Entanglement Dynamics for Two Spins in an Optical Cavity —Field Interaction
Induced Decoherence and Coherence Revival

Xue-Min Bai,1 Chun-Ping Gao,1 Jun-Qi Li,1,2 Ni Liu,1,3 J.-Q. Liang,1,*

1Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006,
China
<EMAIL_ADDRESS>
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###### Abstract

We in this paper study quantum correlations for two neutral spin-particles
coupled with a single-mode optical cavity through the usual magnetic
interaction. Two-spin entangled states for both antiparallel and parallel
spin-polarizations are generated under the photon coherent-state assumption.
Based on the quantum master equation we derive the time-dependent quantum
correlation of Clauser-Horne-Shimony-Holt (CHSH) type explicitly in comparison
with the well known entanglement-measure concurrence. In the two-spin singlet
state, which is recognized as one eigenstate of the system, the CHSH
correlation and concurrence remain in their maximum values invariant with time
and independent of the average photon-numbers either. The correlation varies
periodically with time in the general entangled-states for the low average
photon-numbers. When the photon number increases to a certain value the
oscillation becomes random and the correlations are suppressed below the Bell
bound indicating the decoherence of the entangled states. In the high photon-
number limit the coherence revivals periodically such that the CHSH
correlation approaches the upper bound value at particular time points
associated with the cavity-field period.

OCIS codes: (270.0270) Quantum optics; (270.5580) Quantum electrodynamics;
(270.5585 ) Quantum information and processing.

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##  1 Introduction

The non-locality is one of the most peculiar characteristics of quantum
mechanics without classical correspondence. As a typical example of non-
locality the two-particle entangled-state has become the essential ingredients
in quantum information and computation [1, 2, 3], although it was originally
considered by Einstein, Podolsky, and Rosen to question the completeness of
quantum mechanics. In general, an entangled state for a composite system is
not able to be factorized into a direct product of the individual states of
subsystems. Bell’s inequality (BI) derived in terms of classical statistics is
a criteria of the local correlation for the classical bipartite-model. It
actually provides a quantitative test of the entangled states, which are
fundamentally different from the classical world [4, 5, 6, 7, 8, 9, 10, 11],
while the underlying physics is obscure [12]. The BI and its violation in
agreement with quantum mechanical predictions [13, 14, 15, 16, 17, 18, 19, 20]
have attracted considerable attentions both theoretically and experimentally
in recent years [21, 22, 27, 28, 29, 26, 30, 23, 24, 25, 31]. Indeed, the
violation of BI has been verified undoubtedly in various systems, for example,
with entangled photons [13, 14, 15], trapped ions [16] and switchable
Josephson qubits [17]. Soon after the Bell’s pioneer work the original BI was
modified to various new forms [32, 33, 34, 35, 36, 37, 38], among which a more
suitable inequality for the quantitative test was formulated by Clauser-Horne-
Shimony-Holt (CHSH). The entanglement and BI have been also investigated in
optical cavity with identical atoms [39, 40, 41, 42, 43, 44, 45, 46]. Recently
a loop-free experiment on the violation of BI was reported [47, 48] using the
spins of a nitrogen-vacancy defect in diamond [49, 50, 51, 52, 53, 54].

Nonetheless, most of these current studies are focused on the measurement of
static entangled states. It is of fundamental importance to explore the
dynamic behavior of quantum correlation for the bipartite entangled states and
the related violation of BI in an interacting system. The environment induced
decoherence is considered as one of the most serious obstacle to realize the
quantum operation. The dynamic evolution of quantum correlations for a two-
qubit system has been studied extensively in various dissipative-environments
[55, 56, 57, 58, 59, 60]. Motivated by the experimental test of BI in spin
systems [47, 48, 61, 62] we in the present paper study a dynamic model for two
spins interacting with a cavity-field to see the field-interaction effect on
the entanglement. We demonstrate that general entangled states indeed can be
generated and then derive the quantum correlation in terms of quantum
probability statistics under the assumption of measuring outcome-independence
[63]. The quantum CHSH-correlation denoted by $P_{CHSH}$ results generally in
the violation of BI, $P_{CHSH}<2$, for the entangled states, and the maximum
correlation-value is found as $P_{CHSH}^{\max}=2\sqrt{2}$. The dynamic
evolution of the maximum quantum CHSH-correlation $P_{CHSH}^{\max}$ is
evaluated explicitly with the help of quantum master equation and the result
is compared with the entanglement measure, concurrence. We reveal the field-
interaction induced decoherence and the coherence revival, which is
characterized by the violation of BI.

The paper is organized as follows: we present the model Hamiltonian and
stationary solutions in Sec.II. The general entangled states are generated in
the spin cavity system. In Sec. III, we derive the quantum CHSH-correlation in
terms of quantum statistics for the two-spin entangled states. The quantum
dynamics of density operator is formulated in Sec.IV. The time-evolutions of
$P_{CHSH}^{\max}$ and concurrence are presented in Sec.V. Finally, we in
Sec.VI summarize and discuss the results.

##  2 Stationary solutions and entangled states

Following the recent experiment for the violation of BI with nitrogen-vacancy
defects in diamond [47, 48, 49, 50] we assume that the two spins are subject
to a one-mode optical cavity of frequency $\omega$. The usual Zeeman energy
for two spins in the magnetic component of quantized light field gives rise to
the Hamiltonian

$H=\omega
a^{\dagger}a+ig\sum_{i=1}^{2}(a\sigma_{i}^{+}-a^{\dagger}\sigma_{i}^{-}),$ (1)

with the convention $\hbar=1$ throughout the paper. The Pauli matrices
$\sigma_{i}^{z}$, $\sigma_{i}^{\pm}=\sigma_{i}^{x}\pm i\sigma_{i}^{y}$ are
defined to describe the spin operators. $a$ ($a^{\dagger}$) is the
annihilation (creation) operator of the single-mode cavity field.

We are looking for the stationary solutions of Hamiltonian Eq. (1) in the
optical coherent state $\left|\alpha\right\rangle$ under the semiclassical
approximation. Taking the average in the coherent state
$\left|\alpha\right\rangle$, in which the complex eigenvalue of the photon
annihilation operator, that $a|\alpha\rangle=\alpha|\alpha\rangle$, is
parameterized as $\alpha=\gamma e^{i\phi}$, we have an effective Hamiltonian
with spin operator only

$H_{sp}(\alpha)=\left\langle\alpha\right|H\left|\alpha\right\rangle=\omega\gamma^{2}+\sum_{i=1}^{2}i\gamma
g[e^{i\phi}\sigma_{i}^{+}-e^{-i\phi}\sigma_{i}^{-}].$ (2)

$\gamma^{2}=\langle\alpha|a^{\dagger}a|\alpha\rangle$ denotes the average
photon-number, which plays a important role in the entanglement dynamics. The
spin Hamiltonian $H_{sp}(\alpha)$ can be diagonalized in the two-qubit bases
$|e_{1}\rangle=\left|+,+\right\rangle$,
$|e_{2}\rangle=\left|+,-\right\rangle$,
$|e_{3}\rangle=\left|-,+\right\rangle$, $|e_{4}\rangle=\left|-,-\right\rangle$
with the energy eigenvalues given by

$\displaystyle\varepsilon_{0}(\gamma)$ $\displaystyle=$
$\displaystyle\omega\gamma^{2}-2g\gamma,$
$\displaystyle\varepsilon_{3}(\gamma)$ $\displaystyle=$
$\displaystyle\omega\gamma^{2}+2g\gamma,$
$\displaystyle\varepsilon_{1}(\gamma)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(\gamma)=\omega\gamma^{2},$ (3)

which are functions of parameter $\gamma$. The corresponding eigenstates are
seen to be

$\displaystyle\left|\psi_{0}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}[e^{-i\phi}\left|-,-\right\rangle-e^{i\phi}\left|+,+\right\rangle-i(\left|+,-\right\rangle+\left|-,+\right\rangle)],$
$\displaystyle\left|\psi_{3}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}[e^{-i\phi}\left|-,-\right\rangle-e^{i\phi}\left|+,+\right\rangle+i(\left|+,-\right\rangle+\left|-,+\right\rangle)],$
$\displaystyle\left|\psi_{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(e^{i\phi}\left|+,+\right\rangle+e^{-i\phi}\left|-,-\right\rangle),$
$\displaystyle\left|\psi_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\left|-,+\right\rangle-\left|+,-\right\rangle),$
(4)

The two-spin entangled states indeed can be generated in the optical cavity
under the semiclassical field approximation. $\left|\psi_{1}\right\rangle$ is
the well studied two-spin singlet state in relation with the nonlocality and
BI. While $\left|\psi_{2}\right\rangle$, which is a degenerate state of
$\left|\psi_{1}\right\rangle$, is the entangled state with parallel spin
polarization. The stability of the states $\left|\psi_{0}\right\rangle$,
$\left|\psi_{1}\right\rangle$, and $\left|\psi_{2}\right\rangle$ can be
examined with variational method regarding $\gamma$ as a variational parameter
of energy function. For example, the solution $\gamma=0$ of the energy
extremum equation $\partial$ $\varepsilon_{1}(\gamma)/\partial\gamma=0$ is
obviously the energy minimum of the energy function $\varepsilon_{1}(\gamma)$.
Thus $\left|\psi_{1}\right\rangle$, and $\left|\psi_{2}\right\rangle$ are
stable states of normal phase with zero average photon-number. While
$\left|\psi_{0}\right\rangle$ is the ground state of superradiant phase [64,
65, 66]. We solve in the present paper the dynamic equation of motion for the
given initial entangled states and then study the time-evolution of measuring
outcome-correlations. Particular attention is paid on the field interaction
induced decoherence and coherence revival. For the sake of simplicity we
consider the dynamics associated with the initial states
$\left|\psi_{1}\right\rangle$, and $\left|\psi_{2}\right\rangle$.

##  3 Measuring outcome correlation and Bell’s inequality

BI derived from classical statistics with the assumption of locality, was one
of the first criteria to detect the quantum entanglement. Recently the BIs and
their violations were revisited in terms of quantum probability statistics
with the help of state density-operator [63], which has advantage to formulate
the various forms of inequality and the violation in a unified manner. We
consider a general two-spin entangled state with antiparallel spin-
polarization

$\left|\psi\right\rangle=\sin\xi e^{i\eta}\left|+,-\right\rangle+\cos\xi
e^{-i\eta}\left|-,+\right\rangle,$ (5)

where $\xi$, $\eta$ are two arbitrary angle-parameters. The density operator
can be separated into the local and non-local parts

$\rho_{\psi}=\rho_{\psi}^{lc}+\rho_{\psi}^{nlc}.$ (6)

The local part

$\rho_{\psi}^{lc}=\sin^{2}\xi\left|+,-\right\rangle\langle+,-|+\cos^{2}\xi\left|-,+\right\rangle\langle-,+|,$
(7)

which is the classical probability state, gives rise to BIs. While the non-
local part

$\rho_{\psi}^{nlc}=\sin\xi\cos\xi
e^{-i2\eta}\left|+,-\right\rangle\langle-,+|+\sin\xi\cos\xi
e^{i2\eta}\left|-,+\right\rangle\langle+,-|,$ (8)

which comes from the quantum interference between two components of the
entangled state leads to the violation of BIs. Following Bell the measurements
of two spins are performed independently along arbitrary directions, say
$\mathbf{a}$ and $\mathbf{b}$, respectively. The measuring outcomes fall into
the eigenvalues of projection spin-operators

$\sigma\cdot\mathbf{a}\left|\pm\mathbf{a}\right\rangle=\pm\left|\pm\mathbf{a}\right\rangle$

and

$\sigma\cdot\mathbf{b}\left|\pm\mathbf{b}\right\rangle=\pm\left|\pm\mathbf{b}\right\rangle,$

according to the quantum measurement theory. By solving the above
eigenequations we obtain the explicit form of eigenstates

$\left|+\mathbf{n}\right\rangle=\cos(\frac{\theta_{n}}{2})\left|+\right\rangle+\sin(\frac{\theta_{n}}{2})e^{i\Phi_{n}}\left|-\right\rangle$

and

$\left|-\mathbf{n}\right\rangle=\sin(\frac{\theta_{n}}{2})\left|+\right\rangle-\cos(\frac{\theta_{n}}{2})e^{i\Phi_{n}}\left|-\right\rangle,$

which are known as the spin coherent states [64, 65, 66] of north- and south-
pole gauges. Here, $\mathbf{n}=(\sin\theta_{n}\cos\Phi_{n}$,
$\sin\theta_{n}\sin\Phi_{n}$, $\cos\theta_{n})$ with $\mathbf{n}=\mathbf{a}$,
$\mathbf{b}$ is a unit vector parameterized by the polar and azimuthal angles
($\theta_{n}$ , $\Phi_{n}$). The outcome-independent base vectors of two-spin
measurements are the product eigenstates of operators $\sigma\cdot\mathbf{a}$
and $\sigma\cdot\mathbf{b}$ labeled arbitrarily as

$\left|1\right\rangle\\!=\\!\left|\\!+\mathbf{a,\\!+b}\right\rangle,\\!\left|2\right\rangle\\!=\\!\left|\\!+\mathbf{a,\\!-b}\right\rangle,\\!\left|3\right\rangle\\!=\\!\left|\\!-\mathbf{a,\\!+b}\right\rangle,\\!\left|4\right\rangle\\!=\\!\left|\\!-\mathbf{a,\\!-b}\right\rangle.$
(9)

The measuring outcome correlation-probability for two spins respectively along
directions $\mathbf{a}$ and $\mathbf{b}$ is thus obtained in terms of the
quantum statistical-average under the outcome-independent base vectors Eq. (9)

$P(a,b)=Tr[(\sigma\cdot\mathbf{a})(\sigma\cdot\mathbf{b})\rho_{\psi}].$ (10)

Using the separated density operators Eq. (6) the measuring outcome
correlation-probability Eq. (10) can be also split into the local and non-
local parts

$P(a,b)=P_{lc}(a,b)+P_{nlc}(a,b).$

The local part Eq. (7) gives rise to the Bell correlation [63] of two-
direction ($a,b$) measurements

$P_{lc}(a,b)=-\cos\theta_{a}\cos\theta_{b}.$ (11)

Substituting the the Bell correlation Eq. (11) into the CHSH-correlation-
probability of four-direction measurements defined as

$P_{CHSH}=\left|P(a,b)+P(a,c)+P(d,b)-P(d,c)\right|,$ (12)

we recover the well known CHSH-inequality

$P_{CHSH}^{lc}\\!=\\!\left|\cos\theta_{a}(\cos\theta_{b}\\!+\\!\cos\theta_{c})\\!+\\!\cos\theta_{d}(\cos\theta_{b}\\!-\\!\cos\theta_{c})\right|\\!\leq
2,$ (13)

which was derived originally by CHSH from classical statistics with the
assumption of locality. The CHSH-inequality Eq. (13) is valid for arbitrary
entangled states given by Eq. (5) considering the local part Eq. (7) only.
Including the non-local part Eq. (8) the quantum correlation for the two-
direction measurements is simply a scaler product of the two unit-vectors
$\mathbf{a}$ and $\mathbf{b}$

$P(a,b)=-\mathbf{a\cdot b,}$ (14)

which is valid, however, under the condition $\xi=3\pi/4$, $\eta=n\pi$ with
$n$ being a integer. Thus the arbitrary entangled state Eq. (5) reduces to the
two-spin singlet state $\left|\psi_{1}\right\rangle$ in Eq. (4). With the
quantum correlation Eq. (14) the quantum CHSH-probability Eq. (12) becomes
[63, 67]

$P_{CHSH}=\left|\mathbf{a\cdot(b+c)+d\cdot(b-c)}\right|.$ (15)

When the unit vector $\mathbf{b}$ is perpendicular to $\mathbf{c}$, and
$\mathbf{a}$, $\mathbf{d}$ are respectively parallel to ($\mathbf{b+c}$),
($\mathbf{b-c}$), we obtain the maximum quantum CHSH-probability as

$P_{CHSH}^{\max}=2\sqrt{2},$ (16)

which indicates the maximum violation of CHSH-inequality Eq. (13). Our
formalism shows a direct relation between the decoherence and CHSH-inequality,
which can serves a criteria of coherence for the two parts of entangled state.
Most recently it was proved that for the entangled state of parallel spin
polarization

$\left|\psi\right\rangle=\sin\xi e^{i\eta}\left|+,+\right\rangle+\cos\xi
e^{-i\eta}\left|-,-\right\rangle,$ (17)

the Bell correlation of local part is simply [67]

$P_{lc}(a,b)=\cos\theta_{a}\cos\theta_{b},$

which is different from that of antiparallel spin-polarization Eq. (11) only
by a sign change. The CHSH-inequality for the local part Eq. (13) remains not
changed. The quantum CHSH-correlation-probability Eq. (16) is valid for the
entangled state under the condition $\xi=\pi/4$, $\eta=n\pi$, namely,
$\left|\psi\right\rangle=1/2(\left|+,+\right\rangle+\left|-,-\right\rangle).$

We are going to study the dynamics of CHSH-probability to reveal the field
interaction induced effect on the entanglement. The dynamic evolution of
$P_{CHSH}$ can be obtained from the time dependent density-operator $\rho(t)$
in terms of the quantum probability statistics presented in this paper. For
speciality we consider only the maximum quantum CHSH-probability Eq. (16),
which has been shown to evaluate in a clever way [68, 69] directly from the
state-density operator itself.

##  4 Quantum dynamics with photon coherent state

The dynamic evolution of two-qubit entangled states in the optical cavity can
be evaluated from the Schr$\ddot{o}$dinger equation $i\partial/\partial
t|\psi\rangle=H|\psi\rangle$. To this end we begin with the interaction
picture $i\partial/\partial t|\psi_{i}\rangle=H_{i}|\psi_{i}\rangle$, where
$|\psi_{i}\rangle=R(t)|\psi\rangle$ and the unitary operator is

$R(t)=e^{i\omega a^{\dagger}at}.$

The interaction-picture Hamiltonian is seen to be

$\displaystyle H_{i}$ $\displaystyle=$ $\displaystyle
R(t)HR^{{\dagger}}(t)-iR(t)\frac{\partial}{\partial t}R^{{\dagger}}(t)$ (18)
$\displaystyle=$ $\displaystyle ig\sum_{i=1}^{2}(ae^{-i\omega
t}\sigma_{i}^{+}-a^{\dagger}e^{i\omega t}\sigma_{i}^{-}).$

The entail time-evolution operator of the system reads as

$U(t)=R^{{\dagger}}(t)U_{i}(t),$ (19)

with $U_{i}(t)=\exp(-iH_{i}t)$ being the time evolution operator for the
interaction-picture Hamiltonian Eq. (1). In the two-qubit bases
$|e_{i}\rangle$ ($i=1,2,3,4$), time-evolution operator possesses a matrix form
[70],

$U_{i}(t)\\!=\left(\begin{array}[]{cccc}1\\!-\\!2a\frac{\sin^{2}(gt\sqrt{\frac{S}{2}})}{S}a^{\dagger}&a\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}&a\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}&2a\frac{\sin^{2}(gt\sqrt{\frac{S}{2}})}{S}a\\\
-\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}a^{\dagger}&\cos^{2}(gt\sqrt{\frac{S}{2}})&-\sin^{2}(gt\sqrt{\frac{S}{2}})&\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}a\\\
-\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}a^{\dagger}&-\sin^{2}(gt\sqrt{\frac{S}{2}})&\cos^{2}(gt\sqrt{\frac{S}{2}})&\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}a\\\
2a^{\dagger}\frac{\sin^{2}(gt\sqrt{\frac{S}{2}})}{S}a^{\dagger}&-a^{\dagger}\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}&-a^{\dagger}\frac{\sin(gt\sqrt{2S})}{\sqrt{2S}}&1-2a^{\dagger}\frac{\sin^{2}(gt\sqrt{\frac{S}{2}})}{S}a\end{array}\right),$
(20)

where $S=2a^{\dagger}a+1$.

From the quantum master equation

$i\frac{d\rho(t)}{dt}=[H,\rho(t)],$ (21)

the density operator at time $t$ is found as
$\rho(t)=U(t)\rho(0)U^{\dagger}(t)$, where the initial density operator

$\rho(0)=\rho_{\psi}(0)\rho_{f}(0),$

is the product of the spin part
$\rho_{\psi}(0)=\left|\psi(0)\right\rangle\langle\psi(0)|$ and the cavity-
field part $\rho_{f}(0)=|\alpha\rangle\langle\alpha|$ with $|\alpha\rangle$
being the photon coherent state [64]. We are interested in the field-
interaction effect on the quantum measuring-outcome correlation and thus
assume the initial state being normalized two-spin entangled-state Eq. (5).
The matrix representation of the density operator Eq. (5) in the two-qubit
bases is

$\rho_{\psi}(0)=\left(\begin{array}[]{cccc}0&0&0&0\\\
0&\sin^{2}\xi&\sin\xi\cos\xi e^{i2\eta}&0\\\ 0&\sin\xi\cos\xi
e^{-i2\eta}&\cos^{2}\xi&0\\\ 0&0&0&0\end{array}\right).$ (22)

Using the explicit form of photon coherent state in the Fock space,
$|\alpha\rangle=\sum_{n}\alpha^{n}/\sqrt{n!}\exp\\{-|\alpha|^{2}/2\\}|n\rangle$
, we take the trace of density operator over the photon number state
$|n\rangle$ to obtain the reduced density operator that

$\rho_{r}(t)=\sum\limits_{n=0}^{\infty}\rho_{n}(t),\qquad\rho_{n}(t)=\langle
n|\rho(t)|n\rangle,$

with

$\rho_{n}(t)=\sum\limits_{j,l=0}^{\infty}\langle
n|U(t)|j\rangle\rho_{\psi}(0)\langle
l|U^{\dagger}(t)|n\rangle\frac{\gamma^{j+l}e^{i(j-l)\phi}}{\sqrt{j!l!}}e^{-\gamma^{2}}.$
(23)

The reduced density matrix is written in the two-qubit bases explicitly as

$\rho_{r}(t)=\sum\limits_{n=0}^{\infty}\left(\begin{array}[]{cccc}(\rho_{n})_{11}&(\rho_{n})_{12}&(\rho_{n})_{13}&(\rho_{n})_{14}\\\
(\rho_{n}^{\ast})_{12}&(\rho_{n})_{22}&(\rho_{n})_{23}&(\rho_{n})_{24}\\\
(\rho_{n}^{\ast})_{13}&(\rho_{n}^{\ast})_{23}&(\rho_{n})_{33}&(\rho_{n})_{34}\\\
(\rho_{n}^{\ast})_{14}&(\rho_{n}^{\ast})_{24}&(\rho_{n}^{\ast})_{34}&(\rho_{n})_{44}\end{array}\right).$
(24)

The diagonal elements of the density matrix $\rho_{n}(t)$ are seen to be

$\displaystyle(\rho_{n})_{11}$ $\displaystyle=$ $\displaystyle\langle
e_{1}|\rho_{n}|e_{1}\rangle=\frac{e^{-\gamma^{2}}}{n!}C_{1}^{2}\left(\sin
2\xi\cos 2\eta+1\right),$ $\displaystyle(\rho_{n})_{22}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(C_{2}^{2}\sin^{2}\xi+C_{3}^{2}\cos^{2}\xi+C_{2}C_{3}\sin
2\xi\cos 2\eta),$ $\displaystyle(\rho_{n})_{33}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(C_{2}^{2}\cos^{2}\xi+C_{3}^{2}\sin^{2}\xi+C_{2}C_{3}\sin
2\xi\cos 2\eta),$ $\displaystyle(\rho_{n})_{44}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}C_{4}^{2}\left(\sin 2\xi\cos
2\eta+1\right),$

and the off-diagonal elements are

$\displaystyle(\rho_{n})_{12}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}C_{1}e^{-2i\eta}(C_{2}\sin\xi+C_{3}e^{2i\eta}\cos\xi)(e^{2i\eta}\sin\xi+\cos\xi),$
$\displaystyle(\rho_{n})_{13}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}C_{1}e^{-2i\eta}(C_{2}e^{2i\eta}\cos\xi+C_{3}\sin\xi)(e^{2i\eta}\sin\xi+\cos\xi),$
$\displaystyle(\rho_{n})_{14}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}C_{1}C_{4}\left(\sin 2\xi\cos
2\eta+1\right),$ $\displaystyle(\rho_{n})_{23}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{2n!}\left(C_{2}^{2}e^{2i\eta}\sin
2\xi+2C_{2}C_{3}+C_{3}^{2}e^{-2i\eta}\sin 2\xi\right),$
$\displaystyle(\rho_{n})_{24}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}C_{4}e^{-2i\eta}(C_{2}e^{2i\eta}\sin\xi+C_{3}\cos\xi)(e^{2i\eta}\cos\xi+\sin\xi),$
$\displaystyle(\rho_{n})_{34}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}C_{4}e^{-2i\eta}(C_{2}\cos\xi+C_{3}e^{2i\eta}\sin\xi)(e^{2i\eta}\cos\xi+\sin\xi).$

We have ($\rho_{n})_{ij}=(\rho_{n})_{ji}^{\ast}$, since the state-density
operator $\rho_{n}(t)$ is Hermitian. The time functions are given by

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\gamma^{n+1}\frac{\sin(gt\sqrt{4n+6})}{\sqrt{4n+6}},$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\gamma^{n}\cos^{2}(gt\sqrt{\frac{2n+1}{2}}),$ $\displaystyle
C_{3}$ $\displaystyle=$
$\displaystyle-\gamma^{n}\sin^{2}(gt\sqrt{\frac{2n+1}{2}}),$ $\displaystyle
C_{4}$ $\displaystyle=$
$\displaystyle-\gamma^{n-1}\frac{n\sin(gt\sqrt{4n-2})}{\sqrt{4n-2}}.$

For the entangled initial-state of parallel spin polarization Eq. (17) the
density operator becomes

$\rho_{\psi}(0)=\left(\begin{array}[]{cccc}\sin^{2}\xi&0&0&\sin\xi\cos\xi
e^{2i\eta}\\\ 0&0&0&0\\\ 0&0&0&0\\\ \sin\xi\cos\xi
e^{-2i\eta}&0&0&\cos^{2}\xi\end{array}\right).$ (25)

The diagonal elements of the density matrix $\rho_{n}(t)$ are

$\displaystyle(\rho_{n})_{11}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(D_{1}^{2}\sin^{2}\xi+D_{2}^{2}\cos^{2}\xi+D_{1}D_{2}\sin
2\xi\cos 2\eta),$ $\displaystyle(\rho_{n})_{22}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(D_{3}^{2}\sin^{2}\xi+D_{4}^{2}\cos^{2}\xi+D_{3}D_{4}\sin
2\xi\cos 2\eta),$ $\displaystyle(\rho_{n})_{44}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(D_{5}^{2}\sin^{2}\xi+D_{6}^{2}\cos^{2}\xi+D_{5}D_{6}\sin
2\xi\cos 2\eta),$ $\displaystyle(\rho_{n})_{33}$ $\displaystyle=$
$\displaystyle(\rho_{n})_{22},$

and the off-diagonal elements read

$\displaystyle(\rho_{n})_{12}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(D_{4}\cos\xi+e^{-2i\eta}D_{3}\sin\xi)(D_{2}\cos\xi+e^{2i\eta}D_{1}\sin\xi),$
$\displaystyle(\rho_{n})_{14}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(D_{6}\cos\xi+e^{-2i\eta}D_{5}\sin\xi)(D_{2}\cos\xi+e^{2i\eta}D_{1}\sin\xi),$
$\displaystyle(\rho_{n})_{24}$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma^{2}}}{n!}(D_{6}\cos\xi+e^{-2i\eta}D_{5}\sin\xi)(D_{4}\cos\xi+e^{2i\eta}D_{3}\sin\xi),$
$\displaystyle(\rho_{n})_{13}$ $\displaystyle=$
$\displaystyle(\rho_{n})_{12},$ $\displaystyle(\rho_{n})_{23}$
$\displaystyle=$ $\displaystyle(\rho_{n})_{22},$
$\displaystyle(\rho_{n})_{34}$ $\displaystyle=$
$\displaystyle(\rho_{n})_{24},$

with the time functions being

$\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle\gamma^{n}(1-\frac{2(n+1)\sin^{2}(gt\sqrt{\frac{2n+3\allowbreak}{2}})}{2n+3}),$
$\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle\gamma^{n+2}\frac{2\sin^{2}(gt\sqrt{\frac{2n+3}{2}})}{2n+3},$
$\displaystyle D_{3}$ $\displaystyle=$
$\displaystyle-\gamma^{n-1}\frac{n\sin(gt\sqrt{4n+2})}{\sqrt{4n+2}},$
$\displaystyle D_{4}$ $\displaystyle=$
$\displaystyle\gamma^{n+1}\frac{\sin(gt\sqrt{4n+2})}{\sqrt{4n+2}},$
$\displaystyle D_{5}$ $\displaystyle=$
$\displaystyle\gamma^{n-2}\frac{2n(n-1)\sin^{2}(gt\sqrt{\frac{2n-1}{2}})}{2n-1},$
$\displaystyle D_{6}$ $\displaystyle=$
$\displaystyle\gamma^{n}(1-\frac{2n\sin^{2}(gt\sqrt{\frac{2n-1}{2}})}{2n-1}).$

In the following we are going to investigate measuring outcome-correlations of
two spins in terms of the density operator $\rho_{r}(t)$ to see the field-
interaction induced effect on the entanglement. The dynamic evolution of
quantum CHSH-correlation-particularly $P_{CHSH}^{\max}$ is evaluated in
comparison with the entanglement measure, concurrence.

##  5 Time evolution of quantum correlation

We now study the entanglement dynamics in terms of the time-evolution of
density matrix $\rho_{r}(t)$, from which the time-dependent maximum quantum
CHSH-correlation-probability is derived. To this end we define [68, 69] a
matrix relating to $\rho_{r}(t)$ as

$T_{\rho_{r}(t)}=\rho_{r}(t)\sigma\otimes\sigma.$

A symmetric matrix is given by

$U_{\rho_{r}(t)}=T_{\rho_{r}(t)}^{T}T_{\rho_{r}(t)},$ (26)

where $T_{\rho}^{T}$ is the transposition of $T_{\rho}$. The maximum quantum
CHSH-correlation-probability can be found [68, 69] from the eigenvalues of the
symmetric matrix $U_{\rho_{r}(t)}$

$P_{CHSH}^{\max}(t)=2\sqrt{m(\rho_{r})}.$ (27)

where $m(\rho_{{}_{r}(t)})=\max_{j<k}(u_{j}+u_{k})$ with $u_{j}$ $(j=1,2,3)$
being the eigenvalues of $U_{{}_{\rho_{r}(t)}}$ defined in Eq. (26). In the
absence of the cavity field we have
$P_{CHSH}^{\max}=2\sqrt{m(\rho_{\psi})}=2\sqrt{2}$, which coincides with Eq.
(16). The $P_{CHSH}^{\max}(t)$-value can be used to indicate the entanglement
that when $P_{CHSH}^{\max}(t)<2$, the entanglement collapses due to the field-
interaction induced quantum decoherence.

It is interesting to compare the $P_{CHSH}^{\max}(t)$ with the concurrence,
which is a well known measure of the entanglement [71] for a two-qubit system.
The instantaneous concurrence is defined by [71]

$C(t)=\max\\{0,\Lambda(t)\\},$ (28)

where $\Lambda(t)$ $=$
$\lambda_{1}(t)-\lambda_{2}(t)-\lambda_{3}(t)-\lambda_{4}(t)$.
$\lambda_{i}(t)$ denotes the square root of the instantaneous eigenvalue of
the matrix
$\rho_{r}(t)(\sigma_{y}\otimes\sigma_{y})\rho_{r}^{\ast}(t)(\sigma_{y}\otimes\sigma_{y})$
for $i=1,2,3,4$ in the decreasing order of eigenvalue magnitudes.
$\rho_{r}^{\ast}(t)$ is the complex conjugate of the two-qubit density matrix
$\rho_{r}(t)$.

The maximum CHSH-correlation-probability $P_{CHSH}^{\max}(t)$ is compared with
concurrence $C(t)$, so that to establish a dynamic relation between the
violation of CHSH inequality and the entanglement measure. In the photon
coherent-state the average photon-number $\gamma^{2}$, which, we will see,
affects greatly the entanglement dynamics.

###  5.1 Entangled states of antiparallel spin polarizations

Fig. 1: Color online) The maximum quantum CHSH-probability
$P_{CHSH}^{\max}(t)$ (a) and concurrence $C(t)$ (b) as functions of time $t$
measured in the period $T$ of optic field for the entangled-state of
antiparallel spin-polarization with angle parameters $\eta=0$ and $\xi=\pi/6$
(green solid line), $\pi/4$ (blue dot and dash line) $\pi/3$ (red dash line)
with average photon-number $\gamma^{2}=0.01$ (1), $1$ (2), $15$ (3), $150$
(4). $P_{CHSH}^{\max}(t)=2\sqrt{2}$, $C(t)=1$ not varying with time for the
spin singlet state $\left|\Psi_{1}\right\rangle$. Fig. 2: Color online) Time-
variation curves of $P_{CHSH}^{\max}(t)$ (a) and $C(t)$ (b) for the entangled-
state of antiparallel spin-polarization with angle parameters $\xi=\pi/4$ and
$\eta=\pi/6$ (green solid line), $\pi/4$ (blue dot and dash line) $\pi/3$
(pink dash line) with average photon-number $\gamma^{2}=0.01$ (1), $1$ (2),
$15$ (3), $150$ (4).

The time-variation curves of maximum CHSH-correlation-probability
$P_{CHSH}^{\max}(t)$ (a) and the concurrence $C(t)$ (b) are displayed in Figs.
1-2, for initial entangled states of antiparallel spin-polarizations Eq. (5)
with various average photon-numbers $\gamma^{2}=0.01$ (1), $1$ (2), $15$ (3),
$150$ (4). The time $t$ is measured in the optic-field period $T=2\pi/\omega$
and the atom-filed coupling $g=1$ is in the unit of field frequency $\omega$.
The time-variation curves of $P_{CHSH}^{\max}(t)$ and $C(t)$ have one to one
correspondence with the upper bound value $P_{CHSH}^{\max}=2\sqrt{2}$, which
corresponds to the concurrence $C=1$ indicating the maximum entanglement. The
dynamics is sensitive to the angle parameters $\xi$ and $\eta$ of the
entangled states. For $\eta=0$ the two coefficients of the entangled states
are real and the time-evolution curves are plotted for the parameters
$\xi=\pi/6$ (green solid line), $\pi/4$ (blue dot and dash line), and $3\pi/4$
(red dash line) in Fig. 1. It is remarkable to see a fact that both the
maximum CHSH-correlation-probability and concurrence remain in the maximum
values $P_{CHSH}^{\max}(t)=2\sqrt{2}$, $C(t)=1$ not varying with time when
$\xi=3\pi/4$. In this case the initial state becomes an eigenstate of the
system $\left|\psi_{1}\right\rangle$ i.e. the two-spin singlet state as shown
in Eq. (4), which is independent of the cavity-field parameters $\gamma$ and
$\phi$. The measuring outcome of two-spin correlation operator is invariant in
the Hamiltonian eigenstate $\left|\psi_{1}\right\rangle$, since the density
operator commutes with the Hamiltonian and thus is a conserved quantity. If
the initial state is not the eigenstate with $\xi=\pi/6$, $\pi/4$,
$P_{CHSH}^{\max}(t)$ oscillates with time below the quantum upper-bound
$2\sqrt{2}$. The oscillation behavior depends on the average photon numbers.
In the lower value of average photon-number $\gamma^{2}=0.01$,
$P_{CHSH}^{\max}(t)$ is a periodic oscillation around the classical upper-
bound value of $P_{CHSH}^{lc}=2$ seen in Eq. (13). The oscillation becomes
random for $\gamma^{2}$ tending to $1$. When the average photon-number
increases to $\gamma^{2}=15$, the entanglement collapses seen from Fig. 1(b3),
since we have $P_{CHSH}^{\max}(t)$ $<2$, namely the CHSH inequality Eq. (13)
is satisfied. In other words the quantum interference part $\rho_{\psi}^{nlc}$
disappears due to the field-interaction induced decoherence. In the large
photon-number limit the coherence revives periodically characterized by the
sharp-peak values, which approaches the quantum upper-bound $2\sqrt{2}$ $(1)$
seen from Fig. 1(a4,b4). The discrete peak-positions of $P_{CHSH}^{\max}(t)$
curves are located precisely at $t=n5T$ with $n$ being a integer. When the
superposition coefficients of entangled states become complex with non-
vanishing $\eta$, the periodic coherence-revival basically remains as shown in
Fig. 2 for $\xi=\pi/4$ and various phase angles $\eta$ for small or large
photon-number limit. It is interesting to see a fact that the CHSH inequality
is nearly always violated independent of average photon numbers for the state
with angle parameter $\eta=\pi/3$ (pink dash line), although the
$P_{CHSH}^{\max}(t)$ value is suppressed from the quantum upper-bound. The
field interaction does not lead to the decoherence for this state. While the
entanglement collapses for the states with $\eta=\pi/4$, $\pi/6$ at the
average photon number $\gamma^{2}=15$ (b3). The sharp peak-positions of
$P_{CHSH}^{\max}(t)$ curves become $t=(2n)5T$ in the large photon-number limit
with $\gamma^{2}=150$ (a4). While The peak-height almost shrinks to the
classical bound $2$ at the time points $t=$ $(2n+1)5T$. The concurrence plots
are displayed in (b1) to (b4) showing the same behaviors as
$P_{CHSH}^{\max}(t)$.

###  5.2 Entangled states for parallel spin polarizations

The BI was originally formulated and well studied based on the two-spin
singlet state $\left|\psi_{1}\right\rangle$. It was proved long ago [72] and
verified recently by means of quantum statistics [63] that the BI and its
violation are actually valid for arbitrary two-spin entangled states with
antiparallel polarization [72, 63] given by Eq. (5). Most recently it was
demonstrated that the original formula of BI has to be slightly modified for
the two-spin entangled state of parallel spin polarization, while the CHSH
inequality remains not changed [67]. Now we turn our attention to the dynamics
of the maximum quantum-CHSH-probability $P_{CHSH}^{\max}(t)$ and the
concurrence $C(t)$ for the initial entangled states with parallel spin-
polarizations given by Eq. (17). The $P_{CHSH}^{\max}(t)$ and $C(t)$ curves
are displayed respectively in upper (a) and lower (b) panels in Fig. 3 for the
average photon-numbers $\gamma^{2}=0.01$ (1), $5$ (2), $15$ (3), $150$ (4)
with the state parameters $\eta=0$, $\xi=\pi/3$ (green solid-line), $\pi/4$
(blue dot and dash line), $3\pi/4$ (red dash-line). Again the time evolution
becomes random when the average photon number reaches $\gamma^{2}=1$. When the
average photon-number increases to $\gamma^{2}=15$, the entanglement collapses
only for the initial state with $\xi=3\pi/4$. The CHSH inequality is, however,
violated for the states with $\xi=\pi/6$, $\pi/4$ since $P_{CHSH}^{\max}(t)>2$
in these cases. $P_{CHSH}^{\max}(t)$ remains in the upper quantum bound value
with small oscillation for the state $\xi=\pi/4$ (blue dot-and dash-line),
while the average value of $P_{CHSH}^{\max}(t)$ for the state $\xi=\pi/6$
(green solid line) is slightly lower than the upper bound. In the large
photon-number limit $\gamma^{2}=150$ (a4), the decoherence for the initial
state $\xi=3\pi/4$ revivals periodically at the time points $t=n5T$.
Correspondingly the time-variation curves of $P_{CHSH}^{\max}(t)$ and $C(t)$
are displayed in Fig. 4 for the initial states with complex superposition
coefficients, in which the angle parameters are chosen as $\xi=\pi/4,$
$\eta=\pi/6$ (green solid-line), $\pi/4$ (blue dot- and dash-line), $\pi/3$
(pink dash-line) respectively. At the average photon-number $\gamma^{2}=15$,
the entanglement collapses for the initial states with $\eta=\pi/4$, $\pi/3$
(b3) and revivals at the time points of $t=(n/2)5T$ in the large photon-number
limit $\gamma^{2}=150$ (b4). However, the peak-height of $P_{CHSH}^{\max}(t)$
can approach the upper quantum bound only at $t=(2n)5T$. The
$P_{CHSH}^{\max}(t)$-curve varies with time for the state $\eta=\pi/6$ above
the classical bound and so that the coherence remains.

Fig. 3: Color online) The maximum CHSH correlation-probability
$P_{CHSH}^{\max}(t)$ (upper panel) concurrence $C(t)$ (lower panel) as
functions of time $t$ in the initial entangled states of parallel spin-
polarizations with angle parameters $\eta=0$, $\xi=\pi/3$, $\pi/4$, and
$3\pi/4$ for the average photon-number $\gamma^{2}=0.01$ (1), $1$ (2), $15$
(3), $150$ (4).  Fig. 4: Color online) The time variation of
$P_{CHSH}^{\max}(t)$ (upper panel) and concurrence $C(t)$ (lower panel) in the
entangled states of parallel spin-polarizations with the angle parameters
$\xi=\pi/4,$ $\eta=\pi/6$, $\pi/4$, and $\pi/3$ for the average photon-number
$\gamma^{2}=0.01$ (1), $1$ (2), $15$ (3), $150$ (4).

.

##  6 Conclusion and discussion

We propose a theoretical model for two spins in a single-mode optical cavity
with the ordinary magnetic interaction. Stationary entangled-states are
realized in terms of variational method under the condition of optical
coherent-state. The well studied two-spin singlet state is indeed generated in
this simple model. Besides we also obtain the entangled state with parallel
spin polarizations. The BI and its violation are reformulated in a unified way
by means of quantum probability statistics with the help of state density
operator. This formulation establishes a direct relation between decoherence
and BI, which is obtained by neglecting the quantum interference part of
density operator. Thus the BI indeed can serve as a criteria of decoherence.
The BI in the CHSH form is also valid for the general two-spin entangled
states of parallel spin polarizations so that the entanglement dynamics can be
investigated for arbitrary entangled states with both antiparallel and
parallel spin polarizations. Based on the time-evolution of state-density
operator derived with the quantum master equation the dynamics of maximum
CHSH-correlation-probability $P_{CHSH}^{\max}(t)$ is evaluated in comparison
with the concurrence $C(t)$ of two-spin entangled states. It is interesting to
see a fact that the maximum quantum CHSH-probability as well as the
concurrence does not vary with time in the spin singlet state. This is because
that the density operator, which commutes with Hamiltonian, is a conserved
quantity, and the measuring outcome is invariant with time in this state. In
the general entangled states with antiparallel spin polarizations, the time-
evolutions of $P_{CHSH}^{\max}(t)$ and $C(t)$ are periodic in the weak field
regimen with the average photon number in the order of $\gamma^{2}=0.01$. The
spin flip due to the field interaction leads to the time oscillation of
measuring outcome correlation. When the field intensity increases to the order
of $\gamma^{2}=1$ the time evolution becomes random. The entanglement
collapses at the average photon number $\gamma^{2}=15$ due to the field
induced decoherence, which is characterized by the fulfilling of CHSH-
inequality. The coherence revivals periodically in the large photon-number
limit $\gamma^{2}=150$, such that the sharp peaks of $P_{CHSH}^{\max}(t)$
curves approach the upper quantum-bound $2\sqrt{2}$. The peak positions are
located precisely at $t=n5T$. For the complex superposition-coefficients of
the entangled states the complete decoherence appears for the states
$\eta=\pi/6$, $\pi/4$. Coherence revivals at the time points $t=(2n)5T$, but
not at $t=(2n+1)5T$ , where the peak heights shrink to the classical bound
value $2$. The entanglement dynamics is similar but different in details for
the states with parallel spin-polarizations. The precise period of coherence
revival associated with the cavity-field frequency is an interesting
observation in the large photon-number limit.

## Acknowledgments

This work was supported by the Natural Science Foundation of China (Grants
Nos. 11105087, 61275210, 11275118, 11404198) and Scientific and Technological
Innovation Programs of Higher Education Institutions in Shanxi (STIP) (Grant
No. 2014102).


# Tissue Classification During Needle Insertion Using Self-Supervised
Contrastive Learning and Optical Coherence Tomography

Debayan Bhattacharya1,2, Sarah Latus1, Finn Behrendt1, Florin Thimm1, Dennis
Eggert2,
Christian Betz2, Alexander Schlaefer1 1Institute of Medical Technology and
Intelligent Systems, Hamburg University of Technology, Hamburg, Germany
<EMAIL_ADDRESS>for Ears, Nose and Throat, University
Medical Center Hamburg-Eppendorf, Hamburg, Germany

###### Abstract

Needle positioning is essential for various medical applications such as
epidural anaesthesia. Physicians rely on their instincts while navigating the
needle in epidural spaces. Thereby, identifying the tissue structures may be
helpful to the physician as they can provide additional feedback in the needle
insertion process. To this end, we propose a deep neural network that
classifies the tissues from the phase and intensity data of complex OCT
signals acquired at the needle tip. We investigate the performance of the deep
neural network in a limited labelled dataset scenario and propose a novel
contrastive pretraining strategy that learns invariant representation for
phase and intensity data. We show that with 10% of the training set, our
proposed pretraining strategy helps the model achieve an F1 score of 0.84±0.10
whereas the model achieves an F1 score of 0.60±0.07 without it. Further, we
analyse the importance of phase and intensity individually towards tissue
classification.

###### Index Terms:

self-supervised learning, contrastive learning, optical coherence tomography,
needle navigation

##  I Introduction

Precise positioning of needles is essential for various medical applications.
For example, during epidural anesthesia the physician needs to navigate a
needle to the epidural space, which has a width of a few millimeter. Hence,
identification of the tissue structures passed during insertion is important
to estimate the position of the needle. Image-guided approaches, e.g., using
ultrasound (US) [1], as well as electromagnetic or optical needle pose
tracking have been investigated to assist needle placement. To address
inaccuracies of these external needle navigation methods [2], approaches to
estimate tissue properties from the needle tip have been studied, e.g. by
sensing the forces acting at the needle tip [3, 4] or integrating miniaturized
imaging probes [5]. Recently, the application of optical coherence tomography
(OCT), an imaging modality with high spatial and temporal resolution, has been
proposed. It enables depth-resolved scanning of tissue structures by measuring
the reflections of infrared light. The acquired complex OCT signal can be
split into intensity and phase data, with the latter resolving tissue motion
in sub-pixel range. However, imaging depth of OCT is limited to a few
millimeters and interpretation of image data requires time consuming analysis
by experienced clinician who may not be available every time.

Hence, supervised deep learning methods have been proposed that use OCT
intensity data for tissue classification [5, 6] and boundary tracking [7]
during needle insertion. In addition, deep learning based rupture detection
[8] has been proposed, combining intensity and phase data. Still, manual
labeling of acquired OCT datasets is time consuming and requires additional
information on the needle pose and surrounding tissue structures. These
ground-truth information are accessible for phantom studies with optimized
needle tracking conditions. But gaining ground-truth information during in-
vivo needle insertions is complicated and hampered by decreased imaging
resolutions. This motivates the investigation of self-supervised learning
strategies for tissue classification during needle insertion so as to reduce
the labelling effort.

Self-supervised learning has been proposed to alleviate the problem of time-
consuming labelling by learning visual features from unlabelled dataset [9,
10, 11, 12]. A popular self-supervised learning method SimCLR [13] learns
invariant representation of an image and its augmented version. Using the
InfoNCE loss [14], the network learns invariant representations for the image
and its augmented version. In our case, we have intensity and phase data from
the complex OCT signal. Augmentations such as rotation and random crop may not
be ideal for our data as the augmented views will represent signals that are
unrealistic to generate by needle insertions. As such, meaningful
representations will not be learned in the pretraining step. Instead, to learn
better representations, we align phase and intensity features within a time
window. Empirically, we demonstrate that this pretraining strategy is helpful
for tissue classification especially in limited labelled dataset scenario.

In conclusion, our contributions are three fold. First, we propose a deep
learning algorithm which classifies tissue class by jointly using phase and
intensity data. Second, to circumvent the issues arising from limited labelled
dataset, we propose a novel self-supervised contrastive pretraining strategy
that increases the classification performance in an extremely low labelled
dataset scenario. Third, analogous to the rupture detection study [8], we
investigate the importance of the phase and intensity data on the
classification performance by evaluating only using phase or intensity data.

##  II Materials and Methods

###  II-A Data acquisition and preparation

Figure 1: a) Experimental setup for data acquisition. We use a microcontroller
(E) with attached stepper motor (A) to position soft tissue phantoms (B)
relative to a fixed US probe (C). We insert our OCT needle (D) with a second
stepper motor (F). A logic analyzer (G) records the trigger signals of all
components. Visualisation of OCT b) intensity and c) phase M-scan with
highlighted uncertainty windows (red).

We design an experimental setup to perform reproducible needle insertions in
soft tissue phantoms (Fig. 1). We embed a cleaved optical fiber in an epidural
Touhy needle to acquire one-dimensional scans (A-scans) at the needle tip with
our OCT system (Telesto I, Thorlabs). To track the needle motion relative to
soft tissue structures we additionally apply US imaging with a linear probe
(Cicada, Cephasonics). We use two stepper motors to control the needle and
phantom motion and conduct multiple needle insertions per phantom. We insert
the needle with a velocity of
$1\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$ and simultaneously acquire
OCT A-scans and US images with frequencies of $91\text{\,}\mathrm{kHz}$ and
$60\text{\,}\mathrm{Hz}$, respectively. For data synchronization we record the
trigger signals of the motor driver, OCT and US system. Our soft tissue
phantoms are composed of beef, pork, or turkey meat embedded in gelatin of
$5\text{\,}\mathrm{\char 37\relax}$ density to imitate tissue structures
punctured during epidural punctures similar to [3]. We perform three data
preprocessing steps. First, we display the A-scans sequentially recorded
during insertion side by side over time to obtain M-scans, considering either
OCT intensity or phase data (Fig. 1 (b) and (c)). Second, we average the
M-scans over time with a window size of 1000 A-scans and apply spatial
cropping. Third, we extract the needle position relative to the tissue
structures from the US images to label the OCT M-scans. To consider
inaccuracies for transitions between tissue classes we define an uncertainty
window to be excluded from the labelled dataset.

###  II-B Overall Framework

Fig. 2 illustrates our proposed method. Our proposed method is a deep neural
network with two branches ($f_{\theta}$, $g_{\theta}$), that processes the
intensity and phase data respectively. We split the method in two parts: (i)
Intensity and phase alignment using contrastive pretraining, (ii)
Classification task using $c_{\theta}$ as the classification head.

Figure 2: (a) Our contrastive pretraining strategy (b) Finetuning for tissue
classification task.

(i) Intensity and phase alignment using contrastive pretraining: Our positive
pair consists of intensity and phase data in the same time window $t_{n}$.
Formally, let $x^{phs}_{t}\in R^{H\times W\times 3}$ and $x^{int}_{t}\in
R^{H\times W\times 3}$ be phase and intensity crops at time window $t$
respectively. It is to be noted that the phase and intensity crops are
originally single channel images. The information is copied to the 2 channels
to make it a 3 channel input for $f_{\theta}$ and $g_{\theta}$. We pass
$x^{int}_{t}$ through $f_{\theta}$ and $x^{phs}_{t}$ through $g_{\theta}$.
This results in phase and intensity feature vectors $z$ in hyperspace $R^{D}$,
with $z^{int}_{t}=f_{\theta}(x^{int}_{t})$ and
$z^{phs}_{t}=f_{\theta}(x^{phs}_{t})$ respectively. $sim$ computes the cosine
similarity between two vectors. We train the models using the following
contrastive loss:

$\begin{split}&\mathrm{L_{\mathrm{c}}(i)}=-\mathrm{log}\frac{\mathrm{e}^{\mathrm{sim(z^{int}_{i},z^{phs}_{i})}/\tau}}{\mathrm{e}^{\mathrm{sim(z^{int}_{i},z^{phs}_{i})}/\tau}+S_{1}+S_{2}}\hskip
14.22636pt\text{with}\\\ &S_{1}=\sum\limits_{j\neq
i}\mathrm{e}^{\mathrm{sim(z^{int}_{i},z^{phs}_{j})}/\tau},S_{2}=\sum\limits_{j\neq
i}\mathrm{e}^{\mathrm{sim(z^{int}_{i},z^{int}_{j})}/\tau}.\end{split}$ (1)

(ii) Classification Task: For our experiments, we consider four tissue
classes: gelatin, pork, beef, turkey. Our mini-batch consists of randomly
sampled intensity and phase data crops belonging to a time window $t_{n}$ from
multiple phantom insertions. We train $c_{\theta}$, $f_{\theta}$ and
$g_{\theta}$ using cross-entropy loss.

##  III Experiments

We test the feasibility of our proposed method using our in-house dataset. We
compare our method against models trained using randomly initialised weights
(”Scratch”) and using ImageNet [15] weights in different training set sizes.
We compare all the methods with 10%, 20%, 30%, 60%, 80% and 100% of the
training set. Further, we compare the importance of phase and intensity
features towards tissue classification by solely using $f_{\theta}$ or
$g_{\theta}$.

Dataset split:  We have 34, 14 and 18 needle insertions on soft-tissue phantom
with beef, pork and turkey respectively. We split the needle insertions into
80:10:10 train, validation and test split stratified on tissue class. From
each M-scan, we crop intensity and phase data by passing a rectangular window
of width 256 and height 250 along the time dimension of the M-scans. Examples
of the extracted phase and intensity data are shown in Fig 1 (c) and (d). Note
that the rectangular window does not overlap in area with its last position as
it slides across the time dimension. In total, 269, 169, 64 and 115 phase and
intensity crops of gelatin, beef, pork and turkey are extracted. For our
contrastive pretraining, we use 100% of the training set without labels.

Implementation Detail: $f_{\theta}$ and $g_{\theta}$ are two identical
ResNet18 [16] models. $c_{\theta}$ is a multilayer perceptron with 3 layers
having dimension 1024, 512 and 4 with a ReLu in between when using both
$f_{\theta}$ and $g_{\theta}$ and 512, 512 and 4 when using $f_{\theta}$ or
$g_{\theta}$. We train our network using Adam optimizer [17]. We use a batch
size of 28. D is set to 512 in $R^{D}$ and $\tau=0.1$. We train all our models
for 100 epochs. Our reported metrics are weighted Average Precision (AP) and
F1. All our experiments are trained using 3-fold cross validation and we
report the mean and standard deviation of the metrics.

##  IV Results

The results of our experiments are reported in Tab. I. We report
classification performance in 10%, 20% and 30% of the training set. First, AP
and F1 scores are best for models trained using our pretraining strategy in
all the limited dataset scenarios.

TABLE I: Results in limited labelled dataset scenario

% Training Set | Method | AP | F1
---|---|---|---
10 | Scratch | 0.73±0.07 | 0.60±0.07
| ImageNet | 0.85±0.10 | 0.76±0.1
| Pretrained | 0.94±0.05 | 0.84±0.10
20 | Scratch | 0.88±0.04 | 0.69±0.05
| ImageNet | 0.95±0.03 | 0.84±0.04
| Pretrained | 0.97±0.01 | 0.89±0.03
30 | Scratch | 0.93±0.05 | 0.84±0.05
| ImageNet | 0.97±0.02 | 0.92±0.03
| Pretrained | 0.99±0.006 | 0.95±0.02

Figure 3: (Top) Comparing Average Precision vs % of training set for models
trained from scratch, with ImageNet weights and using our pretraining strategy
(Bottom) Average Precision vs % of training set for models trained using only
phase crop, only intensity crop and using both phase and intensity crop

Fig. 3 (Top) shows that with infusion of data into the training stage, the
three methods converge in performance. Fig. 3 (Bottom) shows the relative
importance of phase and intensity data towards classification. Combination of
phase and intensity shows consistently highest tissue classification
performance throughout. We also observe that phase data are more beneficial
towards intensity.

##  V Discussion and Conclusion

From the results in Tab. I we observe that pretraining using our proposed
strategy is the most beneficial for tissue classification in limited dataset
scenario. We conjecture that the improvement in performance is because the
model learns invariant representations for phase and intensity data and in
doing so, the weights of the network are tuned to our target OCT dataset. The
lowest performances are observed for the models trained from scratch. This can
be attributed to overfitting of the models to the small labelled dataset. With
respect to the models trained using ImageNet weights, we hypothesize that due
to the large differences between ImageNet dataset and our OCT dataset, the
ImageNet weights do not transfer well to our OCT dataset. Considering Fig. 3
(right), the differences in performance when training with intensity or phase
data might be related to the fact that the speckle patterns in the intensity
data are not necessarily clearly distinguishable when the tissue structures
are compressed in front of the needle. In contrast the phase data, capable for
sub-pixel motion detection, might contain tissue dependent features during
compression. Finally, from Fig. 3 we also conclude that with infusion of data,
all the methods achieve competitive performance.

In conclusion, we propose a deep learning model to classify tissue from
complex OCT data. We show that combination of intensity and phase data is most
beneficial and leads to the highest average precision. Further, we propose a
novel self-supervised contrastive pretraining strategy that proves to be
beneficial in low labelled dataset scenarios. We believe that this pretraining
stategy may prove to be helpful in improving the classification performance
for in-vivo needle insertion.

##  VI Compliance with ethical standards

No experiments were performed on humans or animals. No ethics approval was
required.

##  VII Acknowledgments

The authors have no conflicts of interests to report. This work has not been
submitted for publication anywhere else. This work is funded partially by the
i3 initiative of the Hamburg University of Technology. The authors also
acknowledge the partial funding by the Free and Hanseatic City of Hamburg
(Interdisciplinary Graduate School) from University Medical Center Hamburg-
Eppendorf.

## References

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# Before and After:
Machine learning for perioperative patient care

Iuliia Ganskaia, Independent Researcher, Stanislav Abaimov, University of
Rome, Tor Vergata

###### Abstract

For centuries nursing has been known as a job that requires complex manual
operations, that cannot be automated or replaced by any machinery. All the
devices and techniques have been invented only to support, but never fully
replace, a person with knowledge and expert intuition. With the rise of
Artificial Intelligence and continuously increasing digital data flow in
healthcare, new tools have arrived to improve patient care and reduce the
labour-intensive work conditions of a nurse.

This cross-disciplinary review aims to build a bridge over the gap between
computer science and nursing. It outlines and classifies the methods for
machine learning and data processing in patient care before and after the
operation. It comprises of Process-, Patient-, Operator-, Feedback-, and
Technology-centric classifications. The presented classifications are based on
the technical aspects of patient case.

###### Index Terms:

Nursing, Patient care, machine learning, artificial intelligence, data
processing.

##  I Introduction

Artificial Intelligence (AI) has already greatly contributed to the
development of humankind. AI is applied in research, economic and strategic
planning, warfare, entertainment, and in numerous branches of healthcare.
Various awards and funding opportunities across the globe are announced to
boost the growth of AI for health, care, and well-being111Artificial
Intelligence Award in Health and Care Award, NHS.

“Progress in robotics and artificial intelligence should be oriented ”towards
respecting the dignity of the person and of Creation,” \- Pope
Francis222”Pope’s November prayer intention: that progress in robotics and AI
“be human””, Vatican News, November 2020.

Machine learning has already been extensively researched for medical and
pharmacological purposes. Thus, following Maslow’s hierarchy of needs, this
review presents the subject of machine-learning-powered technologies and
methods directly related to patient care before and after the operation.
Preoperative diagnostics and real-life monitoring during the operation are the
essential steps in emergency response, patient treatment. However,
postoperative recovery of the patient takes significantly longer and can be
interrupted by even the smallest changes in the patient’s condition and
behaviour.

Machine learning can be applied to save lives directly and indirectly. A great
number of research papers has already been published about the application of
machine learning for diagnostics and innovative drug discovery. However,
emerging technologies in patient care still remain majorly unexplored.

This review proposes a set of classifications at the technical level of
devices and data processing, that is specific to medical and patient care
equipment, as well as health-related personal data. In addition, the review
lists current and future challenges for the field of study. The review shows,
that there are numerous methods, in which machine learning can improve the
effectiveness of medical care, speed up the recovery process, and reduce
stress in patients.

##  II Background

The development of the new applications requires sufficient knowledge in both
patient care and computer sciences, as patient care has a set of unique
characteristics of human-machine interaction, that are not present in other
fields, such as engineering, entertainment, and marketing.

###  II-A Patient care

In emergency situation, monitoring vital conditions and symptoms before the
operation, is essential in order to determine the correct actions and
administered medications. Pre-existing medical conditions and stress can
affect the behaviour and reaction to medications, anesthetics, and provided
aid in general.

Monitoring and support after the operation is equally important, to ensure
stable and rapid recovery of the patient.

Some of the electronic and connected IoT devices, used for patient care
include: computers, tablets, smart beds, smart life support, smart mobility
vehicles, monitoring devices and sensors, experimental robotic systems [11].

###  II-B Maslow’s hierarchy of needs

Published in 1943 [12], the theory of human motivation and the hierarchy of
needs333Image from Pennsylvania State University, 2015 (see Fig. 1) are still
used in patient care practice. Each person’s needs must be met individually in
order for them to feel satisfied, cared for and cooperative:

  * •

Identify the level of care required for the patient.

  * •

Evaluate patient communication to determine what needs are not being met.

  * •

Adjust your behavior towards your patients to show acknowledgement of those
needs.

  * •

Allow yourself to form a bond with patients in extended care.

Figure 1: Maslow’s hierarchy of needs

###  II-C Machine Learning

Machine learning (ML) is a set of methods and algorithms, that allows a
computerised device to learn from already existing data.

ML can be supervised, unsupervised, semi-supervised, reinforcement learning.
Any type of ML requires data in a form of a data set (N-dimensional matrix or
a database) or a continuous stream of data. The preprocessed data set is used
for the training of the model, and then the trained model is used to identify
or ”predict” the most likely outcomes of the new data (typically not present
in the training data set).

##  III Existing applications and research

Patient care has always been a task, too complex for any machinery to be
trusted with. To ensure the gradual and successful recovery of the patients, a
nurse needs knowledge, flexible decision making, and a degree of expert
intuition.

Earlier reviews addressed the issues of physical assistance in nursing using
robotics systems [11]. Robert [14] presents one of the first non-technical
overviews of potential AI applications for nursing. A 2021 review[1] presented
the way to structure the data for AI-based decision support systems. Another
2021 survey[15] provides recommendations for Education, Practice, Research,
and Leadership, as well as a top-level review of ethical and legal issues, and
future implications of the use of AI for nursing.

###  III-A Smart devices in patient care

The market of Internet-of-Things and connected devices is continuously growing
and it will keep growing over the upcoming decades. Many of those devices have
been programmed and named ”smart devices”. There are several distinct types of
smart devices that are directly relevant for patient care and can be augmented
with machine learning.

Smart beds and smart medical beds

An average person with healthy sleeping habits spends approximately a third of
life sleeping. Post-operative patients spend most of their time in bed to
maximise recovery.

Machine learning is used for the improvement of consumer beds, directly
linking electronics to rest and recovery in humans. Similarly, ML is used for
smart medical beds to maximise patient recovery and monitoring capacity [7].

Emergency response

Machine learning can provide early diagnostics and guidance, playing an
advisory role in emergency response, such as dynamic detection of cerebral
ischemia[13] or discharge prediction with spinal cord injury[4].

Ventilation control

ML is applied for pressure-controlled ventilation [16] for patients with
breathing conditions, especially crucial for the detection of anomalies and
new critical viral infection or kinetic damage of the tissues.

Smart mobility vehicles

A wide range of mobility devices can be used to the patients with limited
abilities during recovery. Those vehicles incluse autonomous power-operated
scooters, as well as limb, back, or full body exoskeletons [18]. Robotic and
advanced bionic prosthetic limbs use machine learning for muscle and neural
signal processing to return or even enhance mobility to patients with
disabilities.

Remote patient monitoring

Remote patient monitoring is the most common application of IoT devices for
healthcare. IoT devices can automatically collect health metrics like heart
rate, blood pressure, temperature, and more from patients who are not
physically present in a healthcare facility, eliminating the need for patients
to travel to the providers, or for patients to collect it themselves.

When an IoT device collects patient data, it forwards the data to a software
application where healthcare professionals and/or patients can view it.
Algorithms may be used to analyze the data in order to recommend treatments or
generate alerts. For example, an IoT sensor that detects a patient’s unusually
low heart rate may generate an alert so that healthcare professionals can
intervene.

A major challenge with remote patient monitoring devices is ensuring that the
highly personal data that these IoT devices collect is secure and private.

In addition to general health data, wearable devices can passively[6] detect
the symptoms of viral infections, such as COVID-19 that can be used for early
warming and epidemic management.

Glucose monitoring

Glucose levels impact the behaviour and well-being of the patient before and
after the operation. Glucose levels can fluctuate and cannot be monitored
continuously with conventional manual methods, even with periodic testing.
Smart IoT devices can address these challenges by providing continuous,
automatic monitoring of glucose levels in patients. Furthermore, such devices
eliminate the need to keep records manually, and they can alert patients when
glucose levels are problematic.

Heart-rate monitoring

Like glucose, monitoring heart rates can be challenging, even for patients who
are present in healthcare facilities. Periodic heart rate checks do not guard
against rapid fluctuations in heart rates, and conventional devices for
continuous cardiac monitoring used in hospitals require patients to be
attached to wired machines constantly, impairing their mobility.

For the past decade, a variety of small IoT devices have become commercially
available in an affordable way for heart rate monitoring, freeing patients to
move around as they like while ensuring that their hearts are monitored
continuously. Guaranteeing ultra-accurate results remains a challenge, but
most modern devices can deliver accuracy rates of about 90 percent or better
and provide sufficient early warming in case of rapid raise of heart rate.

Hand hygiene monitoring

Traditionally, there has never been a verifiable way to ensure that providers
and patients inside a healthcare facility washed their hands properly in order
to minimize the risk of spreading contagion.

With the COVID-19 popularisation of regular hand sensitisation the issue has
become less severe. Furthermore, many hospitals and other health care
operations use IoT devices to alert people to sanitise their hands when they
enter hospital rooms. The devices can even give instructions on how best to
sanitise to mitigate a particular risk for a particular patient. These devices
can reduce infection rates by more than 60 percent in hospitals.

Connected Inhalers

Conditions such as asthma or COPD often involve attacks that come on suddenly,
with little warning. IoT-connected inhalers can help patients by monitoring
the frequency of attacks, as well as collecting data from the environment to
help healthcare providers understand what triggered an attack.

In addition, connected inhalers can alert patients when they leave inhalers at
home, placing them at risk of suffering an attack without their inhaler
present, or when they use the inhaler improperly.

Ingestible sensors

Collecting data from inside the human body is typically a messy and highly
disruptive affair. No no enjoys having a camera or probe stuck into their
digestive tract, for example.

With ingestible sensors, it is possible to collect information from digestive
and other systems in a much less invasive way. They provide insights into
stomach PH levels, for instance, or help pinpoint the source of internal
bleeding.

These devices must be small enough to be swallowed easily. They must also be
able to dissolve or pass through the human body cleanly on their own. Several
companies are hard at work on ingestible sensors that meet these criteria.

Behaviour advisory systems

It is impossible to predict behaviour of a healthy human. It is even harder to
predict the actions of a distressed patient in emergency condition. Modern
technologies can provide smart solutions, that analyse the changes in the
patients condition and yield adaptive guidance and reminders in activities,
such as rest, exercises, hydration, and nutrition.

Connected glasses and contact lenses

In the near future ocular devices can collect additional . Smart contact
lenses provide another opportunity for collecting healthcare data in a
passive, non-intrusive way. They could also, incidentally, include
microcameras that allow wearers effectively to take pictures with their eyes,
which is probably why companies like Google have patented connected contact
lenses.

Whether they are used to improve health outcomes or for other purposes, smart
lenses promise to turn human eyes into a powerful tool for digital
interactions.

Public health

Smart cities have the capacity to monitor health of the general population,
forecast and combat the spread of infectious diseases [3]. Using health data
it is possible to predict emergency admission in specific parts of a city or a
country [8].

Support for patients with disabilities

The care for patients with disabilities, who can have impaired speech, vision,
or hearing, can be improved by the use of devices and sensors, that provide
interpretation and additional information about the sensations and specific
requests.

Training and Evaluation of preparedness

A Data-Driven approach using machine learning can be used to improve training
programs for nurses and for further evaluation of acquired skills [10, 2].

###  III-B Relevant applications of medical AI

Beyond patient care, some medical applications of the ML can be directly
related to patient’s successful recovery.

Drug discovery and toxicology

A large percentage of candidate drugs fail to win regulatory approval. These
failures are caused by insufficient efficacy (on-target effect), undesired
interactions (off-target effects), or unanticipated toxic effects. Research
has explored use of deep learning to predict the biomolecular targets, off-
targets, and toxic effects of environmental chemicals in nutrients, household
products and drugs.

AtomNet is a deep learning system for structure-based rational drug design[17]
AtomNet was used to predict novel candidate biomolecules for disease targets
such as the Ebola virus and multiple sclerosis.

In 2017 graph neural networks were used for the first time to predict various
properties of molecules in a large toxicology data set. In 2019, generative
neural networks were used to produce molecules that were validated
experimentally all the way into mice.

Medical Image Analysis

Deep learning has been shown to produce competitive results in medical
application such as cancer cell classification, lesion detection, organ
segmentation and image enhancement [9, 5].

##  IV Classification

The applications, presented in Section III, can be classified by the level of
interaction with the care process, with the patient, with the operator (i.e.,
authorised care staff), the level of intensity (i.e., from low to high), and
the volume of technological requirements and data volume.

###  IV-A Process-centric

Process-centric approach aims to group the devices by the frequency of data
collection, volume of data, direct or indirect interaction with the patient,
and speed of making decisions.

  * •

Primary - the device collects data in real time and outputs the current state
of the patient.

  * •

Secondary - the device collects data over time to create a recovery profile,
and make forecasts on the potential changes and recovery speed.

  * •

Statistical - data is collected over time from multiple patients, analysed and
further used to advise care staff on similar patients and deviations of the
specific patient from previous similar cases.

###  IV-B Patient-centric

Patient-centric classification is based on the actions, required from the
patient in order to engage with the device.

  * •

none: used by statistical methods

  * •

passive: wear and/or regularly charge the device

  * •

active: perform specific actions while the device is actively collecting data

###  IV-C Operator-centric

Operator-centric classification is based on the actions, required from the
nurses or other care staff:

  * •

none

  * •

passive: the device needs to be only activated or placed in the proximity or
in contact with the patient

  * •

active: the device needs to be activated, and control actions are needed to
ensure the adequate operation of the device.

###  IV-D Feedback-centric

Feedback-centric classification outlines the way that the device interacts
with the patient:

  * •

no-feedback: the device is placed close to or on the patient and performs data
collection

  * •

feedback: the device sends feedback signals to the patient, either visual,
auditary or tactile.

  * •

data: output data is displayed on the screen of the device and/or parsed to an
application or internal webserver.

###  IV-E Technology-centric

Technology-centric classification is based on the technical aspects of device
manifacturing and data processing, and divided into three sub-categories. Size
or scale of the device, volume and origin of data, and level of provided
privacy.

Size and computational power of the electronic device:

  * •

small - a small device, e.g., with sensors

  * •

medium - a device that can be placed at the bedside of the patient

  * •

large - a large device that requires a patient to be placed inside the device

  * •

server - computer or a physical or virtual network of computers, that is used
for data mining and processing

Volume and origin of data:

  * •

Small volume, real-time - e.g., 1-5 variable parameters to monitor the current
state of the recovering post-operative patient

  * •

Large volume, real-time - e.g., full-body analysis for pre-operational
decision making

  * •

Small volume, statistical - e.g., heart-rate, blood pressure, sugar levels,
oxygen levels in blood, to observe changes over time

  * •

Large volume, statistical - e.g., regular full-body analysis over and extended
period of time.

With the constantly growing number of interconnected devices and growing cyber
crime, the concerns of personal privacy are rising every year. Thus, the data
can be classified by the level of detail it provides about the patient:

  * •

anonymous data - cannot be attributed back to the patient, without prior
knowledge

  * •

confidential data - can be attributed back to the patient, but is stored
securely and cannot be accessed without authorisation.

  * •

public records - publicly available data

##  V Future challenges in patient care

The growing importance of machine learning in medical systems cannot be
overstated. By 2025, research predicts that global AI healthcare spending will
equal $36.1 billion444”Artificial Intelligence in Healthcare Spending to Hit
$36B”, December 2018.

”The strategy comes in the week new data shows global investors poured £13.5
billion into more than 1,400 UK private technology firms between January and
June [2021] - more than that achieved in other large tech markets Germany,
France and Israel combined.” 555”New ten-year plan to make the UK a global AI
superpower”, GOV.UK, 22 Sept 2021

###  V-A Post-operative care and general well-being

Post-operative care is more engaged than the general monitoring of well-being
after the ”check out”. However, in both cases machine learning aims to achieve
the same goal - steady recovery of the patient and general well-being.

###  V-B Current applications of machine learning

The vast majority of the published methods do not yet have the capacity for
real-time monitoring of vital data, to ensure the successful advisory role of
ML-based software. Machine learning currently is only used for the
optimisation of data collection and analysis for further linear decision
making.

###  V-C Trust in new technology

The notion of trust in machine learning. Medical staff requires additional
training. Modern machine learning applications have not reached human level of
decision-making accuracy. However, ML allows to analyse large volumes of data
a lot quicker than a human operator, and provide the most likely outcome based
on the previously collected data.

###  V-D Cyber security

With the use of constantly advising Information Technologies, the concerns
about privacy and security equally grow.

Death, caused by a cyber attack on healthcare systems was registered in
September 2020 666Cyber Attack Suspected in German Woman’s Death, The New York
Times, 2020. Another death, caused by a cyber attack, was registered in
2021777Is this the first ransomware death in the USA?, Panda Security, 2021,
and was proven in court, creating a precedent.

###  V-E Ethical concerns

Similar to Lethal Autonomous Weapon Systems, medical system, powered by
machine learning, cause ethical concerns, as they are responsible for human
lives. Global experts have already voiced their concerns in open letters
regarding Autonomous Weapon Systems, advising international organisation and
national governments to ban the use of lethal systems. However, medical
robotic devices can save lives, instead of taking them.

###  V-F Standardisation and Regulation

”The UK National Screening Committee (UK NSC) has approved interim guidance
for those proposing to use AI in breast screening mammography. We will also
shortly produce general guidance about when and how companies should engage
with PHE about their AI products.” 888New guidance for AI in screening, Public
Health England, 2019

###  V-G Future applications of machine learning

Entrusting decision-making to autonomous drug delivery systems can cause
ethical concerns, that need to be addressed through the development of trusted
technologies and legal standards.

###  V-H Ageing technology and cyber security

Similarly to the industrial control systems, medical facilities can accumulate
a combination of outdated and modern technology. In medicine the hardware and
software is directly or indirectly responsible for human lives, the fast-
growing market of smart and IoT devices with potential cyber vulnerabilities
results in high health risks.

##  VI Conclusion

The article presents the review of the applications of Artificial Intelligence
and Machine Learning for patient care. This includes real-time monitoring,
reporting, diagnostics, and decision-making for supervised and unsupervised
robotic first aid.

With the growing population and the increasing complexity of health knowledge,
the world is in desperate need of new tools and methods of healthcare. Machine
learning can and will improve the quality of patient care.

Machine learning can improve the effectiveness of medical care, speed up the
recovery process, and reduce stress in patients. However, machine learning is
a promising technology, which is not sufficiently developed to be fully
entrusted with human lives, and can be only used as an experimental advisory
approach.

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