# Biological computations: limitations of attractor-based formalisms and the
need for transients

Daniel Koch${}^{1^{\dagger}}$, Akhilesh Nandan${}^{1^{\dagger}}$, Gayathri
Ramesan${}^{1^{\dagger}}$ and Aneta Koseska${}^{1,^{\ast}}$

1 Lise Meitner Group Cellular Computations and Learning,
Max Planck Institute for Neurobiology of Behaviour - caesar, Bonn, Germany

†These authors contributed equally.
∗To whom correspondence should be addressed; E-mail:
<EMAIL_ADDRESS>
## Abstract

Living systems, from single cells to higher vertebrates, receive a continuous
stream of non-stationary inputs that they sense, e.g., via cell surface
receptors or sensory organs. Integrating these time-varying, multi-sensory,
and often noisy information with memory using complex molecular or neuronal
networks, they generate a variety of responses beyond simple stimulus-response
association, including avoidance behavior, life-long-learning or social
interactions. In a broad sense, these processes can be understood as a type of
biological computation. Taking as a basis generic features of biological
computations, such as real-time responsiveness or robustness and flexibility
of the computation, we highlight the limitations of the current attractor-
based framework for understanding computations in biological systems. We argue
that frameworks based on transient dynamics away from attractors are better
suited for the description of computations performed by neuronal and signaling
networks. In particular, we discuss how quasi-stable transient dynamics from
ghost states that emerge at criticality have a promising potential for
developing an integrated framework of computations, that can help us
understand how living system actively process information and learn from their
continuously changing environment.

## Introduction

When referring to computations, the associated concept usually reflects the
formal definition of computation adopted during the first half of the 20th
century which was devised with the purpose of answering questions relating
e.g. the extent to which mathematics can be reduced to discrete logical
formulas, and how mathematical proofs or calculations may be automated (?, ?).
For example, a function on the integer numbers is called _computable_ , if an
output integer can be calculated by an algorithm after a finite number of
steps (?). This implies that computation, in an abstract way, refers to a
defined mapping between inputs and outputs. In that broad sense, many basic
processes characteristic of living systems on all scales of organization, from
single cells in tissues to freely-living single-cell organisms and higher
vertebrates, can be defined in terms of computation. While vertebrates and
many other multi-cellular organisms rely on neuronal networks, single cells
and single-cell organisms use protein and/or gene-regulatory networks as
computational entities to integrate multi-dimensional sensory information
(inputs) with memory, generating complex self-organized behavior (output) (?).
A fox that chases a rabbit, for example, uses its neuronal network to
continually processes sensory (visual, auditory, olfactory, tactile etc.)
information which is disrupted and changes over time and space, i.e. when the
rabbit hides behind a bush (Fig. 1a, top). In order to avoid random change in
the running direction in the absence of a visual contact with the rabbit, the
fox integrates the current sensory information with the memory of the last
localization of the rabbit to determine its behavior. Immune cells in our body
that chase invading bacteria, however, face similar challenges as the fox
chasing the rabbit: the cells navigate guided by local chemical cues secreted
by the bacteria that are noisy, disrupted, and vary over time and space, in
order to engulf and degrade the motile invading microbes (Fig. 1a, bottom). To
avoid immediate switching to random migration when signals are disrupted,
single cells, just like the fox chasing a rabbit, require a memory of the
localization of the last encountered chemical signal, as a means of generating
a reliable migration trajectory over long distances. We will refer to such
computations performed by living systems as biological or natural
computations.

This description of processes in living systems through the concept of
computation has inevitably led to a frequent referral to terms such as
’circuitry’, ’computer/machine’, ’execution of programs’ or ’interpretation of
code’. However, their indiscriminate use can easily convey misleading ideas
(and often does) by neglecting fundamental differences between the
computational process in living and engineered systems (?, ?, ?, ?, ?, ?, ?).
For example, computers accept input once at the beginning of the computation
(i.e., do not accept signals while performing a particular task), and
obtaining the result of the computation requires presentation of the complete
input. In contrast, living systems compute in real-time (like the fox and the
immune cell), using incomplete inputs which continuously change across time
and space. The process of natural computation itself is adaptive, while
yielding robust and reproducible responses even in the presence of noise,
whereas computer algorithms account only for robustness and reproducibility of
computations, which is achieved by minimizing noise (in the constituting
computer circuits). Moreover, living systems are characterized by on-the-fly
and life-long learning, features that have not been realized in any man-made
system so far. We therefore discuss in this article the current concepts of
computation from a dynamical systems point of view and argue that they explain
information processing in living systems only to a very limited extent. We
advocate the necessity to base theories of biological computation on
transients which can serve as a dynamical basis through which the majority of
the observed features of natural computation can be captured (?, ?, ?, ?, ?,
?, ?, ?). Given that from a conceptual viewpoint, the challenges, which single
cells and higher vertebrates face are overlapping to a large degree, we
emphasize in this review the necessity for a general theory of natural
computations and learning that is applicable to both neural and aneural
systems.

Figure 1: Examples of natural and machine computations. (a) Top: A predator
hunting its prey integrates time-varying (for e.g. the prey escapes and/or
hides) and multi-sensory (visual, olfactory, auditory etc.) inputs and
continually adapts its response in real-time. A large fraction of the animal
kingdom relies in the brain’s neuronal network to process such information.
Bottom: A leukocyte chasing a bacterium faces similar challenges, yet relies
on networks comprising entities of a completely different nature (i.e.
molecules such as proteins, nucleic acids etc) to solve this task. Despite
large differences in scale, type and phylogeny, the similarity of the tasks
from a information processing perspective implies similar dynamical principles
to underline the computational process. Gray circles: red blood cells.
Dashed/solid lines: Prey(bacteria)/predator(leukocyte) tracks. (b) Turnstile,
an example of a finite-state-machine, and a schematic representation of the
underlying state-dependent computational process. (c) Quasi-potential
landscape depicting the possible dynamical solutions, that are an intrinsic
property of the underlying (signaling/genetic/neuronal) network topology and
node dynamics.

## Results

##  1 Computing with stable attractors: machines, neuronal networks and
cellular signaling

The existing theoretical frameworks for biological computations mainly refer
to Turing-like computations (?). Let us exemplify this by considering an
every-day example of a finite-state machine, a turnstile. A turnstile can be
found in two different states, closed ($S_{1}$) and open ($S_{2}$). It
requires an input, i.e. a coin to switch from the closed to the open state
($S_{1}\rightarrow S_{2}$), and a second input - a push, to return to the
closed state ($S_{2}\rightarrow S_{1}$). If a coin is inserted when the system
is in $S_{2}$ (open), or if its bars are pushed when it is in $S_{1}$
(closed), the machine does not respond (Fig. 1b). For this system, the
computation is realized through the switching between the distinct stable
states ($S_{1}$, $S_{2}$) that are available, such that the specific inputs
(coin, push) are mapped to a defined state of the turnstile. Despite that this
machine has a very limited computational power that does not reflect that of
Turing machines in general, we will use it to demonstrate a line of thought
that (i) the state-dependent computations performed by machines display
limitations for tasks that are relevant for living systems, and (ii) range of
computations performed by living systems lie outside of the domain of
machines, prompting us to propose that broader definition and mechanisms of
computations are necessary to describe biological computations.

Figure 2: Attractor-based computations in cell signaling networks. (a) Cell-
surface receptor module enabling sensing of time-varying growth factor
signals. EGFRp: phosphorylated epidermal growth factor receptor; EGF-EGFRp:
epidermal growth factor (EGF) ligand-bound receptor fraction. (b) Bifurcation
diagram corresponding to (a), and a schematic representation of the underlying
state-dependent computational process for organization in the bistable regime.
SN: sadddle-node bifurcation. (c) EGFRp temporal response to single and
multiple EGF pulses for organization in the bistable regime, and a
corresponding phase-space trajectory (right). Details in (?, ?). (d)
Corresponding quasi-potential landscape description. Note the remodeling of
the landscape in the presence of transient external signals. (e) Schematic
representation of cell migration through disrupted and spatially non-uniform
growth factor field for receptor network organized in the bistable regime. The
direction is determined by the initial signal the cell encounters. Green/red
lines: migration trajectory in presence/absence of signals; arrows denote
temporal evolution.

The computations of the turnstile, in the language of dynamical systems, can
be formalized as attractor-based computation. Turing implicitly used this idea
in his unpublished work on intelligent machines (?), as well as in his seminal
work on self-organization in living systems (?). The idea, however, was
clearly explicated by Hebb (?) and Hopfield (?) for neuronal networks,
receiving formalization by Hirsch and Baird (?): ”as the overall system
evolves in time, each subsystem passes through a sequence of attractors that
are related to specific output of the system, and this sequence is termed as
the computation process”. In this view, the number and type of attractors is
an intrinsic property of the system, determined by the underlying network
topology and nodal dynamics. Thus, restricting only to fixed point dynamics
for simplicity, the possible solutions of the system form a so-called quasi-
potential landscape which can be explicitly calculated for gradient systems
(?). Each valley in the landscape corresponds to a stable state, separated by
unstable states or saddles (Fig. 1c). The external signals induce switching
between the available states, such that a signal is uniquely associated with a
specific valley. This also implies that in absence of a perturbation, the
dynamics would be retained indefinitely in a valley. The same conceptual
framework has been also adopted to study signaling networks in single cells.
Large number of experimental and theoretical studies over the past two decades
have been focused on identifying the underlying protein- or gene-interaction
networks or network modules (?, ?), relating the possible attractors with the
observed phenotypes or responses (?, ?, ?, ?, ?, ?, ?). The question is
however, to which extent attractor-based computations can explain the basic
features of natural computations, the simplest being real-time processing of
non-stationary signals.

##  2 Limitations of the current framework: an example from single-cell
signaling

Let us consider a bistable system as a minimal case of multistability, i.e. as
a system that has multiple, coexisting states/attractors (cf. Box 1). In the
quasi-potential landscape analogy, this corresponds to a landscape with two
valleys, separated by a saddle. Generally, bistability can emerge via double
negative or positive feedback loops, common motifs in gene regulatory (?, ?),
neuronal (?, ?), as well as in signaling networks including, e.g., the
experimentally identified Epidermal growth factor receptor (EGFR) network (?)
(Fig. 2a). For parametric organization in the bistable regime (Fig. 2b), in
this case defined by a certain range of EGFR concentrations on the plasma
membrane, the system has two available stable states: basal ($S_{1}$) and high
EGFR-phosphorylated ($S_{2}$) state. A pulse of epidermal growth factor (EGF)
induces a transition from $S_{1}$ to $S_{2}$, however, the high EGFRp state
will be maintained even after removal of the EGF signal, as $S_{2}$ is also a
stable attractor. This can be seen from the temporal EGFRp profile, as well as
the trajectory of the signaling state of the EGFR network (Fig. 2c). Thus,
addition of subsequent EGF pulses will not lead to further state changes in
the system, implying that the cell will remain unresponsive to upcoming
changes in the environment. Better understanding of these temporal system’s
responses can be gained from a quasi-potential landscape description: as noted
above, the organization in the bistable regime corresponds to a quasi-
potential landscape with two wells, corresponding to $S_{1}$ and $S_{2}$ (Fig.
2d). In absence of a signal, the system resides in $S_{1}$. Upon EGF addition,
the quasi-potential landscape remodels to a single well corresponding to
$S_{2}$, resulting in a robust EGFR phosphorylation. When the EGF is removed,
however, the landscape resets to its former double-well shape, but the system
remains in the $S_{2}$ well and thereby is unresponsive to upcoming signals
(Fig. 1c), just like the turnstile does not respond to a coin when already in
$S_{2}$.

That such state-dependent computation limits responsiveness to time-varying
signals has also been experimentally demonstrated for epithelial cells (?),
implying broad consequences for the mechanistic understanding of how cells
navigate in dynamic spatial-temporal chemical fields, e.g. during wound
healing or embryogenesis (?, ?). We and others have shown using numerical
simulations, for example, that the cellular migration will be ”locked” in the
direction of the initial input, leaving the cell unresponsive to any further
spatial or temporal signal changes (Fig.2e), unless the upcoming signals have
$\sim$7 fold higher amplitude (?, ?). Moreover, this attractor-based framework
of computation also does not explain how cells can resolve two competing
chemoattractant signals (?, ?), as observed for example for leukocyte
navigation (?).

These findings therefore suggest that the attractor-based framework reaches
its limitations for biological computations in single cells: although it
provides an explanation for robustness (i.e. maintaining directional memory
when the signal is disrupted), it cannot capture the flexibility of natural
computations - adaptation to dynamic signals that vary over space and/or time,
and thus processing of dynamic signals in real time. Moreover, the attractor-
based description also faces an additional, formal problem. As depicted in
Fig.2d, under non-stationary cues, the quasi-potential landscape that
underlies the dynamical structure of the system continuously changes (?, ?, ?,
?, ?). Since the number and positions of steady states are not preserved, the
steady states are not formal solutions of the system. It has been therefore
argued across different systems including developmental (?, ?, ?), signaling
(?, ?, ?) or neuronal systems (?, ?, ?, ?), that trajectories or so-called
transient dynamics are better suited to describe natural computations.

##  3 Computing with transients: chaotic itinerancy and heteroclinic networks

Figure 3: Computing with trajectories. (a) Schematic representation of
attraction to, and escape from, a quasi-attractor in chaotic itinerancy (?).
(b) Schematic of a heteroclinic channel consisting of multiple saddles (inset)
connected such that the unstable manifold of the preceding saddle is connected
to the stable manifold of the following in the sequence. Reproducible
trajectory guidance can be achieved in the deterministic case or low noise
intensities (top), whereas moderate noise levels compromise the reliability
(bottom) (?).

Trajectories represent the evolution of the state of the system in phase-
space. In the presence of attractors, a trajectory will converge
asymptotically to a valley in the quasi-potential landscape and remain trapped
in it (switching from $S_{1}$ to $S_{2}$ in the case of the turnstile and the
EGFR system (Figs. 1b, 2d)). However, when the landscape continuously changes,
the trajectory, thus the dynamics of the system can be maintained away from
the attractors (?, ?, ?). In such cases, instabilities have been advocated as
an advantageous substitute to attractors, suggesting that systems can exploit
unstable states to compute (?). One can think of these as instabilities of a
special type that have both stable and unstable directions and are referred to
as quasi-attractors or attractor-ruins (?, ?, ?). The presence of the unstable
directions allows the trajectory to sporadically switch from one quasi-
attractor to another, resulting in the existence of low-dimensional ordered
motion. This can be realized for example in chaotic systems, such as networks
of nonlinear oscillators (asynchronous neuronal networks) or coupled maps (?,
?, ?, ?), and is known as chaotic itinerancy (Fig.3a) (?, ?, ?). The transient
trapping in the vicinity of the quasi-attractor is proposed to enable
robustness via quasi-stability, and since the trajectories do not settle on
any attractor, the computations can potentially adapt to dynamic signals.
Thus, chaotic itinerancy has been proposed in the general context for natural
computation, in particular modeling brain activity, decision making or forming
of cognitive functions (cf. (?) and references within). However, this
description is limited to systems that display chaotic dynamics.

Following a growing body of empirical evidence that a large class of natural
systems display quasi-stable and sequential (but not chaotic) dynamics,
including neuronal firing patterns during olfactory sensing or discrimination
tasks (?, ?, ?), pattern matching during camouflage in animals (?), cellular
signaling systems (?, ?), replicator networks (?) etc., computation with
heteroclinic networks (?, ?, ?, ?, ?) has been proposed as a mechanism that
generates such sequential dynamics. Heteroclinic objects consist of joined
saddles, such that the unstable manifold of the preceding is the stable
manifold for the next saddle in the sequence (Fig. 3b). In such systems,
computations are performed when external input signals typically induce long
‘complex’ orbits that pass near a sequence of several ‘simple’ saddles.
Moreover, in the course of the temporal evolution, the system’s trajectory
slows down and is transiently trapped in the vicinity of the saddle fixed
point, followed by a quick transition to the upcoming saddle in the sequence
(Fig. 3b, top). However, we have recently demonstrated theoretically that the
fidelity of heteroclinic objects to reliably guide trajectories is lost in the
presence of moderate noise intensities (?). Since the saddle has two unstable
directions, noise can easily drive the trajectory away from the path (Fig. 3b,
bottom). This implies that heteroclinic networks are likely not suitable as a
generic dynamical framework for computations in living systems, which are
inherently noisy.

##  4 What can be learned about real-time computations from single cells?

Figure 4: Cellular responsiveness to changing signals is enabled for
organization at criticality. (a) Bifurcation diagram for the EGFR network
depicting organization at criticality. Notations as in Fig. 2b. (b)
Corresponding EGFRp temporal response to single EGF pulse and respective
phase-space trajectory. Inset: quasi-potential landscape depicting the
geometry of the ghost state. (c) Corresponding quasi-potential landscape
changes, showing ”ghost” memory state emergence upon signal removal. (d) EGFRp
profiles depicting the dependence on the frequency of the EGF signal. (e)
Schematic representation of cell migration through disrupted and spatially
non-uniform growth factor field for the EGFR network organized at criticality.
Cell remains responsive and continuously adapts the migration trajectory to
newly encountered signals. Notations as in Fig. 2e. Details in (?). (f)
Schematic of a ghost channel consisting of multiple ghosts connected such that
the unstable direction of the preceding ghost is connected to the stable
direction of the following in the sequence. Reproducible trajectories, and
thus reliable computing can be achieved even in the presence of noise. (g)
Hybrid phase-space object consisted of saddles (circles) and ghost, that can
be utilized to generate context-dependent dynamics. See also (?).

As outlined in the introduction, living systems base their behavioral
responses on noisy, often conflicting and incomplete time-varying signals,
regardless of whether considering a migrating cell following chemotactic cues
or a predator hunting its prey. The question therefore arises whether
understanding the mechanism of real-time computations for example, for a given
single cell task would allow to extract possible dynamical basis on which a
generic framework of natural computations can be set.

We recently explored this question by investigating how epithelial cells
navigate in dynamic EGF chemo-attractant fields. To establish long-distance
migration trajectory, cells would require a dynamic memory to integrate the
environmental signals and to quickly adapt the computations following signal
changes, while simultaneously allowing for robustness in the responses. Using
the EGFR network, we have shown that under physiological conditions, cell do
not implement state-dependent computations to solve this task. In contrast,
the system is actually organized at criticality (?, ?), i.e. exactly at the
border between mono- and bistability (Fig. 4a), where in absence of a growth
factor signals, only one steady state, $S_{1}$, is stable. This organization
turns out to be crucial for epithelial cells to sense time-varying signals.
What is then here the mechanism of computation?

To understand how cells compute in which direction to migrate when signals
change over space and time, we used the quasi-potential landscape description
of the EGFR system in conjunction with the system’s state trajectories. As
mentioned before, the quasi-potential landscape changes under influence of
time-varying EGF signals (the simplest being a single EGF pulse). In this
case, upon signal addition, the landscape remodels such that the $S_{1}$ well
is lost and the $S_{2}$ well is stabilized. This landscape remodeling guides
the system’s state trajectory from $S_{1}$ to $S_{2}$, enabling robust EGFR
phosphorylation. However, when $S_{2}$ is lost upon signal removal, on its
place a region with a shallow slope is formed. Due to the topological
characteristics of this region (three attracting and a single repelling
direction, which canalizes the flow (Figs. 4b)), the trajectory is transiently
trapped in a slow region before transiting rapidly to $S_{1}$. This slow
movement of the trajectory is manifested as maintained high EGFRp for a
transient period of time after EGF removal (Fig. 4b,c). Dynamically, this
corresponds to a ghost state (?, ?). The prolonged EGFRp due to the presence
of the ghost therefore acts as a temporal memory that the cell encountered an
EGF signal. In single epithelial cells, this memory lasts $\sim$40min on
average (?), which generates a slow time-scale in the system, despite the sub-
minute timescale of phosphorylation-dephosporylation events.

Cells then use this memory to integrate signals over time. For example, if the
following EGF pulse is received while the trajectory is in the ghost, the time
the systems stays in the ghost will be prolonged, resulting in longer overall
high EGFRp activity. In contrast, if the pulse is received outside of the
ghost, the system re-sets to $S_{1}$ and generates another loop in phase-
space, resulting in distinct temporal EGFRp profile (Figs. 4d). Thus, ghosts
aid differential integration of time-varying EGF signals depending on their
frequency (?). Since ghosts enable for information to be stored over time,
while still maintaining responsiveness to upcoming signals, they correspond to
a mechanism of a ’working memory’ on the level of single cells. We have shown
both theoretically and experimentally that the working memory of the EGFR
network is necessary for epithelial cells to robustly maintain directional
migration when signals are disrupted, while remaining sensitive (adaptive) to
newly encountering chemoattractant signals (Fig. 4e) (?, ?). From a
computational perspective, this example demonstrates that single cells do not
rely on attractor-based computations, but rather utilize transient states i.e.
ghosts, to maintain information about previously sensed signals, whereas the
quasi-stability of these states enables the integration of complex
environmental signals to determine a robust, yet adaptable response. Ghosts
could therefore potentially serve as an elementary mechanism of natural
computations, as they enable to explain real-time response, flexibility,
robustness, and adaptability of natural computations.

However, to capture complexity of natural computations, such as sequential and
context-dependent responses for example (?, ?), the underlying biochemical
and/or neuronal network likely necessitates composite ghost-based phase-space
objects that can give rise to differential dynamics. In a recent theoretical
work, we have taken the first steps and, based on a geometrical perspective on
ghost states, demonstrated that multiple ghosts can be aligned to form ghost
channels (Fig. 4f) or ghost cycles, in the same way that saddles can form
heteroclinic objects (i.e. Fig. 3b). These structures enable reproducibility
of the dynamics, even under large noise intensity, indicating that they are
likely suitable model for natural computations (?). The hypothesis that ghost
scaffold could underline natural computations is further strengthened with the
fact that ghost cycles, for example, are typical for gene-regulatory network
models (?, ?, ?). Additionally, it is straightforward to describe how hybrid
phase-space objects composed of ghost and saddles for example, can arise in
general neuronal or signaling network models (schematic shown in Fig. 4g).
Such structures bear high resemblance to manifolds that emerge from the
analysis of neuronal dynamics of simple animals (?). Hybrid ghost scaffolds
thereby offer a promising new framework for studying natural computation, as
they potentially reproduce dynamical features observed in living systems at
different scales.

## Computations at criticality as a possible road forward

Living systems are no trivial input-output devices, and even single cells or
’simple’ species are capable of a surprising variety of complex behaviors,
such as memory, learning, hunting, social interactions etc. (?, ?, ?, ?, ?).
This property is inevitably related to the nature of biological computations:
living system generally balance between opposed features, for example
robustness and adaptability. The classical computational frameworks, such as
the attractor-based computations that also reflects the functioning of the
computers and machines (in an abstract sense), on the other hand, account only
for one of these features - the robustness. Thus, a cell that would implement
attractor-based computations will remain unresponsive in a dynamic
physiological environment, as we have argued using the example of the EGFR
network. Computation with unstable states or saddles, on the other hand, does
not fulfill the requirement of reproducible transients, such that robustness
in the responsiveness cannot be guaranteed. The ghost states in contrast,
demonstrate a potential to serve as a basic computational ’unit’ of natural
computations. These states typically occur for organization at criticality,
i.e. at the transition between two distinct dynamical regimes (e.g. between
mono- and bistability).

Indeed, criticality has long been a candidate for a governing principle of
living systems, especially brain dynamics (?, ?, ?, ?, ?). In this context
however, criticality has been discussed as a transition between ordered and a
chaotic regime, where the former refers to the (neuronal) activity fading away
quickly into a featureless attractor and the latter to a regime where slightly
similar initial conditions lead to diverging dynamics. The chaotic regime thus
erases any correlation between potentially related inputs and results in
trajectories without computational significance. The so-called criticality or
’edge of chaos’ separates both behaviours, balancing the stability of the
ordered phase and the versatile dynamics of the chaotic dynamics. The chaotic
itinerancy we have discussed however, can be also realized via critical states
(?), which implies that organization at criticality could potentially provide
a link between different computational ’units’ in a dynamical sense that
satisfy the conditions of natural computations. Defining a framework of
computation with trajectories (?) that relies on a combination of chaotic
itinerancy, ghosts and saddles for example, likely has a tremendous potential
not only for the description of biological computations in neuronal and
signaling networks, but also for understanding the peripheral nervous system,
collective responses at the tissue scale or inter-organ communication.

It becomes evident therefore that there’s a need to broaden the existing
definitions of computation, which predominantly revolve around machine
operations. Specifically, a framework for natural computation must encompass
real-time responsiveness, the simultaneous presence of flexibility and
robustness in responses, the facilitation of anticipatory processes that
preemptively prepare for actions, adaptability in computations, and the
enduring capacity for lifelong learning. These attributes stand as fundamental
characteristics of living systems, demanding a more comprehensive
understanding within the framework of computational theory. To which extent
computing with transients can incorporate learning, and in particular on-the-
fly and life-long learning, is another exciting open question. Classical
frameworks of learning rely on mechanisms that change the network topology
(e.g. Hebbian learning generating a ’hard-wired’ memory or ’memory-as-
attractor’ (?)). However, a study aimed at understanding the mechanisms how
artificial neuronal networks (ANNs) learn tasks dependent on time-varying
inputs, for example a memory-demanding, two-point moving average (of input
pulses given at random times) task, showed that the ANN required a set of slow
points in a plane (which we interpret as a multi-dimensional ghost), rather
than a plane-attractor (?), implying that quasi-stable states likely aid
learning. In another example, using the topology of the neuronal network of C.
elegans and single neuron Ca2+-dynamics recordings, it has been also recently
shown that a consistent behavior in a response to time-dependent inputs is
realized via a ’soft-wired’ memory (?) - a memory that does not require
structural changes in the network connectivity, but relies only on the system
dynamics for encoding. Such a ’soft-wired’ memory could emerge from a ghost
state, as we have argued above. These findings therefore suggest that
transiently stable states, in addition to being utilized by cells, likely also
enable artificial and natural neuronal networks to learn and compute complex
tasks. Thus, identifying a mechanism of learning that relies on transiently
stable states could potentially open the avenue to study both theoretically
and experimentally learning on the level of single cells and single-cell
organisms (?, ?, ?, ?). We therefore suggest that developing an integrated
framework of computations relying on transient quasi-stable dynamics can
potentially help to understand how organisms having neuronal networks, but
also single cells and cellular communities actively process information and
learn from their continuously changing environment to adapt and stabilize
their phenotype.

## Box 1: concepts from nonlinear dynamics

• Phase-space: When studying a system of interest (e.g. a cell, neuronal
circuit, etc.), a mathematical description captures the evolution of the
dynamics of the relevant system variables. These can be for e.g. the
concentrations of proteins in a cell, ionic currents in a single neuron, or
the instantaneous firing rates of individual neurons in a neuronal network.
Together, the space spanned by the system’s variables is called the phase
space. A point in phase space represents a joint state of all variables in the
system. • Phase-space flow: Closely neighboring areas in phase space often
exhibit similar dynamics, allowing to understand the system behavior by
visualizing the direction in which the system moves from a collection of
nearby points via arrows. The resulting vector field indicates the flow in
phase space. • Trajectory in phase-space: A set of vectors that describes the
evolution of the state of the system over time. • Attractors: Abstract objects
in phase-space towards which the system’s trajectories converge. The area or
volume in phase-space from which the trajectories are attracted is called
basin of attraction, implying that small perturbations only lead to short-
lived deviations from the asymptotic dynamics. Different types of attractors
include: a stable fixed point (often also called a steady state), stable limit
cycles (oscillations) or chaotic attractors. • Unstable objects: Unstable
objects include unstable fixed points/limit cycles and saddles. When in the
vicinity of an unstable object, the system’s trajectories diverge, in contrast
to attractors. Saddles are special types of unstable objects that attract the
trajectories from two directions along the stable manifold, and repell them
from two directions along the unstable manifold. • Quasi-potential landscape:
Geometric description of the over-all system’s dynamics, where valleys
correspond to attractors, separated by saddles or unstable states. •
Bifurcation: Parameter value for which a sudden qualitative change in the
dynamics of a system occurs. A bifurcation diagram is a graphical depiction of
the bifurcations in a system using the values of parameters and the variables
plotted against each other. • Criticality: System’s parameter organization at
a transition between two dynamical regimes (in the vicinity of a bifurcation),
for example between a fixed point and a limit cycle. • Ghosts: Phase-space
objects characterized with shallow slope in the quasi-potential landscape that
funnels system’s trajectories towards an unstable direction. • Quasi-
stability: Transient stability of the system’s dynamics by trapping the
trajectory in a specific phase-space area, e.g. due to a ghost state. Note
that the time spent in this area can be considerably longer than the time-
scales of the underlying processes. • Noise: Stochastic, random-like
fluctuations in the system dynamics due to processes like Brownian motion,
variability in gene expression etc.

## Acknowledgments

A.K. acknowledges funding by the Lise Meitner Excellence Programme of the Max
Planck Society, D.K. is funded by a fellowship from the European Molecular
Biology Organization (Grant nr. ALTF 310-2021).

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# Generalized concurrence and partial transpose for pure continuous variable
systems of arbitrary degrees of freedom using Lagrange’s identity and wedge
product

Vineeth S. Bhaskara<EMAIL_ADDRESS>Indian Institute of Technology
Guwahati, Guwahati 781 039, Assam, India  Prasanta K. Panigrahi
<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Kolkata, Mohanpur 741 246, West Bengal, India

###### Abstract

Concurrence, introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022
(1997)], provides an important measure of entanglement for a general pair of
qubits that is strictly positive for entangled states and vanishing for all
separable states. We present an extension of concurrence to general continuous
variable pure states of multiple degrees of freedom by generalizing the
Lagrange’s identity and wedge product framework proposed by Bhaskara et al.
[Quantum Inf. Process. 16, 118 (2017)] for pure discrete variable systems in
arbitrary dimensions. A family of faithful entanglement measures, of which
concurrence is a member, is constructed that admit necessary and sufficient
conditions for separability across arbitrary bipartitions, which is shown as a
particular invariance with connections to the partial transpose, uncovering an
inherent geometry of entanglement. This framework may be useful for the
further extensions to mixed states and entanglement in quantum field theories.

###### pacs:

03.65.Ud, 03.67.-a, 03.67.Mn, 02.10.Ud

Quantum entanglement, having played a fundamental role in quantum information
theory, is also finding its context in deeper questions, including, on the
origins of space-time nature-qc-grav ; raamsdonk , quantum field theories
qftent1 , many-body physics revmodentmany , Berry phase berry , phase-
transitions phasetransit , and quantum gravity raamsdonkgravity . Detecting
the presence of such a resource and quantifying it faithfully for the general
case of continuous variable systems would have far reaching applications
beyond novel quantum computation.

Previous works, including, the extensions of Peres-Horodecki criteria by Simon
simonph , Werner simonextended ; using nonlinear maps on matrices by Giedke et
al. giedke ; criteria based on uncertainity principles by Duan et al. cirac ,
Hillery, Nha and Zubairy hzubairy ; nzubairy ; and others for continuous
variable (CV) systems provided necessary conditions for general CV states with
rich descriptions of the geometry of entanglement. In this paper, we provide a
family of faithful measures of entanglement, as an extension to concurrence
ourpaper1 ; Wootters1 , admitting a necessary and sufficient criterion for
separability across arbitrary bipartitions and degrees of freedom for general
pure CV states, which is shown as an invariance under a particular coordinate
transformation with connections to the partial transpose, exposing an inherent
geometry of entanglement in a similar spirit. We comment on the connections to
other widely used measures in the concluding section, with examples of general
pure Gaussian CV states in Appendix B.

We define separability for pure states in the context of CV systems for future
convenience. Consider, a $n$-degree of freedom quantum system. Let $P|Q$ be a
bipartition across the degrees of freedom of this composite(whole) system
$P\cup Q$, with respective infinite-dimensional Hilbert spaces
$\mathcal{H}_{P}$ and $\mathcal{H}_{Q}$ for the states of the sub-systems $P$
and $Q$, then the state space of the composite system is given by the tensor
product $\mathcal{H}=\mathcal{H}_{P}\otimes\mathcal{H}_{Q}$. If a pure state
$|\psi\rangle\in\mathcal{H}$ of the composite system with
$\rho_{\psi}=|\psi\rangle\langle\psi|$ can be written in the form

$|\psi\rangle=|\phi\rangle\otimes|\chi\rangle,\text{ i.e.,
}\rho_{\psi}=\rho_{\phi}\otimes\rho_{\chi},$

where $|\phi\rangle\in\mathcal{H}_{P}$ and $|\chi\rangle\in\mathcal{H}_{Q}$
are the pure states of the sub-systems $P$ and $Q$ respectively with
$\rho_{\phi}=|\phi\rangle\langle\phi|$ and
$\rho_{\chi}=|\chi\rangle\langle\chi|$, then the system is said to be
separable across the bipartition $P|Q$. Otherwise, the sub-systems $P$ and $Q$
are said to be entangled.

Consider a general $n$-degree of freedom pure CV state $|\psi\rangle$ with the
degrees of freedom taking continuous values and labeled by
{$x_{1},~{}x_{2},~{}\ldots,~{}x_{n}$} in an orthonormal basis with
$\langle\vec{x^{\prime}}|\vec{x}\rangle=\delta(\vec{x}-\vec{x^{\prime}})$ and
$\int|\vec{x}\rangle\langle\vec{x}|~{}\mathrm{d}\vec{x}=1$ as

$|\psi\rangle=\int\phi(x_{1},...,x_{n})~{}~{}|x_{1}\rangle\otimes|x_{2}\rangle\otimes\cdots\otimes|x_{n}\rangle~{}~{}\mathrm{d}\vec{x},$
(1)

with $\langle\psi|\psi\rangle=1$, i.e.,

$\int{\phi}^{*}(x_{1},...,x_{n})\phi(x_{1},...,x_{n})~{}\mathrm{d}\vec{x}=1,$

where $\vec{x}=(x_{1},x_{2},...,x_{n})$,
$\mathrm{d}\vec{x}\equiv\mathrm{d}^{n}x=\mathrm{d}x_{1}~{}\mathrm{d}x_{2}~{}...~{}\mathrm{d}x_{n}$,
and $\delta$ is the Dirac delta function of appropriate dimension. Note that
the limits of the integrals are over the appropriate continuous range of
values for the degrees of freedom (commonly, $-\infty$ to $+\infty$) unless
otherwise specified. By $n$-degree of freedom system one could mean, for
instance, a system of $n$-particles in one spatial dimension, or a system of
$k$-particles in $3D$ where $n=3k$, or a quantum optics system having multiple
modes. The physical state $|\psi\rangle$ exists in an infinite-dimensional
Hilbert space spanned by $\\{|x\rangle\\}$. Note that, unlike the case of
discrete variable (DV) systems, the basis states $\\{|x\rangle\\}$ by
themselves are not normalizable and hence non-physical.

Consider the bipartite separability of a particular set $\mathcal{M}$ of $m$
degrees $(m<n)$ out of the $n$ degrees of freedom of the system. Without any
loss of generality, let the $m$-degrees be labeled by {$1,2,...,m$}, so that
the degrees labeled by {$m+1,m+2,...,n$} represent the rest of $(n-m)$-degrees
of freedom belonging to the complement set $\overline{\mathcal{M}}$. The state
$|\psi\rangle$ is said to be separable across the bipartition
$\mathcal{M}|\overline{\mathcal{M}}$ if and only if $|\psi\rangle$ is
expressible as

$\displaystyle\bigg{[}\int\phi_{\mathcal{M}}$
$\displaystyle(x_{1},...,x_{m})~{}|x_{1}...x_{m}\rangle~{}\mathrm{d}^{m}x\bigg{]}~{}\otimes~{}$
$\displaystyle\bigg{[}\int\phi_{\overline{\mathcal{M}}}(x_{m+1},...,x_{n})~{}|x_{m+1}...x_{n}\rangle~{}\mathrm{d}^{n-m}x\bigg{]},$
(2)

where $\phi_{\mathcal{M}}$ is the normalized pure state of the sub-system
$\mathcal{M}$,
$|x_{1}...x_{m}\rangle\equiv|x_{1}\rangle\otimes\cdots\otimes|x_{m}\rangle$,
$\mathrm{d}^{m}x\equiv\mathrm{d}x_{1}~{}\mathrm{d}x_{2}~{}...~{}\mathrm{d}x_{m}$,
and similarly $\phi_{\overline{\mathcal{M}}}$ is the normalized pure state of
the sub-system $\overline{\mathcal{M}}$,
$|x_{m+1}...x_{n}\rangle\equiv|x_{m+1}\rangle\otimes\cdots\otimes|x_{n}\rangle$,
$\mathrm{d}^{n-m}x\equiv\mathrm{d}x_{m+1}~{}\mathrm{d}x_{m+2}~{}...~{}\mathrm{d}x_{n}$.

One may rewrite the state $|\psi\rangle$, defined in Eq. (1), as

$\int\bigg{[}|x_{1}...x_{m}\rangle~{}\otimes~{}\bigg{(}\int\phi(x_{1},...,x_{n})~{}|x_{m+1}...x_{n}\rangle~{}\mathrm{d}^{n-m}x\bigg{)}~{}\mathrm{d}^{m}x\bigg{]}.$
(3)

By noting that

$\displaystyle\langle
x^{\prime}_{1}x^{\prime}_{2}...x^{\prime}_{m}|\psi\rangle$
$\displaystyle=\int\phi(x_{1},...,x_{n})~{}\underbrace{\langle
x^{\prime}_{1}x^{\prime}_{2}...x^{\prime}_{m}|x_{1}x_{2}...x_{n}\rangle}_{\delta(x_{1}-x^{\prime}_{1},...,x_{m}-x^{\prime}_{m})~{}|x_{m+1}...x_{n}\rangle}~{}\mathrm{d}^{n}x$
(4)
$\displaystyle=\int\phi(x^{\prime}_{1},x^{\prime}_{2},...,x^{\prime}_{m},x_{m+1},...,x_{n})~{}|x_{m+1}...x_{n}\rangle~{}\mathrm{d}^{n-m}x,$
(5)

one may express $|\psi\rangle$, using Eq. (3), as

$|\psi\rangle=\int\bigg{[}|x_{1}...x_{m}\rangle~{}\otimes~{}\bigg{(}\langle
x_{1}x_{2}...x_{m}|\psi\rangle\bigg{)}~{}\mathrm{d}^{m}x\bigg{]}.$ (6)

Observe that, for the separability of $|\psi\rangle$ across
$\mathcal{M}|\overline{\mathcal{M}}$, each of the vectors $\langle
x_{1}x_{2}...x_{m}|\psi\rangle$ in Eq. (6) must be mutually “parallel” for the
$m$-degree of freedom state to factor out, i.e., for each
$\vec{r}=(r_{1},r_{2},...,r_{m})$ one needs

$\langle r_{1}r_{2}...r_{m}|\psi\rangle=c_{\vec{r}\vec{s}}\langle
s_{1}s_{2}...s_{m}|\psi\rangle$

for any $\vec{s}=(s_{1},s_{2},...,s_{m})$ where $c_{\vec{r}\vec{s}}$ is some
complex scalar, for separability. This becomes evident once one choses, say,
$(s_{1},s_{2},...,s_{m})=(0,0,...,0)=\vec{0}$, so that

$\langle
r_{1}r_{2}...r_{m}|\psi\rangle=c(r_{1},r_{2},...,r_{m})~{}\langle\underbrace{00...0}_{m}|\psi\rangle$

where $c(r_{1},r_{2},...,r_{m})$ is some complex scalar. Substituting this
back in Eq. (6), one can see the state becomes separable (with constant $k$
ensuring the normalization of each of the sub-system’s state) as

$\displaystyle|\psi\rangle$
$\displaystyle=\int\bigg{[}|x_{1}...x_{m}\rangle~{}\otimes~{}c(x_{1},x_{2},...,x_{m})~{}\langle
00...0|\psi\rangle~{}\mathrm{d}^{m}x\bigg{]}$
$\displaystyle=\bigg{(}\underbrace{k\int|x_{1}...x_{m}\rangle~{}c(x_{1},x_{2},...,x_{m})~{}\mathrm{d}^{m}x}_{state~{}of~{}\mathcal{M}}\bigg{)}\otimes\underbrace{\frac{1}{k}\langle
00...0|\psi\rangle,}_{state~{}of~{}\overline{\mathcal{M}}}$ (7)

where

$\langle
00...0|\psi\rangle=\int\phi(0,0,...,0,x_{m+1},...,x_{n})~{}|x_{m+1}...x_{n}\rangle~{}\mathrm{d}^{n-m}x,$

and

$\displaystyle k^{2}$
$\displaystyle=\int\phi^{*}(0,...,0,x_{m+1},...,x_{n})\phi(0,...,0,x_{m+1},...,x_{n})~{}\mathrm{d}^{n-m}x,$
$\displaystyle\text{or}~{}\left[\int
c^{*}(x_{1},...,x_{m})~{}c(x_{1},...,x_{m})~{}\mathrm{d}^{m}x\right]^{-1}.$
(8)

Interestingly, therefore, one may note that even if a single “pair” of
elements of the continuous, infinite set of vectors {$\langle
x_{1}x_{2}...x_{m}|\psi\rangle$} over the continuous variables
$x_{1},...,x_{m}$ is not mutually parallel, this adds to the presence of
entanglement. We express this condition for separability using the notion of a
wedge product extended to multivariable complex-valued function spaces based
on the framework proposed in Ref. ourpaper1  for general pure discrete
variable systems in arbitrary dimensions.

In geometric algebra geoalgcam , the wedge product of two vectors is seen as a
particular generalization of cross product to higher dimensions. We construct
such a notion for the case of complex infinite-dimensional vector spaces.
Consider two vectors $\vec{a}$ and $\vec{b}$ in the complex, infinite-
dimensional space as

$\vec{a}=\int
f(\vec{x})~{}|\vec{x}\rangle~{}\mathrm{d}\vec{x},~{}~{}\vec{b}=\int
g(\vec{x})~{}|\vec{x}\rangle~{}\mathrm{d}\vec{x}$

in the continuous orthonormal basis set $\\{|\vec{x}\rangle\\}$ with
$\langle\vec{x^{\prime}}|\vec{x}\rangle=\delta(\vec{x}-\vec{x^{\prime}})$ and
$\int|\vec{x}\rangle\langle\vec{x}|~{}\mathrm{d}\vec{x}=1$ where
$\vec{x}=(x_{1},...,x_{n})$. Then the wedge product of $\vec{a}$ and $\vec{b}$
in the interval $(\vec{t},\vec{u})$ is defined as a bivector in an “exterior”
space with continuous basis set
$\\{|\vec{x}\rangle\wedge|\vec{x^{\prime}}\rangle\\}_{x^{\prime}>x}$,
stipulating that
$|\vec{x}\rangle\wedge|\vec{x^{\prime}}\rangle=-|\vec{x^{\prime}}\rangle\wedge|\vec{x}\rangle$
and $|\vec{x}\rangle\wedge|\vec{x}\rangle=0$, as

$\vec{a}\wedge\vec{b}=\int\limits_{\vec{x}=\vec{t}}^{\vec{u}}~{}\int\limits_{\vec{x^{\prime}}=\vec{x}}^{\vec{u}}\left[f(\vec{x})g(\vec{x^{\prime}})-f(\vec{x^{\prime}})g(\vec{x})\right]|\vec{x}\rangle\wedge|\vec{x^{\prime}}\rangle~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x},$
(9)

where $\vec{t}=(t_{1},t_{2},...,t_{n})$, and
$\vec{u}=(u_{1},u_{2},...,u_{n})$. Therefore, one may note
$\vec{a}\wedge\vec{b}=0\iff\vec{b}=k\vec{a}$, and
$\vec{a}\wedge\vec{b}=-\vec{b}\wedge\vec{a}$, by definition, for some complex
scalar $k$ and vectors $\vec{a},~{}\vec{b}$.

This notion of an extended wedge product allows one to write the separability
condition in a compact and useful form. Since one requires that each of the
vectors in the continuous set {$\langle x_{1}x_{2}...x_{m}|\psi\rangle$} to be
mutually “parallel” for the separability across
$\mathcal{M}|\overline{\mathcal{M}}$, their mutual wedge products must vanish,
equivalently, for separability. This is a necessary and sufficient condition
for separability as noted before. Hence, one may construct a family of
faithful measures of entanglement, parametrized by $f$, $p$, and $q$, across
the bipartition as

$\mathcal{E}_{\mathcal{M}}=\\\ \left[\int\int f\left(\left|\left|\langle
y^{\prime}_{1}y^{\prime}_{2}...y^{\prime}_{m}|\psi\rangle\wedge\langle
y_{1}y_{2}...y_{m}|\psi\rangle\right|\right|_{\text{p}}\right)\mathrm{d}^{m}y~{}\mathrm{d}^{m}y^{\prime}\right]^{1/q},$
(10)

where $f:\mathbb{R}\to\mathbb{R}$ with $f(x)=0$ iff $x=0$ so that
$\mathcal{E}_{\mathcal{M}}=0$ $\iff$ separability, in addition to $f$ being a
monotonic and strictly increasing function in $\mathbb{R}^{+}$ and
$q\in\mathbb{R}^{+}$ so that $\mathcal{E}_{\mathcal{M}}>0$ measures
entanglement faithfully; the $p$-norm is computed in the basis
$\\{|x_{m+1}...x_{n}\rangle\wedge|x^{\prime}_{m+1}...x^{\prime}_{n}\rangle\\}_{x^{\prime}>x}$,
and, from Eq. (5), $\langle
y^{\prime}_{1}y^{\prime}_{2}...y^{\prime}_{m}|\psi\rangle\wedge\langle
y_{1}y_{2}...y_{m}|\psi\rangle\equiv$

$\displaystyle\int\limits_{\vec{x}=-\infty}^{+\infty}~{}\int\limits_{\vec{x^{\prime}}=\vec{x}}^{+\infty}\bigg{[}\phi(y^{\prime}_{1}$
$\displaystyle,y^{\prime}_{2},...,y^{\prime}_{m},x_{m+1},...,x_{n})~{}\phi(y_{1},y_{2},...,y_{m},x^{\prime}_{m+1},...,x^{\prime}_{n})~{}~{}-$
$\displaystyle\phi(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x^{\prime}_{m+1},...,x^{\prime}_{n})~{}\phi(y_{1},y_{2},...,y_{m},x_{m+1},...,x_{n})\bigg{]}~{}|x_{m+1}...x_{n}\rangle\wedge|x^{\prime}_{m+1}...x^{\prime}_{n}\rangle~{}\mathrm{d}^{n-m}x^{\prime}~{}\mathrm{d}^{n-m}x.$
(11)

Therefore, Eq. (10) provides a family of functions parametrized on $f,p,q$,
functional on the wavefunction, which vanish only under the case of
separability, in addition to measuring entanglement faithfully. This
construction may be extended to the DV case ourpaper1  analogously. One
retrieves a generalized version of concurrence, $E_{\mathcal{M}}$, coinciding
with Wootters’s concurrence Wootters1  defined by
$C(\psi)=|\langle\psi|\tilde{\psi}\rangle|$ (where
$|\tilde{\psi}\rangle=\sigma_{y}|\psi^{*}\rangle$,
$\sigma_{y}=\left({\begin{array}[]{cc}0&-i\\\ i&0\\\ \end{array}}\right)$ and
$|\psi^{*}\rangle$ is the complex conjugate of $|\psi\rangle$) for a two-qubit
system in the discrete variable approximation when $f(x)=2x^{2}$, $p=2$, and
$q=2$ is chosen, i.e.,

$E^{2}_{\mathcal{M}}=2\int\int\left|\left|\langle
y^{\prime}_{1}y^{\prime}_{2}...y^{\prime}_{m}|\psi\rangle\wedge\langle
y_{1}y_{2}...y_{m}|\psi\rangle\right|\right|^{2}~{}\mathrm{d}^{m}y~{}\mathrm{d}^{m}y^{\prime},$
(12)

which may be explicitly written in terms of the wavefunction, using Eq. (11),
as

$\displaystyle
E^{2}_{\mathcal{M}}=\int\int\int\int~{}~{}\bigg{|}\phi(y^{\prime}_{1},y^{\prime}_{2},...,$
$\displaystyle
y^{\prime}_{m},x_{m+1},...,x_{n})~{}\phi(y_{1},y_{2},...,y_{m},x^{\prime}_{m+1},...,x^{\prime}_{n})~{}~{}-$
$\displaystyle\phi(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x^{\prime}_{m+1},...,x^{\prime}_{n})~{}\phi(y_{1},y_{2},...,y_{m},x_{m+1},...,x_{n})\bigg{|}^{2}~{}\mathrm{d}^{n-m}x^{\prime}~{}\mathrm{d}^{n-m}x~{}\mathrm{d}^{m}y~{}\mathrm{d}^{m}y^{\prime}.$
(13)

Therefore, the necessary and sufficient condition for separability of
$|\psi\rangle$ across the bipartition $\mathcal{M}|\overline{\mathcal{M}}$ is
given by $E^{2}_{\mathcal{M}}=0$, i.e.,

$\displaystyle\phi$
$\displaystyle(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x_{m+1},...,x_{n})\phi(y_{1},y_{2},...,y_{m},x^{\prime}_{m+1},...,x^{\prime}_{n})=$
$\displaystyle\phi(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x^{\prime}_{m+1},...,x^{\prime}_{n})\phi(y_{1},y_{2},...,y_{m},x_{m+1},...,x_{n}),$
(14)

which may elegantly be rewritten as

$\displaystyle\widetilde{\Phi}(\text{X})$
$\displaystyle=\widetilde{\Phi}(\Lambda_{m}\text{X}),$ (15)

where $\text{X}_{2n\times 1}\equiv\left[\begin{matrix}\vec{x}_{1}\\\
\vec{x}_{2}\end{matrix}\right]$ with {$\vec{x}_{1},\vec{x}_{2}$} being some
general coordinates of the system, and $\widetilde{\Phi}(\text{X})$ being
defined as

$\displaystyle\widetilde{\Phi}(\text{X})\equiv\langle\text{X}|\mathbb{N}_{1}^{\top}|\psi\rangle\langle\text{X}|\mathbb{N}_{2}^{\top}|\psi\rangle=\phi(\mathbb{N}_{1}\text{X})~{}\phi(\mathbb{N}_{2}\text{X}),$
(16)

with $\mathbb{N}_{1}=\left[\begin{matrix}\mathbbm{1}_{n\times n}&0_{n\times
n}\end{matrix}\right]_{n\times 2n}$,
$\mathbb{N}_{2}=\left[\begin{matrix}0_{n\times n}&\mathbbm{1}_{n\times
n}\end{matrix}\right]_{n\times 2n}$,

$\displaystyle\mathbb{M}\equiv\begin{bmatrix}\mathbbm{1}_{m\times m}&\\\
&{0}_{(n-m)\times(n-m)}\end{bmatrix}_{n\times n},\text{ and}$
$\displaystyle\Lambda_{m}\equiv\begin{bmatrix}\mathbbm{1}_{n\times
n}-\mathbb{M}_{n\times n}&\mathbb{M}_{n\times n}\\\ \mathbb{M}_{n\times
n}&\mathbbm{1}_{n\times n}-\mathbb{M}_{n\times n}\end{bmatrix}_{2n\times 2n}$
(17)

($\mathbbm{1}$ being the identity matrix of appropriate dimensions). Hence,
one needs $\Phi$, as defined above, to be invariant under the transformation
$\text{X}\rightarrow\Lambda_{m}\text{X}$ for separability, in which case one
may write down the pure states of the separable sub-systems using Eq. (7).
This is one of the central results of the paper valid for general pure CV
states.

One may realize a connection between the $\Lambda_{m}$ matrix and the partial
transpose operation by defining
$\widetilde{\rho}=|\psi\rangle\langle\psi^{*}|=$

$\displaystyle\int\phi(\mathbb{N}_{1}\text{X})~{}\phi(\mathbb{N}_{2}\text{X})~{}\mathbb{N}_{1}|\text{X}\rangle\langle\text{X}|\mathbb{N}_{2}^{\top}~{}\mathrm{d}\text{X},$
(18)

and noticing that the matrix element could be written as

$\widetilde{\rho}_{\text{X}}=\widetilde{\Phi}(\text{X})~{}\mathbb{N}_{1}|\text{X}\rangle\langle\text{X}|\mathbb{N}_{2}^{\top},$

since $\mathbb{N}_{1}|\text{X}\rangle\equiv|\vec{x}_{1}\rangle$, and
$\mathbb{N}_{2}|\text{X}\rangle\equiv|\vec{x}_{2}\rangle$. Under the partial
transpose operation
$PT\equiv\mathbbm{1}\otimes\text{T}_{\overline{\mathcal{M}}}$, where the
transposition is done on the $\overline{\mathcal{M}}$ sub-system, the matrix
element changes to

$\widetilde{\rho}_{\text{X}}\xrightarrow{PT}\widetilde{\Phi}(\text{X})~{}\mathbb{N}_{1}\Lambda_{m}|\text{X}\rangle\langle\text{X}|\Lambda_{m}^{\top}\mathbb{N}_{2}^{\top},$

which under the condition of separability is also equal to

$\widetilde{\rho}_{\text{X}}\xrightarrow{PT}\widetilde{\Phi}(\Lambda_{m}\text{X})~{}\mathbb{N}_{1}\Lambda_{m}|\text{X}\rangle\langle\text{X}|\Lambda_{m}^{\top}\mathbb{N}_{2}^{\top}=\widetilde{\rho}_{\Lambda_{m}\text{X}},$

by Eq. (15). Therefore under separability, the partial transpose takes
$\text{X}\rightarrow\Lambda_{m}\text{X}$ with respect to the matrix elements
of $\widetilde{\rho}$.

The separability condition and the generalized concurrence may be stated in
terms of $\widetilde{\rho}$ by considering the partial transpose as

$\displaystyle\widetilde{\rho}_{PT}=\int\phi(\mathbb{N}_{1}\text{X})~{}\phi(\mathbb{N}_{2}\text{X})~{}\mathbb{N}_{1}\Lambda_{m}|\text{X}\rangle\langle\text{X}|\Lambda_{m}^{\top}\mathbb{N}_{2}^{\top}~{}\mathrm{d}\text{X}.$
(19)

Since the integration is on X, $\widetilde{\rho}_{PT}$ does not change under
the substitution X$\rightarrow\Lambda_{m}$X. Therefore,

$=\int\phi(\mathbb{N}_{1}\Lambda_{m}\text{X})~{}\phi(\mathbb{N}_{2}\Lambda_{m}\text{X})~{}\mathbb{N}_{1}|\text{X}\rangle\langle\text{X}|\mathbb{N}_{2}^{\top}~{}\mathrm{d}\text{X},$

noting that $\mathrm{d}$X$=\mathrm{d}(\Lambda_{m}$X$)$, and
$\Lambda_{m}^{2}=\mathbbm{1}$. But by Eqs. (15) and (18), under separability,
this is identical to $\widetilde{\rho}$. Therefore, $\widetilde{\rho}$ being
invariant under the partial transposition operation is a necessary and
sufficient criterion for separability, and considering

$\displaystyle\widetilde{\rho}-\widetilde{\rho}_{PT}=$
$\displaystyle\int\bigg{[}\phi(\mathbb{N}_{1}\text{X})\phi(\mathbb{N}_{2}\text{X})-\phi(\mathbb{N}_{1}\Lambda_{m}\text{X})\phi(\mathbb{N}_{2}\Lambda_{m}\text{X})\bigg{]}\mathbb{N}_{1}|\text{X}\rangle\langle\text{X}|\mathbb{N}_{2}^{\top}~{}\mathrm{d}\text{X},$
(20)

it is evident that the concurrence defined in Eq. (13) can be instead
interpreted as

$\displaystyle
E^{2}_{\mathcal{M}}=||\widetilde{\rho}-\widetilde{\rho}_{PT}||^{2}_{\text{HS}},$
(21)

where $||\cdot||_{\text{HS}}$ is the Hilbert-Schmidt norm for matrices. This
would be shown to be related to the Hilbert-Schmidt distance of the reduced
density matrix from the maximally mixed state in the subsequent section using
the Lagrange’s identity.

One may note the dynamics in the phase space by considering the Wigner
transform on both the sides of Eq. (15). Noting that $\vec{x}_{1}$ and
$\vec{x}_{2}$ are independent, the Wigner transform of the LHS is

$\displaystyle\widetilde{W}(\text{X},\text{P})$
$\displaystyle=\left(\frac{1}{\pi}\right)^{2n}\int\text{e}^{2i\text{P}\cdot\text{Y}}\widetilde{\Phi}(\text{X}-\text{Y})\widetilde{\Phi}^{*}(\text{X}+\text{Y})~{}\mathrm{d}\text{Y}$
$\displaystyle=W(\vec{x}_{1},\vec{p}_{1})~{}W(\vec{x}_{2},\vec{p}_{2}),$ (22)

where $\text{Y}\equiv\left[\begin{matrix}\vec{y}_{1}\\\
\vec{y}_{2}\end{matrix}\right]$,
$\text{P}\equiv\left[\begin{matrix}\vec{p}_{1}\\\
\vec{p}_{2}\end{matrix}\right]$, and $W$ is the corresponding Wigner function
of the given pure state $|\psi\rangle$. Under separability, using Eq. (15),
$\widetilde{W}(\text{X},\text{P})$ from Eq. (22) may equivalently be written
as

$\displaystyle\left(\frac{1}{\pi}\right)^{2n}\int\text{e}^{2i\text{P}\cdot\text{Y}}\widetilde{\Phi}(\Lambda_{m}(\text{X}-\text{Y}))\widetilde{\Phi}^{*}(\Lambda_{m}(\text{X}+\text{Y}))~{}\mathrm{d}\text{Y},$
(23)

and since the integration runs on Y, transforming Y$\rightarrow\Lambda_{m}$Y
does not change the integral. Therefore, under separability,
$\widetilde{W}(\text{X},\text{P})$

$\displaystyle=\left(\frac{1}{\pi}\right)^{2n}\int\text{e}^{2i\text{P}\cdot\Lambda_{m}\text{Y}}\widetilde{\Phi}(\Lambda_{m}\text{X}-\text{Y})\widetilde{\Phi}^{*}(\Lambda_{m}\text{X}+\text{Y})~{}\mathrm{d}\text{Y}$
$\displaystyle=\widetilde{W}(\Lambda_{m}\text{X},\Lambda_{m}\text{P}),$ (24)

noting that P$\cdot\Lambda_{m}$Y = $\Lambda_{m}$P$\cdot$Y,
$\mathrm{d}$Y$=\mathrm{d}(\Lambda_{m}$Y$)$, and $\Lambda_{m}^{2}=\mathbbm{1}$.
Therefore, $\widetilde{W}(\xi)$ being invariant under the coordinate
transformation $\xi\rightarrow\left({\begin{array}[]{cc}\Lambda_{m}&0\\\
0&\Lambda_{m}\\\ \end{array}}\right)\xi$, where $\xi_{4n\times
1}=\left[\begin{matrix}\text{X}\\\ \text{P}\end{matrix}\right]$, is a
necessary and sufficient condition for separability.

One may simplify Eq. (13) using an extension of the Lagrange’s identity lars ;
brahma ; linearalg ; oliverpaper  for vectors in $\mathbb{C}^{m}$ ourpaper1
to multivariate complex-valued functions. Consider, two complex-valued
functions in $\mathbb{R}^{n}$, $f(\vec{x})$ and $g(\vec{x})$. Then the
Lagrange’s identity takes the form, analogous to
$\|\overrightarrow{a}\|^{2}\|\overrightarrow{b}\|^{2}-|\overrightarrow{a}\cdot\overrightarrow{b}|^{2}=\|\overrightarrow{a}\wedge\overrightarrow{b}\|^{2}$
for vectors $\vec{a}$, $\vec{b}$ in $\mathbb{C}^{m}$, as

$\displaystyle\left(~{}\int\limits_{\vec{t}}^{\vec{u}}|f(\vec{x})|^{2}~{}\mathrm{d}\vec{x}~{}\right)\left(~{}\int\limits_{\vec{t}}^{\vec{u}}|g(\vec{x})|^{2}~{}\mathrm{d}\vec{x}~{}\right)-\left|~{}\int\limits_{\vec{t}}^{\vec{u}}f(\vec{x}){g^{*}(\vec{x})}~{}\mathrm{d}\vec{x}~{}\right|^{2}=\int\limits_{\vec{x}=\vec{t}}^{\vec{u}}~{}\int\limits_{\vec{x^{\prime}}=\vec{x}}^{\vec{u}}\left|~{}f(\vec{x})g(\vec{x^{\prime}})-f(\vec{x^{\prime}})g(\vec{x})~{}\right|^{2}~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x},$
(25)

where ∗ represents complex conjugation (see Appendix A for proof).

By this identity, one may rewrite the entanglement measure
$E^{2}_{\mathcal{M}}$ constructed in Eq. (13) as

$E^{2}_{\mathcal{M}}=2\int\int\bigg{[}\left(~{}\int|\phi(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x_{m+1},...,x_{n})|^{2}~{}\mathrm{d}^{n-m}x~{}\right)\left(~{}\int|\phi(y_{1},y_{2},...,y_{m},x_{m+1},...,x_{n})|^{2}~{}\mathrm{d}^{n-m}x~{}\right)-\\\
\left|~{}\int\phi(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x_{m+1},...,x_{n})\phi^{*}(y_{1},y_{2},...,y_{m},x_{m+1},...,x_{n})~{}\mathrm{d}^{n-m}x~{}\right|^{2}\bigg{]}~{}\mathrm{d}^{m}y~{}\mathrm{d}^{m}y^{\prime},~{}\text{or}$
(26)

$~{}~{}=2\bigg{[}1-\int\int~{}\left|~{}\int\phi(y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{m},x_{m+1},...,x_{n})\phi^{*}(y_{1},y_{2},...,y_{m},x_{m+1},...,x_{n})~{}\mathrm{d}^{n-m}x~{}\right|^{2}~{}\mathrm{d}^{m}y~{}\mathrm{d}^{m}y^{\prime}\bigg{]},$
(27)

noting the normalization of $\phi$. This may elegantly be written in terms of
$\Phi$ and $\Lambda_{m}$ (defined by Eqs. (16) and (17)) as

$\displaystyle
E^{2}_{\mathcal{M}}=2\left[1-\text{Re}\int\Phi(\text{X})~{}\Phi^{*}(\Lambda_{m}\text{X})~{}\mathrm{d}^{2n}\text{X}\right].$
(28)

Hence, for maximal entanglement, one needs $\Phi(\text{X})$ and
$\Phi(\Lambda_{m}\text{X})$ to be orthogonal, i.e., their inner product must
vanish, so that $E^{2}_{\mathcal{M}}$ takes the maximum value of $2$. On the
contrary, when $\Phi(\text{X})=\Phi(\Lambda_{m}\text{X})$, their inner product
takes the maximum overlap of $1$, thereby, implying separability with
$E^{2}_{\mathcal{M}}=0$. This is one of the important results of the paper on
the geometry of entanglement in CV systems.

Consider the pure state density matrix of the system represented by
$|\psi\rangle$ in Eq. (1) as $\rho=|\psi\rangle\langle\psi|=$

$\displaystyle\int\int\phi(x_{1},...,x_{n})\phi^{*}(x^{\prime}_{1},...,x^{\prime}_{n})|x_{1}...x_{n}\rangle\langle
x^{\prime}_{1}...x^{\prime}_{n}|~{}~{}\mathrm{d}\vec{x}~{}\mathrm{d}\vec{x^{\prime}}.$
(29)

One may define the reduced density matrix $\rho^{\mathcal{M}}$ of
$\mathcal{M}$ by tracing out $\overline{\mathcal{M}}$ as $\rho^{\mathcal{M}}\
\equiv\ \hbox{Tr}_{\overline{\mathcal{M}}}\;(\rho)$

$\displaystyle=\int\langle
k_{m+1}...k_{n}|~{}\rho~{}|k_{m+1}...k_{n}\rangle~{}\mathrm{d}^{n-m}k$
$\displaystyle=\int\int~{}\rho^{\mathcal{M}}_{x,x^{\prime}}~{}|x_{1}x_{2}...x_{m}\rangle\langle
x^{\prime}_{1}x^{\prime}_{2}...x^{\prime}_{m}|~{}~{}\mathrm{d}^{m}x~{}\mathrm{d}^{m}x^{\prime},$
(30)

where $\rho^{\mathcal{M}}_{x,x^{\prime}}=\langle
x_{1}...x_{m}|\rho^{\mathcal{M}}|x^{\prime}_{1}...x^{\prime}_{m}\rangle$

$\displaystyle=\int\phi(x_{1},...,x_{m},k_{m+1},...,k_{n})\phi^{*}(x^{\prime}_{1},...,x^{\prime}_{m},k_{m+1},...,k_{n})\mathrm{d}\vec{k}$
(31)

is the matrix element. Considering $(\rho^{\mathcal{M}})^{2}=$

$\displaystyle\int\int\int\rho^{\mathcal{M}}_{x,y}~{}\rho^{\mathcal{M}}_{y,y^{\prime}}~{}|x_{1}...x_{m}\rangle\langle
y^{\prime}_{1}...y^{\prime}_{m}|~{}\mathrm{d}^{m}x~{}\mathrm{d}^{m}y~{}\mathrm{d}^{m}y^{\prime},$
(32)

one may conclude the “diagonal” elements of $(\rho^{\mathcal{M}})^{2}$ to be

$\displaystyle\langle
k_{1}k_{2}...k_{m}|(\rho^{\mathcal{M}})^{2}|k_{1}k_{2}...k_{m}\rangle=\int\rho^{\mathcal{M}}_{k,y}~{}\rho^{\mathcal{M}}_{y,k}~{}\mathrm{d}^{m}y.$
(33)

Thus, considering

$\displaystyle
2\left[1-\hbox{Tr}\left[\left(\rho^{\mathcal{M}}\right)^{2}\right]\right]$
$\displaystyle=2\left[1-\int\langle
k_{1}...k_{m}|(\rho^{\mathcal{M}})^{2}|k_{1}...k_{m}\rangle~{}\mathrm{d}^{m}k\right]$
$\displaystyle=2\left[1-\int\int\rho^{\mathcal{M}}_{k,y}~{}\rho^{\mathcal{M}}_{y,k}~{}\mathrm{d}^{m}y~{}\mathrm{d}^{m}k\right]$
(34)

and using Eq. (31), one realizes that Eq. (34) equals $E^{2}_{\mathcal{M}}$
from Eq. (27), therefore,
$E^{2}_{\mathcal{M}}=2\left[1-\hbox{Tr}\left[\left(\rho^{\mathcal{M}}\right)^{2}\right]\right]$.
Since for CV systems
$0\leq\hbox{Tr}\left[\left(\rho^{\mathcal{M}}\right)^{2}\right]\leq 1$ adesso1
, therefore, $0\leq E_{\mathcal{M}}\leq\sqrt{2}$. Since
$\hbox{Tr}\left[\left(\rho^{\mathcal{M}}\right)^{2}\right]$ attains the
minimum iff $\rho^{\mathcal{M}}$ is maximally mixed, this condition,
therefore, implies maximal entanglement across
$\mathcal{M}|\overline{\mathcal{M}}$ with $E_{\mathcal{M}}=\sqrt{2}$. Hence,
the family of faithful measures $\\{\mathcal{E}_{\mathcal{M}}\\}_{f,p,q}$,
defined in Eq. (10), also take the maximum value iff $\rho^{\mathcal{M}}$ is
maximally mixed, since any chosen $f$ is monotonic and strictly increasing in
$\mathbb{R}^{+}$. Conversely, one may define a given state $\rho$ to be
maximally mixed iff it maximizes the family of measures
$\\{\mathcal{E}(\rho)\\}_{f,p,q}$.

One may also show that the concurrence and the PPT-criterion are equivalent
conditions for separability that are necessary and sufficient for pure states
noting the relation of the partial transpose with the reduced density matrix.
Consider the partial transpose $\rho_{PT}=$

$\displaystyle\int\phi(\vec{x})\phi^{*}(\vec{y})|x_{1}...x_{m}y_{m+1}...y_{n}\rangle\langle
y_{1}...y_{m}x_{m+1}...x_{n}|~{}\mathrm{d}\vec{x}~{}\mathrm{d}\vec{y}.$ (35)

The square of the partial transpose is related to the reduced density matrix
as, $\rho_{PT}^{2}$

$\displaystyle=\int\int\phi(\vec{x})\phi^{*}(\vec{y})\phi(y_{1}...y_{m}a_{m+1}...a_{n})\phi^{*}(b_{1}...b_{m}x_{m+1}...x_{n})$
$\displaystyle~{}~{}~{}~{}~{}|x_{1}...x_{m}y_{m+1}...y_{n}\rangle\langle
b_{1}...b_{m}a_{m+1}...a_{n}|~{}\mathrm{d}\vec{x}~{}\mathrm{d}\vec{y}~{}\mathrm{d}^{n-m}a~{}\mathrm{d}^{m}b$
$\displaystyle=\left[\int\phi(\vec{x})\phi^{*}(b_{1}...b_{m}x_{m+1}...x_{n})|x_{1}...x_{m}\rangle\langle
b_{1}...b_{m}|~{}\mathrm{d}\vec{x}~{}\mathrm{d}^{m}b\right]$
$\displaystyle\otimes\bigg{[}\int\phi(y_{1}...y_{m}a_{m+1}...a_{n})\phi^{*}(\vec{y})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|y_{m+1}...y_{n}\rangle\langle
a_{m+1}...a_{n}|~{}\mathrm{d}\vec{y}~{}\mathrm{d}^{n-m}a\bigg{]}$
$\displaystyle=\rho^{\mathcal{M}}\otimes\rho^{\overline{\mathcal{M}}~{}\dagger},$
(36)

using Eq. (31). Similarly, noting $\widetilde{\rho}_{PT}=$

$\displaystyle\int\phi(\vec{x})\phi(\vec{y})|x_{1}...x_{m}y_{m+1}...y_{n}\rangle\langle
y_{1}...y_{m}x_{m+1}...x_{n}|~{}\mathrm{d}\vec{x}~{}\mathrm{d}\vec{y},$

one may also evaluate
$\widetilde{\rho}_{PT}\widetilde{\rho}_{PT}^{~{}\dagger}$ to be equal to
$\rho^{\mathcal{M}}\otimes\rho^{\overline{\mathcal{M}}}$. Therefore, noting
that a reduced density matrix is Hermitian, one has

$\displaystyle\rho_{PT}^{2}=\rho^{\mathcal{M}}\otimes\rho^{\overline{\mathcal{M}}}=\widetilde{\rho}_{PT}\widetilde{\rho}_{PT}^{~{}\dagger}$
(37)

for any given $\rho$. Since
$\widetilde{\rho}\equiv|\psi\rangle\langle\psi^{*}|$, we have
$\widetilde{\rho}~{}\widetilde{\rho}^{~{}\dagger}=\rho$. Therefore, under
separability, since $\widetilde{\rho}$ is invariant under partial
transposition, we have
$\widetilde{\rho}_{PT}~{}\widetilde{\rho}_{PT}^{~{}\dagger}=\widetilde{\rho}~{}\widetilde{\rho}^{~{}\dagger}$.
Hence, an equivalent necessary and sufficient condition of separability in
terms of the square of the partial transpose can be written, noting Eq. (37),
as

$\displaystyle\rho_{PT}^{2}=\rho.$ (38)

Considering,
$\hbox{Tr}(\rho_{PT}^{4})=\hbox{Tr}[(\rho^{\mathcal{M}})^{2}\otimes(\rho^{\overline{\mathcal{M}}})^{2}]=(\hbox{Tr}[(\rho^{\mathcal{M}})^{2}])^{2}$,
one may rewrite concurrence as

$\displaystyle E^{2}_{\mathcal{M}}$
$\displaystyle=2\left[1-\sqrt{\hbox{Tr}(\rho_{PT}^{4})}\right]$ (39)
$\displaystyle=2\left[1-\sqrt{\int
W_{PT}^{4}(\vec{x},\vec{p})~{}\mathrm{d}\vec{x}~{}\mathrm{d}\vec{p}}~{}\right],$
(40)

given the Wigner function $W$ of the pure state. For the case of two-degrees
of freedom, therefore, if $\int
W_{PT}^{4}(x_{1},p_{1},x_{2},p_{2})~{}\mathrm{d}\vec{x}~{}\mathrm{d}\vec{p}=1$
for a given pure state, one may write
$W_{PT}(x_{1},p_{1},x_{2},p_{2})=W(x_{1},p_{1},x_{2},-p_{2})$ as shown by
Simon simonph  under separability. Observing that $\hbox{Tr}(\rho_{PT})=1$ and
$\hbox{Tr}(\rho_{PT}^{2})=1$ (by Eq. (37)) for any given pure density matrix
$\rho$, one may note that, when $\rho_{PT}$ is positive semi-definite, the
eigenvalues must be either 0, or 1 with multiplicity one in the DV case. So
any higher powers of $\rho_{PT}$ would also have unit trace. Hence by Eq.
(39), $\rho$ is separable iff $\rho_{PT}$ is positive semi-definite.

We conclude by commenting on the connections to other widely used measures in
the literature to show their equivalence to the concurrence. The Hilbert-
Schmidt distance $D$ between two density matrices $\rho_{1}$ and $\rho_{2}$,
given by
$D^{2}_{\rho_{1}}(\rho_{2})\equiv||{\rho_{1}}-{\rho_{2}}||^{2}_{\text{HS}}=\hbox{Tr}[(\rho_{1}-\rho_{2})^{2}]$,
has been widely used to study the geometry and structure of entanglement with
connections to negativity and PPT-states geometryFrank ; geometryKus ;
geoAaryaman . Consider the case of a general DV system with density matrix
$\rho_{N\times N}$. Taking $\rho_{1}=\frac{1}{N}\mathbbm{1}_{N\times N}$, one
may write the distance of $\rho$ to the maximally mixed state $\rho_{1}$ as

$\mathcal{D}^{2}(\rho)=\hbox{Tr}(\frac{1}{N^{2}}\mathbbm{1}+\rho^{2}-\frac{2}{N}\rho)=\hbox{Tr}(\rho^{2})-\frac{1}{N},$

noting $\hbox{Tr}(\rho)=1$ and $\hbox{Tr}(\mathbbm{1})=N$. In the CV case as
$N\to\infty$, $\mathcal{D}^{2}(\rho)=\hbox{Tr}(\rho^{2})=1-E^{2}(\rho)/2$,
where $E$ is the generalized concurrence. One may, therefore, conversely use
this property to geometrically define a maximally mixed CV state. Moreover,
noting Eq. (21), one has the following identity for pure CV states

$||\widetilde{\rho}-\widetilde{\rho}_{PT}||^{2}_{\text{HS}}+2||{\rho_{\mathcal{M}}}-{\rho_{1}}||^{2}_{\text{HS}}=2,$

given the density matrix $\rho$. On the same note, one can show the
equivalence of the von Neumann entropy as an entanglement measure to the
concurrence. The von Neumann entropy, given a density matrix $\rho$, is
defined as $S=-\hbox{Tr}(\rho\ln\rho)=-\langle\ln\rho\rangle$, where
$\langle\cdot\rangle$ denotes the expectation value. Expanding $S$ around a
pure state $\rho^{2}=\rho$, that is, the non-negative matrix
$\mathbbm{1}-\rho$, and noting that
$E^{2}/2=1-\hbox{Tr}(\rho^{2})=\hbox{Tr}(\rho(\mathbbm{1}-\rho))=\langle\mathbbm{1}-\rho\rangle$,
one infers

$\displaystyle
S=-\langle\ln\rho\rangle=\langle\mathbbm{1}-\rho\rangle+\langle(\mathbbm{1}-\rho)^{2}\rangle/2+\langle(\mathbbm{1}-\rho)^{3}\rangle/3+\ldots,$

and therefore, $S=\frac{E^{2}}{2}+$ residual. Clearly, iff $E=0$, the residual
terms vanish, giving $S=0$; else when $E>0$, the residual remains positive,
giving $S>E^{2}/2$ for any state $\rho$. Therefore, $S$ and $E$ are equivalent
in characterizing separable states, and entanglement among entangled states
faithfully. It is, however, faster computationally to calculate the
concurrence than it is to find the von Neumann entropy, as it does not require
diagonalization of the density matrix.

We hope our work provides new insights into the geometry and structure of
entanglement in general pure systems by providing a family of faithful
entanglement measures, and equivalent forms of necessary and sufficient
conditions for separability across arbitrary bipartitions. We believe the
results hold deep connections to the recent works on the nature of quantum
correlations in many-body systems Zhu16 ; reiter , monogamy of entanglement
monogamy ; gwallen ; kundu , and foundational aspects of quantum mechanics,
including, the uncertainity principle and commutation relations nzubairy ;
hzubairy ; simonextended ; simonph ; cirac , and identical particles home .
One may also extend the result to CV mixed states by considering the convex
roof of $E_{\mathcal{M}}$ over pure state decompositions as proposed by
previous works roland ; uhlmann1 ; uhlmann ; revmodent  in the context of
concurrence for discrete variable states.

##  Appendix A Proof of Lagrange’s identity

Consider RHS of Eq. (25),

$\displaystyle=\int\limits_{\vec{x}=\vec{t}}^{\vec{u}}~{}\int\limits_{\vec{x^{\prime}}=\vec{x}}^{\vec{u}}\left|f(\vec{x})g(\vec{x^{\prime}})-f(\vec{x^{\prime}})g(\vec{x})\right|^{2}~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x}$
$\displaystyle=\frac{1}{2}\int\limits_{\vec{t}}^{\vec{u}}\int\limits_{\vec{t}}^{\vec{u}}~{}\left|f(\vec{x})g(\vec{x^{\prime}})-f(\vec{x^{\prime}})g(\vec{x})\right|^{2}~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x}$
$\displaystyle=\frac{1}{2}\int\limits_{\vec{t}}^{\vec{u}}\int\limits_{\vec{t}}^{\vec{u}}\left(f(\vec{x})g(\vec{x^{\prime}})-f(\vec{x^{\prime}})g(\vec{x})\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(f^{*}(\vec{x})g^{*}(\vec{x^{\prime}})-f^{*}(\vec{x^{\prime}})g^{*}(\vec{x})\right)~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x}$
$\displaystyle=\frac{1}{2}\int\limits_{\vec{t}}^{\vec{u}}\int\limits_{\vec{t}}^{\vec{u}}\bigg{[}|f(\vec{x})|^{2}|g(\vec{x^{\prime}})|^{2}-2~{}\text{Re}\big{(}f(\vec{x})g(\vec{x^{\prime}})f^{*}(\vec{x^{\prime}})g^{*}(\vec{x})\big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+|f(\vec{x^{\prime}})|^{2}|g(\vec{x})|^{2}\bigg{]}~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x}$
$\displaystyle=\bigg{(}~{}\int\limits_{\vec{t}}^{\vec{u}}|f(\vec{x})|^{2}~{}\mathrm{d}\vec{x}~{}\bigg{)}\bigg{(}~{}\int\limits_{\vec{t}}^{\vec{u}}|g(\vec{x})|^{2}~{}\mathrm{d}\vec{x}~{}\bigg{)}~{}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{Re}\int\limits_{\vec{t}}^{\vec{u}}\int\limits_{\vec{t}}^{\vec{u}}f(\vec{x})g(\vec{x^{\prime}})f^{*}(\vec{x^{\prime}})g^{*}(\vec{x})~{}\mathrm{d}\vec{x^{\prime}}~{}\mathrm{d}\vec{x}$
$\displaystyle=\bigg{(}~{}\int\limits_{\vec{t}}^{\vec{u}}|f(\vec{x})|^{2}~{}\mathrm{d}\vec{x}~{}\bigg{)}\bigg{(}~{}\int\limits_{\vec{t}}^{\vec{u}}|g(\vec{x})|^{2}~{}\mathrm{d}\vec{x}~{}\bigg{)}-\left|\int\limits_{\vec{t}}^{\vec{u}}f(\vec{x}){g^{*}(\vec{x})}~{}\mathrm{d}\vec{x}\right|^{2}$
$\displaystyle=\text{LHS}.$

Hence the identity.

##  Appendix B Gaussian CV pure states

We consider the example of a general pure Gaussian CV state to evaluate our
criterion and provide the condition for separability, and analyze concurrence
for the case of a general two-mode Gaussian state.

Consider

$\psi_{1}(x_{1},...,x_{n})=\mathcal{N}_{1}\exp\bigg{(}{-\frac{1}{2}{\big{[}\sum_{k=1}^{n}a_{k}x_{k}^{2}+\sum\limits_{k,j;j>k}^{n}c_{kj}x_{k}x_{j}\big{]}}}\bigg{)},$
(41)

where $\quad a_{k}\in\mathbb{R}$, $a_{k}\geq 0$, $~{}c_{kj}\in\mathbb{C}$, and
$\mathcal{N}_{1}$ is the appropriate normalization term for the wavefunction.
Consider the separability of $m$ degrees labeled by {$x_{1},...,x_{m}$} from
the $n$ available degrees of freedom. By Eq. (14), for separability across the
$m|(n-m)$ bipartition, one needs

$\text{exp}\bigg{(}-\frac{1}{2}\big{[}\sum_{k=1}^{m}a_{k}{y^{\prime}_{k}}^{2}+\sum_{k=m+1}^{n}a_{k}x_{k}^{2}+\sum\limits_{k=1,j=1;j>k}^{k,j=m}c_{kj}y^{\prime}_{k}y^{\prime}_{j}+\\\
\sum\limits_{k=1,j=m+1}^{k=m,j=n}c_{kj}y^{\prime}_{k}x_{j}+\sum\limits_{k=m+1,j=m+1;j>k}^{k=n,j=n}c_{kj}x_{i}x_{j}\big{]}\bigg{)}\times\\\
\text{exp}\bigg{(}-\frac{1}{2}\big{[}\sum_{k=1}^{m}a_{k}{y_{k}}^{2}+\sum_{k=m+1}^{n}a_{k}{x^{\prime}_{k}}^{2}+\sum\limits_{k=1,j=1;j>k}^{k,j=m}c_{kj}y_{k}y_{j}+\\\
\sum\limits_{k=1,j=m+1}^{k=m,j=n}c_{kj}y_{k}x^{\prime}_{j}+\sum\limits_{k=m+1,j=m+1;j>k}^{k=n,j=n}c_{kj}x^{\prime}_{i}x^{\prime}_{j}\big{]}\bigg{)}\\\
=\text{exp}\bigg{(}-\frac{1}{2}\big{[}\sum_{k=1}^{m}a_{k}{y^{\prime}_{k}}^{2}+\sum_{k=m+1}^{n}a_{k}{x^{\prime}_{k}}^{2}+\sum\limits_{k=1,j=1;j>k}^{k,j=m}c_{kj}y^{\prime}_{k}y^{\prime}_{j}+\\\
\sum\limits_{k=1,j=m+1}^{k=m,j=n}c_{kj}y^{\prime}_{k}x^{\prime}_{j}+\sum\limits_{k=m+1,j=m+1;j>k}^{k=n,j=n}c_{kj}x^{\prime}_{i}x^{\prime}_{j}\big{]}\bigg{)}\times\\\
\text{exp}\bigg{(}-\frac{1}{2}\big{[}\sum_{k=1}^{m}a_{k}{y_{k}}^{2}+\sum_{k=m+1}^{n}a_{k}{x_{k}}^{2}+\sum\limits_{k=1,j=1;j>k}^{k,j=m}c_{kj}y_{k}y_{j}+\\\
\sum\limits_{k=1,j=m+1}^{k=m,j=n}c_{kj}y_{k}x_{j}+\sum\limits_{k=m+1,j=m+1;j>k}^{k=n,j=n}c_{kj}x_{i}x_{j}\big{]}\bigg{)},$
(42)

which simplifies to the requirement

$\displaystyle\sum\limits_{k=1,j=m+1}^{k=m,j=n}c_{kj}\left(y^{\prime}_{k}x_{j}+y_{k}x^{\prime}_{j}\right)=\sum\limits_{k=1,j=m+1}^{k=m,j=n}c_{kj}\left(y^{\prime}_{k}x^{\prime}_{j}+y_{k}x_{j}\right),$
(43)

which can only be true for arbitrary values of $y^{\prime}_{i}$, $x_{i}$,
$x^{\prime}_{i}$, $y_{i}$, if and only if

$\displaystyle c_{kj}=0~{}\forall~{}k\in[1,m]\text{ and }j\in[m+1,n],\text{
or}$ $\displaystyle V_{k,j}=V^{\text{T}}_{k,j}=0~{}\forall~{}k\in[1,m]\text{
and }j\in[m+1,n],\text{ or}$
$\displaystyle\mathbb{M}~{}V~{}(\mathbbm{1}_{n\times
n}-\mathbb{M})=\mathbb{M}~{}V^{\text{T}}~{}(\mathbbm{1}_{n\times
n}-\mathbb{M})=0_{n\times n}$ (44)

where $V=\sum^{-1}$ is the inverse of the covariance matrix of the Gaussian,
and $\mathbb{M}$ is as defined in Eq. (17). This is a necessary and sufficient
condition for separability of the general Gaussian wavefunction
$\psi_{1}(\vec{x})$ in $n$-degrees of freedom defined in Eq. (41) across the
bipartition $m|(n-m)$ labeled by $\\{(1,...,m)~{}|~{}({m+1},...,n)\\}$.
Moreover, one may state, conversely, that the system is entangled across the
bipartition if and only if $\exists~{}k,j$ such that $c_{kj}\neq 0$ for some
$k\in[1,m]\text{ and }j\in[m+1,n]$.

(a)

(b)

Figure 1: Squared-concurrence and normalization for the two-mode Gaussian
state in Eq. (45) as a function of $c$ when $a=b=1$, and (a) $c$ is real with
$c\in(-2,+2)$; (b) $c$ is purely imaginary with $c=im$, $m\in\mathbb{R}$.

To analyze concurrence for the case of a general two-mode Gaussian state, as
an example, consider

$\psi_{2}(x_{1},x_{2})=\mathcal{N}_{2}~{}\text{e}^{-\frac{1}{2}(ax_{1}^{2}+bx_{2}^{2}+cx_{1}x_{2})},$
(45)

where $a,b\in\mathbb{R},~{}a,b>0$, $c$ is either purely real or imaginary, and

$\mathcal{N}_{2}=\begin{cases}\left[\frac{2\pi}{\sqrt{4ab-c^{2}}}\right]^{-1/2}&\quad\text{if}~{}c~{}\text{is
real}\\\
\left[\frac{\pi}{\sqrt{ab}}\right]^{-1/2}&\quad\text{if}~{}c=im,~{}m\in\mathbb{R},\end{cases}$

with $i\equiv\sqrt{-1}$. Using Eq. (27), one may compute the concurrence
across the modes as

$E^{2}=\begin{cases}2\left[1-\frac{\sqrt{4ab-c^{2}}}{2\sqrt{ab}}\right]&\quad\text{if}~{}c~{}\text{is
real}\\\
2\left[1-\frac{2\sqrt{ab}}{\sqrt{4ab+m^{2}}}\right]&\quad\text{if}~{}c=im,~{}m\in\mathbb{R}.\end{cases}$

Clearly, $E=0$ iff $c=0$ for both the cases. But the case of maximal
entanglement may not be attained here exactly with $E^{2}=2$ without having
the normalization $\mathcal{N}_{2}$ vanished when $c$ is real. The dependence
of entanglement and normalization on the parameter $c$ when $a=b=1$ for the
two cases is illustrated in the Figures LABEL:sub@fig:real and
LABEL:sub@fig:img. When $c$ is real, one requires $-2\sqrt{ab}<c<+2\sqrt{ab}$
so that the state remains normalizable, and hence physical. For this case, as
$c\to\pm 2\sqrt{ab}$, $E^{2}\to 2$. When $c$ is purely imaginary with
$c=im,~{}m\in\mathbb{R}$, as $m\to\pm\infty$, $E^{2}\to 2$ asymptotically.

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