# Almost orthogonality and Hausdorff interval topologies of atomic lattice
effect algebras

Jan Paseka111This work was supported by by the Grant Agency of the Czech
Republic under the grant No. 201/06/0664 and by the Ministry of Education of
the Czech Republic under the project MSM0021622409<EMAIL_ADDRESS>Zdenka
Riečanová22footnotemark: 2<EMAIL_ADDRESS>Department of Mathematics
and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37
Brno, Czech Republic  Department of Mathematics, Faculty of Electrical
Engineering and Information Technology, Slovak University of Technology,
Ilkovičova 3, SK-812 19 Bratislava, Slovak Republic  Department of
Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China
Wu Junde<EMAIL_ADDRESS>
###### Abstract

We prove that the interval topology of an Archimedean atomic lattice effect
algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost
orthogonal. In such a case $E$ is order continuous. If moreover $E$ is
complete then order convergence of nets of elements of $E$ is topological and
hence it coincides with convergence in the order topology and this topology is
compact Hausdorff compatible with a uniformity induced by a separating
function family on $E$ corresponding to compact and cocompact elements. For
block-finite Archimedean atomic lattice effect algebras the equivalence of
almost orthogonality and s-compact generation is shown. As the main
application we obtain the state smearing theorem for these effect algebras, as
well as the continuity of $\oplus$-operation in the order and interval
topologies on them.

###### keywords:

Non-classical logics , D-posets , effect algebras , MV-algebras , interval and
order topology, states

###### PACS:

02.10.De , 02.40.Pc

###### MSC:

: 03G12, 06F05, 03G25, 54H12, 08A55

fn1fn1footnotetext: This work was supported by the Slovak Resaerch and
Development Agency under the contract No. APVV–0071–06 and the grant
VEGA-1/3025/06 of MŠ SR.

##  1 Introduction, basic definitions and facts

In the study of effect algebras (or more general, quantum structures) as
carriers of states and probability measures, an important tool is the study of
topologies on them. We can say that topology is practically equivalent with
the concept of convergence. From the probability point of view the convergence
of nets is the main tool in spite of that convergence of filters is easier to
handle and preferred in the modern topology. It is because states or
probabilities are mappings (functions) defined on elements but not on subsets
of quantum structures. Note also, that connections between order convergence
of filters and nets are not trivial. For instance, if a filter order converges
to some point of a poset then the associated net need not order converge (see
e.g, [12]).

On the other hand certain topological properties of studied structures
characterize also their certain algebraic properties and conversely. For
instance a known fact is that a Boolean algebra $B$ is atomic iff the interval
topology $\tau_{i}$ on $B$ is Hausdorff (see [20, Corollary 3.4]). This is not
more valid for lattice effect algebras (even MV-algebras). By Frink’s Theorem
the interval topology $\tau_{i}$ on $B$ (more generally on any lattice $L$) is
compact iff it is a complete lattice [5]. In [16] it was proved that if a
lattice effect algebra $E$ (more generally any basic algebra) is compactly
generated then $E$ is atomic.

We are going to prove that on an Archimedean atomic lattice effect algebra $E$
the interval topology $\tau_{i}$ is Hausdorff and $E$ is (o)-continuous if and
only if $E$ is almost orthogonal. Moreover, if $E$ is complete then $\tau_{i}$
is compact and coincides with the order topology $\tau_{o}$ on $E$ and this
compact topology $\tau_{i}=\tau_{o}$ is compatible with a uniformity on $E$
induced by a separating function family on $E$ corresponding to compact and
cocompact elements of $E$.

As the main corollary of that we obtain that every Archimedean atomic block-
finite lattice effect algebra $E$ has Hausdorff interval topology and hence
both topologies $\tau_{i}$ and $\tau_{o}$ are Hausdorff and they coincide. In
this case almost orthogonality of $E$ and s-compact generation by finite
elements of $E$ are equivalent. As an application the state smearing theorem
for these effect algebras is formulated. Moreover, the continuity of
$\oplus$-operation in $\tau_{i}$ and $\tau_{o}$ on them is shown.

######  Definition 1.1.

A partial algebra $(E;\oplus,0,1)$ is called an effect algebra if $0$, $1$ are
two distinct elements and $\oplus$ is a partially defined binary operation on
$E$ which satisfy the following conditions for any $a,b,c\in E$:

(Ei)ii

$b\oplus a=a\oplus b$ if $a\oplus b$ is defined,

(Eii)i

$(a\oplus b)\oplus c=a\oplus(b\oplus c)$ if one side is defined,

(Eiii)

for every $a\in E$ there exists a unique $b\in E$ such that $a\oplus b=1$ (we
put $a^{\prime}=b$),

(Eiv)i

if $1\oplus a$ is defined then $a=0$.

We often denote the effect algebra $(E;\oplus,0,1)$ briefly by $E$. In every
effect algebra $E$ we can define the partial order $\leq$ by putting

$a\leq b$ and  $b\ominus a=c$ iff $a\oplus c$ is defined and $a\oplus c=b$, we
set $c=b\ominus a$ .

If $E$ with the defined partial order is a lattice (a complete lattice) then
$(E;\oplus,0,1)$ is called a lattice effect algebra (a complete lattice effect
algebra).

Recall that a set $Q\subseteq E$ is called a sub-effect algebra of the effect
algebra $E$ if

  1. (i)

$1\in Q$

  2. (ii)

if out of elements $a,b,c\in E$ with $a\oplus b=c$ two are in $Q$, then
$a,b,c\in Q$.

If $Q$ is simultaneously a sublattice of $E$ then $Q$ is called a sub-lattice
effect algebra of $E$.

We say that a finite system $F=(a_{k})_{k=1}^{n}$ of not necessarily different
elements of an effect algebra $(E;\oplus,0,1)$ is $\oplus$-orthogonal if
$a_{1}\oplus a_{2}\oplus\dots\oplus a_{n}$ (written
$\bigoplus\limits_{k=1}^{n}a_{k}$ or $\bigoplus F$) exists in $E$. Here we
define $a_{1}\oplus a_{2}\oplus\dots\oplus a_{n}=(a_{1}\oplus
a_{2}\oplus\dots\oplus a_{n-1})\oplus a_{n}$ supposing that
$\bigoplus\limits_{k=1}^{n-1}a_{k}$ exists and
$\bigoplus\limits_{k=1}^{n-1}a_{k}\leq a^{\prime}_{n}$. An arbitrary system
$G=(a_{\kappa})_{\kappa\in H}$ of not necessarily different elements of $E$ is
$\oplus$-orthogonal if $\bigoplus K$ exists for every finite $K\subseteq G$.
We say that for a $\oplus$-orthogonal system $G=(a_{\kappa})_{\kappa\in H}$
the element $\bigoplus G$ exists iff $\bigvee\\{\bigoplus K\mid K\subseteq G$,
$K$ is finite$\\}$ exists in $E$ and then we put $\bigoplus
G=\bigvee\\{\bigoplus K\mid K\subseteq G\\}$ (we write $G_{1}\subseteq G$ iff
there is $H_{1}\subseteq H$ such that $G_{1}=(a_{\kappa})_{\kappa\in H_{1}}$).

Recall that elements $x$ and $y$ of a lattice effect algebra are called
compatible (written $x\leftrightarrow y$) if $x\vee y=x\oplus(y\ominus(x\wedge
y))$ [13]. For $x\in E$ and $Y\subseteq E$ we write $x\leftrightarrow Y$ iff
$x\leftrightarrow y$ for all $y\in Y$. If every two elements are compatible
then $E$ is called an MV-effect algebra. In fact, every MV-effect algebra can
be organized into an MV-algebra (see [2]) if we extend the partial $\oplus$ to
a total operation by setting $x\oplus y=x\oplus(x^{\prime}\wedge y)$ for all
$x,y\in E$ (also conversely, restricting a total $\oplus$ into partial
$\oplus$ for only $x,y\in E$ with $x\leq y^{\prime}$ we obtain a MV-effect
algebra).

Moreover, in [23] it was proved that every lattice effect algebra is a set-
theoretical union of MV-effect algebras called blocks. Blocks are maximal
subsets of pairwise compatible elements of $E$, under which every subset of
pairwise compatible elements is by Zorn’s Lemma contained in a maximal one.
Further, blocks are sub-lattices and sub-effect algebras of $E$ and hence
maximal sub-MV-effect algebras of $E$. A lattice effect algebra is called
block-finite if it has only finitely many blocks.

Finally note that lattice effect algebras generalize orthomodular lattices
[10] (including Boolean algebras) if we assume existence of unsharp elements
$x\in E$, meaning that $x\wedge x^{\prime}\neq 0$. On the other hand the set
$S(E)=\\{x\in E\mid x\wedge x^{\prime}=0\\}$ of all sharp elements of a
lattice effect algebra $E$ is an orthomodular lattice [8]. In this sense a
lattice effect algebra is a “smeared” orthomodular lattice, while an MV-effect
algebra is a “smeared” Boolean algebra. An orthomodular lattice $L$ can be
organized into a lattice effect algebra by setting $a\oplus b=a\vee b$ for
every pair $a,b\in L$ such that $a\leq b^{\perp}$.

For an element $x$ of an effect algebra $E$ we write ${\mathop{\rm
ord}}(x)=\infty$ if $nx=x\oplus x\oplus\dots\oplus x$ ($n$-times) exists for
every positive integer $n$ and we write ${\mathop{\rm ord}}(x)=n_{x}$ if
$n_{x}$ is the greatest positive integer such that $n_{x}x$ exists in $E$. An
effect algebra $E$ is Archimedean if ${\mathop{\rm ord}}(x)<\infty$ for all
$x\in E$. We can show that every complete effect algebra is Archimedean (see
[22]).

An element $a$ of an effect algebra $E$ is an atom if $0\leq b<a$ implies
$b=0$ and $E$ is called atomic if for every nonzero element $x\in E$ there is
an atom $a$ of $E$ with $a\leq x$. If $u\in E$ and either $u=0$ or
$u=p_{1}\oplus p_{2}\oplus\dots\oplus p_{n}$ for some not necessarily
different atoms $p_{1},p_{2},\dots,p_{n}\in E$ then $u\in E$ is called finite
and $u^{\prime}\in E$ is called cofinite. If $E$ is a lattice effect algebra
then for $x\in E$ and an atom $a$ of $E$ we have $a\leftrightarrow x$ iff
$a\leq x$ or $a\leq x^{\prime}$. It follows that if $a$ is an atom of a block
$M$ of $E$ then $a$ is also an atom of $E$. On the other hand if $E$ is atomic
then, in general, every block in $E$ need not be atomic (even for orthomodular
lattices [1]).

The following theorem is well known.

######  Theorem 1.2.

[25, Theorem 3.3] Let $(E;\oplus,0,1)$ be an Archimedean atomic lattice effect
algebra. Then to every nonzero element $x\in E$ there are mutually distinct
atoms $a_{\alpha}\in{E}$, $\alpha{\in\mathcal{E}}$ and positive integers
$k_{\alpha}$ such that

$x=\bigoplus\\{k_{\alpha}a_{\alpha}\mid\alpha\in{\mathcal{E}}\\}=\bigvee\\{k_{\alpha}a_{\alpha}\mid\alpha\in{\mathcal{E}}\\}$

under which $x\in S(E)$ iff $k_{\alpha}=n_{a_{\alpha}}={\mathop{\rm
ord}}(a_{\alpha})$ for all $\alpha\in{\mathcal{E}}$.

######  Definition 1.3.

(1) An element $a$ of a lattice $L$ is called compact iff, for any $D\subseteq
L$ with $\bigvee D\in L$, if $a\leq\bigvee D$ then $a\leq\bigvee F$ for some
finite $F\subseteq D$.

(2) A lattice $L$ is called compactly generated iff every element of $L$ is a
join of compact elements.

The notions of cocompact element and cocompactly generated lattice can be
defined dually. Note that compact elements are important in computer science
in the semantic approach called domain theory, where they are considered as a
kind of primitive elements.

##  2 Characterizations of interval topologies on bounded lattices

The order convergence of nets ((o)-convergence), interval topology $\tau_{i}$
and order-topology $\tau_{o}$ ((o)-topology) can be defined on any poset. In
our observations we will consider only bounded lattices and we will give a
characterization of interval topologies on them.

######  Definition 2.1.

Let $L$ be a bounded lattice. Let $\mathcal{H}=\\{[a,b]\subseteq L|a,b\in L$
with $a\leq b\\}$ and let
$\mathcal{G}=\\{\bigcup^{n}_{k=1}[a_{k},b_{k}]|[a_{k},b_{k}]\in\mathcal{H},k=1,2,...,n\\}$.
The _interval topology $\tau_{i}$ of $L$_ is the topology of $L$ with
$\mathcal{G}$ as a closed basis, hence with $\mathcal{H}$ as a closed
subbasis.

From definition of $\tau_{i}$ we obtain that $U\in\tau_{i}$ iff for each $x\in
U$ there is $F\in\mathcal{G}$ such that $x\in L\backslash F\subseteq U$.

######  Definition 2.2.

Let $L$ be a poset.

  * (i)

A net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ _order
converges_ (_$(o)$ -converges_, for short) to a point $x\in L$ if there exist
nets $(u_{\alpha})_{\alpha\in\mathcal{E}}$ and
$(v_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ such that

$x\uparrow u_{\alpha}\leq x_{\alpha}\leq v_{\alpha}\downarrow
x,\,{\alpha\in\mathcal{E}}$

where $x\uparrow u_{\alpha}$ means that $u_{\alpha_{1}}\leq u_{\alpha_{2}}$
for every ${\alpha_{1}}\leq{\alpha_{2}}$ and
$x=\bigvee\\{x_{\alpha}\mid\alpha\in\mathcal{E}\\}$. The meaning of
$v_{\alpha}\downarrow x$ is dual.

We write
$x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}$
in $L$.

  * (ii)

A topology $\tau_{0}$ on $L$ is called the _order topology_ on $L$ iff

    1. (a)

for any net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ and $x\in
L$: $x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}a$ in L
$\Rightarrow$ $x_{\alpha}\stackrel{{\scriptstyle\tau_{0}}}{{\rightarrow}}x$,
${\alpha\in\mathcal{E}}$, where
$x_{\alpha}\stackrel{{\scriptstyle\tau_{0}}}{{\rightarrow}}x$ denotes that
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ _converges_ to $x$ in the topological
space $(L,\tau_{0})$,

    2. (b)

if $\tau$ is a topology on $L$ with property (a) then $\tau\subseteq\tau_{o}$.

Hence $\tau_{o}$ is the strongest (finest, biggest) topology on $L$ with
property (a).

Recall that, for a directed set $(\mathcal{E},\leq)$, a subset
$\mathcal{E}^{\prime}\subseteq\mathcal{E}$ is called _cofinal_ in
$\mathcal{E}$ iff for every $\alpha\in\mathcal{E}$ there is
$\beta\in\mathcal{E}^{\prime}$ such that $\alpha\leq\beta$. A special kind of
a subnet of a net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ is net
$(x_{\beta})_{\beta\in\mathcal{E}^{\prime}}$ where $\mathcal{E}^{\prime}$ is a
cofinal subset of $\mathcal{E}$. This kind of subnets works in many cases of
our considerations.

In what follows we often use the following useful characterization of
topological convergence of nets:

######  Lemma 2.3.

For a net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of a topological
space $(X,\tau)$ and $x\in X$:

$x_{\alpha}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\alpha\in\mathcal{E}$ | iff |  for all $\mathcal{E}^{\prime}\subseteq\mathcal{E}$, where $\mathcal{E}^{\prime}$ is cofinal in $\mathcal{E}$ there exist
---|---|---
|  |  $\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$, $\mathcal{E}^{\prime\prime}$ cofinal in $\mathcal{E}^{\prime}$ such that $x_{\gamma}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$.

###### Proof.

$\Rightarrow$: It is trivial.

$\Leftarrow$: Let for every $\mathcal{E}^{\prime}\subseteq\mathcal{E}$, where
$\mathcal{E}^{\prime}$ is cofinal in $\mathcal{E}$ there exist
$\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$,
$\mathcal{E}^{\prime\prime}$ cofinal in $\mathcal{E}^{\prime}$ and
$x_{\gamma}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$,
and let
$x_{\alpha}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\alpha\in\mathcal{E}$.
Then there exist $U(x)\in\tau$ such that for all $\alpha\in\mathcal{E}$ there
exist $\beta_{\alpha}\in\mathcal{E}$ with $\beta_{\alpha}\geq\alpha$ and
$x_{\beta_{\alpha}}\not\in U(x)$. Let
$\mathcal{E}^{\prime}=\\{\beta_{\alpha}\in\mathcal{E}|\alpha\in\mathcal{E},\beta_{\alpha}\geq\alpha,x_{\beta_{\alpha}}\not\in
U(x)\\}$ then
$x_{\beta_{\alpha}}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,{\beta_{\alpha}}\in\mathcal{E}^{\prime}$
and for all cofinal $\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$:
$x_{\gamma}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$.
Hence there exists $\mathcal{E}^{\prime}\subseteq\mathcal{E}$ cofinal in
$\mathcal{E}$ and for all
$\mathcal{E}^{\prime\prime}\subseteq\mathcal{E}^{\prime}$,
$\mathcal{E}^{\prime\prime}$ cofinal in $\mathcal{E}^{\prime}$:
$x_{\gamma}\not\stackrel{{\scriptstyle\tau}}{{\rightarrow}}x,\gamma\in\mathcal{E}^{\prime\prime}$
a contradiction. ∎

Further, let us recall the following well known facts:

######  Lemma 2.4.

Let $L$ be a bounded lattice. Then

  1. (i)

$F\subseteq L$ is $\tau_{0}$-closed iff for every net
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$ and $x\in L$:
$(x_{\alpha}\in
F,x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}x,\alpha\in\mathcal{E})\Rightarrow
x\in F$.

  2. (ii)

For every $a,b\in L$ with $a\leq b$ the interval $[a,b]$ is $\tau_{0}$-closed.

  3. (iii)

$\tau_{i}\subseteq\tau_{o}$.

  4. (iv)

For any net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of a $L$ and
$x\in L$:

$x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}\
\Longrightarrow\
x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}.$

  5. (v)

If $\tau_{i}$ is Hausdorff then $\tau_{0}=\tau_{i}$ (see [4]).

  6. (vi)

The interval topology $\tau_{i}$ of a lattice $L$ is compact iff $L$ is a
complete lattice (see [5]).

Finally, let us note that compact Hausdorff topological space is always
normal. Thus separation axiom $T_{2},T_{3}$ and $T_{4}$ are trivially
equivalent for the interval topology of a complete lattice $L$.

######  Theorem 2.5.

Let $L$ be a complete lattice with interval topology $\tau_{i}$. If
$F\subseteq L$ is a complete sub-lattice of $L$ then

  1. (a)

$\tau^{F}_{i}=\tau_{i}\cap F$ is the interval topology of $F$,

  2. (b)

for any net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $F$ and $x\in
F$:

$x_{\alpha}\stackrel{{\scriptstyle\tau^{F}_{i}}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}\
\Longleftrightarrow\
x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x,{\alpha\in\mathcal{E}}.$

###### Proof.

(a): Let $\mathcal{H}$ and $\mathcal{H}_{F}$ be a closed subbasis of
$\tau_{i}$ and $\tau^{F}_{i}$ respectively. Then evidently $\mathcal{H}\cap
F=\\{[a,b]\cap F|[a,b]\in\mathcal{H}\\}$ is a closed subbasis of $\tau_{i}\cap
F$. Further for $[c,d]_{F}\in\mathcal{H}_{F}$ we have $[c,d]_{F}=\\{x\in
F|c\leq x\leq d\\}=[c,d]\cap F\in\mathcal{H}\cap F$. Conversely, since $F$ is
a complete sub-lattice of $L$, if $[a,b]\in\mathcal{H}$ then $[a,b]\cap
F=\\{x\in F|a\leq x\leq b\\}$ and either $[a,b]\cap F=\emptyset$ or there
exist $c=\wedge\\{x\in F|a\leq x\leq b\\}$ and $d=\vee\\{x\in F|a\leq x\leq
d\\}$ and $[a,b]\cap F=[c,d]_{F}\in\mathcal{H}_{F}$. This proves that
$\tau^{F}_{i}=\tau_{i}\cap F$.

(b): This is an easy consequence of (a). ∎

##  3 Hausdorff interval topology of almost orthogonal Archimedean atomic
lattice effect algebras and their order continuity

The atomicity of Boolean algebra $B$ is equivalent with Hausdorffness of
interval topology on $B$ (see [11], [29] and [20, Corollary 3.4]). This is not
more valid for lattice effect algebras, even also for MV-algebras.

######  Example 3.1.

Let $M=[0,1]\subseteq{\mathbb{R}}$ be a standard MV-effect algebra, i.e., we
define $a\oplus b=a+b$ iff $a+b\leq 1$, $a,b\in M$. Then $M$ is a complete
(o)-continuous lattice with $\tau_{i}=\tau_{o}$ being Hausdorff and with
(o)-convergence of nets coinciding with $\tau_{o}$-convergence. Nevertheless,
$M$ is not atomic.

We have proved in [16] that a complete lattice effect algebra is atomic and
(o)-continuous lattice iff $E$ is compactly generated. Nevertheless, in such a
case, the interval topology on $E$ need not be Hausdorff.

######  Example 3.2.

Let $E$ be a horizontal sum of infinitely many finite chains
$(P_{i},\bigoplus_{i},0_{i},1_{i})$ with at least 3 elements,
$i=1,2,...,n,\dots,$ (i. e., for $i=1,2,...,n,\dots,$, we identify all $0_{i}$
and all $1_{i}$ as well, $\bigoplus_{i}$ on $P_{i}$ are preserved and any
$a\in P_{i}\backslash\\{0_{i},1_{i}\\}$, $b\in
P_{j}\backslash\\{0_{j},1_{j}\\}$ for $i\not=j$ are noncomparable). Then $E$
is an atomic complete lattice effect algebra, $E$ is not block-finite and the
interval topology $\tau_{i}$ on $E$ is compact. Nevertheless, $\tau_{i}$ is
not Hausdorff because e.g., for $a\in P_{i},b\in P_{j},i\not=j$, $a,b$
noncomparable, we have $[a,1]\cap[0,b]=\emptyset$ and there is no finite
family $\mathcal{I}$ of closed intervals in $E$ separating $[a,1],[0,b]$
(i.e., the lattice $E$ can not be covered by a finite number of closed
intervals from $\mathcal{I}$ each of which is disjoint with at least one of
the intervals $[a,1]$ and $[0,b]$). This implies that $\tau_{i}$ is not
Hausdorff by [20, Lemma 2.2]. Further $E$ is compactly generated by finite
elements (hence (o)-continuous). It follows by [16] that the order topology
$\tau_{o}$ on $E$ is a uniform topology and (o)-convergence of nets on $E$
coincides with $\tau_{o}$-convergence.

In what follows we shall need an extension of [26, Lemma 2.1 (iii)].

######  Lemma 3.3.

Let $E$ be a lattice effect algebra, $x,y\in E$. Then $x\wedge y=0$ and $x\leq
y^{\prime}$ iff $kx\wedge ly=0$ and $kx\leq(ly)^{\prime}$, whenever $kx$ and
$ly$ exist in $E$.

###### Proof.

Let $x\leq y^{\prime}$, $x\wedge y=0$ and $2y$ exists in $E$. Then $x\oplus
y=(x\vee y)\oplus(x\wedge y)=x\vee y\leq y^{\prime}$ and hence there exists
$x\oplus 2y=(x\vee y)\oplus y=(x\oplus y)\vee 2y=x\vee y\vee 2y=x\vee 2y$,
which gives that $x\leq(2y)^{\prime}$ and $x\wedge 2y=0$. By induction, if
$ly$ exists then $x\oplus ly=x\vee ly$ and hence $x\leq(ly)^{\prime}$ and
$x\wedge ly=0$.

Now, $x\leq(ly)^{\prime}$ iff $ly\leq x^{\prime}$ and because $x\wedge ly=0$,
we obtain by the same argument as above that $ly\oplus kx=ly\vee kx$, hence
$kx\leq(ly)^{\prime}$ and $ly\wedge kx=0$ whenever $kx$ exists in $E$.

Conversely, $kx\wedge ly=0$ implies that $x\wedge y=0$ and
$kx\leq(ly)^{\prime}$ implies $x\leq kx\leq(ly)^{\prime}\leq y^{\prime}$. ∎

In next we will use the statement of Lemma 3.3 in the following form: For any
$x,y\in E$ with $x\wedge y=0$, $x\not\leq y^{\prime}$ iff
$kx\not\leq(ly)^{\prime}$, whenever $kx$ and $ly$ exist in $E$.

######  Definition 3.4.

Let $E$ be an atomic lattice effect algebra. $E$ is said to be almost
orthogonal if the set $\\{b\in E\mid b\not\leq a^{\prime},b\ \mbox{is an
atom}\\}$ is finite for every atom $a\in E$.

Note that our definition of almost orthogonality coincides with the usual
definition for orthomodular lattices (see e.g. [17, 18]).

######  Theorem 3.5.

Let $E$ be an Archimedean atomic lattice effect algebra. Then $E$ is almost
orthogonal if and only if for any atom $a\in E$ and any integer $l$, $1\leq
l\leq n_{a}$, there are finitely many atoms $c_{1},\dots,c_{m}$ and integers
$j_{1},\dots,j_{m}$, $1\leq j_{1}\leq n_{c_{1}},\dots,1\leq j_{m}\leq
n_{c_{m}}$ such that $j_{k}{}c_{k}\not\leq(la)^{\prime}$ for all
$k\in\\{1,\dots,m\\}$ and, for all $x\in E$, $x\not\leq(la)^{\prime}$ implies
$j_{k_{0}}{}c_{k_{0}}\leq x$ for some ${k_{0}}\in\\{1,\dots,m\\}$.

###### Proof.

$\Longrightarrow$: Assume that $E$ is almost orthogonal. Let $a\in E$ be an
atom, $1\leq l\leq n_{a}$. We shall denote $A_{a}=\\{b\in E\mid b\ \mbox{is}\
\mbox{an}\ \mbox{atom},b\not\leq a^{\prime}\\}$. Clearly, $A_{a}$ is finite
i.e. $A_{a}=\\{b_{1},\dots,b_{n}\\}$ for suitable atoms $b_{1},\dots,b_{n}$
from $E$.

Let $b\in E$ be an atom, $1\leq k\leq n_{b}$ and $kb\not\leq(la)^{\prime}$.
Either $b=a$ or $b\not={}a$ and in this case we have by Lemma 3.3 (iv) that
$b\not\leq a^{\prime}$. Hence either $b=a$ or $b\in A_{a}$. Let us put
$\\{c_{1},\dots,c_{m}\\}=\left\\{\begin{array}[]{l l}A_{a}&\mbox{if}\ a\in
S(E)\\\ A_{a}\cup\\{a\\}&\mbox{otherwise}\end{array}\right.$. In both cases we
have that $a\in\\{c_{1},\dots,c_{m}\\}$.

Now, let $x\in E$ and $x\not\leq(la)^{\prime}$. By Theorem 1.2 there is an
atom $c\in E$ and an integer $1\leq j\leq n_{c}$ such that $jc\leq x$ and
$jc\not\leq(la)^{\prime}$. Either $c=a$ or $c\not\leq a$. In the first case we
have that $j\geq(n_{a}-l+1)$ i.e. $x\geq(n_{a}-l+1)a$. In the second case we
get that $c\not\leq a^{\prime}$ i.e. $c\in A_{a}$ and $x\geq b_{i}$ for
suitable $i\in\\{1,\dots,n\\}$. Hence it is enough to put $j_{k}=1$ if
$c_{k}\in A_{a}$ and $j_{k}=(n_{a}-l+1)$ if $c_{k}=a$.

$\Longleftarrow$: Conversely, let $a\in E$ be an atom. Then there are finitely
many atoms $c_{1},\dots,c_{m}$ and integers $j_{1},\dots,j_{m}$, $1\leq
j_{1}\leq n_{c_{1}},\dots,1\leq j_{m}\leq n_{c_{m}}$ such that
$j_{k}{}c_{k}\not\leq a^{\prime}$ for all $k\in\\{1,\dots,m\\}$ and, for all
$x\in E$, $x\not\leq a^{\prime}$ implies $j_{k_{0}}{}c_{k_{0}}\leq x$ for some
${k_{0}}\in\\{1,\dots,m\\}$. Let us check that
$A_{a}\subseteq\\{c_{1},\dots,c_{m}\\}$. Let $b\in A_{a}$. Then $b\geq
j_{k_{0}}{}c_{k_{0}}\geq c_{k_{0}}$ for some ${k_{0}}\in\\{1,\dots,m\\}$.
Hence $b=c_{k_{0}}$. This yields $A_{a}$ is finite. ∎

######  Lemma 3.6.

Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then, for any atom $a\in E$ and any integer $l$, $1\leq l\leq n_{a}$ there are
finitely many atoms $b_{1},\dots,b_{n}$ and integers $j_{1},\dots,j_{n}$,
$1\leq j_{1}\leq n_{b_{1}},\dots,1\leq j_{n}\leq n_{b_{n}}$ such that

|
$E=[0,(la)^{\prime}]\cup\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$
---|---
and |
|
$[0,(la)^{\prime}]\cap\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)=\emptyset$.

Hence $[0,(la)^{\prime}]$ is a clopen subset in the interval topology.

###### Proof.

Let $a\in E$ be an atom, $1\leq l\leq n_{a}$. By Definition 3.5, let
$\\{j_{1}{}b_{1},\dots,j_{n}{}b_{n}\\}$ be the finite set of non-orthogonal
finite elements to $la$ of the form $j_{k}b_{k}$, $1\leq j_{k}\leq n_{b_{k}}$
minimal such that $b_{1},\dots,b_{n}$ are atoms different from $a$. We put
$D=[0,(la)^{\prime}]\cup\left(\bigcup_{k=1}^{n}[j_{k}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$.
Let us check that $D=E$. Clearly, $D\subseteq E$. Now, let $z\in E$. Then by
Theorem 1.2 there are mutually distinct atoms $c_{\gamma}\in{E}$,
$\gamma{\in\mathcal{E}}$ and integers $t_{\gamma}$ such that

$z=\bigoplus\\{t_{\gamma}c_{\gamma}\mid\gamma\in{\mathcal{E}}\\}=\bigvee\\{t_{\gamma}c_{\gamma}\mid\gamma\in{\mathcal{E}}\\}.$

Either $t_{\gamma}c_{\gamma}\leq(la)^{\prime}$ for all
$\gamma\in{\mathcal{E}}$ and hence $z\in[0,(la)^{\prime}]$ or there exists
$\gamma_{0}\in{\mathcal{E}}$ such that
$t_{\gamma_{0}}c_{\gamma_{0}}\not\leq(la)^{\prime}$. Hence, by almost
orthogonality, either $j_{k_{0}}b_{k_{0}}\leq t_{\gamma_{0}}c_{\gamma_{0}}\leq
z$ for some $k_{0}\in\\{1,\dots,n\\}$ or $(n_{a}+1-l)a\leq
t_{\gamma_{0}}c_{\gamma_{0}}\leq z$. In both cases we get that $z\in D$.

Now, assume that
$y\in{}[0,(la)^{\prime}]\cap\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$.
Then $(n_{a}+1-l)a\leq y\leq(la)^{\prime}$ or $j_{k}{}b_{k}\leq
y\leq(la)^{\prime}$ for some $k\in\\{1,\dots,n\\}$. In any case we have a
contradiction. ∎

######  Proposition 3.7.

Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then, for any not necessarily different atoms $a,b\in E$ and any integers
$l,k$; $1\leq l\leq n_{a}$, $1\leq k\leq n_{b}$, the interval
$[kb,(la)^{\prime}]$ is clopen in the interval topology.

###### Proof.

From Lemma 3.6 we have that $[0,(la)^{\prime}]$ is a clopen subset. Since a
dual of an almost orthogonal Archimedean atomic lattice effect algebra is an
almost orthogonal Archimedean atomic lattice effect algebra as well, we have
that $[kb,1]$ is again clopen in the interval topology. Hence also
$[kb,(la)^{\prime}]$ is clopen in the interval topology. ∎

######  Theorem 3.8.

Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then the interval topology $\tau_{i}$ on $E$ is Hausdorff.

###### Proof.

Let $x,y\in E$ and $x\not=y$. Then (without loss of generality) we may assume
that $x\not\leq y$. Then by [25, Theorem 3.3] there exists an atom $b$ from
$E$ and an integer $k$, $1\leq k\leq n_{b}$ such that $kb\leq x$ and
$kb\not\leq y$. Applying the dual of [25, Theorem 3.3] there exists an atom
$a$ from $E$ and an integer $l$, $1\leq l\leq n_{a}$ such that
$y\leq(la)^{\prime}$ and $kb\not\leq(la)^{\prime}$. Clearly, $x\in[kb,1]$,
$y\in[0,(la)^{\prime}]$.

Assume that there is an element $z\in E$ such that
$z\in[kb,1]\cap[0,(la)^{\prime}]$. Then $kb\leq z\leq(la)^{\prime}$, a
contradiction. Hence by Proposition 3.7, $[kb,1]$ and $[0,(la)^{\prime}]$ are
disjoint open subsets separating $x$ and $y$. ∎

######  Theorem 3.9.

Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then $E$ is compactly generated and therefore (o)-continuous.

###### Proof.

It is enough to check that, for any atom $a\in E$ and any integer $l$, $1\leq
l\leq n_{a}$ the element $la$ is compact in $E$ since any element of $E$ is a
join of such elements (see Theorem 1.2 resp. [25, Theorem 3.3]).

Let $x=\bigvee_{\alpha\in\mathcal{E}}x_{\alpha}$ for some net
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ in $E$, $la\leq x$, i.e.,
$(la)^{\prime}\geq x^{\prime}\downarrow x^{{}^{\prime}}_{\alpha}$.

By Lemma 3.6 we have
$E=[0,(la)^{\prime}]\cup\left(\bigcup_{k=1}^{n}[j_{k}{}b_{k},1]\cup[(n_{a}+1-l)a,1]\right)$,
$[0,(la)^{\prime}]\cap\left(\bigcup_{k=1}^{n}[b_{k},1]\cup[(n_{a}+1-l)a,1]\right)=\emptyset$,
$b_{1},\dots,b_{n}$ are atoms of $E$, $1\leq j_{k}\leq n_{b_{k}}$, $1\leq
k\leq n$.

Since ${\mathcal{E}}$ is directed upwards, there exists a cofinal subset
$\mathcal{E}^{\prime}\subseteq\mathcal{E}$ such that
$x^{{}^{\prime}}_{\beta}\in[0,(la)^{\prime}]$ for all
$\beta\in\mathcal{E}^{\prime}$ or there exists $k_{0}\in\\{1,2,...,n\\}$ such
that $x_{\beta}\in[j_{k_{0}}b_{k_{0}},1]$ for all
$\beta\in\mathcal{E}^{\prime}$ or $x^{{}^{\prime}}_{\beta}\in[(n_{a}+1-l)a,1]$
for all $\beta\in\mathcal{E}^{\prime}$. If
$x^{{}^{\prime}}_{\beta}\in[0,(la)^{\prime}]$ for all
$\beta\in\mathcal{E}^{\prime}$ then clearly $la\leq x_{\beta}$ for all
$\beta\in\mathcal{E}^{\prime}$. If there exists $k_{0}\in\\{1,2,...,n\\}$ such
that $x^{{}^{\prime}}_{\beta}\in[j_{k_{0}}b_{k_{0}},1]$ for all
$\beta\in\mathcal{E}^{\prime}$ or $x^{{}^{\prime}}_{\beta}\in[(n_{a}+1-l)a,1]$
for all $\beta\in\mathcal{E}^{\prime}$ we obtain that
$x^{\prime}\in[j_{k_{0}}b_{k_{0}},1]$ or $x^{{}^{\prime}}\in[(n_{a}+1-l)a,1]$
which is a contradiction with $x^{{}^{\prime}}\in[0,(la)^{\prime}]$. ∎

Let $E$ be an Archimedean atomic lattice effect algebra. We put
${\mathcal{U}}=\\{x\in E\mid x=\bigvee_{i=1}^{n}l_{i}a_{i},a_{1},\dots,a_{n}\
\mbox{are atoms of }\ E,1\leq l_{i}\leq n_{a_{i}},1\leq i\leq n,n\
\mbox{natural}$ $\mbox{number}\\}$ and ${\mathcal{V}}=\\{x\in E\mid
x^{\prime}\in{\mathcal{U}}\\}$. Then by [25, Theorem 3.3], for every $x\in L$,
we have that

$x=\bigvee\\{u\in{\mathcal{U}}\mid u\leq
x\\}=\bigwedge\\{v\in{\mathcal{V}}\mid x\leq v\\}.$

Consider the function family $\Phi=\\{f_{u}\mid u\
\in{\mathcal{U}}\\}\cup\\{g_{v}\mid v\in{\mathcal{V}}\\}$, where
$f_{u},g_{v}:L\to\\{0,1\\}$, $u\in{\mathcal{U}},v\in{\mathcal{V}}$ are defined
by putting $f_{u}(x)=\left\\{\begin{array}[]{r c l}1&\hbox{iff}&u\leq x\\\
0&\hbox{iff}&u\not\leq x\end{array}\right.$
and $g_{v}(y)=\left\\{\begin{array}[]{r c l}1&\hbox{iff}&x\leq v\\\
0&\hbox{iff}&x\not\leq v\end{array}\right.$ for all $x,y\in L$.

Further, consider the family of pseudometrics on $L$:
$\Sigma_{\Phi}=\\{\rho_{u}\mid u\in{\mathcal{U}}\\}\cup\\{\pi_{v}\mid
v\in{\mathcal{V}}\\}$, where $\rho_{u}(a,b)=|f_{u}(a)-f_{u}(b)|$ and
$\pi_{v}(a,b)=|g_{v}(a)-g_{v}(b)|$ for all $a,b\in L$.

Let us denote by $\mathcal{U}_{\Phi}$ the uniformity on $L$ induced by the
family of pseudometrics $\Sigma_{\Phi}$ (see e.g. [3]). Further denote by
$\tau_{\Phi}$ the topology compatible with the uniformity
$\mathcal{U}_{\Phi}$.

Then for every net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $L$

$\begin{array}[]{l}x_{\alpha}\stackrel{{\scriptstyle\tau_{\Phi}}}{{\longrightarrow}}x\hbox{
iff }\varphi(x_{\alpha})\to\varphi(x)\hbox{ for any
}\varphi\in\Phi.\end{array}$

This implies, since $f_{u}$, $u\in{\mathcal{U}}$, and $g_{v}$,
$v\in{\mathcal{V}}$, is a separating function family on $L$, that the topology
$\tau_{\Phi}$ is Hausdorff. Moreover, the intervals
$[u,v]=[u,1]\cap[0,v]=f_{u}^{-1}(\\{1\\})\cap g_{v}^{-1}(\\{1\\})$ are clopen
sets in $\tau_{\Phi}$.

######  Definition 3.10.

Let $E$ be an Archimedean atomic lattice effect algebra. Let $\Phi$ be a
separating function family on $E$ defined above. We will denote by
$\tau_{\Phi}$ the uniform topology on $E$ defined by this function family,
that means for every net $(x_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of
$L$

$\begin{array}[]{l}x_{\alpha}\stackrel{{\scriptstyle\tau_{\Phi}}}{{\longrightarrow}}x\
\hbox{ iff }\ \varphi(x_{\alpha})\to\varphi(x)\hbox{ for any
}\varphi\in\Phi.\end{array}$

######  Theorem 3.11.

Let $E$ be an almost orthogonal Archimedean atomic lattice effect algebra.
Then $\tau_{i}=\tau_{o}=\tau_{\Phi}$.

###### Proof.

Since by Theorem 3.8, $\tau_{i}$ is Hausdorff we obtain by [4] that
$\tau_{i}=\tau_{o}$. Further if $O\in\tau_{o}$ and $x\in O$ then by Theorem
1.2 we have $x=\bigvee\\{u\in{\mathcal{U}}\mid u\leq
x\\}=\bigwedge\\{v\in{\mathcal{V}}\mid x\leq v\\}$, which by [12] implies that
there exist finite sets $F\subseteq{\mathcal{U}}$, $G\subseteq{\mathcal{V}}$
such that $x\in[\bigvee F,\bigwedge G]\subseteq O$. Hence
$\tau_{o}\subseteq\tau_{\Phi}$. By Theorem 3.9 and [16, Theorem 1] we obtain
$\tau_{o}=\tau_{\Phi}$. ∎

######  Theorem 3.12.

Let $E$ be an Archimedean atomic block-finite lattice effect algebra. Then
$\tau_{i}=\tau_{o}$ is a Hausdorff topology.

###### Proof.

As in [18], it suffices to show that for every $x,y\in E$, $x\not\leq y$ there
are finitely many intervals, none of which contains both $x$ and $y$ and the
union of which covers $E$.

By [15], $E$ is a union of finitely many atomic blocks $M_{i}$,
$i=1,2,\dots,n$. Choose $i\in\\{1,2,\dots,n\\}$. If $x,y\in M_{i}$ then there
is an atom $a_{i}\in M_{i}$ and an integer $l_{i}$, $1\leq l_{i}\leq
n_{a_{i}}$ such that $l_{i}a_{i}\leq x$, $l_{i}a_{i}\not\leq y$. Let us put
$k_{i}=n-l_{i}+1$. Since $M_{i}$ is almost orthogonal (the only possible non-
orthogonal $kb$ to $la$ for an atom $a$, $1\leq l\leq n_{a}$ is that $a=b$) we
have by Lemma 3.6 that $M_{i}=([0,(k_{i}a_{i})^{\prime}]\cap
M_{i})\cup([(n_{a_{i}}+1-k_{i})a_{i},1]\cap M_{i})$. Hence
$M_{i}\subseteq[0,(k_{i}a_{i})^{\prime}]\cup[(n_{a_{i}}+1-k_{i})a_{i},1]$. Let
us check that
$[0,(k_{i}a_{i})^{\prime}]\cap[(n_{a_{i}}+1-k_{i})a_{i},1]=\emptyset$. Assume
that $(n_{a_{i}}+1-k_{i})a_{i}\leq z\leq(k_{i}a_{i})^{\prime}$. Then
$(n_{a_{i}}+1-k_{i})a_{i}\leq(k_{i}a_{i})^{\prime}$, a contradiction. Put
$J_{i}=[0,(k_{i}a_{i})^{\prime}]$, $K_{i}=[(n_{a_{i}}+1-k_{i})a_{i},1]$. This
yields $x\in K_{i}$, $y\in J_{i}$, $M_{i}\subseteq J_{i}\cup K_{i}$ and
$J_{i}\cap K_{i}=\emptyset$. Let $x\not\in M_{i}$. Then there exists an atom
$a_{i}\in M_{i}$ that is not compatible with $x$. Let us check that
$x\not\in[0,(a_{i})^{\prime}]\cup[n_{a_{i}}a_{i},1]$. Assume that
$x\in[0,(a_{i})^{\prime}]$ or $x\in[n_{a_{i}}a_{i},1]$. Then
$x\leq(a_{i})^{\prime}$ or $a_{i}\leq n_{a_{i}}a_{i}\leq x$, i.e., in both
cases we get that $x\leftrightarrow a_{i}$, a contradiction. Let us put
$J_{i}=[0,(a_{i})^{\prime}]$, $K_{i}=[n_{a_{i}}a_{i},1]$. As above,
$M_{i}\subseteq J_{i}\cup K_{i}$, $J_{i}\cap K_{i}=\emptyset$ and moreover
$x\notin J_{i}\cup K_{i}$. The remaining case $y\not\in M_{i}$ can be checked
by similar considerations. We obtain
$E=\bigcup_{i=1}^{n}M_{i}\subseteq\bigcup_{i=1}^{n}(J_{i}\cup K_{i})\subseteq
E$ and none of the intervals $J_{i},K_{i}$, $i=1,2,\dots,n$ contains both $x$
and $y$. ∎

##  4 Order and interval topologies of complete atomic block-finite lattice
effect algebras

We are going to show that on every complete atomic block-finite lattice effect
algebra $E$ the interval topology is Hausdorff. Hence both topologies
$\tau_{i}$ and $\tau_{o}$ are in this case compact Hausdorff and they
coincide. Moreover, a necessary and sufficient condition for a complete atomic
lattice algebra $E$ to be almost orthogonal is given.

For the proof of Theorems 4.2 and 4.3 we will use the following statement,
firstly proved in the equivalent setting of D-posets in [19].

######  Theorem 4.1.

[19, Theorem 1.7] Suppose that $(E;\oplus,0,1)$ is a complete lattice effect
algebra. Let $\emptyset\not=D\subseteq E$ be a sub-lattice effect algebra. The
following conditions are equivalent:

  1. (i)

For all nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ such that $x_{\alpha}\in D$
for all ${\alpha\in\mathcal{E}}$

$x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}x\ \mbox{in}\ E\
\mbox{if and only if}\ x\in D\ \mbox{and}\
x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}x\ \mbox{in}\ D.$

  2. (ii)

For every $M\subseteq D$ with $\bigvee M=x$ in $E$ it holds $x\in D$.

  3. (iii)

For every $Q\subseteq D$ with $\bigwedge Q=y$ in $E$ it holds $y\in D$.

  4. (iv)

$D$ is a complete sub-lattice of $E$.

  5. (v)

$D$ is a closed set in order topology $\tau_{o}$ on $E$.

Each of these conditions implies that $\tau_{o}^{D}=\tau_{o}^{E}\cap D$, where
$\tau_{o}^{D}$ is an order topology on $D$.

Important sub-lattice effect algebras are blocks, $S(E)$,
$B(E)=\bigcap\\{M\subseteq E\mid M\ \mbox{block\/ of}\ E\\}$ and
$C(E)=B(E)\cap S(E)$ (see [6, 7, 13, 21, 23]).

######  Theorem 4.2.

Let $E$ be a complete lattice effect algebra. Then for every
$D\in\\{S(E),C(E),B(E)\\}$ or $D=M$, where $M$ is a block of $E$, we have:

  1. (1)

$x_{\alpha}\stackrel{{\scriptstyle\tau^{E}_{i}}}{{\rightarrow}}x$ $\Longleftrightarrow$ $x_{\alpha}\stackrel{{\scriptstyle\tau^{D}_{i}}}{{\rightarrow}}x$, |  for all nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ in $D$ and all $x\in D$.
---|---

  2. (2)

If $\tau_{i}$ is Hausdorff then

$x_{\alpha}\stackrel{{\scriptstyle\tau^{E}_{i}}}{{\rightarrow}}x$ $\Longleftrightarrow$ $x_{\alpha}\stackrel{{\scriptstyle\tau^{D}_{i}}}{{\rightarrow}}x$, |  for all nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ in $D$ and all $x\in E$.
---|---

###### Proof.

The first part of the statement follows by Theorem 2.5 and the fact that if
$E$ is a complete lattice effect algebra then $M$, $S(E)$, $C(E)$ and $B(E)$
are complete sub-lattices of $E$ (see [9, 24]). The second part follows by [4]
since $\tau_{i}$ is Hausdorff implies $\tau_{i}=\tau_{o}$ and by Theorem 4.1.
∎

######  Theorem 4.3.

(i) The interval topology $\tau_{i}$ on every Archimedean atomic MV-effect
algebra $M$ is Hausdorff and $\tau_{i}=\tau_{o}=\tau_{\Phi}$.

(ii) For every complete atomic MV-effect algebra $M$ and for any net
$(x_{\alpha})$ of $M$ and any $x\in M$,

$x_{\alpha}\stackrel{{\scriptstyle\tau_{o}}}{{\longrightarrow}}x$ if and only
if $x_{\alpha}\stackrel{{\scriptstyle(o)}}{{\longrightarrow}}x$ (briefly
$\tau_{o}\equiv(o)$).

Moreover, $\tau_{o}$ is a uniform compact Hausdorff topology on $M$.

(iii) For every atomic block-finite lattice effect algebra $E$, $E$ is a
complete lattice iff $\tau_{i}=\tau_{o}$ is a compact Hausdorff topology.

###### Proof.

(i), (ii): This follows from the fact that every pair of elements of $M$ is
compatible, hence every pair of atoms is orthogonal. Thus for (i) we can apply
Theorem 3.11 and for (ii) we can use (i) and [16, Theorem 2] since $M$ is
compactly generated by finite elements and $\tau_{i}$ is compact.

(iii) From Theorem 3.12 we know that $\tau_{i}=\tau_{o}$ is a Hausdorff
topology. By Lemma 2.4 (vi) the interval topology $\tau_{i}$ on $E$ is compact
iff $E$ is a complete lattice. ∎

In what follows we will need Corollary 4.5 of Lemma 4.4.

######  Lemma 4.4.

Let $E$ be an Archimedean atomic lattice effect algebra. Then

  1. (i)

If $c,d\in E$ are compact elements with $c\leq d^{\prime}$ then $c\oplus d$ is
compact.

  2. (ii)

If $u=\bigoplus G$, where $G$ is a $\oplus$-orthogonal system of atoms of $E$,
and $u$ is compact then $G$ is finite.

###### Proof.

(i) Let $c\oplus d\leq\bigvee D$. Let ${\cal E}=\\{F\subseteq D:F\ \mbox{is
finite}\\}$ be directed by set inclusion and let for every $F\in{\cal E}$ be
$x_{F}=\bigvee F$. Then $x_{F}\uparrow x=\bigvee D$. Since $c\leq\bigvee D$
and $d\leq\bigvee D$ there is a finite subset $F_{1}\subseteq D$ such that
$c\vee d\leq\bigvee F_{1}$. Therefore, for $F\geq F_{1}$, $x_{F}\ominus
c\uparrow x\ominus c,d\leq x\ominus c$. Then there is a finite subset
$F_{2}\subseteq D$, $F_{1}\leq F_{2}$ such that $d\leq x_{F_{2}}\ominus c$.
Hence $c\oplus d\leq x_{F_{2}}$.

(ii) Let $u\in E$, $u=\bigoplus G=\bigvee\\{\bigoplus K\mid K\subseteq G$ is
finite$\\}$ where $G=(a_{\kappa})_{\kappa\in H}$ is a $\oplus$-orthogonal
system of atoms. Clearly if $K_{1},K_{2}\subseteq G$ are finite such that
$K_{1}\subseteq K_{2}$ then $\bigoplus K_{1}\leq\bigoplus K_{2}$.

Assume that $u$ is compact. Hence there are finite $K_{1},K_{2},$
$\dots,K_{n}\subseteq G$ such that $u\leq\bigvee\\{\bigoplus K_{i}\mid
i=1,2,\dots,n\\}$. Let $K_{0}=\bigcup\\{K_{i}\mid i=1,2,\dots,n\\}$. Then
$K_{0}\subseteq G$, $K_{0}$ is finite and $\bigoplus K_{i}\leq\bigoplus
K_{0}$, $i=1,2\dots,n$, which gives that $\bigvee\\{\bigoplus K_{i}\mid
i=1,2,\dots,n\\}\leq\bigoplus K_{0}$. It follows that $u\leq\bigoplus
K_{0}\leq u=\bigvee\\{\bigoplus K\mid K\subseteq G$ is finite$\\}$. Hence
$u=\bigoplus K_{0}$, $K_{0}\subseteq G$ is finite. Further, for every finite
$K\subseteq G\setminus K_{0}$ we have $\bigoplus
K_{0}\subseteq\bigoplus(K_{0}\cup K)=\bigoplus K_{0}\oplus\bigoplus K\leq
u=\bigoplus K_{0}$ , which gives that $\bigoplus K=0$. Hence $K=\emptyset$ and
thus $G\setminus K_{0}=\emptyset$ which gives that $K_{0}=G$.∎

######  Corollary 4.5.

Let $E$ be an o-continuous Archimedean atomic lattice effect algebra. Then
every finite element of $E$ is compact.

###### Proof.

Clearly, by [16, Theorem 7] we know that $E$ is compactly generated.
Therefore, any atom of $E$ is compact. The compactness of every finite element
follows by an easy induction. ∎

######  Theorem 4.6.

Let $E$ be an Archimedean atomic lattice effect algebra. Then the following
conditions are equivalent:

  1. (i)

$\tau_{i}=\tau_{o}=\tau_{\Phi}$.

  2. (ii)

$E$ is o-continuous and $\tau_{i}$ is Hausdorff.

  3. (iii)

$E$ is almost orthogonal.

###### Proof.

(i)$\implies$(ii): Since $\tau_{o}=\tau_{\Phi}$ we have by [16, Theorem 1]
that $E$ is compactly generated and hence o-continuous. The condition
$\tau_{i}=\tau_{\Phi}$ implies that $\tau_{i}$ is Hausdorff because
$\tau_{\Phi}$ is Hausdorff.

(ii)$\implies$(i), (iii): Since $\tau_{i}$ is Hausdorff we obtain
$\tau_{i}=\tau_{o}$ by [4]. Moreover, from [16, Theorem 7] and Corollary 4.5
the (o)-continuity of $E$ implies that $E$ is compactly generated by the
elements from ${\mathcal{U}}$. This gives $\tau_{o}=\tau_{\Phi}$ from [16,
Theorem 1].

Let $a\in E$ be an atom, $1\leq l\leq n_{a}$. Then the interval
$[0,(la)^{\prime}]$ is a clopen set in the order topology
$\tau_{o}=\tau_{\Phi}=\tau_{i}$. Hence there is a finite set of intervals in
$E$ such that $0\in
E\setminus\bigcup_{i=1}^{n}[u_{i},v_{i}]\subseteq[0,(la)^{\prime}]$. Thus
$E\subseteq[0,(la)^{\prime}]\cup\bigcup_{i=1}^{n}[u_{i},v_{i}]\subseteq[0,(la)^{\prime}]\cup\bigcup_{i=1}^{n}[k_{i}{}b_{i},1]$,
where $b_{i}\in E$ are atoms such that $k_{i}{}b_{i}\leq u_{i}$, $1\leq
k_{i}\leq n_{b_{i}}$, $i=1,\dots,n$. This yields that $E$ is almost
orthogonal.

(iii)$\implies$(ii): From Theorems 3.8 and 3.9 we have that $\tau_{i}$ is
Hausdorff and $E$ is compactly generated, hence (o)-continuous. ∎

######  Corollary 4.7.

Let $E$ be a complete atomic lattice effect algebra. Then the following
conditions are equivalent:

  1. (i)

$E$ is almost orthogonal.

  2. (ii)

$\tau_{i}=\tau_{o}=\tau_{\Phi}\equiv(o)$.

  3. (iii)

$E$ is (o)-continuous and $\tau_{i}$ is Hausdorff.

###### Proof.

It follows from Theorems 4.6 and the fact that by (o)-continuity of $E$ [27,
Theorem 8] we have $\tau_{o}\equiv(o)$. ∎

The next example shows that a complete block-finite atomic lattice effect
algebra need not be (o)-continuous and almost orthogonal in spite of that
$\tau_{i}=\tau_{o}$ is a compact Hausdorff topology.

######  Example 4.8.

Let $E$ be a horizontal sum of finitely many infinite complete atomic Boolean
algebras $(B_{i},\bigoplus_{i},0_{i},1_{i})$, $i=1,2,...,n$. Then $E$ is an
atomic complete lattice effect algebra, $E$ is not almost orthogonal, $E$ is
not compactly generated by finite elements (hence $\tau_{o}\not=\tau_{\Phi}$),
$E$ is block-finite, $\tau_{i}=\tau_{o}$ is Hausdorff by Theorem 3.12, and the
interval topology $\tau_{i}$ on $E$ is compact.

##  5 Applications

######  Theorem 5.1.

Let $E$ be a block-finite complete atomic lattice effect algebra. Then the
following conditions are equivalent:

  1. (i)

$E$ is almost orthogonal.

  2. (ii)

$E$ is compactly generated.

  3. (iii)

$E$ is (o)-continuous.

  4. (iv)

$\tau_{i}=\tau_{o}=\tau_{\Phi}\equiv(o)$.

###### Proof.

By Theorem 3.12, $\tau_{i}=\tau_{o}$ is a Hausdorff topology. This by [16,
Theorem 7] gives that (ii) $\Longleftrightarrow$ (iii) and by Corollary 4.7 we
obtain that (i) $\Longleftrightarrow$ (iii) $\Longleftrightarrow$ (iv). ∎

In Theorem 5.1, the assumption that $E$ is atomic can not be omitted. For
instance, every non-atomic complete Boolean algebra is (o)-continuous but it
is not compactly generated, because in such a case $E$ must be atomic by [16,
Theorem 6].

######  Remark 5.2.

If a $\oplus$-operation on a lattice effect algebra $E$ is continuous with
respect to its interval topology $\tau_{i}$ meaning that $x_{\alpha}\leq
y_{\alpha}^{{}^{\prime}}$,
$x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x$,
$y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}y$,
$\alpha\in\mathcal{E}$ implies $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$, then
$\tau_{i}$ is Hausdorff (see [14]). Hence $\oplus$-operation on complete
(o)-continuous atomic lattice effect algebras which are not almost orthogonal
cannot be $\tau_{i}$-continuous, by [14] and Corollary 4.7.

######  Theorem 5.3.

Let $E$ be a block-finite complete atomic lattice effect algebra. Let
$(x_{\alpha})_{\alpha\in\mathcal{E}}$ and
$(y_{\alpha})_{\alpha\in\mathcal{E}}$ be nets of elements of $E$ such that
$x_{\alpha}\leq y_{\alpha}^{{}^{\prime}}$ for all ${\alpha\in\mathcal{E}}$.

If $x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x$,
$y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}y$,
$\alpha\in\mathcal{E}$ then $x\leq y^{{}^{\prime}}$ and $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$. Moreover, $\tau_{i}=\tau_{o}$.

###### Proof.

Since, by Theorem 3.12, $\tau_{i}$ is Hausdorff, we obtain that
$\tau_{i}=\tau_{o}$ by [4]. Let $\\{M_{1},\dots,M_{n}\\}$ be the set of all
blocks of $E$. Further, for every ${\alpha\in\mathcal{E}}$, elements of the
set $\\{x_{\alpha},y_{\alpha},x_{\alpha}\oplus y_{\alpha}\\}$ are pairwise
compatible. It follows that for every ${\alpha\in\mathcal{E}}$ there exists a
block $M_{k_{\alpha}}$ of $E$, ${k_{\alpha}}\in\\{1,\dots,n\\}$ such that
$\\{x_{\alpha},y_{\alpha},x_{\alpha}\oplus y_{\alpha}\\}\subseteq
M_{k_{\alpha}}$. Let ${\mathcal{E}}^{\prime}$ be any cofinal subset of
${\mathcal{E}}$. Since ${\mathcal{E}}^{\prime}$ is directed upwards, there
exists a block $M_{k_{0}}$ of $E$ and a cofinal subset
${\mathcal{E}}^{\prime\prime}$ of ${\mathcal{E}}^{\prime}$ such that
$\\{x_{\beta},y_{\alpha},x_{\beta}\oplus y_{\beta}\\}\subseteq M_{k_{0}}$ for
all $\beta\in{\mathcal{E}}^{\prime\prime}$. Otherwise we obtain a
contradiction with the finiteness of the set $\\{M_{1},\dots,M_{n}\\}$.
Further, by Theorem 2.5, we obtain that $\tau^{M_{k_{0}}}_{i}=\tau_{i}\cap
M_{k_{0}}$, as $M_{k_{0}}$ is a complete sublattice of $E$ (see Theorem 4.2).
It follows that the interval topology $\tau^{M_{k_{0}}}_{i}$ on the complete
MV-effect algebra $M_{k_{0}}$ is Hausdorff. The last by [14, Theorem 3.6]
gives that $x_{\beta}\oplus
y_{\beta}\stackrel{{\scriptstyle\tau^{M_{k_{0}}}_{i}}}{{\rightarrow}}x\oplus
y$, $\beta\in\mathcal{E}^{\prime\prime}$ and hence $x_{\beta}\oplus
y_{\beta}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\beta\in\mathcal{E}^{\prime\prime}$, as $\tau^{M_{k_{0}}}_{i}=\tau_{i}\cap
M_{k_{0}}$. It follows that $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$ by Lemma 2.3. ∎

In [22, Theorem 4.5] it was proved that a block-finite lattice effect algebra
$(E;\oplus,0,1)$ has a MacNeille completion which is a complete effect algebra
$(MC(E);\oplus,0,1)$ containing $E$ as a (join-dense and meet-dense) sub-
lattice effect algebra iff $E$ is Archimedean. In what follows we put
$\widehat{E}=MC(E)$.

######  Corollary 5.4.

Let $E$ be a block-finite Archimedean atomic lattice effect algebra. Then for
any nets $(x_{\alpha})_{\alpha\in\mathcal{E}}$ and
$(y_{\alpha})_{\alpha\in\mathcal{E}}$ of elements of $E$ with $x_{\alpha}\leq
y_{\alpha}^{{}^{\prime}}$, ${\alpha\in\mathcal{E}}$:
$x_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x$,
$y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}y$,
$\alpha\in\mathcal{E}$ implies $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\tau_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$.

###### Proof.

By [20, Lemma 1.1], for interval topologies $\widehat{\tau}_{i}$ on
$\widehat{E}$ and ${\tau_{i}}$ on $E$, we have $\widehat{\tau_{i}}\cap
E={\tau_{i}}$. Thus for $x_{\alpha},y_{\alpha},x,y\in E$ we obtain
$x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle\widehat{\tau}_{i}}}{{\rightarrow}}x\oplus
y$, $\alpha\in\mathcal{E}$ which gives $x_{\alpha}\oplus
y_{\alpha}\stackrel{{\scriptstyle{\tau}_{i}}}{{\rightarrow}}x\oplus y$,
$\alpha\in\mathcal{E}$ by the fact that $\widehat{\tau_{i}}\cap E={\tau_{i}}$.
∎

######  Definition 5.5.

Let $E$ be a lattice.Then

  1. (i)

An element $u$ of $E$ is called strongly compact (briefly s-compact) iff, for
any $D\subseteq E$: $u\leq c\in E$ for all $c\geq D$ implies $u\leq\bigvee F$
for some finite $F\subseteq D$.

  2. (ii)

$E$ is called s-compactly generated iff every element of $E$ is a join of
s-compact elements.

######  Theorem 5.6.

Let $E$ be a block-finite Archimedean atomic lattice effect algebra. Then the
following conditions are equivalent:

  1. (i)

$E$ is almost orthogonal.

  2. (ii)

$\widehat{E}=MC(E)$ is almost orthogonal.

  3. (iii)

$\widehat{E}=MC(E)$ is compactly generated.

  4. (iv)

${E}$ is s-compactly generated.

###### Proof.

By J. Schmidt [30] a MacNeille completion $\widehat{E}$ of $E$ is (up to
isomorphism) a complete lattice such that for every element $x\in\widehat{E}$
there exist $P,Q\subseteq E$ such that
$x=\bigvee_{\widehat{E}}P=\bigwedge_{\widehat{E}}Q$ (taken in
${}_{\widehat{E}}$). Here we identify $E$ with $\varphi(E)$, where
$\varphi:E\to{\widehat{E}}$ is the embedding (meaning that $E$ and
$\varphi(E)$ are isomorphic lattice effect algebras). It follows that $E$ and
${\widehat{E}}$ have the same set of all atoms and coatoms and hence also the
same set of all finite and cofinite elements, which implies that (i)
$\Longleftrightarrow$ (ii).

Moreover, for any $A\subseteq E$ and $u\in E$, we have ($d\in E$, $A\leq d$
implies $u\leq d$) iff $u\leq\bigvee_{\widehat{E}}A$. Then $u$ is s-compact in
$E$ iff $u$ is compact in $\widehat{E}$, which gives (iii)
$\Longleftrightarrow$ (iv).

Finally (ii) $\Longleftrightarrow$ (iii) by Theorem 5.1. ∎

######  Definition 5.7.

Let $E$ be an effect algebra. A map $\omega:E\to[0,1]$ is called a state on
$E$ if $\omega(0)=0$, $\omega(1)=1$ and $\omega(x\oplus
y)=\omega(x)+\omega(y)$ whenever $x\oplus y$ exists in $E$.

######  Theorem 5.8.

(State smearing theorem for almost orthogonal block-finite Archimedean atomic
lattice effect algebras) Let $(E;\oplus,0,1)$ be a block-finite Archimedean
atomic lattice effect algebra. If $E$ is almost orthogonal then:

  1. (i)

$E_{1}=\\{x\in E\mid x\ \mbox{or}\ x^{\prime}\ \mbox{is finite}\\}$ is a sub-
lattice effect algebra of $E$.

  2. (ii)

If there exists an $(o)$-continuous state $\omega$ on $E_{1}$ (or on
$S(E_{1})=S(E)\cap E_{1}$, or on $S(E)$) then there exists an $(o)$-continuous
state $\widetilde{\omega}$ on $E$ extending $\omega$ and an $(o)$-continuous
state $\widehat{\omega}$ on $\widehat{E}=MC(E)=MC(E_{1})$ extending
$\widetilde{\omega}$.

###### Proof.

(i) By Theorem 5.6, ${E}$ is s-compactly generated and thus by [28, Theorem
2.7] $E_{1}$ is a sub-lattice effect algebra of $E$.

(ii) Since ${E}$ is s-compactly generated, we obtain the existence of
$(o)$-continuous extensions $\widetilde{\omega}$ on $E$ and $\widehat{\omega}$
on $\widehat{E}$ by [28, Theorem 4.2]. ∎

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# Detection power of separability criteria based on a correlation tensor: a
case study

Gniewomir Sarbicki  Institute of Physics, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń,
Poland  Giovanni Scala  Dipartimento Interateneo di Fisica, Università degli
Studi di Bari, I-70126 Bari, Italy  INFN, Sezione di Bari, I-70125 Bari, Italy
Dariusz Chruściński  Institute of Physics, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń,
Poland

(August 27, 2024)

###### Abstract

Detection power of separability criteria based on a correlation tensor is
tested within a family of generalized isotropic state in $d_{1}\otimes d_{2}$.
For $d_{1}\neq d_{2}$ all these criteria are weaker than positive partial
transposition (PPT) criterion. Interestingly, our analysis supports the recent
conjecture that a criterion based on symmetrically informationaly complete
positive operator-valued measure (SIC-POVMs) is stronger than realignment
criterion.

detect entanglement, SIC-POVMs, realignment criterion, entanglement witnesses

###### pacs:

33.15.Ta

##  I Introduction

Quantum entanglement defines a key feature of quantum theory. It becomes a
crucial resource for quantum information theory and for modern quantum based
technologies like quantum communication, quantum cryptography, and quantum
calculations HHHH ; QIT . There are several separability criteria developed
during the last two decades which enable to distinguish between separable and
entangled states (cf. GT ; HHHH ; TOPICAL ). In this paper we analyze a
special family of separability criteria based on correlation tensor. The
prominent example is provided be the realignment criterion or computable
cross-norm (CCNR) criterion R1 ; R2 ; R3 . Apart from CCNR this class contains
de Vicente criterion (dV) Vicente , separability criterion derived in Fei  and
recent criterion based on SIC POMVs (ESIC) GUHNE . Interestingly, all these
criteria were unified in GGD1 . It was shown GGD1  that the above criteria are
just special examples of a 2-parameter family of criteria. In this paper we
provide a comparative anaysis of these criteria testing a simple family of
bipartite states in $d_{1}\otimes d_{2}$.

Let us fix notation:
$\mathbb{C}^{\mathrm{d_{1}}}\otimes\mathbb{C}^{\mathrm{d_{2}}}$ be the Hilbert
space of a bipartite system with dimension of subsystems $d_{1}$ and $d_{2}$
respectively. In what follows we assume that $d_{2}\geq d_{1}$. Let
$\\{G_{\alpha}^{(1)}\\}_{\alpha=1}^{d_{1}}$ and
$\\{G_{\beta}^{(2)}\\}_{\beta=1}^{d_{2}}$ be arbitrary orthonormal bases in
$\mathcal{B}(\mathbb{C}^{\mathrm{d_{1}}})$ and
$\mathcal{B}(\mathbb{C}^{\mathrm{d_{2}}})$, namely $\langle
G_{\alpha}^{(1)}|G_{\alpha^{\prime}}^{(1)}\rangle_{{\rm
HS}}=\delta_{\alpha,\alpha^{\prime}}$ and the same for $G_{\beta}^{(2)}$,
where $\langle X|Y\rangle_{\mathrm{HS}}={\rm Tr(X^{\dagger}Y)}$ is the
Hilbert-Schmidt inner product. One defines a correlation matrix

$C_{\alpha\beta}=\langle G_{\alpha}^{(1)}\otimes
G_{\beta}^{(2)}\rangle_{\rho}={\rm Tr\left(\rho G_{\alpha}^{(1)}\otimes
G_{\beta}^{(2)}\right)}.$ (1)

Let us now restrict ourselves to orthonormal bases (referred as canonical
bases) such that $G^{(i)}_{0}=\boldsymbol{1}_{\mathrm{d_{i}}}/\sqrt{d_{i}}$
(the remaining basis elements are then pairwise orthogonal hermitian traceless
operators of norm one). Since now $C^{\mathrm{can}}$ stands for the
correlation matrix in canonical bases. One proves GGD1 ) the following

###### Theorem 1

If $\rho$ is separable, then

$\|D_{x}^{(1)}C^{{\rm
can}}D_{y}^{(2)}\|_{1}\leq\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}},$ (2)

where

$D_{x}^{(1)}=\mathrm{diag}\left\\{x,1,\dots,1\right\\},\qquad
D_{y}^{(2)}=\mathrm{diag}\left\\{y,1,\dots,1\right\\}$ (3)

and

$\mathcal{N}_{x,d_{1}}=\sqrt{\frac{d_{1}-1+x^{2}}{d_{1}}},\,\ \
\mathcal{N}_{y,d_{2}}=\sqrt{\frac{d_{2}-1+y^{2}}{d_{2}}},$ (4)

for arbitrary $x,y\geq 0$.

Interestingly, for $(x,y)=(1,1)$ the above criterion reduces to CCNR.
Moreover, for $(x,y)=(0,0)$, $(x,y)=\sqrt{2/d_{1}},\sqrt{2/d_{2}})$, and
$(x,y)=(\sqrt{d_{1}+1},\sqrt{d_{2}+1})$ one recovers separability criteria
developed in Vicente , Fei  and GUHNE , respectively. The last criterion GUHNE
was constructed in terms of SIC POMVs (ESIC criterion) and it was conjectured
that it is stronger than original CCNR criterion. In this paper we provide a
comparative analysis of CCNR, ESIC and dV criteria for a class of bipartite
states being a generalization of well known isotropic states

$\displaystyle\rho_{p}$ $\displaystyle=\frac{1-p}{d_{1}d_{2}}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}\otimes{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}+p|\psi^{+}_{d_{1}}\rangle\langle\psi^{+}_{d_{1}}|,$
(5)

where $|\psi^{+}_{d_{1}}\rangle=1/\sqrt{d_{1}}\sum_{i=1}^{d_{1}}|e_{i}\otimes
f_{i}\rangle$, $|e_{i}\rangle$ defines orthonormal basis in
$\mathbb{C}^{d_{1}}$ and $|f_{i}\rangle$ defines orthonormal set in
$\mathbb{C}^{d_{2}}$. It is well known GT  that this state is separable if and
only if it is PPT which is equivalent to

$p\leq\frac{1}{d_{2}+1}.$ (6)

In the paper we provide the upper bound for ‘$p$’ implied by condition (2).
Moreover, we show that ESIC criterion detects more entangled state than the
standard CNNR criterion. This supports the conjecture made in GUHNE  that ESIC
is stronger than CCNR. Finally, we analyze so called enhanced realignment
criterion ZZZ  which states that for separable states

$\|\mathcal{R}(\rho_{AB})\|_{1}\leq\sqrt{1-{\rm Tr}\rho_{A}^{2}}\sqrt{1-{\rm
Tr}\rho_{B}^{2}},$ (7)

with $\rho_{A}={\rm Tr}_{B}\rho_{AB}$ and $\rho_{B}={\rm Tr}_{A}\rho_{AB}$
being marginal states. Enhanced realignment criterion turns out to be the
strongest effectively computable simplification of Correlation Matrix
CriterionCOV-1 ; COV-2 ; COV-3  (see also COV-U  for the unifying approach).
In the recent papers GGD2  we showed that this criterion is equivalent to (2)
for all $x,y\geq 0$. Here we show intricate relation of (7) and (2) using a
family of isotropic states.

##  II  $XY$-criterion for isotropic states

The main result of our paper consists in the following

###### Theorem 2

If an isotropic state $\rho_{p}$ is separable, then $p\leq p_{xy}$, where

$\displaystyle p_{xy}$
$\displaystyle=\Gamma\frac{\sqrt{\left(1+\tilde{x}\right)\left(1+\tilde{y}\right)}-\sqrt{\tilde{x}\left(\left(1+\gamma\right)\tilde{y}+\gamma\tilde{x}+\gamma\right)}}{1-\gamma\tilde{x}}$
(8)

with

$\displaystyle\tilde{x}=\frac{x^{2}}{d_{1}-1},\qquad\tilde{y}=\frac{y^{2}}{d_{2}-1},$
(9) $\displaystyle\gamma=$
$\displaystyle\frac{\left(d_{2}-d_{1}\right)}{d_{2}\left(d_{1}-1\right)\left(d_{1}+1\right)^{2}},\quad\Gamma=\frac{d_{1}}{d_{1}^{2}-1}\frac{\sqrt{d_{1}-1}\sqrt{d_{2}-1}}{\sqrt{d_{1}d_{2}}}$

for arbitrary $x,y\geq 0$.

For the proof cf. Appendix. Note, that for $x=y$ and $d_{1}=d_{2}=d$ one finds

$p_{x,x}=\frac{1}{d+1},$ (10)

and hence one recovers the PPT condition (6). In general, however,

$p_{x,y}>\frac{1}{d_{2}+1},$ (11)

and hence this criterion is weaker than PPT condition. In particular, we have
the following bounds for de Vicente criterion ($x=y=0$), realignment criterion
$x=y=1$, and for ESIC criterion ($x=\sqrt{d_{1}+1}$, $y=\sqrt{d_{2}+1}$):

$\displaystyle p_{dV}$
$\displaystyle=\frac{d_{1}}{d_{1}^{2}-1}\frac{\sqrt{d_{1}-1}\sqrt{d_{2}-1}}{\sqrt{d_{1}d_{2}}}$
(12) $\displaystyle p_{R}$
$\displaystyle=\frac{\left(d_{1}^{2}-1\right)d_{2}-\sqrt{d_{1}^{3}d_{2}-3d_{1}d_{2}+d_{2}^{2}+1}}{d_{2}d_{1}^{3}-2d_{1}d_{2}+1}$
(13) $\displaystyle p_{E}$
$\displaystyle=\frac{2\left(d_{1}-1\right)d_{2}-\sqrt{\frac{d_{1}^{3}d_{2}^{2}-2d_{1}d_{2}^{2}+3d_{2}^{2}+\left(d_{1}^{3}-5d_{1}\right)d_{2}+d_{1}+1}{d_{1}+1}}}{d_{1}^{2}d_{2}-d_{1}d_{2}-d_{2}+1}.$
(14)

We skip the expression $p_{F}$ for the criterion from Fei
($x=\sqrt{2/d_{1}}$,$y=\sqrt{2/d_{2}}$) since it is quite complicated. Again,
for $d_{1}=d_{2}$ one has

$p_{dV}=p_{R}=p_{E}=\frac{1}{d+1}.$

###### Proposition 1

For $d_{1}\neq d_{2}$ one has

$p_{E}<p_{R},$ (15)

that is , ESIC criterion detects more entangled isotropic state that
realignment.

For the proof cf. Appendix.

##  III Comparison with Enhanced Realignment Criterion

Separability of $\rho_{p}$ implies $p\leq p_{xy}$. Clearly, $p_{xy}$ depends
on $(x,y)$ and hence the most efficient criterion corresponds to minimal value
of $p_{xy}$. Let us calculate the minimum of the expression (8). One has

$\partial_{\tilde{x}}p_{xy}=0\iff(1+\gamma)\tilde{y}=\tilde{x}-\gamma.$ (16)

This is a necessary condition for minimum. One can check, that substituting
(16) to (8) a constant value $\Gamma/\sqrt{1+\gamma}$ is obtained, hence we
have the whole line $(1+\gamma)\tilde{y}=\tilde{x}-\gamma$ (hyperbola in
$x$,$y$) of minima of $p_{xy}$. One can summarise the above observations in
the following

###### Theorem 3

The minimum of $p_{xy}$ is attained in points of the hyperbola:

$\frac{x^{2}}{d_{1}-1}-(1+\gamma)\frac{y^{2}}{d_{2}-1}=\gamma$ (17)

and the value of the minimum reads as follows

$p_{\rm
min}=\frac{\Gamma}{\sqrt{1+\gamma}}=\sqrt{\frac{d_{2}-1}{d_{2}(d_{1}^{2}+d_{1}-1)-1}}.$
(18)

The enhanced realignment criterion states that if $\rho_{p}$ is separable,
then

$\|\mathcal{R}(\rho_{p}-\rho_{1}\otimes\rho_{2})\|_{1}\leq\sqrt{1-{\rm
Tr}\rho_{1}^{2}}\sqrt{1-{\rm Tr}\rho_{2}^{2}},$ (19)

where

$\rho_{1}=\frac{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}}{d_{1}},\qquad\rho_{2}=\frac{1-p}{d_{1}}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}+\frac{p}{d_{1}}\sum_{i=1}^{d_{1}}|f_{i}\rangle\langle
f_{i}|$ (20)

are local states in the subsystems.

###### Theorem 4

$\rho_{p}$ satisfies (19) if and only if

$p\leq p_{\rm ER},$ (21)

where

$p_{\mathrm{ER}}=\sqrt{\frac{d_{2}-1}{d_{2}\left(d_{1}^{2}+d_{1}-1\right)-1}}.$
(22)

For the proof cf. Appendix. Hence, the enhanced realignment criterion is
equivalent to the to the family of $XY$-criteria minimising $p_{xy}$.
Interestingly, in GGD2  we proved that this equivalence is always realized for
large values of $x$ and $y$, but for isotropic states in Eq. (5) it is enough
consider $x,y$ belonging to the hyperbola in Eq. (17). Indeed, the figure 1
illustrates the hyperbola of minima of $p_{xy}$ (reducing to line if
dimensions are equal) and four characteristic points representing the four
criteria distinguished in the literature.

(a) $\left(\mathrm{d_{1}=d_{2}}=3\right)$

(b) $\left(\mathrm{d_{1}=d_{2}}=3\right)$

(c) $\left(\mathrm{d_{1}=2;\>}d_{2}=20\right)$

(d) $\left(\mathrm{d_{1}=2;\>}d_{2}=20\right)$

Figure 1: Contour plot with the gradient direction and plot of the threshold
function $p_{xy}$ for $\left(\mathrm{d_{1}}=\mathrm{d_{2}}=3\right)$ and
$\left(\mathrm{d_{1}=2;}d_{2}=20\right)$. The colored points are in
$\left(0,0\right)$ in red for Di Vincente criterion; in $\left(1,1\right)$ in
blue for CCNR; in $\left(2/\sqrt{d_{1}},2/\sqrt{d_{2}}\right)$ in green for
criterion in Fei ; $\left(\sqrt{d_{1}+1},\sqrt{d_{2}+1}\right)$ in black for
ESIC. The hyperbola in blue form Eq.(17). The white line in Fig 1c is for
$a=0$ of Eq. (39).

##  IV Conclusions

We provided a comparative analysis of the detection power of a $XY$-family of
separability criteria in the case of generalized isotropic state in
$d_{1}\otimes d_{2}$. Due to the high symmetry of the isotropic state one can
derive analytical formula for the separability bound $p_{xy}$ which for
$d_{1}\neq d_{2}$ is always higher than a PPT bound
$p_{\mathrm{PPT}}=1/(d_{2}+1)$. Interestingly, minimising over $(x,y)$ we
showed that the most efficient $p_{\rm min}$ is exactly the same as the one
derived in term of so called enhanced realignment criterion. Finally, it is
shown that for a family of generalized isotropic states the ESIC criterion
from GUHNE  detects more entangled state than original CCNR criterion. Hence,
it supports a conjecture raised in GUHNE  that ESIC is stronger than CCNR. It
is clear that a similar analysis can be performed for a Werner-like state
Werner

$\rho_{q}=\frac{1-q}{d_{1}d_{2}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}\otimes{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}+\frac{q}{d_{1}}\sum_{i,j=1}^{d_{1}}|e_{i}\rangle\langle
e_{j}|\otimes|f_{j}\rangle\langle f_{i}|.$ (23)

As a resume of this simple analysis, which provides a clear illustration of
the detection power for a family of separability criteria based on correlation
tensor, on figure is shown the general trend for large value of $d_{1}$ and
$d_{2}$.

Figure 2: The difference of the thresholds which realize the equality in Eqs.
30-33 and 22 ($p_{dV}-p_{ER}$ in blue, $p_{E}-p_{ER}$ in red, $p_{F}-p_{ER}$
in green, $p_{R}-p_{ER}$ in orange) varying from $d_{1}=2$ to $d_{2}=100$.
This is an asymptotic pattern for high values of $d_{1},d_{2}$. For low value
of $d_{1},d_{2}$ the pattern is not respected. In particular for low values of
$d_{1},d_{2}$ $p_{E}\leq p_{dV}$ and $p_{R}\leq p_{F}$. For $d_{1}=2,d_{2}=3$
also $p_{R}\leq p_{dV}$.

###### Acknowledgements.

DC and GSa were supported by the Polish National Science Centre project
2018/30/A/ST2/00837. GSc is supported by Istituto Nazionale di Fisica Nucleare
(INFN) through the project “QUANTUM”. We acknowledge the Toruń
Astrophysics/Physics Summer Program TAPS 2018 and the project PROM at the
Nicolaus Copernicus University. GSc thanks S. Pascazio, P. Facchi and F. V.
Pepe for invaluable human and scientific support, for suggestions and
encouragements which led to the realization of the present work.

##  Appendix A Proof of Theorem 2

Introducing a vectorization of an operator Watrous ; Gilchrist
$A=\sum_{i,j}A_{ij}|i\rangle\langle j|$ via
$|A\rangle=\sum_{i,j}A_{ij}|i\rangle\otimes|j\rangle$ one has
$\mathcal{R}(A\otimes B)=|A\rangle\langle B^{*}|$, where the complex
conjugation is taken w.r.t. the basis used for the vectorization. The
resulting matrix $\mathcal{R}(\rho_{p})$ is the correlation tensor
$C(\rho_{p})$ for a choice of bases: $\\{|e_{i}\rangle\langle
e_{j}|\\}_{i,j=1}^{d_{1}}\subset\mathcal{B}(\mathbb{C}^{d_{1}})$ and
$\\{|f_{i}\rangle\langle
f_{j}|\\}_{i,j=1}^{d_{2}}\subset\mathcal{B}(\mathbb{C}^{d_{2}})$. These bases
are orthonormal, but not hermitian, hence the matrix $C$ can have complex
entries, but its singular values and trace norm are the same as the case we
choose hermitian orthonormal bases. One has

$\displaystyle C=$
$\displaystyle\mathcal{R}(\rho_{p})=(1-p)\frac{1}{d_{1}d_{2}}\sum_{i=1}^{d_{1}}\sum_{j=1}^{d_{2}}|e_{i}\otimes
e_{i}\rangle\langle f_{j}\otimes f_{j}|$
$\displaystyle\qquad+\frac{p}{d_{1}}\sum_{i,j=1}^{d_{1}}|e_{i}\otimes
e_{j}\rangle\langle f_{i}\otimes f_{j}|$ (24) $\displaystyle=$
$\displaystyle(1-p)\left|\frac{{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}}{d_{1}}\right\rangle\left\langle\frac{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}}{d_{2}}\right|+\frac{p}{d_{1}}\sum_{i,j=1}^{d_{1}}|e_{i}\otimes
e_{j}\rangle\langle f_{i}\otimes f_{j}|,$

where we used

$|{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}\rangle=\sum_{i=1}^{d_{1}}|e_{i}\otimes e_{i}\rangle\
,\ \ |{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}\rangle=\sum_{i=1}^{d_{2}}|f_{i}\otimes f_{i}\rangle.$

To recast $D_{x}^{\left(1\right)},D_{y}^{\left(2\right)}$ in the not hermitian
basis we sandwich the above matrix with

$\mathbb{D}^{(1)}_{x}={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{d_{1}}\otimes{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}+\frac{x-1}{d_{1}}|{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}\rangle\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}|,$

and

$\mathbb{D}^{(2)}_{y}={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{d_{2}}\otimes{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}+\frac{y-1}{d_{2}}|{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}\rangle\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}|.$

In other words, $D_{x}^{\left(1\right)}$ is obtained replacing the first entry
$1$ with $x$ using the projector $|{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}\rangle\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}|/d_{1}$. Analogously for
$D_{y}^{\left(2\right)}$. Then matrix $C_{xy}$ follows

$\displaystyle C_{xy}$ $\displaystyle=$ $\displaystyle\left({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}\otimes{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}+\frac{x-1}{d_{1}}|{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}\rangle\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}|\right)\Big{(}(1-p)\left|\frac{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}}{d_{1}}\right\rangle\left\langle\frac{{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}}{d_{2}}\right|+\frac{p}{d_{1}}\sum_{i,j=1}^{d_{1}}|e_{i}\otimes
e_{j}\rangle\langle f_{i}\otimes f_{j}|\Big{)}\left({\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}\otimes{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}+\frac{y-1}{d_{2}}|{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}\rangle\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}|\right)$ (25) $\displaystyle=$
$\displaystyle\frac{\left(y-p\right)x}{d_{1}d_{2}}\left|{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}\right\rangle\left\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{2}}}\right|+\frac{p}{d_{1}}\sum_{i,j=1}^{d_{1}}|e_{i}\otimes
e_{j}\rangle\langle f_{i}\otimes
f_{j}|+p\frac{\left(x-1\right)}{d_{1}^{2}}\sum_{i,j=1}^{d_{1}}|e_{i}\otimes
e_{i}\rangle\langle f_{j}\otimes f_{j}|.$

Note that

$\displaystyle\mathbb{D}^{(1)}_{x}C\mathbb{D}^{(2)}_{y}=$ $\displaystyle
U{D}^{(1)}_{x}U^{\dagger}CVD^{(2)}_{y}V^{\dagger}$ $\displaystyle=$
$\displaystyle U{D}^{(1)}_{x}C^{\rm can}D^{(2)}_{y}V^{\dagger}$

where $C^{\rm can}=U^{\dagger}CV,$ defines the canonical correlation tensor.
Hence, one has

$\|C_{xy}\|_{1}=\|{D}^{(1)}_{x}C^{\rm can}D^{(2)}_{y}\|_{1}.$ (26)

To compute the trace-norm of $C_{xy}$ note that

$\displaystyle C_{xy}C_{xy}^{\dagger}=$
$\displaystyle\Bigg{(}\frac{x^{2}\left(y^{2}-p^{2}\right)}{d_{1}^{2}d_{2}}+\frac{p^{2}\left(x^{2}-1\right)}{d_{1}^{3}}\Bigg{)}\left|\bm{1}_{\mathrm{d_{1}}}\right\rangle\left\langle\bm{1}_{\mathrm{d_{1}}}\right|$
$\displaystyle+\frac{p^{2}}{d_{1}^{2}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}\otimes{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{d_{2}}.$ (27)

Now, since
$\left|\bm{1}_{\mathrm{d_{1}}}\right\rangle\left\langle\bm{1}_{\mathrm{d_{1}}}\right|$
commutes with ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}{d_{1}}\otimes{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{2}}$, they share the same set of eigenvectors, therefore
the spectrum $\sigma$ (expressed with the geometric multiplicity of the
eigenvalues) reads

$\displaystyle\sigma\left(C_{xy}C_{xy}^{\dagger}\right)=$
$\displaystyle\left(d_{1}^{2}-1\right)\times\left\\{\frac{p^{2}}{d_{1}^{2}}\right\\}$
$\displaystyle\cup$
$\displaystyle\Bigg{\\{}\frac{x^{2}}{d_{1}d_{2}}\left(y^{2}+p^{2}\frac{d_{2}-d_{1}}{d_{1}}\right)\Bigg{\\}}.$
(28)

Finally we have that separability of an isotropic state with unequal dimension
$\rho_{p}$ implies

$\displaystyle\|C_{xy}\|_{1}=$
$\displaystyle\frac{d_{1}^{2}-1}{d_{1}}p+\frac{x}{\sqrt{d_{1}d_{2}}}\sqrt{y^{2}+p^{2}\frac{d_{2}-d_{1}}{d_{1}}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\frac{d_{1}-1}{d_{1}}+\frac{x^{2}}{d_{1}}}\sqrt{\frac{d_{2}-1}{d_{2}}+\frac{y^{2}}{d_{2}}}.$
(29)

In particular, for $x,y=0$, $x,y=1$, $x=\sqrt{2/d_{1}},y=\sqrt{2/d_{2}}$ and
$x=\sqrt{d_{1}+1},y=\sqrt{d_{2}+1}$, the above condition implies

$\displaystyle
p\frac{d_{1}^{2}-1}{d_{1}}\leq\sqrt{\frac{d_{1}-1}{d_{1}}}\sqrt{\frac{d_{2}-1}{d_{2}}},$
(30)
$\displaystyle\frac{d_{1}^{2}-1}{d_{1}}p+\frac{1}{\sqrt{d_{1}d_{2}}}\sqrt{1+p^{2}\left(\frac{d_{2}}{d_{1}}-1\right)}\leq
1,$ (31)
$\displaystyle\frac{d_{2}\left(d_{1}^{2}-1\right)p+\sqrt{4+2p^{2}\left(\frac{d_{2}}{d_{1}}-1\right)d_{2}}}{\sqrt{d_{1}^{2}-d_{1}+2}\sqrt{d_{2}^{2}-d_{2}+2}}\leq
1,$ (32)
$\displaystyle\frac{d_{1}^{2}-1}{d_{1}}p+\sqrt{\frac{d_{1}+1}{d_{1}d_{2}}}\sqrt{d_{2}+1+p^{2}\left(\frac{d_{2}}{d_{1}}-1\right)}\leq
2.$ (33)

If the state $\rho_{p}$ is separable, then

$\displaystyle 0$
$\displaystyle\geq\frac{d_{1}^{2}-1}{d_{1}}p+\frac{x}{\sqrt{d_{1}d_{2}}}\sqrt{y^{2}+p^{2}\frac{d_{2}-d_{1}}{d_{1}}}-\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}}$
(34)

which obviously implies a weaker condition

$\displaystyle 0$
$\displaystyle\geq\frac{d_{1}^{2}-1}{d_{1}}p-\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}}.$
(35)

Let

$p_{0}:=\frac{d_{1}}{d_{1}^{2}-1}\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}}.$
(36)

It is clear that if $p>p_{0}$, then a weaker condition (35) detects
entanglement of $\rho_{p}$. If $p\leq p_{0}$, then we can use a tighter
inequality (34)

$-\frac{x}{\sqrt{d_{1}d_{2}}}\sqrt{y^{2}+p^{2}\frac{d_{2}-d_{1}}{d_{1}}}\geq\frac{d_{1}^{2}-1}{d_{1}}p-\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}}.$
(37)

Under the assumption that we do not detect entanglement with the weaker
condition (35), both the terms of the inequality are negative, hence we can
square and inverse the inequality:

$\displaystyle\mathcal{F}\left(p\right)$
$\displaystyle=\left(\frac{d_{1}^{2}-1}{d_{1}}p-\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}}\right)^{2}$
$\displaystyle-\frac{x^{2}}{d_{1}d_{2}}\left(y^{2}+p^{2}\frac{d_{2}-d_{1}}{d_{1}}\right)$
$\displaystyle=a_{d_{1},d_{2}}\left(x\right)p^{2}+b_{d_{1},d_{2}}\left(x,y\right)p+c_{d_{1},d_{2}}\left(x,y\right)\geq
0,$ (38)

where

$\displaystyle a\equiv a_{d_{1},d_{2}}\left(x\right)=$
$\displaystyle\frac{\left(d_{1}^{2}-1\right)^{2}}{d_{1}^{2}}-x^{2}\frac{d_{2}-d_{1}}{d_{1}^{2}d_{2}},$
(39) $\displaystyle b\equiv b_{d_{1},d_{2}}\left(x,y\right)=$
$\displaystyle-2\frac{d_{1}^{2}-1}{d_{1}}\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}},$
(40) $\displaystyle c\equiv c_{d_{1},d_{2}}\left(x,y\right)=$
$\displaystyle\left(\mathcal{N}_{x,d_{1}}\mathcal{N}_{y,d_{2}}\right)^{2}-\frac{x^{2}y^{2}}{d_{1}d_{2}}.$
(41)

A direct calculation shows always $\Delta=b^{2}-4ac\geq 0$. Let
$p_{\pm}=\frac{-b\pm\sqrt{\Delta}}{2a}$ be the roots of
$\mathcal{F}\left(p\right)$ . Notice, that $\mathcal{F}\left(p_{0}\right)\leq
0$ (equality only for $x=0$), $c\geq 0$, $b<0$. Hence Vieta’s formulas implies

  1. 1.

$a>0\Longrightarrow p_{xy}^{\left(+\right)}>p_{0}>p_{xy}^{\left(-\right)}>0$

  2. 2.

$a<0\Longrightarrow p_{0}>p_{xy}^{\left(-\right)}>0>p_{xy}^{\left(+\right)}$.

In both cases the solution of the inequality (A) reads as $p\in[0,p_{-}]$ and
due to the continuity of $\mathcal{F}$ the limit formula for $a=0$ agrees with
the solution of the linear inequality.

##  Appendix B Proof of Proposition 1

The thresholds $p_{E}$ and $p_{R}$ are defined as the lowest roots of the
quadratic polynomials:

$\displaystyle
2(d_{1}^{3}d_{2}-2d_{1}d_{2}+1)p_{R}^{2}-4d_{2}(d_{1}^{2}-1)p_{R}$
$\displaystyle+2(d_{1}d_{2}-1)\stackrel{{\scriptstyle df}}{{=}}f_{R}(p_{R})=0$
(42)
$\displaystyle(d_{1}^{3}d_{2}-2d_{1}d_{2}+d_{1}-d_{2}+1)p_{E}^{2}-4d_{2}(d_{1}^{2}-1)p_{E}$
$\displaystyle+(3d_{1}d_{2}-d_{1}-d_{2}-1)\stackrel{{\scriptstyle
df}}{{=}}f_{E}(p_{E})=0$ (43)

We will prove the relation between roots $p_{E}\leq p_{R}$ by showing that all
roots of $f_{R}$ and $f_{E}$ are in a set: $\\{x:f_{E}(x)<f_{R}(x)\\}$. To
show it let us calculate the difference $f_{E}-f_{R}$:

$\displaystyle(f_{E}-f_{R})(x)=-(d_{1}^{3}d_{2}-2d_{1}d_{2}+d_{2}-d_{1}+1)x^{2}$
$\displaystyle+(d_{1}-1)(d_{2}-1)$ (44)

It is positive in the range $[-x_{0},x_{0}]$, where:

$\displaystyle x_{0}=\sqrt{\frac{d_{2}-1}{(d_{1}+1)d_{1}d_{2}-(d_{2}+1)}}.$
(45)

We will show that $f_{R}$ and $f_{E}$ are positive in the above range, showing
that

  1. 1.

$f_{R}(x_{0})=f_{E}(x_{0})$,

  2. 2.

$f_{R}$ is descending in $x_{0}$.

One has

$\displaystyle f_{R}(x_{0})=$ $\displaystyle f_{E}(x_{0})$ $\displaystyle=$
$\displaystyle
d_{2}\frac{2d_{1}^{3}d_{2}+d_{1}^{2}d_{2}-3d_{1}d_{2}-d_{1}^{3}-d_{1}^{2}+2}{(d_{1}+1)d_{1}d_{2}-(d_{2}+1)}$
$\displaystyle-$ $\displaystyle
2d_{2}(d_{1}^{2}-1)\sqrt{\frac{d_{2}-1}{(d_{1}+1)d_{1}d_{2}-(d_{2}+1)}}.$ (46)

Hence, we want to prove that

$\displaystyle
2d_{1}^{3}d_{2}+d_{1}^{2}d_{2}-3d_{1}d_{2}-d_{1}^{3}-d_{1}^{2}+2$
$\displaystyle-2(d_{1}^{2}-1)\sqrt{(d_{2}-1)((d_{1}+1)d_{1}d_{2}-(d_{2}+1))}\geq
0.$ (47)

To do this, we will rewrite the above as:

$\displaystyle(d_{1}^{2}-1)+d_{1}d_{-}\frac{2d_{1}+3}{2d_{1}+2}\geq$
$\displaystyle\sqrt{d_{-}^{2}(d_{1}^{2}+d_{1}-1)+d_{-}d_{1}(2d_{1}+3)(d_{1}-1)+(d_{1}^{2}-1)^{2}},$

where $d_{-}=d_{2}-d_{1}$. After squaring the latter simplifies to:

$d_{1}^{2}d_{-}^{2}(2d_{1}+3)^{2}\geq
4d_{-}^{2}(d_{1}+1)^{2}(d_{1}^{2}+d_{1}-1),$

what finally gives:

$d_{-}^{2}(d_{1}^{2}+4d_{1}+4)=(d_{2}-d_{1})^{2}(d_{1}+2)^{2}\geq 0.$

We prove the second property showing, that the minimum of $f_{R}$ is greater
than $x_{0}$:

$\frac{d_{2}(d_{1}^{2}-1)}{d_{2}d_{1}^{3}-2d_{1}d_{2}+1}>\sqrt{\frac{d_{2}-1}{d_{1}^{2}d_{2}+d_{1}d_{2}-d_{2}-1}}$

While $d_{1}\leq d_{2}$, we can estimate the RHS from above by $1/(d_{1}+1)$
and prove that the inequality holds for the estimation. The latter reduces to:

$0<d_{2}(d_{1}^{2}-1)(d_{1}+1)-(d_{2}d_{1}^{3}-2d_{1}d_{2}+1)=d_{2}d_{1}^{2}-d_{2}-1$

and holds for $d_{1},d_{2}\geq 2$.

One has $f_{E}\geq f_{R}\geq 0$ in $[-x_{0},x_{0}]$ and $f_{E}<f_{R}$ for
$x>x_{0}$. The threshold $p_{E}$ and $p_{R}$ are roots on the left of the
vertices of the parables $f_{E}$ and $f_{R}$ in Eq. (42) and (43)
respectively. Both $p_{E}$ and $p_{R}$ are obviously greater than $x_{0}$
where $f_{E}<f_{R}$. It implies that $f_{E}$ reaches $0$ first and hence
$p_{E}<p_{R}$.

##  Appendix C Proof of Theorem 3

One finds

$\displaystyle\rho_{p}-\rho_{1}\otimes\rho_{2}$ $\displaystyle=$
$\displaystyle=\frac{p}{d_{1}}\sum_{i,j=1}^{d_{1}}$
$\displaystyle\left|e_{i}\right\rangle\left\langle
e_{j}\right|\otimes\left|f_{i}\right\rangle\left\langle
f_{j}\right|-\frac{p}{d_{1}^{2}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{d_{1}}\otimes\sum_{i=1}^{d_{1}}|f_{i}\rangle\langle f_{i}|,$
(48)

and using again a vectorization technique one obtains

$\displaystyle C_{\mathrm{ER}}\equiv$ $\displaystyle
C\left(\rho_{p}-\rho_{1}\otimes\rho_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{p}{d_{1}}\sum_{i,j=1}^{d_{1}}|e_{i}\otimes
e_{j}\rangle\langle f_{i}\otimes
f_{j}|-\frac{p}{d_{1}^{2}}\left|{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}\right\rangle\left\langle{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathrm{d_{1}}}\right|.$ (49)

$C_{\mathrm{ER}}$ stands for enhanced realignment correlation matrix and the
spectrum of $C_{\mathrm{ER}}$$C_{\mathrm{ER}}^{\dagger}$ reads

$\sigma\left(C_{\mathrm{ER}}C_{\mathrm{ER}}^{\dagger}\right)=\left\\{\frac{p^{2}}{d_{1}^{2}}\right\\}\times\left(d_{1}^{2}-1\right)\cup\left\\{0\right\\}.$
(50)

This brings to the condition

$\frac{d_{1}^{2}-1}{d_{1}}p\leq\sqrt{\frac{d_{1}-1}{d_{1}}}\sqrt{\frac{d_{2}-1}{d_{2}}-p^{2}\left(\frac{d_{2}-d_{1}}{d_{1}d_{2}}\right)}.$
(51)

The equality holds for

$p_{\mathrm{ER}}=\sqrt{\frac{d_{2}-1}{d_{2}\left(d_{1}^{2}+d_{1}-1\right)-1}}.$
(52)

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