# Non-Markovianity and entanglement detection

Sourav Chanduka  Center for Computational Natural Science and Bio informatics,
International Institute of Information Technology, Gachibowli, Hyderabad,
India.  Bihalan Bhattacharya<EMAIL_ADDRESS>S. N. Bose National Centre for
Basic Sciences Block - JD, Sector - III, Salt Lake City, Kolkata - 700 106
Rounak Mundra  Center for Computational Natural Science and Bio informatics,
International Institute of Information Technology, Gachibowli, Hyderabad,
India.  Samyadeb Bhattacharya<EMAIL_ADDRESS>Center for Security
Theory and Algorithmic Research, International Institute of Information
Technology, Gachibowli, Hyderabad, India  Indranil Chakrabarty
<EMAIL_ADDRESS>Center for Security Theory and Algorithmic
Research, International Institute of Information Technology, Gachibowli,
Hyderabad, India

###### Abstract

We have established a novel method to detect non-Markovian indivisible quantum
channels using structural physical approximation. We have shown that this
method can be used to detect eternal non -Markovian operations. We have
further established that harnessing eternal non-Markovianity, we can device a
protocol to detect quantum entanglement.

##  I Introduction

The theory of open quantum systems provides a powerful tool to study system-
environment interactions, spawning decoherence, dissipation and other
irreversible phenomena (Alicki and Lendi, 2007; Breuer and Petruccione, 2002).
Inter and Intra qubit properties like entanglement, discord and coherence are
also studied when one of the qubit undergoes decoherence in a system- bath
interaction model (Chakrabarty  _et al._ , 2011; Bellomo  _et al._ , 2007;
Dijkstra and Tanimura, 2010; Bhattacharya  _et al._ , 2017; Mukhopadhyay  _et
al._ , 2017; Lu  _et al._ , 2010; Luo  _et al._ , 2012; Jiang and Luo, 2013).
Recently, much efforts have been devoted to characterize quantum analogue of
non-Markovian (NM) evolutions (Rivas  _et al._ , 2014; Breuer  _et al._ ,
2016; de Vega and Alonso, 2017; Laine  _et al._ , 2010; Rivas  _et al._ ,
2010; Bellomo  _et al._ , 2007; Dijkstra and Tanimura, 2010; Bhattacharya  _et
al._ , 2017; Mukhopadhyay  _et al._ , 2017; Awasthi  _et al._ , 2018; Wolf
_et al._ , 2008; Lu  _et al._ , 2010; Rajagopal  _et al._ , 2010; Luo  _et
al._ , 2012; Jiang and Luo, 2013; Lorenzo  _et al._ , 2013; Dhar  _et al._ ,
2015). It has been shown from various information theoretic and thermodynamic
aspects, that NM can act as a powerful resource (Bhattacharya  _et al._ ,
2017; Mukhopadhyay  _et al._ , 2017; Awasthi  _et al._ , 2018). Therefore,
identification of non-Markovianity in a process is an extremely important area
to study. However, it still remains an onerous task to construct a theory of
distinguishing NM evolutions from its Markovian counterparts along with
proposals of experimentally feasible detection procedures. In most of the
previous literature, non-Markovianity in terms of information backflow and
divisibility breaking of a dynamical map is addressed by geometric or entropic
distance based quantities (Rivas  _et al._ , 2014; Breuer  _et al._ , 2016; de
Vega and Alonso, 2017). These measures, though offers a proper quantification,
are not feasible measures in an experimental scenario. However, from the
theory of entanglement (Horodecki  _et al._ , 2009), we know that hermitian
operators are experimentally measurable and thus linear witnesses possess a
much higher status from this perspective. Recently the present authors have
constructed a convex resource theory of NM (Bhattacharya  _et al._ , 2020),
creating that very opportunity of experimental verification, by exploring the
geometry of NM dynamics in a similar manner of entanglement detection.

Detection of quantum entanglement is one of the most prevalent area of
research in quantum information theory. It was Peres who introduced the
partial transpose criterion (PT) for the detection of entanglement and later
Horodecki et.al proved that the criterion is necessary and sufficient for
$2\otimes 2$ and $2\otimes 3$ quantum system. However, the criterion is
necessary for $d_{1}\otimes d_{2}$ $(d_{1},d_{2}\geq 3)$ dimensions (Peres,
1996; Horodecki  _et al._ , 2001). The phenomena of witnessing entanglement
(Bourennane  _et al._ , 2004; Terhal, 2000; Lewenstein  _et al._ , 2000; Gühne
_et al._ , 2002; Guhne and Toth, 2009; Chruściński and Sarbicki, 2014; Gühne
_et al._ , 2007; Zhang  _et al._ , 2008; Bhattacharya  _et al._ , 2021) is a
stepping stone in the study of quantum information, offering versatile tools
for experimental detection of entangled quantum states. Witnesses are
hermitian operators and hence observables by construction, giving positive
expectation value for all separable states; whereas negativity of the same
signifies the existence of entanglement. Not only detection of entanglement,
witness operators were also used to detect various information processing
tasks like teleportation and super dense coding (Ganguly  _et al._ , 2011;
Patro  _et al._ , 2017; Vempati  _et al._ , 2021)

In this article, we apply this methodology in open system dynamics, to
construct NM witnesses from the similar footings of that of entanglement.
Although, works have been done with the motivation of developing NM witnesses
(Chruściński  _et al._ , 2017; Macchiavello and Rossi, 2013; Orieux  _et al._
, 2013), but in order to construct a proper NM detection theory, we need to
have a convex and compact set of states beholding all Markovian operations.
Though channel state duality (Jamiołkowski, 1972; Choi, 1975a) allows us to
construct the set of states, due to the non-convex nature of divisible
operations (Wolf  _et al._ , 2008; Wolf and Cirac, 2008), constructing a
theory of linear witnesses is not possible in general. We overcome this
difficulty by “small time interval” approximation, whence constructing the
Choi states. This allows us to build a proper framework of linear witnesses
for NM detection.

There are arguments on whether or not the indivisible operations exhaust all
the non- Markovian operations (Milz  _et al._ , 2019). It is established that
all Markovian operations are divisible, whereas the converse is not proven to
be true. However, even if there exists NM operations which are CP-divisible,
such operations can not generate resource back flow, and hence are not
considered as resource (Bhattacharya  _et al._ , 2018). It is to mention that,
in this paper, we will develop the theory of witneesses for NM operations
which are indivisible. It is also very important here to mention that we are
restricted to the set of operations having Lindblad type generators (Lindblad,
1976; Gorini  _et al._ , 1976) of the form
$\dot{\rho}(t)=\mathcal{L}_{t}(\rho(t))=\sum_{\alpha=1}^{n\leq
d_{S}^{2}}\Gamma_{\alpha}(t)\left(L_{\alpha}\rho(t)L_{\alpha}^{\dagger}-\frac{1}{2}L_{\alpha}^{\dagger}L_{\alpha}\rho(t)-\frac{1}{2}\rho(t)L_{\alpha}^{\dagger}L_{\alpha}\right)$,
with $\Gamma_{\alpha}(t)$s as the Lindblad coefficients, $A_{\alpha}$s as the
Lindblad operators for a system with dimension $d_{s}$. For divisible
operations we have $\Gamma_{\alpha}(t)\geq 0,~{}\forall\alpha,t$ (Rivas  _et
al._ , 2010). If the evolution is non-Markovian in the sense that it is
indivisible, the $\Gamma_{\alpha}(t)<0$, for some $\alpha$ at some instant of
time $t$. According to reference (Rivas  _et al._ , 2010), to ensure complete
positivity of the total dynamics, the condition
$\int_{0}^{T}\Gamma_{\alpha}(t)dt\geq 0~{}(\forall\alpha,T)$ must always hold.
However, it has been shown later (Hall  _et al._ , 2014) that this condition
of positivity of all of the Lindblad coefficients over integration, can be
relaxed for certain non-Markovian completely positive operations. For example,
we can consider the following qubit depolarization operation

$\frac{d\rho}{dt}=\sum_{i=1}^{3}\gamma_{i}\left(\sigma_{i}\rho\sigma_{i}-\rho\right),$
(1)

where $\sigma_{i}$ denotes the Pauli $\sigma$ matrices. If we now consider
$\gamma_{1}(t)=\gamma_{2}(t)=1$ and $\gamma_{3}=-\tanh t$, then despite the
fact that one of the Lindblad coefficient is negative throughout and hence
violating the positivity condition upon integration, the concerning dephasing
operation is still completely positive. This kind of operations with negative
coefficients are coined as eternal non-Markovian operations. It has been shown
that these eternal non-Markovian operations cannot be detected by either the
divisibility based non-Markovianity measure or the information backflow based
measure (Hall  _et al._ , 2014). In this article, we show that our method of
non-Markovianity witness can successfully detect such eternal non-Markovian
operations. We also find that eternal non-Markovian operations can give rise
to positive but not completely positive maps and hence presents us with an
opportunity to detect entanglement via the usage of non-Markovianity.

The paper is constructed in the following manner. In section II, we discuss
the non-Markovianity witnessing and the application of structural physical
approximation in that purpose. Then we discuss the detection of eternal non-
Markovianity by our method. In section III, we then propose our method of
entanglement detection by non-Markovianity and then finally we conclude in
section IV with some possible implications.

##  II Witnessing non-Markovianity and structural physical approximation

In this section, we construct the process of witnessing non-Markovianity,
based upon a previous formulation (Bhattacharya and Bhattacharya, 2018) and
further extending it. However, before going into the main results of this
work, we first elucidate the properties of the set of Choi states for
divisible operations. Then we develop the theory of linear NM witnessing. We
further consider the geometry of the set of Choi states, to identify the
possibility of generalized non-linear witnesses for NM detection. Then we
conclude with stating the possible implications.

Divisible operations and structure of Choi states:Let us consider a quantum
channel $\Lambda(t_{2},t_{1})$ which takes quantum state $\rho(t_{1})$ at time
$t_{1}$ to $\rho(t_{2})$ at time $t_{2}$. Moreover let us consider that the
channel has Lindblad type generator i.e. the channel can be expressed as
$\Lambda(t_{2},t_{1})\equiv\exp\left(\int_{t_{1}}^{t_{2}}\mathcal{L}_{t}dt\right)$.
Let $\mathbb{D}_{\mathcal{C}}$ represents the set of all such channels. It is
well known that the quantum channels are in one to one correspondence with the
corresponding Choi states. Exploiting this channel-state duality, we have an
isomorphism between $\mathbb{D}_{\mathcal{C}}$ and the set of corresponding
Choi states $\mathbb{C}_{\mathcal{A}}$. Now, given a map
$\Lambda_{\mathcal{M}}(t_{3},t_{1})$, it is called divisible if we can find
another linear map $\Lambda_{\mathcal{M}}(t_{3},t_{2})$ on the state space
such that

$\Lambda_{\mathcal{M}}(t_{3},t_{1})=\Lambda_{\mathcal{M}}(t_{3},t_{2})\circ\Lambda_{\mathcal{M}}(t_{2},t_{1})$

with $t_{3}>t_{2}>t_{1}\forall t_{1},t_{2},t_{3}$. Sometimes the map
$\Lambda_{\mathcal{M}}(t_{3},t_{2})$ is called the propagator of the dynamics.
A given map is called complete positive divisible or CP-divisible if the
propagator map $\Lambda_{\mathcal{M}}(t_{3},t_{2})$ is also completely
positive. In this case the Choi state
$\mathcal{C}_{\mathcal{M}}(t_{2},t_{1})=\mathbb{I}\otimes\Lambda(t_{2},t_{1})|\phi\rangle\langle\phi|$
is a valid density matrix for every instant of time, with
$||\mathcal{C}_{\mathcal{M}}(t_{2},t_{1})||_{1}=1$, $\forall t_{1},t_{2}$.
Here $\mathbb{I}$ stands for the identity map acting on one subsystem and
$|\phi\rangle$ corresponds to a maximally entangled state in $d_{s}^{2}$
dimension and $||\cdot||_{1}=Tr[\sqrt{(\cdot)^{\dagger}(\cdot)}]$ is the trace
norm. Breaking of CP-divisibility of a dynamics is a signature of non-
Markovian backflow of information (Rivas  _et al._ , 2010; Laine  _et al._ ,
2010). Hence the CP divisible channels can be considered as memoryless
Markovian channels. Therefore the set
$\mathbb{F}_{\mathcal{M}}\subset\mathbb{C}_{\mathcal{A}}$ can be defined as
$\mathbb{F}_{\mathcal{M}}=\\{\mathcal{C}_{\mathcal{M}}(t_{2},t_{1})~{}|~{}~{}||\mathcal{C}_{\mathcal{M}}(t_{2},t_{1})||_{1}=1,\forall
t_{1},t_{2}\\}$. This includes the Choi states for all CP-divisible
operations. We recall the fact that $\mathbb{F}_{\mathcal{M}}$ is not convex
in general. The convexity can be induced by considering $t_{2}=t_{1}+\epsilon$
and imposing the constraint of $\epsilon\rightarrow 0$ (Bhattacharya  _et al._
, 2020). The physical meaning of this approximation lies in observing the
state in snapshots. Note that
$\mathbb{F}_{\mathcal{M}}^{\epsilon}=\\{\mathcal{C}_{\mathcal{M}}(t_{1}+\epsilon,t_{1})~{}|~{}~{}||\mathcal{C}_{\mathcal{M}}(t_{1}+\epsilon,t_{1})||_{1}=1,\forall
t_{1},\epsilon\\}$

Non-Markovianity Witness: Let us consider the construction of NM witness using
the techniques of entanglement theory.

Definition: A hermitian operator $W$ is said to be a NM witness if it
satisfies following criteria:

  1. 1.

$Tr(W\mathcal{C}_{\mathcal{M}})$ $\geqslant$ 0 $\forall$
$\mathcal{C}_{\mathcal{M}}$ $\in$ $\mathbb{F}_{\mathcal{M}}^{\epsilon}$.

  2. 2.

There exists atleast one NM Choi state $\mathcal{C}_{\mathcal{N}}$ such that
$Tr(W\mathcal{C}_{\mathcal{N}})$ $<$ 0.

It is clear from the definition, that a single witness can not detect all
NMCS. The witness will depend on the NMCS, which one wishes to detect.

Structural Physical Approximation: In this part, we formulate the protocol for
experimental detection of information back flow. The structural physical
approximation (SPA)(Horodecki, 2001; Horodecki and Ekert, 2002) of a positive
map is a convex mixture of a depolarizing map with the given map, so that the
resulting map is complete positive. Consider a NCP map $\mathcal{M}$ acting on
the Hilbert space $\mathcal{H}_{d}$ of dimension $d$. the following
approximate map

$\mathcal{S}(\cdot)=\left[p\Theta+(1-p)\mathcal{M}\right](\cdot),$

with depolarizing map $\Theta(\rho)=\frac{\mathbb{I}_{d}}{d}$ in
$\mathcal{H}_{d}$, can always be completely positive and hence experimentally
implementable, over certain threshold value of $p$. Here $\mathbb{I}_{d}$
stands for identity operator on dimension d. An algorithm to find the optimal
SPA map for a given positive map has been prescribed in (Horodecki, 2001).
Following the prescription , the optimal SPA map corresponding to the map
$\mathcal{M}$ is given by

$\mathcal{S}^{opt}=p^{*}\Theta+(1-p^{*})\mathcal{M^{\prime}}$ (2)

where $p^{*}=\frac{\lambda
dd^{{}^{\prime}}\beta_{\Lambda_{\alpha}}^{-1}}{\lambda
dd^{{}^{\prime}}\beta_{\mathcal{M}}^{-1}+1}$, $\lambda$ is the minimum
eigenvalue of
$\chi=[\mathbb{I}\otimes\mathcal{M}](|\phi_{d}\rangle\langle\phi_{d}|)$ , and
$\mathcal{M^{\prime}}=\beta_{\mathcal{M}}^{-1}\mathcal{M}$ is the re-scaling
of the original map. Here $d$ and $d^{\prime}$ are the input and output
dimension of the map $\mathcal{M}$, $|\phi_{d}\rangle$ denotes the maximally
entangled state in dimension d and $\beta_{\mathcal{M}}$ is a rescaling
parameter. Assuming the trace preservation property of the map, the value of
$\beta_{\mathcal{M}}=1$.

Information backflow witness for qubit systems:- In the following, we
construct information backflow witnesses for two dimensional systems. We also
give some particular examples for specific qubit dynamical maps. But before
going into further study, we prove the following proposition, which will be
utilized frequently, in the later study.

Proposition 1: For any arbitrary dimensional system, the following identity is
always true when $G$ is Harmitian.

$Tr\left[|\alpha\rangle\langle\alpha|\mathbb{I}\otimes\mathcal{N}(\rho)\right]=Tr\left[\mathbb{I}\otimes\mathcal{N}(|\alpha\rangle\langle\alpha|)\rho\right],$

where $|\alpha\rangle$ is any arbitrary pure state in $d$ dimension and

$\begin{array}[]{ll}\mathcal{N}(\rho)=\rho+\\\ \Gamma(t)\left[\left(G\rho
G^{\dagger}-\frac{1}{2}(G^{\dagger}G\rho+\rho
G^{\dagger}G)\right)\right].\end{array}$

###### Proof.

Consider the operation

$\mathcal{R}(\rho)=\rho+\left[\Gamma(t)\left(\mathcal{A}\rho\mathcal{A}^{\dagger}-\frac{1}{2}(\mathcal{A}^{\dagger}\mathcal{A}\rho+\rho\mathcal{A}^{\dagger}\mathcal{A}\right)\right],$

with $\mathcal{A}$ being any Hermitian operator on the Hilbert space
$\mathcal{H}_{d}\otimes\mathcal{H}_{d}$. Now using the property
$Tr[AB]=Tr[BA]$ for any two matrices $A$ and $B$ and
$Tr[\rho]=Tr[|\alpha\rangle\langle\alpha|]=1$, we show that

$Tr[|\alpha\rangle\langle\alpha|\mathcal{R}(\rho)]=Tr[\rho\mathcal{R}(|\alpha\rangle\langle\alpha|)].$

Since $\mathbb{I}\otimes\mathcal{N}(\rho)$ falls into the category of
$\mathcal{R}(\rho)$ with $\mathcal{A}=\mathbb{I}_{d}\otimes G$, this proves
the proposition. ∎

Therefore for the case of unital operations with Harmitian Lindblad operators,
we can device our method to detect information backflow witness.

Construction of witness:- We now move on to construct the witness operator for
detecting non-Makovianity, on the backdrop of SPA protocol. Let us consider
the maximally entangled state $\sigma$. Therefore the operator
$\tilde{\sigma}=\mathcal{S}^{\mathcal{N}}(\sigma)$ is always a positive semi
definite matrix, possessing all the properties of a quantum state. The minimum
eigenvalue $\lambda_{min}$ determines, whether the original map is indivisible
or not. If the minimum eigenvalue $\lambda_{min}<\omega$, the operation is
indivisible. The threshold value $\omega$ for a given dimensional system is to
be determined by the protocol presented in Ref. (Horodecki, 2001). We now
define an operator
$\mathcal{O}=\tilde{\sigma}-\nu\mathbb{I}\otimes\mathcal{N}(\sigma)$, where
$\nu$ is a suitable chosen real number. Let us now consider $|\tau\rangle$, to
be the eigenvector corresponding to the minimum eigenvalue of
$\tilde{\sigma}$. We choose $\nu$ such that

$\langle\tau|\mathcal{O}|\tau\rangle=Tr[(\tilde{\sigma}-\nu\mathbb{I}\otimes\mathcal{N}(\sigma))|\tau\rangle\langle\tau|]=\omega.$

Note that a hermitian operator $\mathcal{W}$ is a NM witness if it produces
non negative expectation on all Markovian Choi states and there exists atleast
one NM Choi matrix on which $\mathcal{W}$ gives negative expectation.
Considering the hermitian operator
$\mathcal{W}=\nu\mathbb{I}\otimes\mathcal{N}(|\tau\rangle\langle\tau|)$ and

Using Proposition 1, we get

$Tr[\mathcal{W}\sigma]=Tr[\nu\mathbb{I}\otimes\mathcal{N}(|\tau\rangle\langle\tau|)\sigma]=Tr[\tilde{\sigma}|\tau\rangle\langle\tau|]-\omega.$

If we consider $\mu_{min}$ to be the minimum eigenvalue of $\tilde{\sigma}$,
the we have $Tr[\mathcal{W}\sigma]=\mu_{min}-\omega$. It is then straight
forward to check that $Tr[\mathcal{W}_{\mathcal{M}}\sigma]\geq 0$, for all
divisible operations $\mathcal{W}_{\mathcal{M}}$ and
$Tr[\mathcal{W}_{\mathcal{N}}\sigma]<0$ for indivisible operations
$\mathcal{W}_{\mathcal{N}}$. Therefore,

$\mathcal{W}=\nu\mathbb{I}\otimes\mathcal{N}(|\tau\rangle\langle\tau|),$ (3)

acts as a witness operator, detecting whether the operation under
consideration is NM or not. Since $\omega$ is the threshold value for
positivity, the witness $\mathcal{W}$ is optimal. Following Ref.(Horodecki,
2001), we now describe the protocol to find the threshold value $\omega$.
Consider the SPA map $\mathcal{S}^{\mathcal{N}}(\sigma)$ of the entangled
state $\sigma$.The error term can be written as

$\Delta(\sigma)=\mathcal{S}^{\mathcal{N}}(\sigma)-\gamma_{\sigma}\mathbb{I}\otimes\mathcal{N}(\sigma),$

where the error term satisfies the invariance condition
$\Delta(\sigma)=\delta_{\sigma}\mathbb{I}$. Here $\gamma_{\sigma}$ and
$\delta_{\sigma}$ are in general state dependent quantity, whose optimal value
gives us $\nu$ and $\omega$ respectively. Therefore the SPA map can be be
written as

$\mathcal{S}^{\mathcal{N}}(\sigma)=\delta_{\sigma}\mathbb{I}+\gamma_{\sigma}\mathbb{I}\otimes\mathcal{N}(\sigma),$

with the functions $\gamma_{\sigma},~{}\delta_{\sigma}\geq 0$. Now following
the protocol described in Ref. (Horodecki, 2001), we find that the optimal
decomposition can be written as

$\mathcal{S}^{\mathcal{N}}_{opt}(\sigma)=\omega\mathbb{I}+\nu\mathbb{I}\otimes\mathcal{N}(\sigma),$
(4)

with

$\nu=\frac{1}{|\lambda_{-}|d^{2}+1},~{}~{}\omega=\frac{|\lambda_{-}|d^{2}}{|\lambda_{-}|d^{2}+1}.$

Let us now illustrate the above mentioned procedure with the example of qubit
dephasing dynamics.

Qubit dephasing channels :- Let us first consider qubit dephasing channels
having Lindblad type generators of the form

$\mathcal{L}_{D}(\cdot)=\Gamma_{D}(t)\left(\sigma_{z}(\cdot)\sigma_{z}-(\cdot)\right),$

with $\sigma_{i}~{}~{}(i=1,2,3)$ being the Pauli matrices and $\Gamma_{D}(t)$
is Lindblad coefficient. This generator will correspond to an indivisible
dephasing operation, if the dephasing parameter has got some negative regions.
Therefore under the small time interval approximation, the map corresponding
to this dephasing operation will be
$\mathcal{N}_{D}(\cdot)=(\mathbb{I}+\epsilon\mathcal{L}_{D})(\cdot)$. The
negative eigenvalue will be equal to $\Delta=-\epsilon|\Gamma_{D}(t)|$,
corresponding to the eigenvector $|\lambda_{D}^{-}\rangle=(-1,0,0,1)^{T}$.

For this map, if we apply the SPA map as given in (2), we find that the
optimal value of mixing parameter is

$p_{min}^{D}=\frac{4\epsilon|\Gamma_{D}(t)|}{1+4\epsilon|\Gamma_{D}(t)|}.$ (5)

Let us now construct the witness operator corresponding to this evolution. We
consider the SPA map $S^{\mathcal{N}}_{D}(\rho)$ with $p>p_{min}^{D}$. In
accordance with the theory of entanglement, we consider

$\omega=\frac{4\epsilon|\Gamma_{D}(t)|}{1+4\epsilon|\Gamma_{D}(t)|}~{}~{}~{}~{}~{}\nu=\frac{1}{1+4\epsilon|\Gamma_{D}(t)|}$

Since the protocol we have described is valid for any state, we consider
$\sigma=|\lambda_{D}^{-}\rangle\langle\lambda_{D}^{-}|$.

##  III Eternal non-Markovianity and entanglement detection

In this section we shall discuss an important class of non-Markovian
operations and its connection with entanglement detection. Markovianity is
closely connected with complete positive divisibility. CP indivisibility
implies non-Markovianity in general. But as mentioned in the introduction
section, there are examples of dynamics which is completely positive through
out the whole time interval but nature of which is non-markovian. Such non-
Markovian operations are known as eternal non-Markovian operations. Such non-
Markovianity can not be detected by trace distance measure or Bures distance
based measure or entanglement measure of non-Markovianity. in this section we
shall illustrate eternal non-Markovianity and its connection with entanglement
detection using example.

Let us consider the operation with the Lindblad equation:

$\Lambda_{dep}(\rho)=\gamma_{x}(\sigma_{x}\rho\sigma_{x}-\rho)+\gamma_{y}(\sigma_{y}\rho\sigma_{y}-\rho)+\gamma_{z}(\sigma_{z}\rho\sigma_{z}-\rho)$

This is actually a qubit depolarising operation. Now consider
$\gamma_{x}=\gamma_{y}=1$ and $\gamma_{z}=-\tanh t$.
To witness this non-Markovianity, we use a witness operator,
$\mathcal{W}_{z}=|\chi\rangle\langle\chi|$ where $\chi=(-1,0,0,1)^{T}$. This
type of non-Markovianity can not be detected by Rivas-Huelga-Plenio (RHP)
measure (Rivas  _et al._ , 2010). We would like to construct the SPA of the
corresponding map. Following the prescription of (Horodecki, 2001) we
calculate the value of $\omega$ and $\nu$ and we find that

$\omega=\frac{4\epsilon\tanh t}{1+4\epsilon\tanh
t}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\nu=\frac{1}{1+4\epsilon\tanh t}$

Entanglement Detection: It is well known that a positive but not completely
positive maps play an important role in detecting entanglement. A quantum
state $\sigma$ is said to be separable if and only if for every positive map
$\Lambda$, the operator $\mathbb{I}\otimes\Lambda(\sigma)$ is a positive
semidefinite. Certainly this criterion for detecting entanglement is extremely
hard in practice. As mentioned in the introduction for two qubit states and
qubit qutrit states transposition map plays an universal role to detect
entanglement. But from two qutrit system onward existence of PPT entanglement
which is considered to be an weak form of entanglement is very hard to detect.
Transposition map can not detect such entanglement. Efforts from both
mathematics and physics community (Stinespring, 1955; Arveson, 1969; Choi,
1975b; Størmer, 1982; Woronowicz, 1976; Cho  _et al._ , 1992; Ha, 2003;
Majewski and Marciniak, 2001; Sarbicki and Chruściński, 2012; Miller and
Olkiewicz, 2015; Bhattacharya  _et al._ , 2021; Chruscinski and Sarbicki,
2012) have been able to shade light on the theory of positive maps and high
dimensional entanglement. Still the the study of positive maps and
entanglement detection is an active area of research as the structure of
positive maps is not all clear even in low dimensions. Here we detect
entanglement in a two qubit state using a family of positive maps. It is true
that the same could have been done using transposition map but the main reason
to present the example of the map is to show the connection between non-
Markovianity and entanglement detection. It is well known that the Lindblad
form of a map plays an important role in non-Markovianity. Let us consider a
general two parameter family of maps motivated from the Lindblad structure,

$\Lambda(\rho)=(\mathbb{I}+\mathcal{L})(\rho)$

where

$\mathcal{L}(\rho)=\gamma_{1}(\sigma_{x}\rho\sigma_{x}-\rho)+\gamma_{1}(\sigma_{y}\rho\sigma_{y}-\rho)+\gamma_{2}(\sigma_{z}\rho\sigma_{z}-\rho)$

It can be shown that the above map is positive for
$0\leqslant\gamma_{1}\leqslant\frac{1}{2}$ and $0\leqslant\gamma_{2}<1$ and
not completely positive whenever $\gamma_{2}\geqslant 0$. Hence it can be used
to detect entanglement. It is well known that the Werner class of states given
by

$\mathbb{W}_{p}=p|\psi^{-}\rangle\langle\psi^{-}|+(1-p)\frac{\mathbb{I}_{2}\otimes\mathbb{I}_{2}}{4}$

is entangled for $p>\frac{1}{3}$. We have found that for the special case of
the above mentioned positive map if $\gamma_{1}=\frac{1}{2}$ and
$\gamma_{2}=\frac{1}{2}$, it can detect the full range of entangled state in
the Werner class. Moreover it is interesting to note that even if we consider
general $\gamma_{1}$ and $\gamma_{2}$ in the range where the above map is
positive but not completely positive; it can detect the entanglement for the
full range of the entangled Werner class.

##  IV Conclusion

In this work, we have first constructed a novel method to witness non-
Markovian quantum channels which are indivisible, by using a the SPA protocol.
We have further observed that a special type of non-Markovian channels namely
the eternal non-Markovian channels generate non complete positive maps within
its dynamical evolution and hence can be utilized to detect entangled states.
This gives us a unique avenue of research which may lead us to dynamical
detection of entanglement in much more complex quantum systems.

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# How accurately must cells measure the concentrations of signaling molecules?

Marianne Bauera,b and William Bialek,b,c aDepartment of Bionanoscience, Kavli
Institute of Nanoscience Delft, Technische Universiteit Delft, Van der Maasweg
9, 2629 HZ Delft, the Netherlands
bJoseph Henry Laboratories of Physics and Lewis–Sigler Institute for
Integrative Genomics, Princeton University, Princeton, NJ 08544 USA
cCenter for Studies in Physics and Biology, Rockefeller University, 1230 York
Avenue, New York, NY 10065 USA

###### Abstract

Information of relevance to the organism often is represented by the
concentrations of particular molecules inside a cell. As outside observers we
can now measure these concentrations precisely, but the cell’s own mechanisms
must be noisier. We show that, in certain limits, there is a universal
tradeoff between the information capacity of these concentration measurements
and the amount of relevant information that is captured, a version of the
information bottleneck problem. This universal tradeoff is confirmed,
quantitatively, in an analysis of the positional information encoded by the
“gap genes” in the developing fly embryo.

In the physics laboratory, and in engineered devices, we are used to
information being represented by electrical or optical signals. While the
brain also uses electrical signaling, inside living cells information often is
represented by the concentrations of particular molecules. The absolute
concentrations of these molecules, and even their total number, can be quite
small. As a result there has been considerable interest in understanding the
physical limits to this molecular signaling berg+purcell_77 ;
bialek+setayeshgar_05 ; bialek+setayeshgar_08 ; endres+wingreen_09 ;
mora+wingreen_10 ; kaizu+al_14 ; mora+al_15 ; tenWoldereview ;
mora+nemenman_19 , the strategies that cells can use to maximize information
in the face of these limits tkacik+al_08 ; tostevin+wolde_09 ;
tkacik+walczak_11 ; siggia+vergassola_13 ; becker+al_15 ; wolde+al_16 , and
the implications for cellular function.

One approach to understanding signaling via molecular concentrations is to
explore increasingly realistic models of the microscopic events ackers+al_82 ;
ptashne+gann_02 ; bintu+al_05 ; hnisz+al_17 ; morrison+al_21 ; zoller+al_22 ;
MartinezCorral . Recently we suggested a different approach, in which we ask
more abstractly about the implications of noise in the response, or more
precisely about the limited information capacity of the cell’s “measurements”
of concentration bauer+al_21 ; bauer_22 . To formalize the problem, we imagine
that some relevant signal $x$ is represented by the concentrations of $K$
different molecules, which we write as
${\mathbf{g}}\equiv\\{g_{1},\,g_{2},\,\cdots,\,g_{K}\\}$. The cell does not
have access to the exact values of ${\mathbf{g}}$, but only to some variables
that constitute intermediates in the response. As an example, if the molecules
act by binding to particular sites along the cell’s DNA and thereby regulating
the expression of downstream genes, the intermediate variable might be the
average occupancy of these binding sites over some relevant time window, or
the state of the enhancers built out of groups of these binding sites
ptashne+gann_02 ; bintu+al_05 ; furlong+levine_18 . Independent of molecular
details, this intermediate variable can carry only a limited amount of
information about the real concentrations; in this sense it is a “compressed”
representation cover+thomas_91 , and we will refer to this representation as
$C$.

Different mechanisms inside the cell will generate different mappings
${\mathbf{g}}\rightarrow C$, and in general we expect this mapping to be
noisy, so it will be described by a probability distribution
$P(C|{\mathbf{g}})$. What would be useful for the cell is to capture as much
information as possible about the relevant variable $x$, subject to the
constraint that the information about $\mathbf{g}$ is limited. This means that
the best mapping ${\mathbf{g}}\rightarrow C$ is one that maximizes

${\cal U}=I(C;x)-\lambda I(C;{\mathbf{g}}),$ (1)

where $\lambda$ is a Lagrange multiplier to implement the constraint on
information about $\mathbf{g}$. This is an example of the information
bottleneck problem bialek+al_99 , which arises in contexts ranging from text
classification slonim+tishby_00  to the analysis of deep networks
tishby_zaslavsky_77 ; saxe ; alemi  and neural coding schneidman+al_03 ;
buesing+maass_10 .

Optimizing $\cal U$ will define a bounding curve, which shows the minimum
$I(C;{\mathbf{g}})$ needed to reach a criterion level of $I(C;x)$. If the
cell’s measurements of concentration become more precise, then
$I(C;{\mathbf{g}})$ becomes larger, but since $C$ depends on $\mathbf{g}$ and
not directly on $x$ we always have $I(C;x)\leq I({\mathbf{g}};x)$. The
question is how close the cell can come to capturing all this available
information given limits on the precision of its response.

The standard approach to solving the information bottleneck problem is to
assume that $C$ is discrete, so that the mapping ${\mathbf{g}}\rightarrow C$
becomes a kind of clustering, and the optimal $P(C|{\mathbf{g}})$ obeys a
self–consistent equation that can be solved iteratively. There is a separate
bounding curve for each choice of the cardinality $||C||$, and the full
structure emerges as we let $||C||\rightarrow\infty$. We have applied this
approach to the representation of positional information by the gap gene
expression levels in the early fly embryo bauer+al_21 . The results showed,
for example, that the optimal mappings for individual genes are close to an
intuitive thresholding model, but that optimal compression of multiple genes
depends crucially on combinatorial interactions among these molecules, as seen
experimentally, for example in the regulation of the pair rule genes by the
gap genes. But this discussion seemed to depend on details of the genetic
network in the fly embryo, and missed the possibility that there is something
more universal about the tradeoff between $I(C;{\mathbf{g}})$ and $I(C;x)$.
Our goal is to uncover this (asymptotically) universal tradeoff, which emerges
in an intuitive limiting regime.

In many cases the relation between $\mathbf{g}$ and $x$ is smooth if noisy. To
be concrete, we write the mean concentrations at each $x$ as $\langle
g_{\mu}\rangle_{x}$, which are smooth functions of $x$, and fluctuations
around these means are described by a covariance matrix $\langle\delta
g_{\mu}\delta g_{\nu}\rangle_{x}$. To first approximation, the probability
distribution $P({\mathbf{g}}|x)$ can be seen as Gaussian, fully described by
these conditional moments. If $\langle g_{\mu}\rangle_{x}$ is smooth, it is
reasonable to think that the compressed variable $C$ also will have a smooth
relation to other variables. Further, although a single measurement of
concentration might be noisy, one can imagine that the noise level is smaller
if we think about the encoding of the relevant variable $x$. These
observations suggest that we search for optima in which $C$ is a continuous
variable, and that at appropriate points we can take a small noise limit.
Concretely, let us assume that

$P(C|{\mathbf{g}})={1\over\sqrt{2\pi\sigma_{C}^{2}({\mathbf{g}})}}\exp\left[-{{(C-\bar{C}({\mathbf{g}}))^{2}}\over{2\sigma_{C}^{2}({\mathbf{g}})}}\right],$
(2)

so that optimization of $\cal U$ now corresponds to finding the optimal
functions $\bar{C}({\mathbf{g}})$ and $\sigma_{C}({\mathbf{g}})$.

With the Gaussian approximation of Eq (2) we can immediately write
cover+thomas_91 ; bialek_12

$I(C;{\mathbf{g}})=S(C)-{1\over 2}{\bigg{\langle}}\log_{2}\left[2\pi
e\sigma_{C}^{2}({\mathbf{g}})\right]{\bigg{\rangle}},$ (3)

where $S(C)$ is the entropy of the variable $C$. Importantly this entropy is
finite in the limit of small noise, since $C$ must be tied, even if
implicitly, to the relevant variable $x$. In particular, if the effective
noise in estimating $x$ from $\mathbf{g}$ is small compared with the scale on
which the distribution $P(x)$ varies, then this relationship becomes nearly
deterministic tkacik+al_08 , and we can write

$\displaystyle P(C)$ $\displaystyle=$ $\displaystyle
P(x){\bigg{|}}{{d\bar{C}}\over{dx}}{\bigg{|}}^{-1}$ (4) $\displaystyle S(C)$
$\displaystyle=$ $\displaystyle
S(x)+{\bigg{\langle}}\log_{2}{\bigg{|}}{{d\bar{C}}\over{dx}}{\bigg{|}}{\bigg{\rangle}},$
(5)

where the dependence of $\bar{C}$ on $x$ is through $\mathbf{g}$,

${{d\bar{C}}\over{dx}}=\sum_{\mu}{{d\langle
g_{\mu}\rangle_{x}}\over{dx}}{{\partial\bar{C}}\over{\partial
g_{\mu}}}{\bigg{|}}_{{\mathbf{g}}=\langle{\mathbf{g}}\rangle_{x}},$ (6)

again working in a small noise limit.

To compute the information which $C$ conveys about $x$ we need the
distribution

$P(C|x)=\int d{\mathbf{g}}P(C|{\mathbf{g}})P({\mathbf{g}}|x),$ (7)

which in general is complicated, but if noise is small we can again make a
Gaussian approximation. The variance of $C$ at fixed $x$ has two components,
one from the variance at fixed $\mathbf{g}$, and one from the (co)variance of
$\mathbf{g}$ at fixed $x$, $\langle\delta g_{\mu}\delta g_{\nu}\rangle_{x}$:

$\displaystyle\langle(\delta C)^{2}\rangle_{x}$ $\displaystyle=$
$\displaystyle\sigma_{C}^{2}({\mathbf{g}}){\bigg{|}}_{{\mathbf{g}}=\langle{\mathbf{g}}\rangle_{x}}+A$
(8) $\displaystyle A$ $\displaystyle=$
$\displaystyle\sum_{\mu\nu}{{\partial\bar{C}}\over{\partial
g_{\mu}}}\langle\delta g_{\mu}\delta
g_{\nu}\rangle_{x}{{\partial\bar{C}}\over{\partial
g_{\nu}}}{\bigg{|}}_{{\mathbf{g}}=\langle{\mathbf{g}}\rangle_{x}},$ (9)

With this we have

$I(C;x)=S(C)-{1\over 2}{\bigg{\langle}}\log_{2}\left[2\pi e\langle(\delta
C)^{2}\rangle_{x}\right]{\bigg{\rangle}},$ (10)

and all the ingredients needed to express the objective function $\cal U$.

We notice that $\cal U$ is a local functional of $\bar{C}({\mathbf{g}})$ and
$\sigma_{C}^{2}({\mathbf{g}})$, so we can use the calculus of variations in a
familiar way. Optimizing with respect to $\sigma_{C}^{2}({\mathbf{g}})$ is
especially straightforward, and we find

$\displaystyle{{\partial{\cal U}}\over{\partial\sigma_{C}^{2}({\mathbf{g}})}}$
$\displaystyle=$ $\displaystyle 0$ (11)
$\displaystyle\Rightarrow\sigma_{C}^{2}({\mathbf{g}})$ $\displaystyle=$
$\displaystyle{\lambda\over{1-\lambda}}A,$ (12)

with $A$ from Eq (9). This makes sense, since it tells us that the precision
of encoding $\mathbf{g}$ in $C$ should be related to the scale of the
fluctuations in $\mathbf{g}$ when the relevant variable $x$ is fixed. At this
optimum we have

$\displaystyle I(C;{\mathbf{g}})$ $\displaystyle=$ $\displaystyle S(x)-{1\over
2}{\bigg{\langle}}\log_{2}\left[{{2\pi e}\over
B}{\lambda\over{1-\lambda}}\right]{\bigg{\rangle}}$ (13) $\displaystyle
I(C;x)$ $\displaystyle=$ $\displaystyle S(x)-{1\over
2}{\bigg{\langle}}\log_{2}\left[{{2\pi e}\over
B}{1\over{1-\lambda}}\right]{\bigg{\rangle}},$ (14)

with $B=(1/A)[{{d\bar{C}}/{dx}}]^{2}$. Optimizing $\cal U$ now is equivalent
to maximizing $B$ with respect to the function $\bar{C}({\mathbf{g}})$.

Notice that if there is only one variable $g$, then

$\displaystyle A$ $\displaystyle=$ $\displaystyle
A_{1}={\bigg{|}}{{d\bar{C}}\over{dg}}{\bigg{|}}^{2}\langle(\delta
g)^{2}\rangle$ (15) $\displaystyle B$ $\displaystyle=$ $\displaystyle
B_{1}={1\over{A_{1}}}{\bigg{|}}{{d\bar{C}}\over{dx}}{\bigg{|}}^{2}.$ (16)

Since are working in the limit where noise is small,

$\displaystyle{{d\bar{C}}\over{dx}}$ $\displaystyle=$
$\displaystyle{{d\bar{C}}\over{dg}}\cdot{{d\langle g\rangle_{x}}\over{dx}}$
(17) $\displaystyle\Rightarrow B$ $\displaystyle=$
$\displaystyle{1\over{\langle(\delta g)^{2}\rangle}}{\bigg{|}}{{d\langle
g\rangle_{x}}\over{dx}}{\bigg{|}}^{2}={1\over{\sigma_{x}^{2}({g})}},$ (18)

where in the last step we recognize $\sigma_{x}$ as the error bar in
estimating $x$ from the vector of concentration $g$ dubuis+al_13 ;
tkacik+al_15 ; this identification itself is correct only in the low noise
limit. In this case $B$ is determined independent of our choice for
${\bar{C}}({\mathbf{g}})$, and we can go further to show that the expectation
values which appear in Eq (13, 14) are related to the available information
$I(g;x)$. Perhaps surprisingly, we will see that with multiple variables we
arrive at the same answer by optimizing the function
${\bar{C}}({\mathbf{g}})$.

Because $\cal U$ depends on the derivatives of $\bar{C}$ with respect to the
components of the vector $\mathbf{g}$, the Euler–Lagrange equation
$\delta{\cal U}/\delta\bar{C}=0$ corresponds as usual to the divergence of
some vector $\mathbf{V}$ vanishing. But if the derivatives $d\langle
g_{\mu}\rangle_{x}/dx$ vanish at extreme values of the relevant variable $x$,
then $\mathbf{V}$ itself must vanish. The result is that we can write an
explicit, self–consistent expression for the gradient of $\bar{C}$,

$\displaystyle{{\partial\bar{C}({\mathbf{g}})}\over{\partial g_{\mu}}}$
$\displaystyle=$ $\displaystyle D\sum_{\nu}\left[\left(\langle\delta g\delta
g\rangle_{x}\right)^{-1}\right]_{\mu\nu}{{d\langle
g_{\nu}\rangle_{x}}\over{dx}}$ (19) $\displaystyle D$ $\displaystyle=$
$\displaystyle{{\nabla_{g}\bar{C}\cdot\langle\delta g\delta
g\rangle_{x}\cdot\nabla_{g}\bar{C}}\over{\nabla_{g}\bar{C}\cdot\nabla_{x}\langle
g\rangle}}.$ (20)

This is complicated, but yields a simple expression for $B$,

$\displaystyle B$ $\displaystyle=$ $\displaystyle\sum_{\mu\nu}{{d\langle
g_{\mu}\rangle}\over{dx}}\left[\left(\langle\delta g\delta
g\rangle_{x}\right)^{-1}\right]_{\mu\nu}{{d\langle g_{\nu}\rangle}\over{dx}}$
(21) $\displaystyle=$ $\displaystyle{1\over{\sigma_{x}^{2}({\mathbf{g}})}},$
(22)

where $\sigma_{x}({\mathbf{g}})$ now is the error bar in estimating $x$ from
the entire vector of concentrations $\mathbf{g}$ dubuis+al_13 ; tkacik+al_15 .
Then we can identify the information that the concentrations $\mathbf{g}$
provide about $x$,

$I({\mathbf{g}};x)=S(x)-{1\over 2}{\bigg{\langle}}\log_{2}\left[2\pi
e\sigma_{x}^{2}({\mathbf{g}})\right]{\bigg{\rangle}}.$ (23)

Finally, putting the different terms together we find that along the bounding
curve

$\displaystyle I(C;x)$ $\displaystyle=$ $\displaystyle
I({\mathbf{g}};x)-{1\over 2}\log_{2}\left({1\over{1-\lambda}}\right)$ (24)
$\displaystyle I(C;{\mathbf{g}})$ $\displaystyle=$ $\displaystyle
I({\mathbf{g}};x)-{1\over 2}\log_{2}\left({\lambda\over{1-\lambda}}\right).$
(25)

We see that with $0<\lambda<1$, the information $I(C;x)$ that is captured
about the relevant variable is less than the available information
$I({\mathbf{g}};x)$, as it must be. As $\lambda\rightarrow 0$ this gap closes,
but at the expense of requiring an increasing information capacity in the
mapping ${\mathbf{g}}\rightarrow C$. Taken together, Eqs (24) and (25) define
the bounding curve in the information plane, as shown in Fig 1.

Figure 1:  (A) The four gap genes Hb, Kr, Kni and Gt are expressed in varying
concentrations $g$ along the anterior–posterior axis of the fly embryo $x$
petkova+al_19 ; $L\sim 0.5\,{\rm mm}$. (B) The information plane $I(C;x)$ vs
$I(C;g)$ for optimized $C$ when $g$ is each of the four gap gene expressions
separately, some exemplary pairs, and all gap gene expressions together;
computations follow the methods of Ref bauer+al_21 . (C) Collapse of the data
as predicted from Eqs (24) and (25), with the boundary curve from the
universal tradeoff, Eqs (24) and (25), in black.

The result in Eqs (24) and (25) is surprising, because all details of the
underlying system have disappeared. This asymptotically universal bounding
curve suggests there is a tradeoff between the capacity of the cell to measure
the concentration of signaling molecules and the resulting ability to capture
relevant information, independent of molecular mechanisms, at least in some
regime. Are real cells in this regime?

The gap genes form a network that is crucial to the early events of fly
development nusslein-vollhard+wieschaus_80 ; jaeger_11 . These four genes take
inputs from primary maternal morphogens and in turn drive the striped patterns
of pair–rule gene expression. The local concentrations of gap gene proteins
provide enough information to specify position to $\sim 1\%$ accuracy along
the anterior–posterior axis, and this is the precision with which the stripes
are positioned dubuis+al_13 ; Liu2013 ; antonetti+al_18 . The algorithm that
achieves optimal readout of this positional information predicts,
quantitatively, the distortions of the pair–rule stripes in mutant flies where
individual maternal inputs are deleted petkova+al_19 .

The concentrations of the gap gene proteins ($\mathbf{g}$) encode information
about position ($x$), providing an example of the problem we have been
discussing; Fig 1A shows the means and standard deviations of $\mathbf{g}$ as
a function of $x$. With four gap genes, we can analyze the information that
they carry individually, or in groups; Figure 1B shows the numerical solutions
to the information bottleneck problem—minimizing $\cal U$ in Eq (1)—for
individual genes, exemplary combinations of two genes, and the full set of
four genes together bauer+al_21 ; bauer_22 . Figure 1C shows that these curves
collapse when shifted by the mutual information $I({\mathbf{g}};x)$, as
predicted in Eqs (24, 25). Notice that these shifts vary by up to $\sim
2.5\,{\rm bits}$ across the different groups. We see that these real examples
follow the predictions of the universal tradeoff (in black) quite accurately.
The case with all four genes is closest to the theory, which makes sense since
this is the case where our small noise approximation is most accurate.

The fact that a real system follows the universal tradeoff in Eqs (24) and
(25) invites us to consider the implications. We notice that the theoretical
prediction is close to being a line of unit slope, $I(C;x)\sim
I(C;{\mathbf{g}})$, ending in saturation at $I(C;x)=I({\mathbf{g}};x)$. We can
never have $I(C;x)>I(C;{\mathbf{g}})$, and it is interesting that, under
fairly general conditions, it is possible to approach this maximal efficiency:
if noisy mechanisms can keep only $I$ bits of information about the signaling
molecule concentrations, then it is possible for all of these $I$ bits to be
relevant for the organism, even when the mappings among signals are
complicated, as in the pattern of gap gene expression vs position; this is not
true in general NgampruetikornSchwab . We can see how close this tradeoff is
to the diagonal $I(C;x)=I(C;{\mathbf{g}})$ by calculating $I(C;x)$ at the
point $I(C;{\mathbf{g}})=I({\mathbf{g}},x)$; from Eq (25) this happens at
$\lambda=0.5$, where Eq (24) predicts that $I(C;x)=I({\mathbf{g}},x)-0.5\,{\rm
bits}$. Thus, in the regime we are considering, sensors with “just enough”
capacity to transmit all the information provided by the signaling molecules
can come close to deploying all this capacity for the relevant information.

On the other hand, it is worth emphasizing that no real mechanism can extract
all of the information that is available about $x$ from a perfect measurement
of $\mathbf{g}$. If the available information is $I$, then to get within
$\epsilon\,{\rm bits}$ Eq (24) tell us that we need a mechanism with capacity
$I(C;{\mathbf{g}})\sim I-(1/2)\log_{2}(2\epsilon\ln 2)$ at small $\epsilon$.
If the cell needs to make a binary decision, then making errors with small
probability $q$ is equivalent to losing $\epsilon\sim q\log_{2}(1/eq)\,{\rm
bits}$. This means, for example, that if the initial signals $\mathbf{g}$ are
just sufficient to provide one bit of information, the cell would need to read
these signals with $\sim 2.5\,{\rm bits}$ of accuracy in order to keep errors
below $\sim 1\%$. The requirements are even more stringent if we imagine that
the initial signal $\mathbf{g}$ is processed through several layers. The
perhaps surprising conclusion is that mechanisms with one bit of information
capacity are not sufficient for cells to make reliable binary decisions.

###### Acknowledgements.

We thank our experimental colleagues T Gregor, MD Petkova, and EF Wieschaus,
whose results inspired these ideas. This work was supported in part by the
National Science Foundation through the Center for the Physics of Biological
Function (PHY–1734030), by fellowships from the Simons Foundation and the John
Simon Guggenheim Memorial Foundation (WB), and by a start-up grant from the
Bionanoscience Department at TU Delft (MB).

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