w_{k}>0,\;\;\sum_{k}w_{k}=1).$
(22)

More generally, by applying appropriate URs, necessary conditions for
separability of various kinds of partitions in multimode states were written
in terms of variances of linear combinations of canonical quadrature operators
LF2003 . Their violations are sufficient for genuine multipartite
entanglement. Note that the line of research proposed in Ref. LF2003  led to
the understanding of Gaussian cluster states LWG2007 .

The enhancement of URs for separable states can be efficiently displayed by
using an important theorem proven by Hofmann and Takeuchi HT  regarding the
LURs as a tool for entanglement detection on any bipartite quantum state. Let
us denote by $\hat{\cal A}_{k},\;\hat{\cal B}_{k}$ some observables at our
choice acting on Alice’s and Bob’s side, respectively. The incompatible
observables of each party obey the sum-type LURs

$\displaystyle\sum_{k}\left(\Delta{\cal A}_{k}\right)^{2}\geqq
C_{A},\qquad\sum_{k}\left(\Delta{\cal B}_{k}\right)^{2}\geqq C_{B},$ (23)

where $C_{A}$ and $C_{B}$ denote their local positive uncertainty limits. The
theorem of Hofmann and Takeuchi HT  asserts that, in any separable bipartite
state, the total sum-form uncertainty of the nonlocal observables

$\displaystyle\hat{\cal M}_{k}:=\hat{\cal
A}_{k}\otimes{\hat{I}}_{k}+{\hat{I}}_{k}\otimes\hat{\cal B}_{k}$ (24)

satisfies the inequality

$\displaystyle\sum_{k}\left(\Delta{\cal M}_{k}\right)^{2}\geqq C_{A}+C_{B}.$
(25)

In other words, for any separable bipartite state, the total sum uncertainty
of the nonlocal observables (24) is minorized by the sum of the local
uncertainty limits of both parties. The bound specified by the LURs in Eq.(25)
is the expression of a strong separability condition G2006 . For discrete-
variable systems, the sum UR (25) can generally be formulated in terms of CMs
built with the incompatible constituents of appropriate nonlocal observables
$\hat{\cal M}_{k}$ G1 ; G2 ; NHa1 . Violation of any sum UR of the form (25)
is therefore a signature of entanglement.

In continuous-variable systems, we have the privilege to employ in this
approach the one-mode canonical observables $\hat{q}_{j}$ and $\hat{p}_{j}$,
whose non-commutativity generates the Heisenberg uncertainty relations that
are at the heart of quantum mechanics since its early days. The system of
$(N\,\text{vs}\,1)$ modes we are dealing with is a perfect test bed for an
elegant application of the Hofmann-Takeuchi inequality (25).

We now focus on a separable $(N\,\text{vs}\,1)$-mode state
${\hat{\rho}}_{sep}$, with a partitioned CM ${\mathcal{V}}$, Eq. (12), and
choose as nonlocal observables $\hat{\cal M}_{k}$ the pair of generalized Reid
operators (6):

$\displaystyle\hat{\cal M}_{1}={\hat{Q}}(\bm{\alpha}),\qquad\hat{\cal
M}_{2}={\hat{P}}_{\pm}(\bm{\beta}),$ (26)

whose one-party components are respectively:

$\displaystyle\hat{\cal
A}_{1}=\sum_{j=1}^{N}{\alpha}_{j}{\hat{q}}_{j},\qquad\hat{\cal
B}_{1}=-{\alpha}_{N+1}{\hat{q}}_{N+1};$ $\displaystyle\hat{\cal
A}_{2}=\sum_{j=1}^{N}{\beta}_{j}{\hat{p}}_{j},\qquad\hat{\cal
B}_{2}=\pm{\beta}_{N+1}{\hat{p}}_{N+1},$
$\displaystyle({\alpha}_{j}>0,\;\;{\beta}_{j}>0:\;j=1,\dots,N+1).$ (27)

When inserting into the inequality (25) the sum-type LURs (23) associated to
the commutators

$\displaystyle\left[\hat{\cal A}_{1},\hat{\cal
A}_{2}\right]=i\,C_{A}{\hat{I}}_{A},\qquad
C_{A}:=\sum_{j=1}^{N}{\alpha}_{j}{\beta}_{j},$ $\displaystyle\left[\hat{\cal
B}_{1},\hat{\cal B}_{2}\right]=\mp i\,C_{B}{\hat{I}}_{B},\qquad
C_{B}:={\alpha}_{N+1}{\beta}_{N+1},$ (28)

we get a pair of sum-form necessary conditions of separability:

$\displaystyle\left[\Delta Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{\pm}({\bm{\beta}})\right]^{2}\geqq\sum_{l=1}^{N+1}{\alpha}_{l}{\beta}_{l}.$
(29)

The upper Eq.(29) has a larger separability bound than the upper physicality
bound in the corresponding sum-form UR (9). Interestingly, the lower
separability condition (29) coincides with the lower sum-form UR (9).
Therefore, in order to be separable, an $(N\,\text{vs}\,1)$-mode state
${\hat{\rho}}$ has to obey both conditions (29) for any values of the involved
parameters $\\{{\bm{\alpha}},{\bm{\beta}}\\}$.

The separability conditions (29) are always met when the absolute minima of
the functions

$\displaystyle{\Sigma}_{\pm}({\mathcal{V}};\,{\bm{\alpha}},{\bm{\beta}}):=\frac{\left[\Delta
Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{\pm}({\bm{\beta}})\right]^{2}}{\sum_{l=1}^{N+1}{\alpha}_{l}{\beta}_{l}}$
(30)

with respect to all their variables
${\alpha}_{j}>0,\;{\beta}_{j}>0,\;(j=1,\dots,N+1)$ are greater than or at
least equal to $1$. Let us denote the corresponding minimum points
$\\{\bm{\alpha}_{\pm},\bm{\beta}_{\pm}\\}$ in order to write concisely the
EPR-like necessary conditions of separability (29):

$\displaystyle{\Sigma}_{\pm}({\mathcal{V}};\,{\bm{\alpha}}_{\pm},{\bm{\beta}}_{\pm})\geqq
1.$ (31)

Except for the simplest case $N=1$ PT2018 , an analytic evaluation of such a
minimum is hampered by algebraic difficulties. Nevertheless, in view of the
formulas (10), we apply Euler’s theorem on homogeneous functions to the
fractions (30) and find the identities:

$\displaystyle\sum_{j=1}^{N+1}{\alpha}_{j}\frac{\partial}{\partial{\alpha}_{j}}\left[{\Sigma}_{\pm}({\mathcal{V}};\,{\bm{\alpha}},{\bm{\beta}})\right]=-\sum_{j=1}^{N+1}{\beta}_{j}\frac{\partial}{\partial{\beta}_{j}}\left[{\Sigma}_{\pm}({\mathcal{V}};\,{\bm{\alpha}},{\bm{\beta}})\right]$
$\displaystyle=\frac{\left[\Delta Q({\bm{\alpha}})\right]^{2}-\left[\Delta
P_{\pm}({\bm{\beta}})\right]^{2}}{\sum_{l=1}^{N+1}{\alpha}_{l}{\beta}_{l}}.$
(32)

Accordingly, any $(N\,\text{vs}\,1)$-mode separable state ${\hat{\rho}}_{sep}$
fulfills the conditions

$\displaystyle\left[\Delta Q(\bm{\alpha}_{\pm})\right]^{2}=\left[\Delta
P_{\pm}(\bm{\beta}_{\pm})\right]^{2}\geqq\frac{1}{2}\sum_{l=1}^{N+1}({\alpha}_{l})_{\pm}({\beta}_{l})_{\pm}.$
(33)

The difference between the necessary conditions of separability (33) for an
$(N\,\text{vs}\,1)$-mode state ${\hat{\rho}}$ arises from the opposite signs
of the ${\hat{p}}_{N+1}$ terms in the nonlocal observables
${\hat{P}}_{\pm}({\bm{\beta}})$ introduced in Eq. (6). As shown by Simon Simon
, the change of sign $p_{N+1}\to-p_{N+1}$ in the Wigner function
$W(q_{1},\dots,q_{N+1},p_{1},\dots,p_{N+1})$ of the $(N\,\text{vs}\,1)$-mode
state ${\hat{\rho}}$ amounts to its transformation into the Wigner function of
the partially transposed operator ${\hat{\rho}}^{{\rm PT}(B)}$ with respect to
Bob’s party:

$W^{{\rm PT}(B)}(q_{1},\dots,q_{N},q_{N+1},p_{1},\dots,p_{N},p_{N+1})$
$=W(q_{1},\dots,q_{N},q_{N+1},p_{1},\dots,p_{N},-p_{N+1}).$

By virtue of the decomposition (13), the CM of the partially transposed
operator ${\hat{\rho}}^{{\rm PT}(B)}$ is positive definite:
${\mathcal{V}}^{{\rm PT}(B)}>0.$ Moreover, in view of Ref. LSA2018 , it
fulfils the Robertson-Schrödinger matrix UR:

$\displaystyle{\mathcal{V}}^{{\rm PT}(B)}+\frac{i}{2}J\geq 0.$ (34)

As a matter of fact, Eq. (34) is a consequence of the Peres theorem Peres ,
which asserts that any bipartite separable state ${\hat{\rho}}$ is a PPT
state: this simply means that its partial transpose is positive,
${\hat{\rho}}^{{\rm PT}(B)}\geq 0$. Therefore, Eq. (33) displays the
quantumness requirements for the couple of $(N\,\text{vs}\,1)$-mode states
${\hat{\rho}}$ and ${\hat{\rho}}^{{\rm PT}(B)}$.

The two-mode states $(N=1)$ are important in their own right. They are present
in many areas, generating a productive research in quantifying bipartite
quantum correlations especially in the Gaussian scenario ARL2014 ; PTH2001 ;
PTH2003 ; PT2008 ; PT2016 . In the two-mode case, the EPR-like observables (6)
depend on two pairs of positive parameters,

${\bm{\alpha}}:=\\{{\alpha}_{1},\,{\alpha}_{2}\\},\qquad{\bm{\beta}}:=\\{{\beta}_{1},\,{\beta}_{2}\\},$

as follows:

$\displaystyle{\hat{Q}}({\bm{\alpha}}):={\alpha}_{1}{\hat{q}}_{1}-{\alpha}_{2}{\hat{q}}_{2},\quad{\hat{P}}_{\pm}({\bm{\beta}}):={\beta}_{1}{\hat{p}}_{1}\pm{\beta}_{2}{\hat{p}}_{2}.$
(35)

Clearly, Reid’s pair of nonlocal observables (1) built with two independent
parameters, $\lambda$ and $\mu$, as well as the single-parameter ones employed
by Duan et al. in Ref.Duan  are particular forms of EPR-like observables (35).

Let us consider an arbitrary separable two-mode state ${\hat{\rho}}_{sep}$,
whose $4\times 4$ CM ${\mathcal{V}}$, Eq. (12), is always characterized by
four standard-form parameters Duan :

$\displaystyle
b_{1}:=\sigma(q_{1},q_{1})=\sigma(p_{1},p_{1}),\;\;\;b_{2}:=\sigma(q_{2},q_{2})=\sigma(p_{2},p_{2}),$
$\displaystyle c:=\sigma(q_{1},q_{2}),\;\;\;d:=\sigma(p_{1},p_{2}).$ (36)

With no loss of generality, one can choose:

$b_{1}\geqq b_{2}\geqq\frac{1}{2},\qquad c\geqq|d|.$

Then the $2\times 2$ submatrices in the partition (2.7) of the CM
${\mathcal{V}}$ are diagonal:

$\displaystyle{\mathcal{V}}_{A}=b_{1}I_{2},\qquad{\mathcal{V}}_{B}=b_{2}I_{2},\qquad{\mathcal{C}}=\left(\begin{matrix}c\;&\;0\\\
\\\ 0\;&\;d\end{matrix}\right),$ (37)

where $I_{2}$ is the $2\times 2$ identity matrix. We specialize Eq. (10) to
get the variances of the EPR-like observables (35):

$\displaystyle\left[\Delta
Q({\bm{\alpha}})\right]^{2}=b_{1}{\alpha}_{1}^{2}+b_{2}{\alpha}_{2}^{2}-2c\,{\alpha}_{1}{\alpha}_{2},$
$\displaystyle\left[{\Delta
P}_{\pm}({\bm{\beta}})\right]^{2}=b_{1}{\beta}_{1}^{2}+b_{2}{\beta}_{2}^{2}\pm
2d\,{\beta}_{1}{\beta}_{2}.$ (38)

The next step is to evaluate the absolute minima of the functions (30) written
for two-mode states,

$\displaystyle{\Sigma}_{\pm}({\mathcal{V}};\,{\bm{\alpha}},{\bm{\beta}}):=\frac{\left[\Delta
Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{\pm}({\bm{\beta}})\right]^{2}}{{\alpha}_{1}{\beta}_{1}+{\alpha}_{2}{\beta}_{2}},$
(39)

with respect to their four positive variables
$\\{{\bm{\alpha}},{\bm{\beta}}\\}.$ These minima are found to be equal to the
doubles of the smallest symplectic eigenvalues ${\kappa}_{-}^{\rm PT}$ and
${\kappa}_{-}$ VW2002  of the CMs ${\mathcal{V}}^{\rm PT}$ and
${\mathcal{V}}$, respectively:

$\displaystyle{\Sigma}_{+}({\mathcal{V}};\,{\bm{\alpha}}_{+},{\bm{\beta}}_{+})=2{\kappa}_{-}^{{\rm
PT}},\quad{\Sigma}_{-}({\mathcal{V}};\,{\bm{\alpha}}_{-},{\bm{\beta}}_{-})=2{\kappa}_{-}.$
(40)

In order to derive the formulas (40) one needs the symplectic eigenvalues
${\kappa}_{\pm}$ and ${\kappa}_{\pm}^{\rm PT}$ expressed in terms of the
standard-form parameters (36) PT2018a :

$\displaystyle({\kappa}_{\pm})^{2}=\frac{1}{2}\left(b_{1}^{2}+b_{2}^{2}+2cd\pm\sqrt{\Delta}\right),$
(41)

where $\Delta$ is the discriminant of a quadratic trinomial:

$\displaystyle\Delta=\left(b_{1}^{2}+b_{2}^{2}+2cd\right)^{2}-4\det({\mathcal{V}})$
$\displaystyle=\left(b_{1}^{2}-b_{2}^{2}\right)^{2}+4(b_{1}c+b_{2}d)(b_{2}c+b_{1}d)\geqq
0.$ (42)

Similarly,

$\displaystyle\left({\kappa}_{\pm}^{\rm
PT}\right)^{2}=\frac{1}{2}\left(b_{1}^{2}+b_{2}^{2}-2cd\pm\sqrt{{\Delta}^{\rm
PT}}\right),$ (43)

with the discriminant

$\displaystyle{\Delta}^{\rm
PT}=\left(b_{1}^{2}+b_{2}^{2}-2cd\right)^{2}-4\det({\mathcal{V}})$
$\displaystyle=\left(b_{1}^{2}-b_{2}^{2}\right)^{2}+4(b_{1}c-b_{2}d)(b_{2}c-b_{1}d)\geqq
0.$ (44)

Although recalled in Ref. PT2018  just in the special case of separable two-
mode Gaussian states, Eqs. (41)-(44) hold for any separable two-mode state,
either Gaussian or non-Gaussian. It is worth mentioning that they imply the
following sign rule:

$\displaystyle\operatorname{sgn}\left({\kappa}_{-}^{\rm
PT}-{\kappa}_{-}\right)=\operatorname{sgn}(d).$ (45)

According to Eq. (40), the EPR-like necessary conditions of separability (31)
acquire in the two-mode case their simplest form:

$\displaystyle{\kappa}_{-}^{\rm
PT}\geqq\frac{1}{2},\qquad{\kappa}_{-}\geqq\frac{1}{2}.$ (46)

The first above inequality manifestly illustrates Peres’ criterion of
inseparability Peres , while the second one is just the quantumness
requirement for the given separable two-mode state ${\hat{\rho}}_{sep}$. Note
that the first minimal normalized UR sum in Eq. (40) was derived in Ref.
PT2018  by using a special set of independent variables, in the framework of a
comprehensive analysis.

##  IV Uncertainty-relation criterion for EPR steering

Our aim in this section is to establish the sum-form URs valid for unsteerable
two-party states. To this end, we briefly recall the meaning of the steering
scenario, as emerging from its interpretation in Refs.W2 ; W3 ; Pusey ; SNC ;
QVB ; UMG ; PW ; Adesso2015 ; Kiukas . Accordingly, two separate observers,
Alice and Bob, share a bipartite state ${\hat{\rho}}_{AB}$ of a composite
quantum system, such that its Hilbert space is the tensor product ${\cal
H}_{A}\otimes{\cal H}_{B}$. Alice performs a local unknown measurement $x$
whose positive operators $\hat{M}_{a|x}$ add up to identity:
$\sum_{a}\hat{M}_{a|x}={\hat{I}}_{A}$. Here $a$ denotes a possible outcome of
the measurement chosen by Alice on her side, having the probability $p(a|x)$.
The conditional unnormalized state on Bob’s side,

$\displaystyle{\hat{\sigma}}^{B}(a|x)={\rm
Tr}_{A}\left[{\hat{\rho}}_{AB}\left({\hat{M}}_{a|x}\otimes{\hat{I}}_{B}\right)\right],$
(47)

has the trace ${\rm Tr}_{B}\left[{\hat{\sigma}}^{B}(a|x)\right]=p(a|x)$. Bob’s
collection of states $\\{{\hat{\sigma}}^{B}(a|x)\\}_{a,x}$ built with all the
possible pairs $\\{a,x\\}$ is called an assemblage Pusey . Let us write the
sum rule

$\displaystyle\sum_{a}{\hat{\sigma}}^{B}(a|x)={\rm
Tr}_{A}\left({\hat{\rho}}_{AB}\right)=:{\hat{\rho}}_{B},$ (48)

giving the reduced state at Bob’s hand, prior to Alice’s measurement. Equation
(4.2) shows that Bob’s reduced state ${\hat{\rho}}_{B}$ is not modified
whatever the nonselective measurement $x$ made by Alice.

When all the conditional states (47) have the special structure

$\displaystyle{\hat{\sigma}}^{B}_{\rm
us}(a|x)=\sum_{\lambda}p(a|x,\lambda)q(\lambda){\hat{\rho}}_{B}(\lambda),$
(49)

then the assemblage $\\{{\hat{\sigma}}^{B}_{\rm us}(a|x)\\}_{a,x}$ is termed
unsteerable. In Eq.(4.3), $\\{{\hat{\rho}}_{B}(\lambda)\\}_{\lambda}$ is a
preexisting ensemble of local hidden states (LHSs), each one depending on the
local hidden variable (LHV) $\lambda$ and having the associated weight
$q(\lambda)$. The weights are normalized, i.e, they sum to one:
$\sum_{\lambda}q(\lambda)=1$. Further, $p(a|x,\lambda)$ is the probability of
the outcome $a$ in Alice’s measurement $x$, provided that Bob’s party is
described by the LHS $\\{{\hat{\rho}}_{B}(\lambda)\\}$. The trace of the
operator (4.3) acting on the Hilbert space ${\cal H}_{B}$ is just the
probability of the outcome $a$ in Alice’s measurement $x$:

$\displaystyle p(a|x)=\sum_{\lambda}p(a|x,\lambda)q(\lambda).$ (50)

On the other hand, the initial Bob’s reduced state has the convex
decomposition

$\displaystyle{\hat{\rho}}_{B}=\sum_{\lambda}q(\lambda){\hat{\rho}}_{B}(\lambda).$
(51)

When the local measurement $x$ performed by Alice has the outcome $a$, then it
changes Bob’s reduced state (51) into an updated convex combination of LHSs:

$\displaystyle{\hat{\rho}}_{B}(a|x)=\frac{1}{p(a|x)}{\hat{\sigma}}^{B}_{\rm
us}(a|x)=\sum_{\lambda}\frac{p(a|x,\lambda)q(\lambda)}{p(a|x)}{\hat{\rho}}_{B}(\lambda).$
(52)

The key quantity in the analysis of unsteerability is the joint probability of
a pair of outcomes $\\{a,\,b\\}$ in two successive measurements, $x$,
performed by Alice, and $y$, made by Bob:

$\displaystyle P\left(a|x,\,b|y,\,{\hat{\rho}}_{AB}\right)={\rm
Tr}_{B}\left[{\hat{\sigma}}^{B}(a|x){\hat{M}}_{b|y}\right].$ (53)

In view of Eq. (4.3), for any unsteerable assemblage
$\left\\{{\hat{\sigma}}^{B}_{\rm us}(a|x)\right\\}_{a,x}$, Eq. (4.7) acquires
a specific form W1 ; W2 :

$\displaystyle
P\left(a|x,\,b|y,\,{\hat{\rho}}_{AB}\right)=\sum_{\lambda}q(\lambda)p(a|x,\lambda)P\left[b|y,\,{\hat{\rho}}_{B}(\lambda)\right].$
(54)

Thus, the joint probability (53) has the asymmetric structure (54) whenever
the measurement performed by Alice cannot steer (pilot) Bob’s party.
Essentially, Eq. (54) can be interpreted in terms of a LHS model for Bob and a
LHV model for Alice, which is correlated with Bob’s reduced state (51) via the
conditional probabilities $p(a|x,\lambda)$. The lack of such a model even for
a single pair $\\{a,x\\}$ of the assemblage means steering from Alice to Bob
W1 ; W2 ; W3 .

It can be directly seen that any measurement performed by Alice on a separable
state, Eq. (22), leads to an unsteerable assemblage (49). It follows that
entanglement is a necessary condition for EPR steerability. For instance, an
unsteerable assemblage can be obtained from the separable state
${\hat{\rho}}_{AB}=\sum_{\lambda}q(\lambda)\left(|\lambda\rangle\langle\lambda|\right)_{A}\otimes{\hat{\rho}}_{B}(\lambda)$.
Here, Alice performs projective measurements of mutually commuting observables
whose common eigenvectors are generically denoted $|\lambda\rangle$. Moreover,
it was proven that unsteerable assemblages can be obtained by joint local
measurements QVB ; UMG ; Kiukas . Joint measurability is an extension of
commutativity to general measurements described by positive operator valued
measures (POVMs). Note that observation of one-way EPR steering requires the
presence of both entanglement and incompatible measurements. The connection
between steering and incompatibility was recently extended to continuous-
variable systems Kiukas ; Kiukas2 . To conclude this discussion, an
unsteerable assemblage of the type (49) at Bob’s party is provided by Alice
via local measurements of commuting or jointly measurable observables.

The main question we have now to answer is as follows: how might URs be
written analogously to the Hofmann-Takeuchi inequality (25) such as to be
specifically valid for two-party unsteerable states? The above-sketched
tableau of one-way EPR steering offers an idea for answering this question.
First, let us remark that the structure (49) of an unsteerable assemblage from
Alice to Bob imposes the existence of a local uncertainty limit $C_{B}$.
Second, this is not the case for Alice’s side, where no uncertainty relation
is available since her measurements are unknown. In other words, our lack of
knowledge about the nature of Alice’s measurements amounts to ignoring the
uncertainty limit $C_{A}$ of her party NHa1 ; Zhen ; Zhen1 . We therefore
suggest that the right necessary condition for unsteerabilty of Bob’s reduced
state through Alice’s local measurement i s the sum-form UR

$\displaystyle\sum_{k}\left(\Delta{\cal M}_{k}\right)^{2}\geqq C_{B},$ (55)

which is weaker than the similar one for separability, Eq. (25). Any violation
of the inequality (55) is thus a pertinent signature of steering from Alice to
Bob.

Accordingly, with the EPR-like observables (6), one gets the sum-form UR which
is compulsory for any $(N\,\text{vs}\,1)$-mode state which is unsteerable
($us$) from Alice to Bob:

$\displaystyle\left[\Delta Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{+}({\bm{\beta}})\right]^{2}\geqq{\alpha}_{N+1}{\beta}_{N+1}.$ (56)

Similarly, for the same state, the necessary condition of unsteerability from
Bob to Alice reads:

$\displaystyle\left[\Delta Q(\bm{\alpha})\right]^{2}+\left[\Delta
P_{+}(\bm{\beta})\right]^{2}\geqq{\sum_{j}^{N}\alpha_{j}\beta_{j}}.$ (57)

Conditions (56) and (57) display the asymmetry of steering. They have to be
fulfilled for all the positive parameters $\\{{\bm{\alpha}},{\bm{\beta}}\\}$.
If the conditions (56) or (57) are not met for some values of these
parameters, then the $(N\,\text{vs}\,1)$-mode state is steerable from Alice to
Bob or from Bob to Alice, respectively.

In the next section we intend to find the extremal normalized URs derived from
Eqs. (56) and (57). Analytically, these distinct extremization problems for
the two ways of steering depend on $2N+2$ parameters and appear to be less
complicated than those of the functions (30) for separability.

##  V EPR-like uncertainty limits for steering

We carry out here the extremization of the two properly normalized uncertainty
sums required for the unsteerability of a $(N\,\text{vs}\,1)$-mode state in
both possible ways of steering.

###  V.1 Steering from Alice to Bob

Guided by the UR (56) we define the normalized sum

$\displaystyle{\Sigma}^{(A\to B)}_{\rm
us}({\bm{\alpha}},{\bm{\beta}}):=\frac{\left[\Delta
Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{+}({\bm{\beta}})\right]^{2}}{{\alpha}_{N+1}{\beta}_{N+1}}\geqq 1.$ (58)

On account of the expressions (10) of the variances involved in the above
function, we first write its extremization conditions with respect to the
variables $\alpha_{j},\beta_{j}$:

$\frac{\partial{\Sigma}^{(A\to B)}_{\rm
us}}{\partial\alpha_{j}}=0,\;\;\;\frac{\partial{\Sigma}^{(A\to B)}_{\rm
us}}{\partial\beta_{j}}=0,\;\;\;(j=1,\dots,N).$

They amount to a couple of independent systems of $N$ linear equations:

$\displaystyle\sum_{k=1}^{N}\alpha_{k}\sigma(q_{j},q_{k})-\alpha_{N+1}\sigma(q_{j},q_{N+1})=0,$
(59)

and

$\displaystyle\sum_{k=1}^{N}\beta_{k}\sigma(p_{j},p_{k})+\beta_{N+1}\sigma(p_{j},p_{N+1})=0.$
(60)

The system (59) with the variables $\alpha_{j},\;\;(j=1,\dots,N)$ has the
coefficient matrix ${\cal V}_{A}^{(q)}$, while the system (60) with the
variables $\beta_{j},\;\;(j=1,\dots,N)$ has the coefficient matrix ${\cal
V}_{A}^{(p)}$.

We start to minimize the function (58) by using the conditions (59) and (60):

$\displaystyle{\Sigma}^{(A\to B)}_{\rm
us}=\frac{1}{{\beta}_{N+1}}\left[{\alpha}_{N+1}\sigma(q_{N+1},q_{N+1})-\sum_{j=1}^{N}{\alpha}_{j}\sigma(q_{j},q_{N+1})\right]$
$\displaystyle+\frac{1}{{\alpha}_{N+1}}\left[{\beta}_{N+1}\sigma(p_{N+1},p_{N+1})+\sum_{j=1}^{N}{\beta}_{j}\sigma(p_{j},p_{N+1})\right].$

When inserting into Eq.(LABEL:sum1) the solutions $\alpha_{j}$ and $\beta_{j}$
of the linear systems (59) and (60), written with Cramer’s rule, we get two
fractions whose numerators are proportional to the Laplace expansions of
$\det({\cal V}^{(q)})$ and $\det({\cal V}^{(p)})$ along the $(N+1)$-th columns
of their matrices. We find thus the function

$\displaystyle{\Sigma}^{(A\to B)}_{\rm us}(\epsilon)=\epsilon\frac{\det{({\cal
V}^{(q)})}}{\det({\cal V}_{A}^{(q)})}+\frac{1}{\epsilon}\frac{\det{({\cal
V}^{(p)})}}{\det({\cal V}_{A}^{(p)})},$ (62)

of a single positive variable, $\epsilon:={\alpha}_{N+1}/{\beta}_{N+1}$. Its
unique minimum point,

$\epsilon_{m}=\left[\frac{\det{({\cal V}^{(p)})}}{\det{({\cal
V}^{(q)})}}\frac{\det({\cal V}_{A}^{(q)})}{\det({\cal
V}_{A}^{(p)})}\right]^{\frac{1}{2}},$

gives, via Eq.(14), the minimal uncertainty sum

$\displaystyle\min_{\\{\bm{\alpha},\bm{\beta}\\}}{\Sigma}^{(A\rightarrow
B)}_{\rm us}=2\,\sqrt{\frac{\det({\mathcal{V}})}{\det({\mathcal{V}}_{A})}}.$
(63)

In view of the URs (56) satisfied by unsteerable states from Alice to Bob we
get the following necessary condition of unsteerability:

$\displaystyle\frac{\det({\mathcal{V}})}{\det({\mathcal{V}}_{A})}\geqq\frac{1}{4}.$
(64)

Therefore, in order to be unsteerable from Alice, who operates on an $N$-mode
state, to Bob, who holds a single-mode state, an $(N\,\text{vs}\,1)$-mode
state has to obey the condition (64). Its violation is a signature of
steerability of any $(N\,\text{vs}\,1)$-mode state and an expression of the
EPR-paradox as stated by Reid in Ref.Reid1 .

###  V.2 Steering from Bob to Alice

For the same setting of $(N\,\text{vs}\,1)$-mode we exploit the alternative
way of unsteerability (57) by using the condition imposed to the normalized
sum-form UR

$\displaystyle{\Sigma}^{(B\to A)}_{\rm
us}(\bm{\alpha},\bm{\beta}):=\frac{\left[\Delta
Q({\bm{\alpha}})\right]^{2}+\left[\Delta
P_{+}({\bm{\beta}})\right]^{2}}{\sum_{j=1}^{N}\alpha_{j}\beta_{j}}\geqq 1.$
(65)

It is convenient to start from its extremization conditions with respect to
the variables $\alpha_{N+1}$ and $\beta_{N+1}$:

$\frac{\partial{\Sigma}^{(B\to A)}_{\rm
us}}{\partial\alpha_{N+1}}=0,\qquad\frac{\partial{\Sigma}^{(B\to A)}_{\rm
us}}{\partial\beta_{N+1}}=0,$

which read, respectively:

$\displaystyle\alpha_{N+1}{\sigma(q_{N+1},q_{N+1})}={\sum_{k=1}^{N}\alpha_{k}\sigma(q_{k},q_{N+1})}$
(66)

and

$\displaystyle\beta_{N+1}{\sigma(p_{N+1},p_{N+1})}=-\sum_{k=1}^{N}\beta_{k}\sigma(p_{k},p_{N+1}).$
(67)

Taking account of Eqs. (66) and (67), the variances (10) acquire reduced forms
that are characteristic for an $N$-mode state:

$\displaystyle\left[{\Delta
Q}({\bm{\alpha}}^{\prime})\right]^{2}:=\sum_{j=1}^{N}\sum_{k=1}^{N}\alpha_{j}\alpha_{k}\sigma^{\prime}(q_{j},q_{k}),$
$\displaystyle\left[{\Delta
P}({\bm{\beta}}^{\prime})\right]^{2}:=\sum_{j=1}^{N}\sum_{k=1}^{N}\beta_{j}\beta_{k}\sigma^{\prime}(p_{j},p_{k}),$
$\displaystyle\text{with}\;\;\;{\bm{\alpha}}^{\prime}:=\\{{\alpha}_{1},\dots,{\alpha}_{N}\\},\quad{\bm{\beta}}^{\prime}:=\\{{\beta}_{1},\dots,{\beta}_{N}\\}.$
(68)

Here we have denoted:

$\displaystyle\sigma^{\prime}(q_{j},q_{k}):=\sigma(q_{j},q_{k})$
$\displaystyle-\sigma(q_{j},q_{N+1})[{\sigma(q_{N+1},q_{N+1})}]^{-1}\sigma(q_{N+1},q_{k}),$
$\displaystyle\sigma^{\prime}(p_{j},p_{k}):=\sigma(p_{j},p_{k})$
$\displaystyle-\sigma(p_{j},p_{N+1})[{\sigma(p_{N+1},p_{N+1})}]^{-1}\sigma(p_{N+1},p_{k}),$
$\displaystyle(j,k=1,\dots,N).$ (69)

Therefore, the unsteerability condition (65) takes a simpler form:

$\displaystyle\min_{\\{{\bm{\alpha}},{\bm{\beta}}\\}}{\Sigma}^{(B\rightarrow
A)}_{\rm
us}=\min_{\\{{\bm{\alpha}}^{\prime},{\bm{\beta}}^{\prime}\\}}\frac{\left[\Delta
Q({\bm{\alpha}}^{\prime})\right]^{2}+\left[\Delta
P({\bm{\beta}}^{\prime})\right]^{2}}{\sum_{j=1}^{N}{\alpha}_{j}{\beta}_{j}}\geqq
1.$ (70)

To see its significance, let us introduce two nonlocal operators built with
the same positive parameters ${\bm{\alpha}}^{\prime}$ and
${\bm{\beta}}^{\prime}$ as in Eq. (68):

$\displaystyle{\hat{Q}}({\bm{\alpha}}^{\prime}):=\sum_{j=1}^{N}{\alpha}_{j}\hat{q}_{j},\qquad{\hat{P}}({\bm{\beta}}^{\prime}):=\sum_{j=1}^{N}{\beta}_{j}\hat{p}_{j}.$
(71)

Equations (68) appear to specify the variances of the observables (71) for the
class of $N$-mode states which share the CM with the only non-vanishing
entries (69).

Indeed, this matrix is precisely the Schur complement of the $2\times 2$ CM
${\mathcal{V}_{B}}$ of the one-mode state held by Bob according to the
partition (12):

$\displaystyle{\mathcal{V}}/{\mathcal{V}_{B}}={\mathcal{V}}_{A}-{\mathcal{C}}\left({\mathcal{V}_{B}}\right)^{-1}{\mathcal{C}}^{\rm
T}.$ (72)

Recall LSA2018  that the Schur complement (72) satisfies the physicality
condition

$\displaystyle{\mathcal{V}}/{\mathcal{V}_{B}}+\frac{i}{2}J_{A}\geq 0,$ (73)

which qualifies it as a bona fide CM of an $N$-mode state Simon ; Simon0 .

By the same token, the inequality (70) holds for the above-mentioned class of
$N$-mode states. In other words, Eqs. (70) and (73) are two equivalent
necessary conditions of unsteerability from Bob to Alice. They are
simultaneously obtained by taking the minimum of the normalized sum-form UR
(65).

Furthermore, we employ the Aitken factorization formula HJ  for the
partitioned CM (12):

$\displaystyle{\mathcal{V}}={\mathcal{T}}\,{\mathcal{D}}\,{\mathcal{T}}^{\rm
T}.$ (74)

The matrix product (74) consists of a central factor, namely, the positive
definite block-diagonal matrix,

$\displaystyle{\mathcal{D}}=\left({\mathcal{V}}/{\mathcal{V}_{B}}\right)\oplus{\mathcal{V}}_{B},$
(75)

sandwiched between a pair of unimodular triangular matrices,

$\displaystyle{\mathcal{T}}=\left(\begin{matrix}I_{A}\;&\;{\mathcal{C}}{\mathcal{V}}_{B}^{-1}\\\
\\\ 0\;&\;I_{B}\end{matrix}\right),\quad{\mathcal{T}}^{\rm
T}=\left(\begin{matrix}I_{A}\;&\;0\\\ \\\
{\mathcal{V}}_{B}^{-1}{\mathcal{C}}^{\rm T}\;&\;I_{B}\end{matrix}\right).$
(76)

The Aitken formula (74) is therefore a LDU decomposition, exhibiting the
Tcongruence of the CM (12) to the direct sum ${\mathcal{D}}$, Eq. (75), via
the upper triangular matrix ${\mathcal{T}}$, Eq. (76). The above discussion is
valid for any bipartite $(N\,\text{vs}\,M)$ CM. A straightforward consequence
of Eqs. (74)-(76) is the Schur determinantal formula:

$\displaystyle\det({\mathcal{V}}/{\mathcal{V}_{B}})=\frac{\det({\mathcal{V}})}{\det\left({\mathcal{V}}_{B}\right)}.$
(77)

Equations (74)-(76) clearly imply the equivalence between the matrix
inequality (73) and the following one:

$\displaystyle{\mathcal{V}}+\frac{i}{2}J_{A}\oplus 0_{B}\geq 0.$ (78)

To sum up, we have proven that two necessary conditions of unsteerability from
Bob to Alice, Eqs. (70) and (78), are fully equivalent, because each of them
is equivalent to Eq. (73). This is the main result of the current subsection.

Owing to Williamson’s theorem Williamson , the physicality condition (73)
implies the inequality

$\displaystyle\det\left({\mathcal{V}}/{\mathcal{V}_{B}}\right)\geqq\frac{1}{2^{2N}}.$
(79)

By combining Eq. (79) with the Schur determinantal formula (77), one gets a
numerical necessary condition of unsteerability from Bob to Alice:

$\displaystyle\frac{\det({\mathcal{V}})}{\det\left({\mathcal{V}}_{B}\right)}\geqq\frac{1}{2^{2N}}.$
(80)

Except for the two-mode states $(N=1)$, this is weaker than the matrix
condition (78).

The inequality (79) is analog to the necessary condition of unsteerability
from Alice to Bob (64), written with Schur’s formula (77) as

$\displaystyle\det\left({\mathcal{V}}/{\mathcal{V}_{A}}\right)\geqq\frac{1}{4},$
(81)

Note, however, that the numerical inequality (81) is the only one which is
equivalent to the matrix necessary conditions of unsteerability from Alice to
Bob similar to the equivalent matrix inequalities (73) and (78):

$\displaystyle{\mathcal{V}}/{\mathcal{V}_{A}}+\frac{i}{2}J_{B}\geq
0\;\iff\;{\mathcal{V}}+0_{A}\oplus\frac{i}{2}J_{B}\geq 0.$ (82)

The above extremizations of the normalized sums of EPR uncertainties do not
allow us to find explicitly the corresponding minima. Nevertheless, violation
of at least one of the similar conditions (64) and (80) is sufficient for the
one-way steerability of any $(N\,\text{vs}\,1)$-mode state. Such violations
are therefore signatures of steerability from Alice to Bob and, respectively,
from Bob to Alice.

##  VI Separability and unsteerability criteria for
${\bm{(}}{\bm{N}}\,\text{vs}\,{\bm{1}}{\bm{)}}$-mode
Gaussian states revisited

The class of Gaussian states (GSs) is a very special one in the whole set of
continuous-variable multimode states. Any $n$-mode GS ${\hat{\rho}}_{G}$ is
completely determined by a displacement vector in the phase space,
$u\in{\mathbb{R}}^{2n}$, and the CM ${\mathcal{V}}\in M_{2n}({\mathbb{R}})$,
which is subjected to the quantumness requirement (15). This characteristic
property of the GSs is the main reason for the central role they play in
quantum optics, as well as in quantum information theory.

###  VI.1 Separability criteria for Gaussian states

Let us start with the case of the two-mode GSs. As recalled in Sec. III, the
first EPR-type inequality in Eq. (31),

$\displaystyle{\kappa}_{-}^{\rm PT}\geqq\frac{1}{2},$ (83)

is manifestly equivalent to the Peres-Simon PPT necessary condition of
separability Peres ; Simon . Moreover, in Ref. Simon , Simon proved that the
physicality requirement (83) for the existence of the two-mode GS
${\hat{\rho}}_{G}^{\rm PT}$ is also a criterion (sufficient condition) for the
separability of a given two-mode GS ${\hat{\rho}}_{G}$. Accordingly, the
symplectic eigenvalue ${\kappa}_{-}^{\rm PT}$ is a PPT-type separability
indicator at hand for any two-mode GS. Furthermore, its equivalence with a
less convenient EPR-type separability indicator was explicitly proven PT2018b
.

The next accomplishment is the treatment of the separability problem for
$(N\,\text{vs}\,1)$-mode GSs by Werner and Wolf in Ref. WW2001 . Their main
result is that the PPT property of such a GS implies its separability.
Although this important Werner-Wolf theorem generalizes to an arbitrary $N$
the remarkable Simon’s one for $N=1$ Simon , it does not provide any analytic
separability indicator for $N>1$. A second complementary result obtained in
Ref. WW2001  is that the PPT property and separability are not equivalent
concepts for any $(N\,\text{vs}\,M)$-mode GS with $M>1$. Indeed, the authors
get an example of a $(2\,\text{vs}\,2)$-mode GS which is PPT, but entangled.
Such a bipartite state is termed bound entangled H4 . A comprehensive analysis
of both above-mentioned results is recently developed in Ref. LSA2018 .

Coming back to the present work, we stress that, by virtue of the Werner-Wolf
theorem, the EPR-like necessary conditions of separability (33) are also
sufficient ones. They could therefore be employed to get computable
separability indicators for $(N\,\text{vs}\,1)$-mode GSs.

###  VI.2 Unsteerability criteria for Gaussian states

We recall the noteworthy results concerning the special case of Gaussian
unsteerability established by Jones, Wiseman, and Doherty in Ref. W2 . Let us
consider a bipartite $(N\,\text{vs}\,M)$-mode GS ${\hat{\rho}}_{G}$ shared by
Alice and Bob, whose CM ${\mathcal{V}}$ has an adequate partition of the type
(12). Based on the assumption that Alice makes only Gaussian measurements, the
necessary and sufficient condition for the Alice-to-Bob unsteerability of the
GS ${\hat{\rho}}_{G}$ found in Refs. W1 ; W2  is

$\displaystyle{\mathcal{V}}+0_{A}\oplus\frac{i}{2}J_{B}\geq 0.$ (84)

Obviously, a similar necessary and sufficient condition holds for the Bob-to-
Alice unsteerability of the GS ${\hat{\rho}}_{G}$:

$\displaystyle{\mathcal{V}}+\frac{i}{2}J_{A}\oplus 0_{B}\geq 0.$ (85)

As shown in Refs. Adesso2 ; Nha ; LSA2018 , the unsteerability inequality (84)
is equivalent to the physicality requirement

$\displaystyle{\mathcal{V}}/{\mathcal{V}_{A}}+\frac{i}{2}J_{B}\geq 0$ (86)

for the Schur complement of the CM ${\mathcal{V}}_{A}$ in a partition of the
type (12),

$\displaystyle{\mathcal{V}}/{\mathcal{V}_{A}}={\mathcal{V}}_{B}-{\mathcal{C}}^{\rm
T}{\left(\mathcal{V}_{A}\right)}^{-1}{\mathcal{C}}.$ (87)

Accordingly, the condition (84) of Alice-to-Bob Gaussian unsteerability holds
no matter whether Alice’s party does or does not fulfill the requirement of
quantumness

$\displaystyle{\mathcal{V}_{A}}+\frac{i}{2}J_{A}\geq 0.$ (88)

Note that this aspect of steering of Bob’s party by Alice’s measurement is
also at the heart of our UR treatment according to Eqs. (55) and (56).

The inequality (86) implies a numerical necessary condition of Alice-to-Bob
Gaussian unsteerability,

$\displaystyle\frac{\det({\mathcal{V}})}{\det\left({\mathcal{V}}_{A}\right)}\geqq\frac{1}{2^{2M}},$
(89)

which is similar to the condition (80) for Bob-to-Alice unsteerability. Except
for the case $M=1$, the necessary condition (89) is weaker than the matrix
inequality (86), so that it is not a sufficient condition for Alice-to-Bob
Gaussian unsteerability.

In this work our interest is restricted to bipartite $(N\,\text{vs}\,1)$-mode
states. By using suitable EPR-like URs, we have derived the necessary matrix
conditions (84) and (85) for their one-way unsteerability, irrespective of
their Gaussian or non-Gaussian nature. Concerning the well-known sufficiency
of these conditions for GSs and Gaussian measurements W1 ; W2 , we just take
advantage of the above discussion to stress that the ratio
${\det({\mathcal{V}})}/{\det\left({\mathcal{V}}_{A}\right)}$ is a valuable
indicator of Gaussian unsteerability from Alice to Bob, when compared to the
number $1/4$, as pointed out by Eq. (64).

##  VII Summary and discussion

A main issue we are concerned with in the present work is to give a unified
description of both entanglement and EPR steering for a class of bipartite
multimode states of continuous-variable systems by using an efficient
theoretical method. Looking at the impressive volume of recent work on
steering matters one can notice that the question on the relationship between
criteria of entanglement and EPR steering is scarcely addressed. Accordingly,
the present work on inseparability and steering conditions for two-party
multimode quantum states is built on the idea of the fundamental role the sum-
form URs of nonlocal EPR-like observables play in defining and describing
these two types of quantum correlations.

A widespread classification of quantum correlations termed as entanglement,
EPR steering, and Bell nonlocality, from a quantum-information perspective was
given by Wiseman et al. W1 ; W2 ; W3 . Adopting here this fruitful point of
view, we extend the quantum-mechanical line of reasoning initiated by Reid for
a two-mode state Reid1  by applying it to the whole class of two-party
$(N\,\text{vs}\,1)$-mode states. A parallel analysis of their entanglement and
steering makes use of the same pair of nonlocal EPR-like observables
introduced in Eq. (6). Essentially, we take two steps. First, we employ the
theorem of Hofmann and Takeuchi HT  to write the uncertainty relations with
the sum of variances of the EPR-like observables (6) for separable states, Eq.
(29), and, respectively, for unsteerable ones, Eqs. (56) and (57).

The second step consists in looking for the extremal normalized sums of
uncertainties with respect to the involved parameters
$\bm{\alpha},\bm{\beta}$. We start with the necessary conditions of
separability (31). Although not exploited analytically for $N>1$, they exhibit
the PPT property, as expected from Peres’ general theorem Peres .
Nevertheless, in the case of two-mode states $(N=1)$, we readily recover the
minimum $2{\kappa}_{-}^{\rm PT}$ for $d<0$ PT2018 , as specified in Eqs. (40)
and (45).

Then we focus on the steering scenario which, unlike entanglement, is known to
be asymmetric with respect to the interchange of the two parties. The
necessary condition of unsteerability from Alice to Bob, Eq. (64), is the only
numerical one to be equivalent to matrix inequalities having the form (82). By
contrast, when $N>1$, the equivalent matrix inequalities (73) and (78),
requested by the Bob-to-Alice unsteerability, are stronger than the numerical
one, Eq. (80). However, the similar inequalities (64) and (80) are valuable
since their violation allows one to recognize the steerability from Alice to
Bob and, respectively, from Bob to Alice.

We finally recall the privileged position of the two-party
$(N\,\text{vs}\,1)$-mode GSs, for whom the above necessary conditions
regarding both kinds of quantum correlations are equally sufficient ones. We
start with Simon’s seminal work Simon , who essentially proved that the first
inequality (46), ${\kappa}_{-}^{\rm PT}\geqq 1/2,$ is a criterium of
separability for two-mode GSs. The result of Werner and Wolf WW2001  that the
PPT property of any $(N\,\text{vs}\,1)$-mode GS implies its separability is
the maximal generalization of Simon’s theorem for $N=1$ Simon . However, by
contrast to the latter, it provides no analytic separability indicator for
$N>1$.

Then we apply the unsteerability criterion for two-party
$(N\,\text{vs}\,M)$-mode GSs, under Gaussian measurements, found by Wiseman et
al. in Refs. W1 ; W2 . Accordingly, the matrix inequalities (84) and (85),
written for a bipartite $(N\,\text{vs}\,1)$-mode GS ${\hat{\rho}}_{G}$ shared
by Alice and Bob, are necessary and sufficient conditions of its
unsteerability from Alice to Bob and, respectively, from Bob to Alice. We
emphasize that the necessity part is proven here with no reference to the
Gaussian or non-Gaussian nature of the state, as well as to that of the one-
party measurements involved.

Let us make a comment about three sets of $(N\,\text{vs}\,1)$-mode states
having the same CM ${\mathcal{V}}$ in the standard form (13): the set ${\cal
S}$ of separable states, as well as the sets ${\cal U}^{(A\to B)}$ and ${\cal
U}^{(B\to A)}$ of unsteerable states from Alice to Bob and from Bob to Alice,
respectively. All the states belonging to ${\cal S}$ fulfill the EPR-like
necessary conditions of separability (31), while those belonging to the sets
${\cal U}^{(A\to B)}$ and ${\cal U}^{(B\to A)}$ satisfy the necessary matrix
conditions of unsteerability (82) and, respectively, (78). However, by virtue
of the Werner-Wolf theorem WW2001 , for the GSs in ${\cal S}$, defined up to a
phase-space translation, the conditions of separability (31) are, in addition,
sufficient ones. Similarly, in view of a theorem proven by Wiseman et al. W2 ,
for the GSs belonging to the sets ${\cal U}^{(A\to B)}$ and ${\cal U}^{(B\to
A)}$ and under one-party Gaussian measurements, the matrix conditions of
unsteerability (82) and (78) are also sufficient ones. These remarks are just
some aspects of the special role played by the GSs among all
$(N\,\text{vs}\,1)$-mode states with the same CM. The best known example of
extremal property of the GSs within the multimode states with a given CM is
that they attain the maximum von Neumann entropy Holevo1999 . Other relevant
examples are mentioned and discussed in Ref. extrem .

We finally emphasize that we have developed in this work a unified treatment
of inseparability and steering of two-party $(N\,\text{vs}\,1)$-mode states,
based on specific sum-type URs of suitable EPR-like observables. Extremization
of the associate normalized sums of variances provides the right necessary
conditions for separability and one-way unsteerability. In the Gaussian
framework, these conditions are, at the same time, the well-known sufficient
ones.

## Acknowledgment

This work was supported by the Funding Agency CNCS-UEFISCDI of the Romanian
Ministry of Research and Innovation through grant No. PN-III-P4-ID-
PCE-2016-0794.

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