# Bipartite entanglement and the arrow of time

Markus Frembs<EMAIL_ADDRESS>Centre for Quantum Dynamics, Griffith
University,
Yugambeh Country, Gold Coast, QLD 4222, Australia

###### Abstract

We provide a new perspective on the close relationship between entanglement
and time. Our main focus is on bipartite entanglement, where this connection
is foreshadowed both in the positive partial transpose criterion due to Peres
[A. Peres, Phys. Rev. Lett., 77, 1413 (1996)] and in the classification of
quantum within more general non-signalling bipartite correlations [M. Frembs
and A. Döring, http://arxiv.org/abs/2204.11471]. Extracting the relevant
common features, we identify a necessary and sufficient condition for
bipartite entanglement in terms of a compatibility condition with respect to
time orientations in local observable algebras, which express the dynamics in
the respective subsystems. We discuss the relevance of the latter in the
broader context of von Neumann algebras and the thermodynamical notion of time
naturally arising within the latter.

##  I Introduction

The connection between quantum entanglement and the arrow of time has been the
subject of numerous research enterprises, some recent ones include
PageWootters1983 ; JenningsRudolph2010 ; LinMarcolliOoguriStoica2015 ;
Susskind2016 ; Racorean2019 . Here, we mainly focus on bipartite entanglement.
It has been surmised that the operation of partial transposition in the
positive partial transpose criterion for bipartite entanglement due to Peres
Peres1996  (henceforth referred to as the ‘PPT criterion’111Sometimes the
criterion is also referred to as the Peres-Horodecki criterion Peres1996 ;
Horodeckisz1996 ) is related to time reversal Horodeckisz2009 . Yet, this
relationship seems not to have been made precise before.

A related area of research, where time unexpectedly enters the picture, is the
classification of quantum from non-signalling bipartite correlations. More
precisely, the present author and Andreas Döring have recently shown that
quantum states are characterised by a compatibility condition with respect to
time orientations—roughly, the unitary evolution—in local observable algebras
FrembsDoering2022a .

We review the basics of and extract some key insights from these results in
the following subsections. Building on those, in Sec. II.1 we prove a
necessary and sufficient criterion for bipartite entanglement. In Sec. II.2 we
show that this, too, can be recast as a compatibility condition with respect
to time orientations. Our work opens up various directions for future
research, including practical considerations of our entanglement criterion as
well as its generalisation to von Neumann algebras ConnesRovelli1994  (see
Sec. II.3 and Sec. III).

###  I.1 The PPT criterion

Peres noted that the operation of partial transposition transforms separable
states into separable states. In turn, any bipartite quantum state222We will
identify states $\sigma\in\mathcal{S}(\mathcal{A})$ on $\mathcal{A}$ with
their respective density matrices via $\sigma(a)=\mathrm{tr}[\rho a]$ for all
$a\in\mathcal{A}$.
$\rho=\sum_{ij}c_{ij}\rho_{\mathcal{A},i}\otimes\rho_{\mathcal{B},j}\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$
with $c_{ij}\in\mathbb{C}$, $\rho_{\mathcal{A},i}\in\mathcal{S}(\mathcal{A})$,
$\rho_{\mathcal{B},j}\in\mathcal{S}(\mathcal{B})$, whose partial transpose
$\rho^{T_{\mathcal{A}}}:=\sum_{ij}c_{ij}\rho^{T}_{\mathcal{A},i}\otimes\rho_{\mathcal{B},j}$
has at least one negative eigenvalue, is necessarily entangled Peres1996 .
Throughout, we write $\mathcal{A}=\mathcal{L}(\mathcal{H}_{\mathcal{A}})$ for
$\mathrm{dim}(\mathcal{H}_{\mathcal{A}})$ finite. The partial-transpose
criterion is necessary and sufficient in low dimensions,
$\mathrm{dim}(\mathcal{H}_{\mathcal{A}})=2$ and
$\mathrm{dim}(\mathcal{H}_{\mathcal{B}})=2,3$, but is merely sufficient in
higher dimensions Horodeckisz1996 . Driven mainly by practical considerations
the result has been sharpened in various ways (see Horodeckisz2009  and
references therein). This development, while rich and still active, has
overshadowed the physical significance of Peres’ insight. In contrast, here we
will only be concerned with the conceptual importance, leaving the practical
value of our work for future study.

Pure and purified mixed states. We recall the following simple fact.

######  Proposition 1.

The PPT criterion is necessary and sufficient for pure bipartite states.

###### Proof.

Let $|\psi\rangle\in\mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}$
and consider the Schmidt decomposition
$|\psi\rangle=\sum_{i}\alpha_{i}|ii\rangle$, $\alpha_{i}\in\mathbb{R}_{+}$,
with density matrix
$\rho_{\psi}=|\psi\rangle\langle\psi|=\sum_{ij}\alpha_{i}\alpha_{j}|i\rangle\langle
j|\otimes|i\rangle\langle j|\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$.
Note that $\rho_{\psi}$ is separable if and only if the sum in the Schmidt
decomposition collapses to a single term. Applying partial transposition on
system $\mathcal{A}$ we obtain
$\rho^{T_{\mathcal{A}}}_{\psi}=\sum_{ij}\alpha_{i}\alpha_{j}|j\rangle\langle
i|\otimes|i\rangle\langle j|$. It is easy to see that this operator has a
negative eigenvalue for every pair of non-zero coefficients
$\alpha_{i},\alpha_{j}\neq 0$. ∎

This is of course well known. The following observation is equally
straightforward: we can apply Prop. 1 to any bipartite state by considering
purifications. What is more, a version of the Schrödinger-HJW theorem assures
independence of the choice of purification Schroedinger1936 ;
HughstonJoszaWootters1993  (see also Kirkpatrick2003 ). The PPT criterion thus
becomes necessary and sufficient with respect to purifications. Of course, for
pure states there are easier ways to check whether a state is entangled or
separable.333For instance, note that we have used the Schmidt rank in Prop.1.
Nevertheless, as we will see below the criterion works _because_ it works on
the level of purifications. This is best expressed in terms of channels.

Transposition vs Hermitian adjoint. Note that transposition reverses the order
of matrix multiplication: let $a\in M_{k\times l}(\mathbb{C})$, $b\in
M_{l\times m}(\mathbb{C})$, then

$((ab)_{ki})^{T}=(ab)_{ik}=\sum_{j=1}^{l}a_{ij}b_{jk}=\sum_{j=1}^{l}(b_{kj})^{T}(a_{ji})^{T}=(b^{T}a^{T})_{ki}\;.$

This fact is somewhat left implicit from the perspective of bipartite states
$\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$. In order to make it
explicit, we identify a bipartite state $\rho$ with its quantum channel
$\phi_{\rho}:\mathcal{A}\rightarrow\mathcal{B}$ under the Choi-Jamiołkowski
isomorphim Jamiolkowski1972 ; Choi1975  (see also, Frembs2022b ): recall that
every quantum channel $\phi:\mathcal{A}\rightarrow\mathcal{B}$, i.e., every
completely positive linear map, determines a bipartite state $\rho_{\phi}$ (up
to normalisation) by

$\rho_{\phi}=\sum_{ij}E_{ij}\otimes\phi(E_{ij})\;,$ (1)

where $E_{ij}$ is the matrix with $1$ in the entry $(i,j)$ and $0$ elsewhere.

Conversely, every bipartite state
$\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$ corresponds to the quantum
channel

$\phi_{\rho}(a)=\mathrm{tr}_{\mathcal{H}_{\mathcal{A}}}[\rho(a^{T}\otimes
1_{\mathcal{B}})]\;.$ (2)

Clearly, with respect to the choice of basis in Eq. (2) we have
$(E_{ij})^{T}=E_{ij}^{*}$. This allows us to replace transposition with the
(Hermitian) adjoint.

######  Lemma 1.

Let $\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$, let $\phi_{\rho}$ be
the map under the linear isomorphism in Eq. (2), and let
$(\Phi_{\rho},v,\mathcal{K})$ be a Stinespring dilation of $\phi_{\rho}$,
i.e., $\phi_{\rho}=v^{*}\Phi_{\rho}v$ with
$v:\mathcal{H}_{\mathcal{B}}\rightarrow\mathcal{K}$ linear and
$\Phi_{\rho}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ a
$C^{*}$-homomorphism Stinespring1955 . Then
$\phi_{\rho^{T_{\mathcal{A}}}}=\phi^{*}_{\rho}=v^{*}\Phi^{*}_{\rho}v$.666Notably,
this is different than so-called co-positive maps, i.e., maps
$\phi^{T}:=T_{\mathcal{B}}\circ\phi$, where
$\phi:\mathcal{A}\rightarrow\mathcal{B}$ is a completely positive map: by a
similar computation we obtain $\phi_{\rho^{T_{\mathcal{A}}}}=\phi^{T}_{\rho}$,
yet, $\phi^{T}_{\rho}\neq v^{*}\Phi^{T}_{\rho}v$ in general.777We remark that
$\phi^{*}:=*\circ\phi$ denotes the adjoint of the image of the channel $\phi$,
not its (Heisenberg) dual.

###### Proof.

We have

$\displaystyle\sum_{ij}E_{ij}\otimes\phi_{\rho^{T_{\mathcal{A}}}}(E_{ij})=\rho^{T_{\mathcal{A}}}=(\rho^{*})^{T_{\mathcal{A}}}=\sum_{ij}E_{ij}\otimes\phi^{*}_{\rho}(E_{ij})=\sum_{ij}E_{ij}\otimes
v^{*}\Phi^{*}_{\rho}(E_{ij})v\;,$

where we used Eq. (1) in the first and third step and $\rho^{*}=\rho$ in the
second. ∎

Partial transposition therefore assumes a more natural interpretation in terms
of the adjoint operation on the local system $\mathcal{B}$ (see also
Frembs2022b ). In Sec. II.3 we will see that this encodes a difference between
time orientations on the system $\mathcal{B}$. The latter also play a crucial
role in selecting quantum from more general non-signalling bipartite
correlations FrembsDoering2022a .

###  I.2 Quantum from non-signalling correlations

It is instructive to view the problem of entanglement classification from the
broader perspective of classifying quantum from non-signalling correlations.
In general, non-signalling distributions are far from being quantum
PopescuRohrlich1994 . Considering product quantum observables, a Gleason-type
argument restricts non-signalling bipartite correlations to normalised linear
functionals that are positive on pure tensors (POPT), yet not necessarily
positive KlayRandallFoulis1987 ; Wallach2002 ; BarnumEtAl2010 ;
FrembsDoering2022b . To further single out quantum correlations among the
latter, various additional physical principles have been proposed, see e.g.
Popescu2014 . While successful in some instances, none has been shown to
recover the quantum state space in general NavascuesGuryanovaHobanAcin2015 .

Recently, a classification of quantum states from more general non-signalling
bipartite correlations has been obtained: quantum states correspond with those
correlations, which satisfy (i) an extension of the no-signalling principle to
dilations, and (ii) a relative consistency condition between the canonical
unitary evolution in the respective subsystems (see Def. 1, Thm. 2 and Def. 2,
Thm. 3 in FrembsDoering2022a ). Correlations under (i) (but not necessarily
(ii)) correspond with decomposable maps under the Choi-Jamiołkowski
isomorphism in Eq. (2).888We remark that for
$\mathrm{dim}(\mathcal{H}_{\mathcal{A}})=2$ and
$\mathrm{dim}(\mathcal{H}_{\mathcal{B}})=2,3$ every positive map
$\phi:\mathcal{A}\rightarrow\mathcal{B}$ is decomposable Stormer1963 ;
Woronowicz1976 , which implies necessity of the PPT criterion in those
dimensions Horodeckisz1996  (see also Thm. 2 below).

Recall that a linear map $\phi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$
is decomposable if there exists a Hilbert space $\mathcal{K}$, a bounded
linear operator $v:\mathcal{H}\rightarrow\mathcal{K}$, and a Jordan
$*$-homomorphism $\Phi$, i.e.,
$\Phi(aa^{\prime}+a^{\prime}a)=\Phi(a)\Phi(a^{\prime})+\Phi(a^{\prime})\Phi(a)$
and $\Phi^{*}(a)=\Phi(a^{*})$ for all $a,a^{\prime}\in\mathcal{A}$ (for
details, see Sec. II.2), such that $\phi=v^{*}\Phi v$. Such maps are more
general than quantum channels
$\phi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$, which are of similar
form: $\phi=v^{*}\Phi v$ with $\Phi$ a $C^{*}$-homomorphism. By Stinespring’s
theorem Stinespring1955 , the latter is equivalent to $\phi$ being completely
positive: if $x_{ij}\in
M_{n}(\mathcal{A})_{+}=(M_{n}(\mathbb{C})\otimes\mathcal{A})_{+}$, then
$\phi(x_{ij}):=\mathrm{id}_{M_{n}(\mathbb{C})}\otimes\phi(x_{ij})\in
M_{n}(\mathcal{B}(\mathcal{H}))_{+}$. Similarly, decomposable maps can be
characterised by a weaker positivity condition Stormer1982 : if $x_{ij}\in
M_{n}(\mathcal{A})_{+}$ _and_ $x_{ji}\in M_{n}(\mathcal{A})_{+}$, then
$\phi(x_{ij})\in M_{n}(\mathcal{B}(\mathcal{H}))_{+}$. Let
$\mathcal{S}_{D}(\mathcal{A}\otimes\mathcal{B})$ denote the class of bipartite
states corresponding to decomposable maps under the Choi-Jamiołkowski
isomorphism. Interestingly, (the weaker positivity condition in)
$\mathcal{S}_{D}(\mathcal{A}\otimes\mathcal{B})$ is preserved under partial
transposition (see also Frembs2022b ).

######  Proposition 2.

Let $\rho\in\mathcal{S}_{D}(\mathcal{A}\otimes\mathcal{B})$, i.e., $\rho$
corresponds to a decomposable map under the Choi-Jamiołkowski isomorphism in
Eq. (1). Then
$\rho^{T_{\mathcal{A}}}\in\mathcal{S}_{D}(\mathcal{A}\otimes\mathcal{B})$.

###### Proof.

By a similar argument to the one in Lm. 1,
$\phi_{\rho^{T_{\mathcal{A}}}}=\phi^{*}_{\rho}=v^{*}\Phi^{*}_{\rho}v$, where
$\Phi_{\rho}$ is a Jordan $*$-homomorphism, hence,
$\Phi^{*}_{\rho}:=*\circ\Phi_{\rho}=\Phi_{\rho}\circ*$. But then so is
$\Phi^{*}_{\rho}$: for all $a_{1},a_{2}\in\mathcal{A}$,

$\Phi^{*}_{\rho}(\\{a_{1},a_{2}\\})=\Phi_{\rho}(\\{a_{1},a_{2}\\}^{*})=\Phi_{\rho}(\\{a_{1}^{*},a_{2}^{*}\\})=\\{\Phi_{\rho}(a_{1}^{*}),\Phi_{\rho}(a_{2}^{*})\\}=\\{\Phi^{*}_{\rho}(a_{1}),\Phi^{*}_{\rho}(a_{2})\\}\;.$

Consequently, $\phi_{\rho^{T_{\mathcal{A}}}}$ is decomposable and
$\rho^{T_{\mathcal{A}}}\in S_{D}(\mathcal{A}\otimes\mathcal{B})$. ∎

Summarising this motivational prologue, in Sec. I.1 we remarked that the PPT
criterion becomes necessary and sufficient when applied to _purifications_ ,
and used the Choi-Jamiołkowski isomorphism to translate the criterion _from
bipartite states to bipartite channels_. In particular, we recast partial
transposition into the (Hermitian) adjoint in Lm. 1, which further suggests to
lift the PPT criterion from the level of bipartite quantum channels $\phi$ to
$C^{*}$-homomorphisms $\Phi$ in Stinespring dilations $\phi=v^{*}\Phi v$.

More generally, in Sec. I.2 we considered the PPT criterion with respect to
dilations of decomposable maps. Prop. 2 shows that the partial transpose
preserves the respective positivity condition of linear functionals
corresponding to such maps. _We deduce that the PPT criterion is sensitive
precisely to the difference between decomposable and completely positive
maps._ In the context of classifying quantum from non-signalling bipartite
correlations, this is achieved by enforcing a compatibility condition with
respect to the relative time orientation between systems $\mathcal{A}$ and
$\mathcal{B}$ FrembsDoering2022a . Building on this motivation, in the next
section we identify a necessary and sufficient condition for bipartite
separability in terms of a compatibility condition with respect to _different
time orientations_ between $\mathcal{A}$ and $\mathcal{B}$.

##  II Entanglement and the arrow of time

In this main part, we combine the insights gained in previous sections. In
Sec. II.1, we identify a necessary and sufficient criterion for bipartite
entanglement (Thm. 1). In Sec. II.2 we employ the structure theory of Jordan
and (associative) $C^{*}$-algebras to translate this criterion into a
compatibility condition between canonical time orientations on local
subsystems (Thm. 2). Finally, in Sec. II.3 we interpret our results in light
of the intrinsic flow of time in von Neumann algebras by means of Tomita-
Takesaki theory, Connes cocyles, and the background-independent
thermodynamical arrow of time ConnesRovelli1994 .

###  II.1 A necessary and sufficient criterion for bipartite entanglement

The PPT criterion translates between bipartite states and bipartite channels
as follows:

$\displaystyle\rho^{T_{\mathcal{A}}}\mathrm{\ positive}$
$\displaystyle\stackrel{{\scriptstyle\mathrm{\ \ \ Choi^{\prime}s\ theorem}\ \
\ \ }}{{\Longleftrightarrow}}$
$\displaystyle\phi_{\rho^{T_{\mathcal{A}}}}\mathrm{\ completely\ positive}$
$\displaystyle\stackrel{{\scriptstyle\ \ \ \ \ \ \ \ \mathrm{Lm.~{}}\ref{lm:
transposition vs *}\ \ \ \ \ \ \ \ }}{{\Longleftrightarrow}}$
$\displaystyle\phi^{*}_{\rho}\mathrm{\ completely\ positive}$
$\displaystyle\stackrel{{\scriptstyle\mathrm{\ Stinespring^{\prime}s\
theorem}}}{{\Longleftrightarrow}}$
$\displaystyle\phi^{*}_{\rho}=(v^{\prime})^{*}\Phi^{\prime}_{\rho}v^{\prime},\mathrm{\
where\ }(v^{\prime},\Phi^{\prime}_{\rho},\mathcal{K}^{\prime})\mathrm{\ is\ a\
Stinespring\ dilation}$

Now since $\phi_{\rho}$ is completely positive, it also has a Stinespring
dilation $\phi_{\rho}=v^{*}\Phi_{\rho}v$. _We may thus strengthen the PPT
criterion as follows: rather than $\phi_{\rho}^{*}$ admitting any Stinespring
dilation, we ask when $v^{\prime}=v$, $\Phi^{\prime}_{\rho}=\Phi^{*}_{\rho}$,
i.e., when $\Phi^{*}_{\rho}$ is a $C^{*}$-homomorphism._

Note first that this condition does not depend on the choice of Stinespring
dilation.

######  Lemma 2.

Let $\phi=v_{1}\Phi_{1}v_{1}^{*}=v_{2}\Phi_{2}v_{2}^{*}$ be two Stinespring
dilations of $\phi$. Then
$\Phi_{1}^{*}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K}_{1})$ is a
$C^{*}$-homomorphism if and only if
$\Phi_{2}^{*}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K}_{2})$ is a
$C^{*}$-homomorphism.

###### Proof.

There is a partial isometry $W:\mathcal{K}_{1}\rightarrow\mathcal{K}_{2}$
defined by $W\Phi_{1}v_{1}|\psi\rangle=\Phi_{2}v_{2}|\psi\rangle$ for all
$|\psi\rangle\in\mathcal{H}_{\mathcal{B}}$ such that
$\Phi_{1}=W^{*}\Phi_{2}W$.999Note that for minimal Stinespring dilations $W$
is unitary. Hence, $\Phi^{*}_{1}=W^{*}\Phi^{*}_{2}W$ and the claim follows. ∎

Next, we have the following important characterisation.

######  Lemma 3.

Let $\Phi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ be a
$C^{*}$-homomorphism. Then
$\Phi^{*}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ is a
$C^{*}$-homomorphism if and only if
$\Phi(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$ is a commutative
subalgebra,101010It is interesting to note that Bell’s theorem holds for
states over $C^{*}$-algebras as long as one of them is commutative Baez1987 .
We discuss the relation with Bell’s theorem and Bell nonlocality elsewhere
DoeringFrembs2019a . equivalently, $\Phi=\Phi|_{V}$ for $V\subset\mathcal{A}$
a commutative subalgebra.

###### Proof.

If $\Phi^{*}$ is a $C^{*}$-homomorphism, then for all
$a_{1},a_{2}\in\mathcal{A}$

$\Phi(a_{1})\Phi(a_{2})=\Phi(a_{1}a_{2})=\Phi^{*}((a_{1}a_{2})^{*})=\Phi^{*}(a_{2}^{*}a_{1}^{*})=\Phi^{*}(a_{2}^{*})\Phi^{*}(a_{1}^{*})=\Phi(a_{2})\Phi(a_{1})\;,$

hence, $\Phi(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$ is a commutative
subalgebra. Conversely, if $\Phi(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$
is a commutative subalgebra, then for all $a_{1}^{*},a_{2}^{*}\in\mathcal{A}$

$\Phi^{*}(a_{2}^{*}a_{1}^{*})=\Phi^{*}((a_{1}a_{2})^{*})=\Phi(a_{1}a_{2})=\Phi(a_{1})\Phi(a_{2})=\Phi(a_{2})\Phi(a_{1})=\Phi^{*}(a_{2}^{*})\Phi^{*}(a_{1}^{*})\;.$

Hence, $\Phi^{*}$ is a $C^{*}$-homomorphism.

Since $\Phi(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$ is a commutative
subalgebra, there exists a maximal commutative subalgebra
$V\subset\mathcal{A}$ such that $\Phi(V)=\Phi(\mathcal{A})$. We show that
$\Phi(a)=0$ for all $a\perp V$ (with respect to the Hilbert-Schmidt inner
product $(a,a^{\prime}):=\mathrm{tr}[a^{\prime*}a]$). Without loss of
generality, we may assume that $V$ is the commutative subalgebra generated by
diagonal matrices. We want to show that $\Phi(a)=0$ for all $a\in V^{\perp}$.
The latter implies $\mathrm{tr}[a]=0$, hence, $a=[b,c]$ for some
$b,c\in\mathcal{A}$ Shoda1936 . We have
$\Phi(a)=\Phi([b,c])=[\Phi(b),\Phi(c)]=0$, since $\Phi$ is a homomorphism and
$\Phi(\mathcal{A})$ is commutative. Consequently, $\Phi$ acts non-trivially
only on $V$ and is zero otherwise. ∎

The following key result shows that this property is equivalent to
separability.

######  Theorem 1.

Let $\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$, let $\phi_{\rho}$ be
the map under the isomorphism in Eq. (2) and let
$\phi_{\rho}=v^{*}\Phi_{\rho}v$ be a Stinespring dilation of $\phi_{\rho}$.
Then $\rho$ is separable if and only if
$\phi^{*}_{\rho}=v^{*}\Phi^{*}_{\rho}v$ is a Stinespring dilation of
$\phi_{\rho}^{*}$, i.e., if and only if $\Phi^{*}_{\rho}$ is a
$C^{*}$-homomorphism.

###### Proof.

By Lm. 2, we can choose the Stinespring dilation
$\phi_{\rho}=v^{*}\Phi_{\rho}v$ to be of the following simple form:
$v:\mathcal{H}_{\mathcal{B}}\rightarrow\mathcal{K}$ for
$\mathcal{K}=\mathcal{H}_{\mathcal{B}}\otimes\mathcal{H}_{\mathcal{A}}$111111For
a general Stinespring dilation, one needs
$\mathcal{K}=\mathcal{H}_{\mathcal{B}}\otimes\mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{A}}$
(cf. Stinespring1955 ; Naimark1943 ). However, as it turns out in the case
that both $\Phi$ and $\Phi^{*}$ are both $C^{*}$-homomorphisms, $\mathcal{K}$
can be reduced by one factor of $\mathcal{H}_{\mathcal{A}}$. and
$\Phi_{\rho}=1_{\mathcal{B}}\otimes\mathrm{id}_{\mathrm{supp}(\Phi_{\rho})}$.121212This
representation allows to interpret a quantum channel $\phi$ as a coarse-
grained unitary bipartite channel on the target and some unknown ancillary
system, after tracing out the latter. In particular, generalised measurements
can be understood as projective measurements on a larger system Ozawa1984 . In
fact, if $\Phi^{*}_{\rho}$ is a $C^{*}$-homomorphism, then
$\phi_{\rho^{T_{\mathcal{A}}}}=\phi^{*}_{\rho}=v^{*}\Phi^{*}_{\rho}v$ is a
Stinespring dilation (see Lm. 1).131313In particular, $\rho^{T_{\mathcal{A}}}$
is positive in this case. By Lm. 3,
$\Phi_{\rho}(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$ is a commutative
subalgebra and there exists a commutative subalgebra $V\subset\mathcal{A}$
such that
$\Phi_{\rho}(V)=\Phi_{\rho}(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$. We
thus have $\Phi_{\rho}(a)=\Phi_{\rho}|_{V}(a)=1_{\mathcal{B}}\otimes
p_{V}ap_{V}$, where $p_{V}$ denotes the projection onto $V$.

Next, let $(|\xi_{k}\rangle)_{k}$ be a basis of $\mathcal{H}_{\mathcal{A}}$
such that $|\xi_{k}\rangle\langle\xi_{k}|=p_{k}$ for all one-dimensional
projections $p_{k}\in\mathcal{P}_{1}(V)$, and let $(|e_{i}\rangle)_{i}$ be an
orthonormal basis of $\mathcal{H}_{\mathcal{B}}$. We can decompose
$v:\mathcal{H}_{\mathcal{B}}\rightarrow\mathcal{K}$ by its action on basis
states,
$v(|e_{j}\rangle):=\sum_{ij}c^{k}_{ij}|e_{i}\rangle\otimes|\xi_{k}\rangle$
with $c^{k}_{ij}\in\mathbb{C}$, hence,
$v=\sum_{ijk}c^{k}_{ij}|e_{i}\rangle\langle
e_{j}|\otimes|\xi_{k}\rangle=\sum_{k}X_{k}\otimes|\xi_{k}\rangle$ with
$X_{k}=\sum_{ij}c^{k}_{ij}|e_{i}\rangle\langle e_{j}|$. Consequently, for all
$a=\sum_{r}a_{r}p_{r}\in V$

$\displaystyle\phi_{\rho}(a)=v^{*}\Phi_{\rho}|_{V}(a)v=v^{*}(1_{\mathcal{B}}\otimes
a)v=\sum_{kl}X_{l}^{*}X_{k}\ \langle\xi_{l}|\sum_{r}a_{r}p_{r}|\xi_{k}\rangle$
$\displaystyle=\sum_{k}E_{k}\
\mathrm{tr}_{\mathcal{H}_{\mathcal{A}}}[|\xi_{k}\rangle\langle\xi_{k}|\ a]\;.$
(3)

Clearly, $E_{k}=X_{k}^{*}X_{k}\in\mathcal{B}_{+}$, hence, (after normalisation
$E_{k}\rightarrow E_{k}/\mathrm{tr}[E_{k}]$,
$|\xi_{k}\rangle\langle\xi_{k}|\rightarrow\mathrm{tr}[E_{k}]|\xi_{k}\rangle\langle\xi_{k}|$)
$\phi_{\rho}$ is an entanglement-breaking channel,141414Entanglement-breaking
channels are also called _measure-prepare channels_ or _in Holevo form_ (see
HorodeckiShorRuskai2003 ; Holevo1998 ). equivalently $\rho$ is a separable
state HorodeckiShorRuskai2003 .

Conversely, let $\rho$ be a separable state.151515Clearly,
$\rho^{T_{\mathcal{A}}}$ is positive, equivalently
$\phi_{\rho^{T_{\mathcal{A}}}}$ is completely positive in this case. Then
$\phi_{\rho}$ is an entanglement-breaking channel HorodeckiShorRuskai2003 ,
i.e., there exist states $E_{k}\in\mathcal{S}(\mathcal{B})$ and positive
operators $F_{k}\geq 0$ such that

$\phi_{\rho}(a)=\sum_{k=1}^{K}E_{k}\
\mathrm{tr}_{\mathcal{H}_{\mathcal{A}}}[F_{k}a]\;.$

We may extend $F_{k}$ to a positive operator-valued measure (POVM)
$(F_{k})_{k=0}^{K}$, $F_{k}\in\mathcal{A}_{+}$ by setting
$F_{0}:=1-\sum_{k=1}^{K}F_{k}$ such that $\sum_{k=0}^{K}F_{k}=1$. By Naimark’s
theorem Naimark1943 ; Stinespring1955 , $F$ admits a dilation
$F=\tilde{v}^{*}\pi\tilde{v}$, where
$\tilde{v}:\mathcal{H}_{\mathcal{A}}\rightarrow\mathcal{H}_{\tilde{\mathcal{A}}}$
is a linear map and $(\pi_{k})_{k=0}^{K}$,
$\pi_{k}\in\mathcal{P}(\mathcal{H}_{\tilde{\mathcal{A}}})$ a projection-valued
measure (PVM). Consequently, $\phi_{\rho}(a)=\tilde{\phi}_{\rho}(a)-E_{0}\
\mathrm{tr}[F_{0}a]$, where e.g. $E_{0}\propto\mathbbm{1}$ and

$\displaystyle\tilde{\phi}_{\rho}(a)=\sum_{k=0}^{K}E_{k}\
\mathrm{tr}_{\mathcal{H}_{\mathcal{A}}}[\tilde{v}^{*}\pi_{k}\tilde{v}a]=\sum_{k=0}^{K}E_{k}\
\mathrm{tr}_{\mathcal{H}_{\mathcal{A}}}[\tilde{v}^{*}\pi_{k}\tilde{\Phi}_{\rho}(a)\tilde{v}]=\sum_{k=0}^{K}E_{k}\
\mathrm{tr}_{\mathcal{H}_{\tilde{\mathcal{A}}}}[\pi_{k}\tilde{\Phi}_{\rho}(a)]\;.$

Here,
$\tilde{\Phi}_{\rho}:\mathcal{A}\rightarrow\tilde{\mathcal{A}}=\mathcal{B}(\mathcal{H}_{\tilde{\mathcal{A}}})$,
$\tilde{\Phi}_{\rho}(a):=\tilde{v}a\tilde{v}^{*}$ is the natural embedding
under the isometry $\tilde{v}$ (that is, $\tilde{v}^{*}\tilde{v}=1$), and we
used that $\tilde{\Phi}_{\rho}\tilde{v}\tilde{v}^{*}=\tilde{\Phi}_{\rho}$,
where
$\tilde{v}\tilde{v}^{*}\in\mathcal{P}(\mathcal{H}_{\tilde{\mathcal{A}}})$,
$(\tilde{v}\tilde{v}^{*})\mathcal{H}_{\tilde{\mathcal{A}}}\cong\mathcal{H}_{\mathcal{A}}$
in the last step. Note that
$(a,a^{\prime}):=\mathrm{tr}_{\mathcal{H}_{\tilde{\mathcal{A}}}}[a^{\prime*}a]$
defines an inner product on $\tilde{\mathcal{A}}$. It follows that we can
restrict the action of $\tilde{\Phi}_{\rho}$ to the pre-image of the
commutative subalgebra
$W:=\langle\pi_{k}\rangle_{k=0}^{K}\subset\tilde{\mathcal{A}}$, spanned by the
projections $\pi_{k}$. More precisely, we define
$\Phi_{\rho}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ for
$\mathcal{K}=\mathcal{H}_{\mathcal{B}}\otimes\mathcal{H}_{\tilde{\mathcal{A}}}$
by

$\Phi_{\rho}(a)=\begin{cases}1_{\mathcal{B}}\otimes\tilde{\Phi}_{\rho}&\mathrm{for\
all\ }a\in\tilde{\Phi}^{-1}_{\rho}(W)\\\ 0&\mathrm{otherwise}\end{cases}\;.$

Clearly, $\Phi_{\rho}$ is a $C^{*}$-homomorphism since
$\tilde{\Phi}_{\rho}|_{\tilde{\Phi}^{-1}_{\rho}(W)}$ is. Moreover,
$\Phi_{\rho}(\mathcal{A})\subset\mathcal{B}(\mathcal{K})$ is a commutative
subalgebra by construction, hence, $\Phi^{*}_{\rho}$ is a $C^{*}$-homomorphism
by Lm. 3. Finally, we obtain Stinespring dilations
$\phi_{\rho}=v^{*}\Phi_{\rho}v$ and $\phi^{*}_{\rho}=v^{*}\Phi^{*}_{\rho}v$ as
before: define $v=\sum_{k=1}^{K}X_{k}\otimes|\xi_{k}\rangle$ with
$X_{k}^{*}X_{k}=E_{k}$ and with $(|\xi_{k}\rangle)_{k=0}^{K}$ the orthonormal
basis of $\mathcal{H}_{\tilde{\mathcal{A}}}$, corresponding to the commutative
subalgebra $W\subset\tilde{\mathcal{A}}$, i.e.,
$|\xi_{k}\rangle\langle\xi_{k}|=\pi_{k}$. ∎

We have used the fact that a state is separable if and only if its
corresponding quantum channel in Eq. (2) is a measure-prepare channel
HorodeckiShorRuskai2003 . The latter requires a decomposition of $v$ as in Eq.
(3). Clearly, such a decomposition exists for any commutative subalgebra
$V\subset\mathcal{A}$. In turn, Thm. 1 shows that such a decomposition exists
for all of $\mathcal{A}$ if and only if $\Phi_{\rho}(\mathcal{A})$ is a
commutative subalgebra.

Moreover, Thm. 1 sheds new light on the reason why the PPT criterion is not
necessary in general: let $\phi_{\rho}=v^{*}\Phi_{\rho}v$ be a Stinespring
dilation and let $\phi^{*}_{\rho}=\phi_{\rho^{T_{\mathcal{A}}}}$ be completely
positive, this does not imply that $\phi^{*}_{\rho}=v^{*}\Phi^{*}_{\rho}v$ is
a Stinespring dilation for $\phi^{*}_{\rho}$.

We record the following corollary of Thm. 1.

######  Corollary 1.

Let $\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$, let $\phi_{\rho}$ the
map under the isomorphism in Eq. (2), and let $\phi_{\rho}=v^{*}\Phi_{\rho}v$
be a Stinespring dilation. Then $\rho$ is separable if and only if
$\phi_{\rho}=\phi_{\rho}|_{V}$ for a commutative subalgebra
$V\subset\mathcal{A}$.

###### Proof.

This follows immediately from Lm. 2, Thm. 1, and Lm. 3. ∎

We surmise that Thm. 1—especially in the form of Cor. 1—entails improvements
of existing protocols for practical verification of entanglement, e.g. in the
form of semi-definite linear programmes in DohertyParriloSpedalieri2002 ;
DohertyParriloSpedalierei2004 . We leave this as an exciting direction for
future research. In the remainder, we focus on the physical content of Thm. 1
in terms of the arrow of time.

###  II.2 Entanglement and time orientations

Comparing Prop. 2 with Thm. 1, it is natural to study the difference between
Jordan $*$-homomorphisms and $C^{*}$-homomorphisms. To this end, we first
review some basic facts about Jordan algebras and their dynamics in terms of
one-parameter groups of automorphisms, before proving a reformulation of Thm.
1 in terms of local time orientations.

Jordan algebras. We recall that an abstract Jordan algebra $\mathcal{J}$ is an
algebra over a field with a product that satisfies $a\circ b=b\circ a$ and
$(ab)(aa)=(a(b(aa))$ for all $a,b\in\mathcal{J}$.161616For an extensive study
of Jordan algebras, see McCrimmon2004 .. Given an associative algebra
$\mathcal{A}$ one obtains a Jordan algebra $\mathcal{J}(\mathcal{A})$ by
symmetrisation. If $\mathcal{J}=\mathcal{J}(\mathcal{A})$ for an associative
algebra $\mathcal{A}$, then the Jordan algebra is called _special_ , otherwise
it is called _exceptional_.171717The prototypical exceptional Jordan algebra
is the so-called Albert algebra $H_{3}(\mathbb{O})$ Albert1934 . In
particular, every $C^{*}$\- (and von Neumann)181818Recall that a
$C^{*}$-algebra is an involutive Banach algebra (closed in norm) satisfying
the defining $C^{*}$-property, $||x^{*}x||=||x||^{2}$. A von Neumann algebra
is a $C^{*}$-algebra closed in the weak operator topology. algebra defines a
JB(W) algebra: a JB(W) algebra is a (weakly closed) Jordan algebra that is
also a Banach space ($||a\circ b||\leq||a||\cdot||b||$) such that
$||a^{2}||=||a||^{2}\leq||a^{2}+b^{2}||$. For simplicity, here we only
consider matrix algebras over the complex numbers,
$\mathcal{A}=M_{n}(\mathbb{C})$, $n\in\mathbb{N}$. In this case, the set of
Hermitian matrices $H_{n}(\mathbb{C})$ under the anti-commutator
$\\{a,b\\}:=ab+ba$ defines a real Jordan algebra
$\mathcal{J}(\mathcal{A})_{\mathrm{sa}}:=(H_{n}(\mathbb{C}),\\{\cdot,\cdot\\})$.
We denote its complexification by
$\mathcal{J}(\mathcal{A})=\mathcal{J}(\mathcal{A})_{\mathrm{sa}}+i\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$.

Crucially, Jordan products are commutative. As such the Jordan algebra
$\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$ is the same as the Jordan algebra
$\mathcal{J}(\mathcal{A}^{\mathrm{op}})_{\mathrm{sa}}$ of the opposite algebra
$\mathcal{A}^{\mathrm{op}}$, i.e., the algebra obtained from $\mathcal{A}$ by
reversing the order of composition (matrix multiplication),

$\displaystyle\begin{split}\mathcal{A}&:=\\{a\in\mathcal{A}\mid\forall
a_{1},a_{2}\in\mathcal{A}:\
a_{1}\cdot_{+}a_{2}=\frac{1}{2}\\{a_{1},a_{2}\\}+\frac{1}{2}[a_{1},a_{2}]\\}\;,\\\
\mathcal{A}^{\mathrm{op}}&:=\\{a\in\mathcal{A}\mid\forall
a_{1},a_{2}\in\mathcal{A}:\
a_{1}\cdot_{-}a_{2}=\frac{1}{2}\\{a_{1},a_{2}\\}-\frac{1}{2}[a_{1},a_{2}]\\}\;.\end{split}$
(4)

The difference between the associative algebras $\mathcal{A}$ and
$\mathcal{A}^{\mathrm{op}}$ is the anti-symmetric part or commutator. In order
to extract from this a notion of time directionality, we relate commutators to
(infinitesimal) symmetries of $\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$.

Time orientations. Dynamics is naturally expressed in terms of one-parameter
groups of Jordan automorphisms $\mathbb{R}\ni
t\mapsto\mathrm{Aut(\mathcal{J}(\mathcal{A})_{\mathrm{sa}})}$. Recall that for
$\mathcal{A}=\mathcal{B}(\mathcal{H}_{A})$ (in particular, for
$\mathcal{A}=M_{n}(\mathbb{C})$) every such one-parameter group is given by
conjugation with a unitary or anti-unitary operator by Wigner’s theorem
Wigner1931_GruppentheorieUndQM ; Bargmann1964 . In fact, Wigner’s theorem
holds on the level of Jordan algebras LandsmanLindenhovius2018 ;
DoeringFrembs2019a . In $\mathcal{A}$, we obtain one-parameter groups of the
form

$e^{t\mathrm{ad}(ia_{1})}(a_{2})=e^{ita_{1}}a_{2}e^{-ita_{1}}\quad\quad\forall
t\in\mathbb{R},a_{1},a_{2}\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}\;.$ (5)

If we interpret $a_{1}$ as the Hamiltonian of the system, then Eq. (5) is just
the standard expression for unitary evolution, in which $t$ plays the role of
a _time parameter_. More generally, Eq. (5) defines a one-parameter group for
every element $a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$. In particular,
note that for every $a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$ and
$\lambda\in\mathbb{R}_{+}$ also $\lambda
a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$. Hence, we cannot give physical
meaning to the absolute value of $t$ without first specifying a Hamiltonian
$a$.

Nevertheless, the sign of $t$ carries physical meaning independent of the
choice of $a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$. To see this, we remark
that inherent in Eq. (5) is the canonical identification between self-adjoint
operators (observables) and generators of Jordan automorphisms (symmetry
generators), $a\mapsto\mathrm{ad}(ia)$ for all
$a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$ GrginPetersen1974 ;
AlfsenShultz1998 ; Baez2020 . Moreover, note that changing this identification
to $a\mapsto\mathrm{ad}(-ia)$ results in a sign change for the parameter $t$
in Eq. (5), equivalently to a change in the commutator and thus to a change in
the order of composition from $\mathcal{A}$ to $\mathcal{A}^{\mathrm{op}}$ in
Eq. (4) (see also Lm. 4 below).

In contrast, in $\mathcal{J}(\mathcal{A})$ there is no canonical
identification between self-adjoint operators and generators of Jordan
automorphisms AlfsenShultz1998 ; Hanche-OlsenStormer1984JordanOA .
Consequently, in $\mathcal{J}(\mathcal{A})$ we cannot interpret the sign of
the parameter $t$ in the corresponding one-parameter groups independently of
the choice of Hamiltonian $a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$. By
comparison, lifting $\mathcal{J}(\mathcal{A})$ to $\mathcal{A}$ thus equips
the latter with an intrinsic direction of time, mediated by the identification
$a\mapsto\mathrm{ad}(ia)$.191919Generalising the mapping
$a\mapsto\mathrm{ad}(ia)$, AlfsenShultz1998a  characterise those maps which
lift JB(W) to $C^{*}$ (von Neumann) algebras. By their physical
interpretation, such maps are called _dynamical correspondences_. To emphasise
this distinction, we define the _canonical time orientation
$\Psi_{\mathcal{A}}$_ on $\mathcal{J}(\mathcal{A})$ by202020The notion of time
orientation was introduced in Doering2014 ; FrembsDoering2022a .

$\Psi_{\mathcal{A}}:=\mathrm{Ad}:\mathbb{R}\times\mathcal{J}(\mathcal{A})_{\mathrm{sa}}\ni(t,a)\mapsto
e^{t\mathrm{ad}(ia)},$ (6)

and call $\mathcal{A}_{+}:=(\mathcal{J}(\mathcal{A}),\Psi_{\mathcal{A}})$ the
observable Jordan algebra together with its canonical time orientation.
Similarly, we define the _reverse time orientation_ by

$\Psi^{*}_{\mathcal{A}}:=*\circ\Psi_{\mathcal{A}}:\mathbb{R}\times\mathcal{J}(\mathcal{A})_{\mathrm{sa}}\ni(t,a)\mapsto
e^{-t\mathrm{ad}(ia)}\;,$ (7)

and set
$\mathcal{A}_{-}:=(\mathcal{J}(\mathcal{A}),\Psi^{*}_{\mathcal{A}})$.212121By
Thm. 23 in AlfsenShultz1998a , these are the only time orientations on
$\mathcal{J}(\mathcal{A})$, deriving from $\mathcal{A}$ and
$\mathcal{A}^{\mathrm{op}}$, respectively.

Entanglement and time orientation. Returning to Thm. 1, we are interested in
the difference between Jordan $*$-homomorphism and $C^{*}$-homomorphism.
Recall that a _Jordan $*$-homomorphism_
$\Phi:\mathcal{J}(\mathcal{A})\rightarrow\mathcal{J}(\mathcal{B})$ is a linear
map preserving the Hermitian adjoint, $*\circ\Phi=\Phi\circ*$, equivalently,
$\Phi|_{\mathcal{J}(\mathcal{A})_{\mathrm{sa}}}:\mathcal{J}(\mathcal{A})_{\mathrm{sa}}\rightarrow\mathcal{J}(\mathcal{B})_{\mathrm{sa}}$,
and the Jordan product, i.e.,
$\Phi(\\{a_{1},a_{2}\\})=\\{\Phi(a_{1}),\Phi(a_{2})\\}$ for all
$a_{1},a_{2}\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$. Consequently,
$\Phi:\mathcal{J}(\mathcal{A})\rightarrow\mathcal{J}(\mathcal{B})$ lifts to a
$C^{*}$-homomorphism $\Phi:\mathcal{A}\rightarrow\mathcal{B}$ if and only if
it preserves commutators, $\Phi([a_{1},a_{2}])=[\Phi(a_{1}),\Phi(a_{2})]$.
Using Eq. (6), we re-express this condition in terms of one-parameter groups
of Jordan automorphisms.

######  Lemma 4.

Let $\Phi:\mathcal{J}(\mathcal{A})\rightarrow\mathcal{J}(\mathcal{B})$ be a
Jordan $*$-homomorphism. Then $\Phi:\mathcal{A}\rightarrow\mathcal{B}$ lifts
to a $C^{*}$-homomorphism if and only if it preserves the canonical time
orientations $\Psi_{\mathcal{A}}$ and $\Psi_{\mathcal{B}}$,

$\forall t\in\mathbb{R},a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}:\
\Phi\circ\Psi_{\mathcal{A}}(t,a)=\Psi_{\mathcal{B}}(t,\Phi(a))\circ\Phi\;.$
(8)

###### Proof.

Clearly, a $C^{*}$-homomorphism $\Phi$ preserves Eq. (8). Conversely, by
differentiation,

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}\left(\Phi\circ\Psi_{\mathcal{A}}(t,a_{1})\right)(a_{2})=\left.\frac{d}{dt}\right|_{t=0}\left(\Psi_{\mathcal{B}}(t,\Phi(a_{1}))\circ\Phi\right)(a_{2})$
$\displaystyle\Leftrightarrow\quad$
$\displaystyle\Phi\left(\left.\frac{d}{dt}\right|_{t=0}e^{ita_{1}}a_{2}e^{-ita_{1}}\right)=\left.\frac{d}{dt}\right|_{t=0}e^{it\Phi(a_{1})}\Phi(a_{2})e^{-it\Phi(a_{1})}$
$\displaystyle\Leftrightarrow\quad$
$\displaystyle\Phi([a_{1},a_{2}])=[\Phi(a_{1}),\Phi(a_{2})]\;.$

for all $a_{1},a_{2}\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$. $\Phi$ thus
preserves commutators, hence, is a $C^{*}$-homomorphism. ∎

Assume $\Phi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ in Eq. (8) is
part of a Stinespring dilation $\phi_{\rho}=v^{*}\Phi v$ for the image of a
bipartite state $\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$ under the
isomorphism in Eq. (2). Since $\mathcal{B}$ arises from
$\mathcal{B}(\mathcal{K})$ by restriction under $v$, the time orientation
$\Psi_{\mathcal{B}}$ on $\mathcal{B}$ uniquely lifts to a time-orientation
$\Psi^{\prime}_{\mathcal{B}}$ on $\mathcal{B}(\mathcal{K})$.

This motivates the following definition, which first appeared in the context
of classifying quantum states from non-signalling bipartite correlations
FrembsDoering2022a  (see also Sec. I.2).

######  Definition 1.

Let $\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$. $\rho$ is called
_time-oriented with respect to
$\mathcal{A}_{-}\cong(\mathcal{J}(\mathcal{A}),\Psi^{*}_{\mathcal{A}})$_ and
$\mathcal{B}_{+}=(\mathcal{J}(\mathcal{B}),\Psi_{\mathcal{B}})$ if and only if
$\Phi_{\rho}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ in
$\phi_{\rho}=v^{*}\Phi_{\rho}v$ preserves time orientations
$\Psi^{*}_{\mathcal{A}}=*\circ\Psi_{\mathcal{A}}$ and
$\Psi^{\prime}_{\mathcal{B}}$,

$\forall t\in\mathbb{R},a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}:\
\Phi_{\rho}\circ\Psi^{*}_{\mathcal{A}}(t,a)=\Psi^{\prime}_{\mathcal{B}}(t,\Phi(a))\circ\Phi_{\rho}\;.$

We remark that the appearance of the reverse time orientation
$\Psi^{*}_{\mathcal{A}}$ in Def. 1 is a consequence of the identification of
bipartite quantum states and quantum channels via Choi’s theorem Choi1975
(for more details, see Frembs2022b ). Def. 1 is the missing piece of physical
data to identify bipartite non-signalling distributions with quantum states
FrembsDoering2022a . Together, this shows that quantum states encode
information about the relative time orientation between subsystems.

This is a genuine quantum effect. What is more, it is intimately related with
entanglement: in fact, Def. 1 allows us to reformulate the separability
criterion in Thm. 1 in terms of time orientations.

######  Theorem 2.

A bipartite state $\rho\in\mathcal{S}(\mathcal{A}\otimes\mathcal{B})$ is
separable if and only if it is time-oriented with respect to
$\mathcal{A}_{-}=(\mathcal{J}(\mathcal{A}),\Psi^{*}_{\mathcal{A}})$ and
$\mathcal{B}_{+}=(\mathcal{J}(\mathcal{B}),\Psi_{\mathcal{B}})$ as well as
$\mathcal{A}_{-}$ and
$\mathcal{B}_{-}=(\mathcal{J}(\mathcal{B}),\Psi^{*}_{\mathcal{B}})$.

###### Proof.

By Thm. 1, $\rho$ is separable if and only if $\Phi_{\rho}$ and
$\Phi^{*}_{\rho}$ are $C^{*}$-homomorphisms for any Stinespring dilation
$\phi_{\rho}=v^{*}\Phi_{\rho}v$. Since $C^{*}$-homomorphisms preserve time
orientations by Lm. 4, $\rho$ is time-oriented with respect to both
$\mathcal{A}_{-}$ and $\mathcal{B}_{+}$ as well as $\mathcal{A}_{-}$ and
$\mathcal{B}_{-}$.222222Note that since
$\Phi_{\rho}:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{K})$ is Hermiticity-
preserving, i.e., $\Phi^{*}_{\rho}(a)=\Phi_{\rho}(a^{*})$ for all
$a\in\mathcal{A}$, $\Phi_{\rho}$ preserves time orientations
$\Psi_{\mathcal{A}}$ and $\Psi_{\mathcal{B}}$ if and only if it preserves time
orientations $\Psi^{*}_{\mathcal{A}}$ and $\Psi^{*}_{\mathcal{B}}$; similarly
$\Phi_{\rho}$ preserves time orientations $\Psi^{*}_{\mathcal{A}}$ and
$\Psi_{\mathcal{B}}$ if and only if it preserves time orientations
$\Psi_{\mathcal{A}}$ and $\Psi^{*}_{\mathcal{B}}$.

Conversely, $\rho$ is time-oriented with respect to $\mathcal{A}_{-}$ and
$\mathcal{B}_{+}$ by Thm. 3 in FrembsDoering2022a , i.e.,

$\forall t\in\mathbb{R},a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}:\
\Phi_{\rho}\circ\Psi^{*}_{\mathcal{A}}(t,a)=\Phi_{\rho}\circ\Psi_{\mathcal{A}}(-t,a)=\Psi^{\prime}_{\mathcal{B}}(t,\Phi(a))\circ\Phi_{\rho}\;.$
(9)

where
$\Psi^{*}_{\mathcal{A}}(t,a)=*\circ\Psi_{\mathcal{A}}(t,a)=\Psi_{\mathcal{A}}(-t,a)$
by Eq. (7). In particular, $\Phi_{\rho}$ in $\phi_{\rho}=v^{*}\Phi_{\rho}v$ is
a $C^{*}$-homomorphism Stinespring1955 . If $\rho$ is also time-oriented with
respect to $\mathcal{A}_{-}$ and $\mathcal{B}_{-}$, then by Def. 1,

$\displaystyle\begin{split}&\forall
t\in\mathbb{R},a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}:\
\Phi_{\rho}\circ\Psi^{*}_{\mathcal{A}}(t,a)=\Psi^{\prime*}_{\mathcal{B}}(t,\Phi(a))\circ\Phi_{\rho}\\\
\Longleftrightarrow\quad&\forall
t\in\mathbb{R},a\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}:\
\Phi_{\rho}\circ\Psi_{\mathcal{A}}(t,a)=\Psi^{\prime}_{\mathcal{B}}(t,\Phi(a))\circ\Phi_{\rho}\;.\end{split}$
(10)

Differentiating Eq. (9) and Eq. (10) yields
$[\Phi_{\rho}(a_{1}),\Phi_{\rho}(a_{2})]=-[\Phi_{\rho}(a_{1}),\Phi_{\rho}(a_{2})]=0$
for all $a_{1},a_{2}\in\mathcal{J}(\mathcal{A})_{\mathrm{sa}}$ (cf. Lm. 4). It
follows that $\Phi_{\rho}(\mathcal{A})\subset\mathcal{B}$ is a commutative
subalgebra, by Lm. 3 $\Phi^{*}_{\rho}$ is therefore a $C^{*}$-algebra
homomorphism and by Thm. 1 $\rho$ is separable. ∎

###  II.3 Time orientations and the arrow of time

In this section, we embed the classification of bipartite entanglement in
terms of compatibility with time orientations in local observable algebras
(Thm. 2) into a wider context. We especially focus on time orientations as a
complex structure on $\mathcal{J}(\mathcal{A})$, as well as their role within
the intrinsic, thermodynamic arrow of time in von Neumann algebras.

Time orientations and complex structure. Following Doering2014 ;
FrembsDoering2022a , we have expressed the difference between
$\mathcal{J}(\mathcal{A})$ and $\mathcal{A}$ in terms of time orientations in
Def. 1. As exponentials of dynamical correspondences AlfsenShultz1998a , time
orientations highlight the double role played by self-adjoint operators: as
observables and generators of dynamics in quantum mechanics GrginPetersen1974
; AlfsenShultz1998 . This perspective has some appeal when considering
axiomatic reconstructions of quantum mechanics and possible generalisations
they suggest.

For instance, note that a ‘quantum formalism’ can be defined also over the
real instead of the complex numbers (see e.g. Stueckelberg1960 ;
HardyWootters2011 ). More generally, several results aiming to reconstruct
quantum mechanics arrive at the level of (special) Jordan algebras
corresponding to associative algebras over the real, complex and quaterionic
numbers (e.g. JordanVNWigner1934 ; GrginPetersen1974 ; Koecher1999 ; Soler1995
). In this context, it is interesting to note that time orientations define a
complex structure on (the order derivations of) $\mathcal{J}(\mathcal{A})$
Connes1974 ; AlfsenShultz1998a . Compare this with Eq. (6), where we used the
complex structure of the associative algebra $\mathcal{A}$ implicitly to
define the canonical time orientation $\Psi_{\mathcal{A}}$. In this way,
dynamical correspondences can be seen as a justification for the prominence of
complex numbers in quantum mechanics Baez2012 .232323For the intimate
relationship between dynamical correspondences and Noether’s theorem, see
Baez2020 . By Thm. 2, these arguments are further inherently connected with
quantum entanglement.

Outlook: intrinsic dynamics and thermal time. One of the deepest insights into
the emergence of time from purely algebraic considerations arises in infinite
dimensions and the structure theory of (hyperfinite) von Neumann algebras
Connes1975 ; Connes1985 ; Haagerup1987 ; Haagerup2016 . The latter heavily
rests on the foundational insights by Tomita and Takesaki Takesaki1970 ;
TakesakiII .

Given a von Neumann algebra $\mathcal{N}$ and a faithful normal state
$\omega\in\mathcal{S}(\mathcal{N})$, $\omega$ becomes a cyclic and separating
vector $\Omega$ in its Gelfand-Naimark-Segal (GNS) representation. The
operator defined by $S_{\omega}a\Omega:=a^{*}\Omega$ for all $a\in\mathcal{N}$
is closable, hence, has a polar decomposition
$S_{\omega}=J_{\omega}\Delta^{\frac{1}{2}}_{\omega}$, where $J_{\omega}$ is an
anti-unitary involution and $\Delta_{\omega}$ is a self-adjoint, positive
operator. The fundamental results of _Tomita-Takesaki theory_ are summarised
in the statements $J_{\omega}\mathcal{N}J_{\omega}=\mathcal{N}^{\prime}$,
where $\mathcal{N}^{\prime}$ is the commutant of $\mathcal{N}$, and
$\Delta_{\omega}^{it}\mathcal{N}\Delta_{\omega}^{-it}=\mathcal{N}$ for all
$t\in\mathbb{R}$ Takesaki1970 . The latter implies that every faithful normal
state $\omega\in\mathcal{S}(\mathcal{N})$ defines a one-parameter group of
automorphisms
$\sigma^{\omega}:\mathbb{R}\rightarrow\mathrm{Aut}(\mathcal{N})$,
$\sigma^{\omega}_{t}(a)\mapsto\Delta^{it}a\Delta^{-it}$, called the modular
automorphism group of $\omega$.

Crucially, $S_{\omega}$ and thus $\sigma^{\omega}$ are state-dependent since
they are defined with respect to the support of the state $\omega$. Despite
this fact, the difference between $\sigma^{\omega}_{t}$ and
$\sigma^{\omega^{\prime}}_{t}$ is merely an inner automorphism
$\sigma^{\omega}_{t}(a)=u_{t}\sigma^{\omega^{\prime}}_{t}(a)u^{-1}_{t}$ for
all $a\in\mathcal{N}$, where the unitaries $(u_{t})_{t\in\mathbb{R}}$ satisfy
_Connes’ cocycle condition_
$\sigma^{\omega}_{s+t}=u_{s}\sigma^{\omega}_{t}(u_{t})$ Connes1973 . As a
consequence, $\mathcal{N}$ carries an _intrinsic, i.e., state-independent
notion of dynamics_ , given by (the subgroup of) the automorphism group
generated by the $\sigma^{\omega}$. In contrast, one can also study the
operators $S_{\omega}$ in JBW algebras. However, without the existence of a
dynamical correspondence (equivalently, time orientation), a JBW algebra
cannot distinguish between the one-parameter families of automorphisms
$\sigma^{\omega}_{t}$ and $\sigma^{\omega}_{-t}$ HaagerupHanche-Olsen1984 .

What is more, the intrinsic dynamics in von Neumann algebras is further
exemplified in the study of statistical mechanics in a background-independent
setting Rovelli1993a ; Rovelli1993b . Here,
$\omega\in\mathcal{S}(\mathcal{N})$ is understood as a state in thermodynamic
equilibrium. In the setting of quantum statistical mechanics such states are
characterised by the KMS condition HaagHugenholtzWinnink1967 . It is a
remarkable fact that $\sigma^{\omega}$ satisfies the KMS condition for every
faithful normal state $\omega\in\mathcal{S}(\mathcal{N})$ Takesaki1970 . In
contrast, no analogue of this condition holds for Jordan algebras
HaagerupHanche-Olsen1984 . In effect, time orientations in von Neumann
algebras allow to interpret time from a thermodynamical standpoint, encoded in
a state of thermodynamic equilibrium ConnesRovelli1994 .

The crucial role played by time orientations (equivalently, dynamical
correspondences) in von Neumann algebras and in Thm. 2 is hardly coincidental.
In particular, it is tempting to ‘explain’ the intrinsic dynamics and
(thermodynamic) origin of the arrow of time more fundamentally in terms of the
entanglement structure of a given faithful normal state. To this end, one
would like to generalise Thm. 2 to the setting of general von Neumann
algebras. We leave this and similar considerations for future work.

##  III Conclusion

We found a necessary and sufficient criterion for bipartite entanglement using
Stinespring dilations in Thm. 1. The latter adopts a clear physical meaning in
terms of a compatibility condition with respect to time orientations (Def. 1)
on the respective local observable algebras in Thm. 2. Moreover, we
highlighted the key role time orientations play within the broader picture of
the intrinsic flow of time in von Neumann algebras.

More explicitly, our results are motivated from and bear close resemblance
with the PPT criterion Peres1996 ; Horodeckisz1996 . As such, it would be
interesting to study the practical relevance of Thm. 1. For example, it seems
possible that existing results on marginal extension problems, e.g.
DohertyParriloSpedalieri2002 ; DohertyParriloSpedalierei2004 , can be
strengthened using Cor. 1.

#### Acknowledgements.

I thank Andreas Döring and Eric G. Cavalcanti for discussions. This work was
supported by grant number FQXi-RFP-1807 from the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation, and ARC Future Fellowship FT180100317.

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