###### Abstract

Identifying the physiological processes in the central nervous system that
underlie our conscious experiences has been at the forefront of cognitive
neuroscience. While the principles of classical physics were long found to be
unaccommodating for a causally effective consciousness, the inherent
indeterminism of quantum physics, together with its characteristic dichotomy
between quantum states and quantum observables, provides a fertile ground for
the physical modeling of consciousness. Here, we utilize the Schrödinger
equation, together with the Planck–Einstein relation between energy and
frequency, in order to determine the appropriate quantum dynamical timescale
of conscious processes. Furthermore, with the help of a simple two-qubit toy
model we illustrate the importance of non-zero interaction Hamiltonian for the
generation of quantum entanglement and manifestation of observable
correlations between different measurement outcomes. Employing a quantitative
measure of entanglement based on Schmidt decomposition, we show that quantum
evolution governed only by internal Hamiltonians for the individual quantum
subsystems preserves quantum coherence of separable initial quantum states,
but eliminates the possibility of any interaction and quantum entanglement.
The presence of non-zero interaction Hamiltonian, however, allows for
decoherence of the individual quantum subsystems along with their mutual
interaction and quantum entanglement. The presented results show that quantum
coherence of individual subsystems cannot be used for cognitive binding
because it is a physical mechanism that leads to separability and non-
interaction. In contrast, quantum interactions with their associated
decoherence of individual subsystems are instrumental for dynamical changes in
the quantum entanglement of the composite quantum state vector and manifested
correlations of different observable outcomes. Thus, fast decoherence
timescales could assist cognitive binding through quantum entanglement across
extensive neural networks in the brain cortex.

###### keywords:

brain cortex; conscious experience; quantum coherence; quantum entanglement;
quantum interaction

13 5 773 Academic Editor: Wiesław Leonski 23 March 2021 19 April 2021 29 April
2021 https://doi.org/10.3390/sym13050773 Quantum Information in Neural Systems
Quantum Information in Neural Systems Danko D. Georgiev  Danko D. Georgiev
Georgiev, D.D. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

We exist in the universe and, consequently, obey its physical laws whatever
those physical laws may be Duhem (1954). The principles of classical physics
Susskind and Hrabovsky (2013); Landau and Lifshitz (1960); Kibble and
Berkshire (2004); Morin (2008); Strauch (2009), however, describe a
deterministic clockwork world that is unable to accommodate causally effective
conscious experiences Georgiev (2013, 2017, 2019). This leads to
insurmountable problems with the theory of evolution Darwin (2006); Dawkins
(2004); Hodge (2009), including a lack of explanation of how consciousness
could be tolerated by natural selection James (1879, 1890); Popper and Eccles
(1983). Fortunately, the discovery of quantum mechanics in 1920s provided a
radically different picture of the physical world in which the fabric of
reality is comprised of quantum probability amplitudes for potential physical
events Feynman (1948, 2014); Feynman et al. (2013), whose actual occurrence is
decided indeterministically by the inherent propensity of quantum systems to
produce a definite physical outcome upon measurement Dirac (1967); von Neumann
(1955); Susskind and Friedman (2014); Mermin (1985). This creates a dichotomy
between what exists in the form of quantum states, and what can be observed in
the form of quantum observables. Since consciousness exists but is not
observable, reductive identification of consciousness with quantum information
contained in quantum brain states explains the inner privacy of conscious
experiences Georgiev (2020a) and justifies the use of bra-ket notation for
writing mental states Georgiev (2020b). For example, there are dozens of
papers on Schrödinger’s cat, Wigner’s friend or Everett’s many worlds/many
minds interpretation of quantum mechanics that write
$|\raisebox{-1.0pt}{$\ddot{\smile}$}\rangle$ or
$|\raisebox{-1.0pt}{$\ddot{\frown}$}\rangle$ to indicate mental states of
happiness or sadness, or to indicate the occurrence or nonoccurrence of
certain conscious experiences Tegmark (2000); Everett III (1957); Albert
(1992); Lockwood (1996); Hemmo and Pitowsky (2003); Wigner (1995); Squires
(1990); Barrett (1995); Whitaker (2000); Fuchs and Schack (2013); Frauchiger
and Renner (2018); Pusey (2018). The reductive identification of consciousness
with quantum information implies that the mental states have to satisfy the
axioms of vectors in Hilbert space and obey the temporal dynamics prescribed
by the Schrödinger equation with a physically admissible Hamiltonian. This
approach allows one to obtain mathematically precise answers to questions
about the nature of consciousness using the tools of quantum information
theory.

In this work, we address a number of important questions related to the
quantum dynamics of consciousness, which has to be governed by the Schrödinger
equation. The level of difficulty and accessibility of the theoretical
discourse is carefully adjusted for the needs of a broad audience comprised of
interdisciplinary researchers whose main expertise is in biomedical sciences
but are keen to learn the basics of quantum information theory. The general
outline of the presentation is as follows—in Section 2, we introduce concisely
the basic postulates of quantum mechanics. Next, in Section 3, we construct a
simple two-qubit toy model whose quantum dynamics illustrates the various
quantum concepts defined in Hilbert space. Then, in Section 4, we utilize the
Planck–Einstein relation, which connects energy and frequency, in order to
characterize the appropriate quantum timescale for conscious processes
involving interaction energies that exceed the energy of the thermal noise.
Next, we elaborate on the concepts of quantum entanglement in Section 5 and
quantum coherence in Section 6, with an emphasis on how these concepts relate
to the process of decoherence and the presence or absence of quantum
interactions. In Sections 7 and 8, we derive the main results with regard to
the importance of internal Hamiltonians or interaction Hamiltonians for the
temporal evolution of initially separable quantum states into quantum
entangled states. In Section 9, we examine the differential effects of quantum
coherence and quantum entanglement on cognitive binding. Finally, we wrap up
the presentation by discussing how the various quantum information theoretic
notions relate to conscious processes that extend across neural networks in
the brain cortex.

##  2 Basic Postulates of Quantum Mechanics

Applying quantum mechanics to study any problem in natural sciences requires a
minimal familiarity with the fundamental quantum mechanical postulates Dirac
(1967); von Neumann (1955); Susskind and Friedman (2014). In order to make the
present exposition self-contained, we will concisely summarize those
postulates, known as Dirac–von Neumann axioms Bhaumik (2020), before we use
them to construct and analyze a simple two-qubit toy model. (State) The
quantum physical state of a closed system is completely described by a unit
state vector $|\Psi\rangle$ in complex-valued Hilbert space $\mathcal{H}$.
(Composition) The Hilbert space of a composite quantum system comprised from
$k$ components is a tensor product of the state spaces of the component
subsystems
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\ldots\otimes\mathcal{H}_{k}$.
(Observables) To every observable physical property $A$ there exists an
associated Hermitian operator $\hat{A}=\hat{A}^{\dagger}$, which acts on the
Hilbert space of states $\mathcal{H}$. The eigenvalues $\lambda_{A}$ of the
operator $\hat{A}$ are the possible values of the observable physical
property. (Born rule) The expectation value $\langle\hat{A}\rangle$ of a
measured quantum observable $\hat{A}$ is given by the inner product with the
current quantum state $|\Psi\rangle$ of the physical system, namely
$\langle\hat{A}\rangle=\langle\Psi|\hat{A}|\Psi\rangle$. (Dynamics) The time
evolution of a closed physical system obeys the Schrödinger equation

$\imath\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat{H}|\Psi\rangle,$ (1)

where the Hamiltonian $\hat{H}=\hat{H}^{\dagger}$ is the observable
corresponding to the total energy of the system.

With the use of the matrix exponential function Bellman (1987), the general
solution of the Schrödinger equation could be written in the form

$|\Psi(t)\rangle=e^{-\frac{\imath}{\hbar}\hat{H}t}|\Psi(0)\rangle,$ (2)

where $|\Psi(0)\rangle$ is the initial quantum state at time $t=0$. Thus, the
Hamiltonian $\hat{H}$ is the generator of time translation that evolves the
initial quantum state forward in time. Exactly because of this great
importance of the Hamiltonian for the ensuing quantum dynamics, throughout
this work we will be interested to delineate the differences in action of
_internal Hamiltonians_ , which operate only on the reduced Hilbert subspaces
of component subsystems, and _interaction Hamiltonians_ , which operate on the
tensor product Hilbert space of the composite system.

The Schrödinger equation, which comprises the core of quantum theory
Schrödinger (1926, 1928); Berezin and Shubin (1991), is linear and obeys the
superposition principle Georgiev (2017). This means that given any two
solutions $|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ of the Schrödinger
equation, we can construct an infinite number of solutions, which are linear
combinations of the given two, namely,
$|\Psi_{s}\rangle=\alpha_{1}|\Psi_{1}\rangle+\alpha_{2}|\Psi_{2}\rangle$.
Thus, the linearity of the Schrödinger equation could be viewed as the
underlying reason why the quantum states of physical systems form a Hilbert
space. Furthermore, the Schrödinger equation is unitary, which is an essential
ingredient in the proofs of important quantum no-go theorems (such as the no-
cloning theorem Wootters and Zurek (1982)) that characterize the distinctive
properties of quantum information. Therefore, there is a precise mathematical
sense in which a physical theory of consciousness should be considered
_quantum_ only if its basic tenets originate from the Schrödinger equation
Georgiev (2013, 2017), but not from putative violations of the Schrödinger
equation.

##  3 Minimal Quantum Toy Model

To substantiate the abstract quantum concepts in Hilbert space Akhiezer and
Glazman (1993); Birman and Solomjak (1986), we will construct a toy model
whose quantum dynamics could be solved analytically and plotted for visual
inspection. Because we will study quantum interactions, we need at least two
quantum systems. Furthermore, because we would like the model to be
biologically relevant (Figure 1), we will consider the quantum dynamics of
electrons Dirac (1928a, b); Thaller (1992), which are elementary particles
with spin-$\frac{1}{2}$ thereby acting as minimal quantum bits of information,
or qubits Hayashi et al. (2015). The simplest possible composite quantum
system that is capable of capturing the process of entanglement and
decoherence is one composed of two non-identical interacting qubits in a
uniform static magnetic field $\vec{B}$ aligned in the $z$-direction.

###  3.1 Hamiltonian of the Toy Model

Let $\mathcal{H}_{A}$ be the two-dimensional complex Hilbert space of the
first qubit $A$ and $\mathcal{H}_{B}$ be the two-dimensional complex Hilbert
space of the second qubit $B$. Then according to the Dirac–von Neumann axioms
of quantum mechanics Dirac (1967); von Neumann (1955), the composite two-qubit
quantum system resides in a four-dimensional complex Hilbert space
$\mathcal{H}_{AB}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $\otimes$
denotes the tensor product (also known as Kronecker product of matrices). The
eigenvectors of the observable $\hat{S}_{z,i}$ describing the spin component
along the $z$-axis for the individual Hilbert spaces can be written as
$\left\\{|\uparrow_{z}\rangle_{i},|\downarrow_{z}\rangle_{i}\right\\}$ where
$i=A,B$. The corresponding eigenvalues appear as scaling factors in the
relations

$\displaystyle\hat{S}_{z,i}|$
$\displaystyle\uparrow_{z}\rangle_{i}=\frac{\hbar}{2}|\uparrow_{z}\rangle_{i}$
(3) $\displaystyle\hat{S}_{z,i}|$
$\displaystyle\downarrow_{z}\rangle_{i}=-\frac{\hbar}{2}|\downarrow_{z}\rangle_{i},$
(4)

where the reduced Planck constant is $\hbar=1.055\times 10^{-34}$ J s. With
the use of the Pauli spin matrices, $\hat{\sigma}_{x}$, $\hat{\sigma}_{y}$ and
$\hat{\sigma}_{z}$, we can define all three different spin components as
$\hat{S}_{x,i}=\frac{\hbar}{2}\hat{\sigma}_{x}$,
$\hat{S}_{y,i}=\frac{\hbar}{2}\hat{\sigma}_{y}$ and
$\hat{S}_{z,i}=\frac{\hbar}{2}\hat{\sigma}_{z}$ in the respective Hilbert
spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$.

Figure 1: Different levels of organization of physical processes within the
central nervous system. At the microscopic scale, the brain cortex is composed
of neurons, which form neural networks. The morphology of the rendered
pyramidal neuron (NMO_09565) from layer 5 of rat motor cortex
(http://NeuroMorpho.Org; accessed on 19 April 2021) reflects the functional
specialization of cable-like neuronal projections (dendrites and axon). At the
nanoscale, the electric activity of neurons is generated by voltage-gated ion
channels, which are inserted in the neuronal plasma membrane. As an example of
ion channel is shown a single voltage-gated K+ channel composed of four
protein $\alpha$-subunits. Each subunit has six $\alpha$-helices traversing
the plasma membrane. The 4th $\alpha$-helix is positively charged and acts as
voltage sensor. At the picoscale, individual elementary electric charges
within the protein voltage sensor could be modeled as qubits represented by
Bloch spheres. For the diameter of each qubit is used the Compton wavelength
of electron. Consecutive magnifications from micrometer ($\upmu$m) to
picometer (pm) scale are indicated by $\times$ symbol.

In the composite four-dimensional complex Hilbert space $\mathcal{H}_{AB}$, we
can explicitly write quantum states or quantum observables in matrix form
using a chosen basis. For the description of the two-qubit toy model, we will
adopt the spin $zz$ basis,
$\left\\{|\uparrow_{z}\uparrow_{z}\rangle,|\uparrow_{z}\downarrow_{z}\rangle,|\downarrow_{z}\uparrow_{z}\rangle,|\downarrow_{z}\downarrow_{z}\rangle\right\\}$,
where for economy of notation we have used ordering from left to right to
indicate which subsystem $i=A,B$ is referred to, namely

$\displaystyle|\uparrow_{z}\uparrow_{z}\rangle$
$\displaystyle\equiv|\uparrow_{z}\rangle_{A}|\uparrow_{z}\rangle_{B}$ (5)
$\displaystyle|\uparrow_{z}\downarrow_{z}\rangle$
$\displaystyle\equiv|\uparrow_{z}\rangle_{A}|\downarrow_{z}\rangle_{B}$ (6)
$\displaystyle|\downarrow_{z}\uparrow_{z}\rangle$
$\displaystyle\equiv|\downarrow_{z}\rangle_{A}|\uparrow_{z}\rangle_{B}$ (7)
$\displaystyle|\downarrow_{z}\downarrow_{z}\rangle$
$\displaystyle\equiv|\downarrow_{z}\rangle_{A}|\downarrow_{z}\rangle_{B}.$ (8)

Then, the individual spin observables in $\mathcal{H}_{AB}$ are given by
product operators

$\hat{S}_{x,A}=\frac{\hbar}{2}\hat{\sigma}_{x}\otimes\hat{I}_{B}=\frac{\hbar}{2}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\otimes\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)=\frac{\hbar}{2}\left(\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\end{array}\right)$ (9)
$\hat{S}_{x,B}=\frac{\hbar}{2}\hat{I}_{A}\otimes\hat{\sigma}_{x}=\frac{\hbar}{2}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\otimes\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)=\frac{\hbar}{2}\left(\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}\right)$ (10)
$\hat{S}_{y,A}=\frac{\hbar}{2}\hat{\sigma}_{y}\otimes\hat{I}_{B}=\frac{\hbar}{2}\left(\begin{array}[]{cc}0&-\imath\\\
\imath&0\end{array}\right)\otimes\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)=\frac{\hbar}{2}\left(\begin{array}[]{cccc}0&0&-\imath&0\\\
0&0&0&-\imath\\\ \imath&0&0&0\\\ 0&\imath&0&0\end{array}\right)$ (11)
$\hat{S}_{y,B}=\frac{\hbar}{2}\hat{I}_{A}\otimes\hat{\sigma}_{y}=\frac{\hbar}{2}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\otimes\left(\begin{array}[]{cc}0&-\imath\\\
\imath&0\end{array}\right)=\frac{\hbar}{2}\left(\begin{array}[]{cccc}0&-\imath&0&0\\\
\imath&0&0&0\\\ 0&0&0&-\imath\\\ 0&0&\imath&0\end{array}\right)$ (12)
$\hat{S}_{z,A}=\frac{\hbar}{2}\hat{\sigma}_{z}\otimes\hat{I}_{B}=\frac{\hbar}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\otimes\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)=\frac{\hbar}{2}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{array}\right)$ (13)
$\hat{S}_{z,B}=\frac{\hbar}{2}\hat{I}_{A}\otimes\hat{\sigma}_{z}=\frac{\hbar}{2}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)\otimes\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)=\frac{\hbar}{2}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right).$ (14)

The magnetic moments of the two qubits are
$\vec{\mu}_{1}=\gamma_{1}\vec{S}_{1}$ and
$\vec{\mu}_{2}=\gamma_{2}\vec{S}_{2}$, where the gyromagnetic ratios are
$\gamma_{1}=g_{1}\frac{q_{1}}{2m_{1}}$ and
$\gamma_{2}=g_{2}\frac{q_{2}}{2m_{2}}$, with $g$-factors $g_{1}$ and $g_{2}$,
electric charges $q_{1}$ and $q_{2}$, and masses $m_{1}$ and $m_{2}$.

For a qubit that is realized by a spinning electron, we have

$\vec{\mu}=\gamma_{e}\vec{S}=g_{e}\frac{q_{e}}{2m_{e}}\vec{S}=g_{e}\frac{\mu_{B}}{\hbar}\vec{S},$
(15)

where the electron $g$-factor is $g_{e}=-2.00231930436256$ and the Bohr
magneton is $\mu_{B}=\frac{q_{e}\hbar}{2m_{e}}\approx 9.27\times 10^{-24}$
J/Tesla.

If the internal Hamiltonians for each of the two qubits $A$ and $B$ are
respectively

$\displaystyle\hat{H}_{A}$ $\displaystyle=-\mu_{1}\cdot\vec{B}$ (16)
$\displaystyle\hat{H}_{B}$ $\displaystyle=-\mu_{2}\cdot\vec{B}$ (17)

and the interaction between the two qubits is given by

$\hat{H}_{\textrm{int}}=\xi\left(\vec{S}_{1}\cdot\vec{S}_{2}\right)$ (18)

we can write the total Hamiltonian as

$\hat{H}=\hat{H}_{A}+\hat{H}_{B}+\hat{H}_{\textrm{int}}.$ (19)

After substitution and expanding the inner products, we have

$\displaystyle\hat{H}$
$\displaystyle=-\mu_{1}\cdot\vec{B}-\mu_{2}\cdot\vec{B}+\xi\left(\vec{S}_{1}\cdot\vec{S}_{2}\right)$
$\displaystyle=\Omega_{1}\vec{S}_{z,A}+\Omega_{2}\vec{S}_{z,B}+\xi\left(\hat{S}_{x,A}\hat{S}_{x,B}+\hat{S}_{y,A}\hat{S}_{y,B}+\hat{S}_{z,A}\hat{S}_{z,B}\right)$
$\displaystyle=\frac{\hbar}{2}\Omega_{1}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{array}\right)+\frac{\hbar}{2}\Omega_{2}\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{array}\right)+\frac{\hbar^{2}}{4}\xi\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-1&2&0\\\ 0&2&-1&0\\\ 0&0&0&1\end{array}\right)$ (32)
$\displaystyle=\left(\begin{array}[]{cccc}E_{+}+E_{s}&0&0&0\\\
0&E_{-}-E_{s}&2E_{s}&0\\\ 0&2E_{s}&-E_{-}-E_{s}&0\\\
0&0&0&-E_{+}+E_{s}\end{array}\right),$ (37)

where $E_{+}=\frac{1}{2}\hbar\left(\Omega_{1}+\Omega_{2}\right)$,
$E_{-}=\frac{1}{2}\hbar\left(\Omega_{1}-\Omega_{2}\right)$ and
$E_{s}=\frac{\hbar^{2}}{4}\xi$. The corresponding angular frequencies are
$\omega_{+}=\frac{1}{2}\left(\Omega_{1}+\Omega_{2}\right)$,
$\omega_{-}=\frac{1}{2}\left(\Omega_{1}-\Omega_{2}\right)$ and
$\omega_{s}=\frac{\hbar}{4}\xi$.

###  3.2 Energy Eigenstates and Eigenvalues of the Toy Model

The eigenvalues and eigenvectors of the Hamiltonian correspond to physical
energies that enter in the general solution of the Schrödinger equation
Susskind and Friedman (2014). The energy eigenvalues of the concrete toy
Hamiltonian (37) are found to be

$\displaystyle E_{1}$ $\displaystyle=E_{+}+E_{s}$ (38) $\displaystyle E_{2}$
$\displaystyle=-E_{s}-\sqrt{E_{-}^{2}+4E_{s}^{2}}$ (39) $\displaystyle E_{3}$
$\displaystyle=-E_{s}+\sqrt{E_{-}^{2}+4E_{s}^{2}}$ (40) $\displaystyle E_{4}$
$\displaystyle=-E_{+}+E_{s}.$ (41)

Their corresponding energy eigenvectors expressed in the spin $zz$ basis are
given by

$\displaystyle|E_{1}\rangle$ $\displaystyle=|\uparrow_{z}\uparrow_{z}\rangle$
(42) $\displaystyle|E_{2}\rangle$
$\displaystyle=-C_{1}|\uparrow_{z}\downarrow_{z}\rangle+C_{2}|\downarrow_{z}\uparrow_{z}\rangle$
(43) $\displaystyle|E_{3}\rangle$
$\displaystyle=C_{2}|\uparrow_{z}\downarrow_{z}\rangle+C_{1}|\downarrow_{z}\uparrow_{z}\rangle$
(44) $\displaystyle|E_{4}\rangle$
$\displaystyle=|\downarrow_{z}\downarrow_{z}\rangle,$ (45)

where for economy of notation we have set

$\displaystyle C_{1}$
$\displaystyle=\sin\left[\frac{1}{2}\arccos\left(\frac{E_{-}}{\sqrt{E_{-}^{2}+4E_{s}^{2}}}\right)\right]=\frac{1}{\sqrt{2}}\sqrt{1-\frac{E_{-}}{\sqrt{E_{-}^{2}+4E_{s}^{2}}}}$
(46) $\displaystyle C_{2}$
$\displaystyle=\cos\left[\frac{1}{2}\arccos\left(\frac{E_{-}}{\sqrt{E_{-}^{2}+4E_{s}^{2}}}\right)\right]=\frac{1}{\sqrt{2}}\sqrt{1+\frac{E_{-}}{\sqrt{E_{-}^{2}+4E_{s}^{2}}}}.$
(47)

The mathematical relationship between eigenvectors and their eigenvalues
allows for easy computation of the action of the Hamiltonian operator, namely

$\hat{H}|E_{n}\rangle=E_{n}|E_{n}\rangle.$ (48)

Hence, if we express the initial composite two-qubit quantum state
$|\Psi\rangle$ in the energy basis

$|\Psi\rangle=\hat{I}|\Psi\rangle=\sum_{n}\mathcal{\hat{P}}\left(E_{n}\right)|\Psi\rangle=\sum_{n}|E_{n}\rangle\langle
E_{n}|\Psi\rangle=\sum_{n}\alpha_{n}|E_{n}\rangle,$ (49)

we can propagate it in time using the Schrödinger equation

$\imath\hbar\frac{\partial}{\partial
t}|\Psi\rangle=\hat{H}|\Psi\rangle=\hat{H}\sum_{n}\alpha_{n}|E_{n}\rangle=\sum_{n}E_{n}\alpha_{n}|E_{n}\rangle.$
(50)

Explicitly rewriting the quantum state in the energy basis gives a system of
differential equations

$\imath\hbar\frac{\partial}{\partial t}\left(\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\\\ \vdots\\\
\alpha_{n}\end{array}\right)=\left(\begin{array}[]{c}E_{1}\alpha_{1}\\\
E_{2}\alpha_{2}\\\ \vdots\\\ E_{n}\alpha_{n}\end{array}\right).$ (51)

Before we solve these differential equations, we can divide both sides by
$\imath\hbar$ and work only with the angular frequencies

$\frac{\partial}{\partial t}\left(\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}\\\ \vdots\\\
\alpha_{n}\end{array}\right)=-\imath\left(\begin{array}[]{c}\omega_{1}\alpha_{1}\\\
\omega_{2}\alpha_{2}\\\ \vdots\\\ \omega_{n}\alpha_{n}\end{array}\right),$
(52)

where $\omega_{1}=E_{1}/\hbar$, $\omega_{2}=E_{2}/\hbar$, $\ldots$,
$\omega_{n}=E_{n}/\hbar$. Quantum physicists usually take the approach of
working in units in which $\hbar=1$, hence energy and angular frequency become
equivalent by the Planck–Einstein relation $E=\hbar\omega$. However, our
chosen approach is more general as it does not require fixing the physical
units. Also, compared to energy, the angular frequencies are more appropriate
for characterizing the dynamic timescale of the composite quantum system.

###  3.3 Quantum Dynamics of the State Vector

The solutions of the Schrödinger Equation (52) are easily found in the form

$\alpha_{n}(t)=\alpha_{n}(0)e^{-\imath\omega_{n}t},$ (53)

with the immediate interpretation that $\alpha_{n}(0)$ is the initial quantum
probability amplitude of the state $|E_{n}\rangle$ at $t=0$.

Then, the general solution of the time-dependent Schrödinger equation becomes

$\displaystyle|\Psi(t)\rangle$
$\displaystyle=\sum_{n}\alpha_{n}(0)e^{-\imath\omega_{n}t}|E_{n}\rangle$ (54)
$\displaystyle=\alpha_{1}(0)e^{-\imath\omega_{1}t}|E_{1}\rangle+\alpha_{2}(0)e^{-\imath\omega_{2}t}|E_{2}\rangle+\alpha_{3}(0)e^{-\imath\omega_{3}t}|E_{3}\rangle+\alpha_{4}(0)e^{-\imath\omega_{4}t}|E_{4}\rangle,$
(55)

where from Equations (38)–(41) we obtain

$\displaystyle\omega_{1}$ $\displaystyle=\omega_{+}+\omega_{s}$ (56)
$\displaystyle\omega_{2}$
$\displaystyle=-\omega_{s}-\sqrt{\omega_{-}^{2}+4\omega_{s}^{2}}$ (57)
$\displaystyle\omega_{3}$
$\displaystyle=-\omega_{s}+\sqrt{\omega_{-}^{2}+4\omega_{s}^{2}}$ (58)
$\displaystyle\omega_{4}$ $\displaystyle=-\omega_{+}+\omega_{s}.$ (59)

The coefficients of the energy eigenstates also can be fully re-written in
terms of the angular frequencies

$\displaystyle C_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}\sqrt{1-\frac{\omega_{-}}{\sqrt{\omega_{-}^{2}+4\omega_{s}^{2}}}}$
(60) $\displaystyle C_{2}$
$\displaystyle=\frac{1}{\sqrt{2}}\sqrt{1+\frac{\omega_{-}}{\sqrt{\omega_{-}^{2}+4\omega_{s}^{2}}}}.$
(61)

Thus, the general quantum state $|\Psi(t)\rangle$ at any time $t$ is expressed
as

$\displaystyle|\Psi(t)\rangle$
$\displaystyle=\alpha_{1}(0)e^{-\imath\omega_{1}t}|\uparrow_{z}\uparrow_{z}\rangle-\left(C_{1}\alpha_{2}(0)e^{-\imath\omega_{2}t}-C_{2}\alpha_{3}(0)e^{-\imath\omega_{3}t}\right)|\uparrow_{z}\downarrow_{z}\rangle$
$\displaystyle+\left(C_{2}\alpha_{2}(0)e^{-\imath\omega_{2}t}+C_{1}\alpha_{3}(0)e^{-\imath\omega_{3}t}\right)|\downarrow_{z}\uparrow_{z}\rangle+\alpha_{4}(0)e^{-\imath\omega_{4}t}|\downarrow_{z}\downarrow_{z}\rangle.$
(62)

The initial energy quantum probability amplitudes can be further re-written in
their exponential form as

$\alpha_{i}(0)=\left|\alpha_{i}(0)\right|e^{\imath\varphi_{i}},$ (63)

where the principal arguments are

$\varphi_{i}\equiv\textrm{Arg}\left[\alpha_{i}(0)\right].$ (64)

The quantum state $|\Psi(t)\rangle$ describes what exists in the quantum
world. The quantum state, however, is not an observable entity Busch (1997).
In order to determine what can be measured, we need to consider the
expectation values of quantum observables.

##  4 Quantum Dynamic Timescale

To determine the relevant dynamic timescale, we need to account for Landauer’s
principle, according to which there is a minimum possible amount of energy
$E_{\min}$ required to erase one bit of information Landauer (1961) or to
transmit one bit successfully across a noisy quantum channel affected by
thermal noise Levitin (1998); Georgiev et al. (2020)

$E_{\min}=k_{B}T\ln 2\approx 2.97\times 10^{-21}\quad\textrm{J},$ (65)

where $k_{B}=1.38\times 10^{-23}$ J/K is the Boltzmann constant and $T=310$ K
is the physiological body temperature of humans. From the Planck–Einstein
relation $E=\hbar\omega$, we can determine the minimal angular frequency

$\omega_{\min}=\frac{k_{B}T\ln 2}{\hbar}\approx 2.8\times
10^{13}\quad\textrm{rad}/\textrm{s}.$ (66)

This means that there will be $\frac{\omega_{\min}}{2\pi}\times
10^{-12}\approx 4.5$ full rotations for time period of 1 ps. The time period
for 1 full rotation is $\frac{2\pi}{\omega_{\min}}\approx 0.22$ ps.

As an alternative calculation, consider the interaction of the Bohr magneton
with an external magnetic field of 3 Tesla, which is routinely used for
functional magnetic resonance imaging (fMRI) of the human brain, to obtain an
interaction energy

$\frac{1}{2}g_{e}\mu_{B}B\approx 2.78\times 10^{-23}\quad\textrm{J},$ (67)

that is about 100 times smaller than the Landauer’s limit. The corresponding
angular frequency is

$\Omega_{1}=\Omega_{2}=\frac{g_{e}\mu_{B}B}{2\hbar}\approx 2.64\times
10^{11}\quad\textrm{rad}/\textrm{s}.$ (68)

This means that there will be $\frac{\Omega_{1}}{2\pi}\times 10^{-10}\approx
4.2$ full rotations for time period of 100 ps. The time period for 1 full
rotation is $\frac{2\pi}{\omega_{\textrm{int}}}\approx 23.8$ ps. One possible
interpretation of the latter result is that the MRI experimental technique is
by a factor of $\approx$100 slower that the predicted quantum characteristic
time of neural processes related to consciousness.

The main conclusion is that for typical energies that are comparable to the
thermal energy, the rotation of the state vector in Hilbert space occurs with
time period at a subpicosecond timescale. For energies that are an order of
magnitude lower than the thermal energy, the dynamic timescale is picoseconds,
for energies that are two orders of magnitude lower than the thermal energy,
the dynamic timescale is tens of picoseconds, and so on. Therefore, the
Planck–Einstein relation $E=\hbar\omega$ that appears in the Schrödinger
equation fixes the dynamic timescale of quantum processes to picosecond
timescale or faster, which is in the realm of quantum chemistry Georgiev
(2017, 2020b). For complex biomolecules such as proteins, which catalyze life-
sustaining physiological activities, key functional role is played by hydrogen
bonds whose energy is an order of magnitude lower compared to covalent bonds.
Consequently, the quantum transport of energy in protein $\alpha$-helices due
to conformational bending of hydrogen bonded peptide groups occurs at
picosecond timescale Georgiev and Glazebrook (2019a, b, 2020), as opposed to
femtosecond electron transport due to destruction or creation of covalent
bonds Marcus (1956); Vos et al. (1993); Beck and Eccles (1992); Beck (1996);
Beck and Eccles (1998). Quantum theories of consciousness that require
operation at longer timescales, such as milliseconds, are incompatible with
the essentials established by the Planck–Einstein relation and the Schrödinger
equation.

Here, we would like to emphasize that there are contrived methods based on
quantum beats that could realize destructive quantum interference in order to
slow down the quantum dynamics of the expectation value of some quantum
observables, however, those methods are not evolutionary plausible as they
reduce the computational power of the brain. To appreciate the meaning of this
criticism, consider the operation of computer processors realized on silicon
chips: a processor that operates at MHz (microsecond timescale) can perform
only $10^{6}$ operations per second, whereas a processor that operates at GHz
(nanosecond timescale) can perform $10^{9}$ operations per second. The faster
the processor is, the more computational power it has. That is why modern
computing technologies aim toward processors operating at THz (picosecond
timescale), where quantum effects become dominant, rather than the slower
timescale of milliseconds or even seconds, where the computation can be
performed manually e.g., by sliding beads on abacus.

##  5 Quantum Entanglement

Quantum entanglement entails interdependence between different component
subsystems, which form a composite quantum system Schrödinger (1935). Although
the quantum entanglement could be manifested in the form of correlations
between observable outcomes, the converse is not true, namely the lack of
correlations between observable outcomes does not imply the lack of
entanglement. This is because quantum entanglement is defined for the quantum
state, which is a vector in Hilbert space, whereas the probabilities for
occurrence of definite measurement outcomes pertain to quantum observables,
which are operators on the Hilbert space.

Incompatible (non-commuting) observables cannot be measured simultaneously
with a single experimental setting of the measurement apparatus. This means
that in the act of quantum measurement only a set of compatible (commuting)
quantum observables can be determined to have a definite value. Those quantum
observables that are not measured do not have a definite value and
consequently there are no actual measurement outcomes from which could be
extracted correlations. If one is careful to distinguish between what is
actual and what is counterfactual, quantum mechanics allows for exact
prediction of the possible correlations that would have been observed had the
necessary quantum measurements been performed. (Separable state) A bipartite
composite quantum state $|\Psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
is called separable if and only if it can be written as a tensor product
Gudder (2020)

$|\Psi\rangle=|\psi\rangle_{A}\otimes|\psi\rangle_{B}.$ (69)

(Entangled state) A bipartite composite quantum state
$|\Psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is called entangled if
and only if it cannot be written as a tensor product Gudder (2020)

$|\Psi\rangle\neq|\psi\rangle_{A}\otimes|\psi\rangle_{B}.$ (70)

(Schmidt decomposition) Let $\left\\{|i\rangle_{A}\right\\}$ be a basis for
$\mathcal{H}_{A}$ and $\left\\{|j\rangle_{B}\right\\}$ be a basis for
$\mathcal{H}_{B}$. Then, every bipartite composite quantum state
$|\Psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ can be expressed in
$\left\\{|i\rangle_{A}|j\rangle_{B}\right\\}$ basis as follows

$|\Psi\rangle=\sum_{i}\sum_{j}c_{ij}|i\rangle_{A}|j\rangle_{B}.$ (71)

Next, construct the matrix $\hat{C}=\left(c_{ij}\right)$ and perform singular
value decomposition in the form

$\hat{C}=\hat{U}\hat{\Lambda}\hat{V}^{\dagger},$ (72)

where $\hat{U}$ and $\hat{V}^{\dagger}$ are unitary matrices, and
$\hat{\Lambda}$ is a diagonal matrix with non-negative singular values sorted
in descending order $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{s}\geq
0$ also referred to as Schmidt coefficients. Thus, the bipartite composite
quantum state becomes

$|\Psi\rangle=\sum_{s}\lambda_{s}|\psi\rangle_{A}\otimes|\psi\rangle_{B},$
(73)

where $|\psi\rangle_{A}=\hat{U}|i\rangle_{A}$ and
$|\psi\rangle_{B}=\hat{V}^{\dagger}|j\rangle_{B}$ is the Schmidt basis
Miszczak (2011). The state is entangled if its Schmidt rank is greater than 1.
Otherwise, the state is separable.

The singular values of a Hermitian matrix $\hat{A}=\hat{A}^{\dagger}$ are the
absolute values of the eigenvalues of $\hat{A}$.

###### Proof.

Every Hermitian matrix has a complete set of eigenvectors, and all of its
eigenvalues are real. This allows spectral decomposition

$\hat{A}=\hat{U}\hat{\Lambda}\hat{U}^{\dagger},$ (74)

where $\hat{U}$ is unitary matrix and $\hat{\Lambda}=\hat{\Lambda}^{\dagger}$
is a diagonal matrix with real entries. Decompose the diagonal matrix as
$\hat{\Lambda}=\left|\hat{\Lambda}\right|\textrm{sign}\hat{\Lambda}$. The
matrix $\hat{V}^{\dagger}=\textrm{sign}\hat{\Lambda}\hat{U}^{\dagger}$ is
unitary because $\hat{U}^{\dagger}$ is unitary. Therefore, the singular value
decomposition of $\hat{A}$ is

$\hat{A}=\hat{U}\left|\hat{\Lambda}\right|\hat{V}^{\dagger}.$ (75)

∎

The entanglement number $e\left(\Psi\right)$ of a quantum state $|\Psi\rangle$
is determined by the Schmidt coefficients using the formula Gudder (2020)

$e\left(\Psi\right)=\sqrt{1-\sum_{s}\lambda_{s}^{4}}.$ (76)

The Hermitian matrix $\hat{C}\hat{C}^{\dagger}$ is positive semidefinite,
namely $\langle x|\hat{C}\hat{C}^{\dagger}|x\rangle\geq 0$ for all vectors
$x$. The eigenvalues of the Hermitian matrix
$\hat{C}\hat{C}^{\dagger}=\hat{U}\hat{\Lambda}\hat{V}^{\dagger}\hat{V}\hat{\Lambda}\hat{U}^{\dagger}=\hat{U}\hat{\Lambda}^{2}\hat{U}^{\dagger}$
are equal to $\lambda_{s}^{2}$. Therefore, an efficient way to compute the
entanglement number without recourse to singular value decomposition is to use
Gudder (2020)

$e\left(\Psi\right)=\sqrt{1-\textrm{Tr}\left(\hat{C}\hat{C}^{\dagger}\hat{C}\hat{C}^{\dagger}\right)}.$
(77)

The entanglement number could be normalized in the interval $[0,1]$ as follows

$\tilde{e}\left(\Psi\right)=\frac{e\left(\Psi\right)}{e\left(\Psi\right)_{\max}},$
(78)

where $e\left(\Psi\right)_{\max}$ is the maximal possible value, which is
determined by the dimension $n$ of the Hilbert space of the smallest
subsystem, namely
$n=\min\left[d\left(\mathcal{H}_{A}\right),d\left(\mathcal{H}_{B}\right)\right]$.
In the maximally entangled state, the Schmidt decomposition has $n$
eigenvalues equal to $\frac{1}{\sqrt{n}}$, resulting in

$e\left(\Psi\right)_{\max}=\sqrt{1-n\frac{1}{n^{2}}}=\sqrt{\frac{n-1}{n}}.$
(79)

The normalized quantum entanglement number $e\left(\Psi\right)$ in the toy
model could be computed from the eigenvalues of $\hat{C}\hat{C}^{\dagger}$ to
be

$\displaystyle\tilde{e}\left(\Psi\right)$
$\displaystyle=2\Big{\\{}\left|\alpha_{1}^{2}(0)\alpha_{4}^{2}(0)\right|+\left|\alpha_{2}^{2}(0)\alpha_{3}^{2}(0)\right|+C_{1}^{2}C_{2}^{2}\left(\left|\alpha_{2}^{4}(0)\right|-4\left|\alpha_{2}^{2}(0)\alpha_{3}^{2}(0)\right|+\left|\alpha_{3}^{4}(0)\right|\right)$
$\displaystyle\quad-2C_{+}C_{-}\left|\alpha_{1}(0)\alpha_{2}(0)\alpha_{3}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-\omega_{2}-\omega_{3}+\omega_{4}\right)t-\varphi_{1}+\varphi_{2}+\varphi_{3}-\varphi_{4}\right]$
$\displaystyle\quad+2C_{1}C_{2}\left|\alpha_{1}(0)\alpha_{2}^{2}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-2\omega_{2}+\omega_{4}\right)t-\varphi_{1}+2\varphi_{2}-\varphi_{4}\right]$
$\displaystyle\quad-2C_{1}C_{2}\left|\alpha_{1}(0)\alpha_{3}^{2}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-2\omega_{3}+\omega_{4}\right)t-\varphi_{1}+2\varphi_{3}-\varphi_{4}\right]$
$\displaystyle\quad-2C_{1}C_{2}C_{+}C_{-}\left(\left|\alpha_{2}^{3}(0)\alpha_{3}(0)\right|-\left|\alpha_{2}(0)\alpha_{3}^{3}(0)\right|\right)\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]$
$\displaystyle\quad-2C_{1}^{2}C_{2}^{2}\left|\alpha_{2}^{2}(0)\alpha_{3}^{2}(0)\right|\cos\left[2\left(\omega_{2}-\omega_{3}\right)t-2\left(\varphi_{2}-\varphi_{3}\right)\right]\Big{\\}}^{\frac{1}{2}}.$
(80)

##  6 Quantum Coherence

Quantum coherence and decoherence are frequently mentioned in discussions on
the feasibility of quantum approaches to consciousness Tegmark (2000, 2015).
Because the visibility of quantum interference patterns requires quantum
superpositions Qureshi (2019a, b), a careless wording may say that quantum
coherence is indicative of present quantum superpositions, whereas decoherence
is indicative of absent quantum superpositions. Unfortunately, such statements
cannot be literally correct and would appear to be based on misunderstanding
the vector nature of quantum states. Mathematically, a vector can always be
decomposed into a superposition of other vectors, which means that the concept
of quantum superposition is not an absolute property, but a basis-dependent
one Georgiev (2017). Therefore, it is meaningless to talk about the presence
or absence of quantum superpositions without explicitly stating the basis in
which those superpositions are considered. (Quantum coherence) The $\ell_{1}$
norm of coherence $\mathcal{C}$ is a basis-dependent quantitative measure of
quantum coherence of an $n\times n$ dimensional density matrix $\hat{\rho}$
defined with the use of the sum of all off-diagonal moduli Bera et al. (2015)

$\mathcal{C}=\frac{1}{n-1}\sum_{i\neq j}\left|\rho_{ij}\right|.$ (81)

The quantum coherence is bounded in the interval $0\leq\mathcal{C}\leq 1$.
Consider a qubit in the state $|\uparrow_{z}\rangle$. The density matrix of
the qubit in spin $z$ basis is

$\hat{\rho}=|\uparrow_{z}\rangle\langle\uparrow_{z}|=\left(\begin{array}[]{cc}1&0\end{array}\right)\left(\begin{array}[]{c}1\\\
0\end{array}\right)=\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right).$
(82)

All off-diagonal elements of the density matrix $\hat{\rho}$ expressed in spin
$z$ basis are zero, hence $\mathcal{C}=0$, indicating that the state is not
superposed in that basis. However, the same density matrix re-written in spin
$x$ basis becomes

$\hat{\rho}=\frac{1}{\sqrt{2}}\left(|\uparrow_{x}\rangle+|\downarrow_{x}\rangle\right)\frac{1}{\sqrt{2}}\left(\langle\uparrow_{x}|+\langle\downarrow_{x}|\right)=\frac{1}{2}\left(\begin{array}[]{cc}1&1\end{array}\right)\left(\begin{array}[]{c}1\\\
1\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right).$ (83)

All off-diagonal elements of the density matrix $\hat{\rho}$ expressed in spin
$x$ basis are $\frac{1}{2}$, hence $\mathcal{C}=1$, indicating that the state
is maximally superposed in that basis. It should be emphasized that the
disappearance of quantum coherence merely by change of basis of a pure state
Streltsov et al. (2018) is not the main physical phenomenon studied by
decoherence theory. Instead, decoherence refers to the loss of quantum purity
and conversion of pure quantum states into mixed quantum states Finkelstein
(1993); Zeh (1997, 2000); Zurek (2003). In this latter process, the coherent
quantum superpositions, which can manifest visible quantum interference
patterns under suitable choice of measurement basis, become converted into
incoherent quantum superpositions, which cannot manifest visible quantum
interference patterns. In general, the loss of visibility of interference
patterns is not abrupt, but occurs gradually with the loss of quantum purity.
(Quantum purity) The quantum purity $\gamma$ of an $n\times n$ dimensional
density matrix $\hat{\rho}$ is defined as

$\gamma=\textrm{Tr}\left(\hat{\rho}^{2}\right).$ (84)

The quantum purity is bounded in the interval $\frac{1}{n}\leq\gamma\leq 1$. A
maximally mixed state with minimal purity $\gamma=\frac{1}{2}$ is realized by
a qubit with density matrix

$\hat{\rho}=\frac{1}{2}\left(|\uparrow_{z}\rangle\langle\uparrow_{z}|+|\downarrow_{z}\rangle\langle\downarrow_{z}|\right)=\frac{1}{2}\left(|\uparrow_{x}\rangle\langle\uparrow_{x}|+|\downarrow_{x}\rangle\langle\downarrow_{x}|\right)=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right).$ (85)

The maximally mixed state is incoherent in every basis, meaning that the
$\ell_{1}$ norm of coherence $\mathcal{C}$ is zero in every basis. As a
result, quantum measurement in any basis will find the spin with equal
probability being in either of the two possible directions, up or down.
Furthermore, quantum superpositions may still be present, even though the
maximally mixed state will not manifest visible quantum interference patterns
in any basis. For that latter reason, such quantum superpositions are called
incoherent. To understand where the incoherent quantum superpositions reside,
we have to consider the quantum mechanical concept of purification.
(Purification) Every mixed state given by a density matrix
$\hat{\rho}=\sum_{i}p_{i}|i\rangle\langle i|$ acting on finite-dimensional
Hilbert space $\mathcal{H}_{A}$ with basis $\left\\{|i\rangle\right\\}$ can be
viewed as the reduced state of some pure state
$|\Psi\rangle\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where
$\mathcal{H}_{B}$ is another copy of $\mathcal{H}_{A}$ with basis
$\left\\{|i^{\prime}\rangle\right\\}$.

###### Proof.

Every density matrix $\hat{\rho}\in\mathcal{H}_{A}$ can be spectrally
decomposed in terms of its eigenvalues $p_{i}$ and corresponding set of
orthonormal eigenvectors $\left\\{|i\rangle\right\\}$. Working with the basis
$\left\\{|i\rangle\right\\}$ massively simplifies the calculations because the
density matrix $\hat{\rho}$ becomes diagonal in that basis. Then, consider the
state $|\Psi\rangle=\sum_{i}\sqrt{p_{i}}|i\rangle\otimes|i^{\prime}\rangle$.
Explicit calculation of the partial trace gives

$\displaystyle\hat{\rho}$
$\displaystyle=\textrm{Tr}_{B}|\Psi\rangle\langle\Psi|$
$\displaystyle=\textrm{Tr}_{B}\left(\sum_{i}\sqrt{p_{i}}|i\rangle\otimes|i^{\prime}\rangle\sum_{j}\sqrt{p_{j}}\langle
j|\otimes\langle j^{\prime}|\right)$
$\displaystyle=\textrm{Tr}_{B}\sum_{i}\sum_{j}\sqrt{p_{i}p_{j}}|i\rangle\langle
j|\otimes|i^{\prime}\rangle\langle j^{\prime}|$
$\displaystyle=\sum_{i}\sum_{j}\delta_{ij}\sqrt{p_{i}p_{j}}|i\rangle\langle
j|$ $\displaystyle=\sum_{i}p_{i}|i\rangle\langle i|.$ (86)

∎

Consider two qubits $A$ and $B$ comprising the pure maximally entangled (anti-
correlated) quantum state

$|\Psi\rangle_{AB}=\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{z}\rangle+|\downarrow_{z}\uparrow_{z}\rangle\right).$
(87)

The composite density matrix is pure

$\hat{\rho}_{AB}=\hat{\rho}_{AB}^{2}=\frac{1}{2}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&1&1&0\\\ 0&1&1&0\\\ 0&0&0&0\end{array}\right),$ (88)

whereas the reduced density matrices of each qubit are maximally mixed

$\hat{\rho}_{A}=\textrm{Tr}_{B}\left(\hat{\rho}_{AB}\right)=\hat{\rho}_{B}=\textrm{Tr}_{A}\left(\hat{\rho}_{AB}\right)=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right).$ (89)

Thus, the quantum superposition is seen as non-zero off-diagonal elements only
in the composite density matrix $\hat{\rho}_{AB}$, but not in the reduced
density matrices $\hat{\rho}_{A}$ and $\hat{\rho}_{B}$.

Consider two qubits $A$ and $B$ comprising the pure maximally entangled
(correlated) quantum state

$|\Psi\rangle_{AB}=\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{z}\rangle+|\downarrow_{z}\downarrow_{z}\rangle\right).$
(90)

Again, the composite density matrix is pure

$\hat{\rho}_{AB}=\hat{\rho}_{AB}^{2}=\frac{1}{2}\left(\begin{array}[]{cccc}1&0&0&1\\\
0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&1\end{array}\right),$ (91)

whereas the reduced density matrices of each qubit are maximally mixed

$\hat{\rho}_{A}=\textrm{Tr}_{B}\left(\hat{\rho}_{AB}\right)=\hat{\rho}_{B}=\textrm{Tr}_{A}\left(\hat{\rho}_{AB}\right)=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right).$ (92)

The reduced density matrices $\hat{\rho}_{A}$ and $\hat{\rho}_{B}$ are exactly
the same as in the case when the composite state is given by (87). Comparison
of the the composite states (87) and (90) shows that the measurement outcomes
of the $z$-components of the two spins will be correlated or anti-correlated,
respectively. Therefore, knowing only the reduced density matrices of the
components does not provide a complete description of the composite system,
whereas knowing the composite state vector does. In the context of decoherence
theory, _quantum coherence_ is used with the intended meaning of _maximal
purity_. Keeping this clarification in mind, it could be said that the
composite two-qubit system is quantum coherent because its purity is maximal,
$\gamma=1$, whereas each of the component qubits is incoherent because its
purity is minimal, $\gamma=\frac{1}{2}$. Thus, quantum entanglement leads to
decoherence and the two processes go hand by hand. Conversely, the two
component qubits cannot be individually maximally coherent (pure) and quantum
entangled at the same time. Instead, if each of the two qubits is in a pure
state, then the composite state is separable. Consider each of the two qubits
$A$ and $B$ being in coherent quantum superposition
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\rangle+|\downarrow_{z}\rangle\right)$.
The composite density matrix is pure

$\hat{\rho}_{AB}=\hat{\rho}_{AB}^{2}=\frac{1}{4}\left(\begin{array}[]{cccc}1&1&1&1\\\
1&1&1&1\\\ 1&1&1&1\\\ 1&1&1&1\end{array}\right)$ (93)

and the reduced density matrices of each qubit are also pure

$\hat{\rho}_{A}=\hat{\rho}_{A}^{2}=\textrm{Tr}_{B}\left(\hat{\rho}_{AB}\right)=\hat{\rho}_{B}=\hat{\rho}_{B}^{2}=\textrm{Tr}_{A}\left(\hat{\rho}_{AB}\right)=\frac{1}{2}\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right).$ (94)

The separability of the composite state follows from the purity of the
components

$|\Psi\rangle_{AB}=\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\rangle+|\downarrow_{z}\rangle\right)\otimes\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\rangle+|\downarrow_{z}\rangle\right)=|\uparrow_{x}\rangle\otimes|\uparrow_{x}\rangle.$
(95)

From the preceding examples, it should be clear that quantum mechanics
contains two very different kinds of relationships between composite systems
and component systems. For separable states, both the composite system and the
component systems have pure quantum states. This implies that it is possible
to write state vectors for both the composite system and the component
systems. For quantum entangled states, it is only the composite system that
has a pure quantum state, whereas the component systems are necessarily
described by mixed quantum states. Because mixed quantum states can only be
represented in the form of a density matrix, but not a quantum state vector,
this implies that the components of quantum entangled states cannot be
completely described in isolation. Indeed, the reduced density matrices that
describe the components of quantum entangled states can be used only for
computing the quantum probabilities of outcomes from local measurements
performed upon the given component. However, the reduced density matrices are
useless for computing existing correlations between different components
Georgiev (2017).

##  7 Measurement of Quantum Observables

So far, we have discussed the importance of the state vector $|\Psi(t)\rangle$
of the composite system and have solved the Schrödinger equation for the two-
qubit toy model. Although, the state vector $|\Psi(t)\rangle$ is not an
observable Busch (1997), it allows us to determine the expectation values of
any physical observable $\hat{A}$ that could be measured at any point in time
$t$. The important thing to keep in mind is that non-commuting quantum
observables are incompatible with each other and cannot be measured at the
same time. This means that without knowing which observable is actually
measured, the quantum dynamics of the expectation value of any quantum
observable $\langle\hat{A}\rangle=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle$ is to
be considered as counterfactual, namely, it describes probability
distributions of physical events that could have happened provided that the
quantum observable $\hat{A}$ were measured. Thus, quantum observables do not
necessarily reflect what exists, but only what could be observed Georgiev
(2020a, b).

###  7.1 Quantum Observables in Spin $zz$ Basis

The probabilities for measuring the composite system in each of the spin $zz$
basis states from the set
$\left\\{|\uparrow_{z}\uparrow_{z}\rangle,|\uparrow_{z}\downarrow_{z}\rangle,|\downarrow_{z}\uparrow_{z}\rangle,|\downarrow_{z}\downarrow_{z}\rangle\right\\}$
are given by the expectation values of the corresponding projection operators
according to the Born rule Born (1955), namely

$\displaystyle\textrm{Prob}\left(\uparrow_{z}\uparrow_{z}\right)$
$\displaystyle=\langle\Psi(t)|\hat{\mathcal{P}}\left(\uparrow_{z}\uparrow_{z}\right)|\Psi(t)\rangle=\langle\Psi(t)|\uparrow_{z}\uparrow_{z}\rangle\langle\uparrow_{z}\uparrow_{z}|\Psi(t)\rangle=\left|\langle\uparrow_{z}\uparrow_{z}|\Psi(t)\rangle\right|^{2}$
(96) $\displaystyle\textrm{Prob}\left(\uparrow_{z}\downarrow_{z}\right)$
$\displaystyle=\langle\Psi(t)|\hat{\mathcal{P}}\left(\uparrow_{z}\downarrow_{z}\right)|\Psi(t)\rangle=\langle\Psi(t)|\uparrow_{z}\downarrow_{z}\rangle\langle\uparrow_{z}\downarrow_{z}|\Psi(t)\rangle=\left|\langle\uparrow_{z}\downarrow_{z}|\Psi(t)\rangle\right|^{2}$
(97) $\displaystyle\textrm{Prob}\left(\downarrow_{z}\uparrow_{z}\right)$
$\displaystyle=\langle\Psi(t)|\hat{\mathcal{P}}\left(\downarrow_{z}\uparrow_{z}\right)|\Psi(t)\rangle=\langle\Psi(t)|\downarrow_{z}\uparrow_{z}\rangle\langle\downarrow_{z}\uparrow_{z}|\Psi(t)\rangle=\left|\langle\downarrow_{z}\uparrow_{z}|\Psi(t)\rangle\right|^{2}$
(98) $\displaystyle\textrm{Prob}\left(\downarrow_{z}\downarrow_{z}\right)$
$\displaystyle=\langle\Psi(t)|\hat{\mathcal{P}}\left(\downarrow_{z}\downarrow_{z}\right)|\Psi(t)\rangle=\langle\Psi(t)|\downarrow_{z}\downarrow_{z}\rangle\langle\downarrow_{z}\downarrow_{z}|\Psi(t)\rangle=\left|\langle\downarrow_{z}\downarrow_{z}|\Psi(t)\rangle\right|^{2}.$
(99)

Explicit calculation for the observable quantum probabilities gives

$\displaystyle\textrm{Prob}\left(\uparrow_{z}\uparrow_{z}\right)$
$\displaystyle=\left|\alpha_{1}(0)\right|^{2}$ (100)
$\displaystyle\textrm{Prob}\left(\uparrow_{z}\downarrow_{z}\right)$
$\displaystyle=C_{1}^{2}\left|\alpha_{2}(0)\right|^{2}-2C_{1}C_{2}\left|\alpha_{2}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]+C_{2}^{2}\left|\alpha_{3}(0)\right|^{2}$
(101) $\displaystyle\textrm{Prob}\left(\downarrow_{z}\uparrow_{z}\right)$
$\displaystyle=C_{2}^{2}\left|\alpha_{2}(0)\right|^{2}+2C_{1}C_{2}\left|\alpha_{2}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]+C_{1}^{2}\left|\alpha_{3}(0)\right|^{2}$
(102) $\displaystyle\textrm{Prob}\left(\downarrow_{z}\downarrow_{z}\right)$
$\displaystyle=\left|\alpha_{4}(0)\right|^{2}.$ (103)

It is easy to see that the four quantum probabilities sum up to unity since
$C_{1}^{2}+C_{2}^{2}=1$ and $\sum_{n}\left|\alpha_{n}(0)\right|^{2}=1$.

In order to perform computer simulations, one can plug in directly different
quantum probability amplitudes $\alpha_{n}(0)$ for the initial superposition
of energy states $|\Psi(0)\rangle=\sum_{n}\alpha_{n}(0)|E_{n}\rangle$ and then
observe the quantum dynamics that follows. In the case when the initial
quantum state $|\Psi(0)\rangle$ is given in a different basis, such as an
eigenvector of the evolving quantum observable, then one first needs to
determine the initial quantum probability amplitudes $\alpha_{n}(0)$ for the
energy states using inner products

$|\Psi(0)\rangle=\hat{I}|\Psi(0)\rangle=\sum_{n}|E_{n}\rangle\langle
E_{n}|\Psi(0)\rangle=\sum_{n}\alpha_{n}(0)|E_{n}\rangle$ (104)

and then proceed with the computer simulation. As an explicit example, we
present the initial energy quantum probability amplitudes for each of the spin
$zz$ basis states in the energy basis

$\displaystyle|\uparrow_{z}\uparrow_{z}\rangle$ $\displaystyle=|E_{1}\rangle$
(105) $\displaystyle|\uparrow_{z}\downarrow_{z}\rangle$
$\displaystyle=-C_{1}|E_{2}\rangle+C_{2}|E_{3}\rangle$ (106)
$\displaystyle|\downarrow_{z}\uparrow_{z}\rangle$
$\displaystyle=C_{2}|E_{2}\rangle+C_{1}|E_{3}\rangle$ (107)
$\displaystyle|\downarrow_{z}\downarrow_{z}\rangle$
$\displaystyle=|E_{4}\rangle.$ (108)

Once we have the initial quantum probability amplitudes in the energy basis,
we can plot the quantum dynamics of the expectation values of the quantum
observables given in Equations (100)–(103).

In the presence of non-zero interaction Hamiltonian, the quantum dynamics of
the composite state vector $|\Psi(t)\rangle$ is able to undergo cycles of
quantum entanglement and disentanglement depending on the choice of the
initial state vector $|\Psi(0)\rangle$. From Equations (100)–(103) describing
the time evolution of the quantum probabilities, it can be concluded that if
the initial state is $|\Psi(0)\rangle=|\uparrow_{z}\uparrow_{z}\rangle$ then
its associated observable represented by the expectation value of the
projection operator
$\hat{\mathcal{P}}\left(\uparrow_{z}\uparrow_{z}\right)=|\uparrow_{z}\uparrow_{z}\rangle\langle\uparrow_{z}\uparrow_{z}|$
does not evolve but stays constant
$\langle\Psi(t)|\hat{\mathcal{P}}\left(\uparrow_{z}\uparrow_{z}\right)|\Psi(t)\rangle=1$
at all times (Figure 2A). From (62), it can be seen that the quantum evolution
of the quantum state is to rotate in the Hilbert space with a pure phase
$e^{-\imath\omega_{1}t}|\uparrow_{z}\uparrow_{z}\rangle$, which leaves the
state separable at all times. It is worth emphasizing that the quantum state
evolves, whereas the expectation value of the projection operator does not.
This illustrates clearly the fact that in the quantum world what is observed
is not what exists, namely, the quantum observables (represented by matrix
operators) are not quantum states (represented by state vectors). Similarly,
if the initial state is $|\Psi(0)\rangle=|\downarrow_{z}\downarrow_{z}\rangle$
then its associated observable represented by the expectation value of the
projection operator
$\hat{\mathcal{P}}\left(\downarrow_{z}\downarrow_{z}\right)=|\downarrow_{z}\downarrow_{z}\rangle\langle\downarrow_{z}\downarrow_{z}|$
does not evolve but stays constant
$\langle\Psi(t)|\hat{\mathcal{P}}\left(\downarrow_{z}\downarrow_{z}\right)|\Psi(t)\rangle=1$
at all times. Again, the quantum evolution of the quantum state is to rotate
in the Hilbert space with a pure phase
$e^{-\imath\omega_{4}t}|\downarrow_{z}\downarrow_{z}\rangle$ thereby leaving
the state separable at all times (Figure 2D). Interesting quantum dynamics
results when the initial state is
$|\Psi(0)\rangle=|\uparrow_{z}\downarrow_{z}\rangle$ or
$|\Psi(0)\rangle=|\downarrow_{z}\uparrow_{z}\rangle$. In such cases, the
expectation values of the corresponding projection operators evolve in time
and there are observable quantum interference effects that can be recorded by
external measurement devices (Figure 2B,C). Furthermore, the state repeatedly
gets entangled, when
$\textrm{Prob}\left(\uparrow_{z}\downarrow_{z}\right)=\textrm{Prob}\left(\downarrow_{z}\uparrow_{z}\right)=\frac{1}{2}$,
followed by disentanglement, when one of the latter two probabilities becomes
unit and the other becomes zero. Thus, initially separable composite quantum
states are able to get quantum entangled if the interaction Hamiltonian is
non-zero.

Figure 2: Expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{z}\uparrow_{z}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{z}\downarrow_{z}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{z}\uparrow_{z}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{z}\downarrow_{z}\right)$ corresponding to
probabilities of obtaining the given measurement outcomes for the $z$-spin
components of the two qubits. The initial state $|\Psi(0)\rangle$ at $t=0$ is
$|\uparrow_{z}\uparrow_{z}\rangle$ in panel (A),
$|\uparrow_{z}\downarrow_{z}\rangle$ in panel (B),
$|\downarrow_{z}\uparrow_{z}\rangle$ in panel (C) and
$|\downarrow_{z}\downarrow_{z}\rangle$ in panel (D). The internal Hamiltonians
were modeled with $\Omega_{1}=\Omega_{2}=0.3$ rad/ps. The interaction
Hamiltonian was non-zero with $\omega_{s}=0.3$ rad/ps. The amount of quantum
entanglement at each moment of time was measured using the normalized
entanglement number $e(\Psi)/e(\Psi)_{\max}$.

In order to see what the role of the interaction Hamiltonian is, we can turn
it off by setting it to zero. In the absence of interaction Hamiltonian, the
quantum dynamics of the composite state vector $|\Psi(t)\rangle$ is no longer
able to undergo cycles of quantum entanglement and disentanglement (Figure 3).
Instead, the initially separable quantum states $|\Psi(0)\rangle$ remain
separable at all times, as indicated by the zero normalized entanglement
number, $e(\Psi)/e(\Psi)_{\max}=0$. Furthermore, the expectation values stay
constant 1 for the particular projector
$\hat{\mathcal{P}}\left(\uparrow_{z}\uparrow_{z}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{z}\downarrow_{z}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{z}\uparrow_{z}\right)$ or
$\hat{\mathcal{P}}\left(\downarrow_{z}\downarrow_{z}\right)$ that corresponds
to the initial state $|\Psi(0)\rangle$ (Figure 3A–D). Thus, the presence of
non-zero interaction Hamiltonian is essential for the generation of quantum
entanglement starting from an initially separable quantum state
$|\Psi(0)\rangle$.

Figure 3: Expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{z}\uparrow_{z}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{z}\downarrow_{z}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{z}\uparrow_{z}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{z}\downarrow_{z}\right)$ corresponding to
probabilities of obtaining the given measurement outcomes for the $z$-spin
components of the two qubits. The initial state $|\Psi(0)\rangle$ at $t=0$ is
$|\uparrow_{z}\uparrow_{z}\rangle$ in panel (A),
$|\uparrow_{z}\downarrow_{z}\rangle$ in panel (B),
$|\downarrow_{z}\uparrow_{z}\rangle$ in panel (C) and
$|\downarrow_{z}\downarrow_{z}\rangle$ in panel (D). The internal Hamiltonians
were modeled with $\Omega_{1}=\Omega_{2}=0.3$ rad/ps. The interaction
Hamiltonian was zero with $\omega_{s}=0$ rad/ps. The amount of quantum
entanglement at each moment of time was measured using the normalized
entanglement number $e(\Psi)/e(\Psi)_{\max}$.

By pairwise comparison of the computer plots performed with or without non-
zero interaction Hamiltonian, it can be seen that there is a difference in the
quantum dynamics if the initial state is $|\uparrow_{z}\downarrow_{z}\rangle$
as shown in Figures 2B and 3B, or $|\downarrow_{z}\uparrow_{z}\rangle$ as
shown in Figures 2C and 3C. Yet, there is no difference in the quantum
dynamics if the initial state is $|\uparrow_{z}\uparrow_{z}\rangle$ as shown
Figures 2A and 3A, or $|\downarrow_{z}\downarrow_{z}\rangle$ as shown in
Figures 2D and 3D. The explanation for these findings is that the quantum
dynamics for spin $zz$ states is governed by the presence of non-zero off-
diagonal elements in the interaction Hamiltonian in the $zz$ basis

$\hat{H}_{\textrm{int}}=\frac{\hbar^{2}}{4}\xi\left(\begin{array}[]{cccc}1&0&0&0\\\
0&-1&2&0\\\ 0&2&-1&0\\\ 0&0&0&1\end{array}\right).$ (109)

The first and fourth rows, which correspond respectively to the states
$|\uparrow_{z}\uparrow_{z}\rangle$ and $|\downarrow_{z}\downarrow_{z}\rangle$,
already contain off-diagonal zeroes, hence the zeroing of the interaction
Hamiltonian does not change anything. On the other hand, only the second and
third rows, which correspond respectively to the states
$|\uparrow_{z}\downarrow_{z}\rangle$ and $|\downarrow_{z}\uparrow_{z}\rangle$
contain non-zero off-diagonal elements (coherences). Zeroing of the
interaction Hamiltonian removes those non-zero off-diagonal elements, which in
turn prevents the two qubits from interacting with each other and getting
quantum entangled.

###  7.2 Complementary Observables in Spin $xx$ Basis

The non-commutativity of quantum operators results in incompatibility of the
corresponding observables, which cannot be measured with a single experimental
setting. Instead, measurements of incompatible quantum observables require
different mutually incompatible settings of the employed experimental
apparatus.

Suppose that we want to measure the two spins in $x$-direction. We need to
consider the relations

$\displaystyle|\uparrow_{z}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\uparrow_{x}\rangle+|\downarrow_{x}\rangle\right)$
(110) $\displaystyle|\downarrow_{z}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\uparrow_{x}\rangle-|\downarrow_{x}\rangle\right).$
(111)

Multiplying out the tensor products

$\displaystyle|\uparrow_{z}\uparrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\rangle+|\downarrow_{x}\rangle\right)\left(|\uparrow_{x}\rangle+|\downarrow_{x}\rangle\right)$
(112) $\displaystyle|\uparrow_{z}\downarrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\rangle+|\downarrow_{x}\rangle\right)\left(|\uparrow_{x}\rangle-|\downarrow_{x}\rangle\right)$
(113) $\displaystyle|\downarrow_{z}\uparrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\rangle-|\downarrow_{x}\rangle\right)\left(|\uparrow_{x}\rangle+|\downarrow_{x}\rangle\right)$
(114) $\displaystyle|\downarrow_{z}\downarrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\rangle-|\downarrow_{x}\rangle\right)\left(|\uparrow_{x}\rangle-|\downarrow_{x}\rangle\right),$
(115)

we obtain

$\displaystyle|\uparrow_{z}\uparrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\uparrow_{x}\rangle+|\uparrow_{x}\downarrow_{x}\rangle+|\downarrow_{x}\uparrow_{x}\rangle+|\downarrow_{x}\downarrow_{x}\rangle\right)$
(116) $\displaystyle|\uparrow_{z}\downarrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\uparrow_{x}\rangle-|\uparrow_{x}\downarrow_{x}\rangle+|\downarrow_{x}\uparrow_{x}\rangle-|\downarrow_{x}\downarrow_{x}\rangle\right)$
(117) $\displaystyle|\downarrow_{z}\uparrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\uparrow_{x}\rangle+|\uparrow_{x}\downarrow_{x}\rangle-|\downarrow_{x}\uparrow_{x}\rangle-|\downarrow_{x}\downarrow_{x}\rangle\right)$
(118) $\displaystyle|\downarrow_{z}\downarrow_{z}\rangle$
$\displaystyle=\frac{1}{2}\left(|\uparrow_{x}\uparrow_{x}\rangle-|\uparrow_{x}\downarrow_{x}\rangle-|\downarrow_{x}\uparrow_{x}\rangle+|\downarrow_{x}\downarrow_{x}\rangle\right).$
(119)

Upon setting $C_{\pm}=C_{2}\pm C_{1}$ and substitution of (116)–(119) into
(62), we obtain the quantum state in spin $xx$ basis

$\displaystyle|\Psi(t)\rangle$
$\displaystyle=\frac{1}{2}\left[\alpha_{1}(0)e^{-\imath\omega_{1}t}+C_{-}\alpha_{2}(0)e^{-\imath\omega_{2}t}+C_{+}\alpha_{3}(0)e^{-\imath\omega_{3}t}+\alpha_{4}(0)e^{-\imath\omega_{4}t}\right]|\uparrow_{x}\uparrow_{x}\rangle$
$\displaystyle\quad+\frac{1}{2}\left[\alpha_{1}(0)e^{-\imath\omega_{1}t}+C_{+}\alpha_{2}(0)e^{-\imath\omega_{2}t}-C_{-}\alpha_{3}(0)e^{-\imath\omega_{3}t}-\alpha_{4}(0)e^{-\imath\omega_{4}t}\right]|\uparrow_{x}\downarrow_{x}\rangle$
$\displaystyle\quad+\frac{1}{2}\left[\alpha_{1}(0)e^{-\imath\omega_{1}t}-C_{+}\alpha_{2}(0)e^{-\imath\omega_{2}t}+C_{-}\alpha_{3}(0)e^{-\imath\omega_{3}t}-\alpha_{4}(0)e^{-\imath\omega_{4}t}\right]|\downarrow_{x}\uparrow_{x}\rangle$
$\displaystyle\quad+\frac{1}{2}\left[\alpha_{1}(0)e^{-\imath\omega_{1}t}-C_{-}\alpha_{2}(0)e^{-\imath\omega_{2}t}-C_{+}\alpha_{3}(0)e^{-\imath\omega_{3}t}+\alpha_{4}(0)e^{-\imath\omega_{4}t}\right]|\downarrow_{x}\downarrow_{x}\rangle.$
(120)

The probabilities of spin $xx$ observables are

$\displaystyle\textrm{Prob}\left(\uparrow_{x}\uparrow_{x}\right)$
$\displaystyle=\frac{1}{4}\left|\alpha_{1}(0)\right|^{2}+\frac{1}{2}C_{-}\left|\alpha_{1}(0)\alpha_{2}(0)\right|\cos\left[\left(\omega_{1}-\omega_{2}\right)t-\varphi_{1}+\varphi_{2}\right]$
$\displaystyle\quad+\frac{1}{2}C_{+}\left|\alpha_{1}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{1}-\omega_{3}\right)t-\varphi_{1}+\varphi_{3}\right]$
$\displaystyle\quad+\frac{1}{2}\left|\alpha_{1}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-\omega_{4}\right)t-\varphi_{1}+\varphi_{4}\right]+\frac{1}{4}C_{-}^{2}\left|\alpha_{2}(0)\right|^{2}$
$\displaystyle\quad+\frac{1}{2}C_{+}C_{-}\left|\alpha_{2}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]$
$\displaystyle\quad+\frac{1}{2}C_{-}\left|\alpha_{2}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{2}-\omega_{4}\right)t-\varphi_{2}+\varphi_{4}\right]+\frac{1}{4}C_{+}^{2}\left|\alpha_{3}(0)\right|^{2}$
$\displaystyle\quad+\frac{1}{2}C_{+}\left|\alpha_{3}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{3}-\omega_{4}\right)t-\varphi_{3}+\varphi_{4}\right]+\frac{1}{4}\left|\alpha_{4}(0)\right|^{2}$
(121) $\displaystyle\textrm{Prob}\left(\uparrow_{x}\downarrow_{x}\right)$
$\displaystyle=\frac{1}{4}\left|\alpha_{1}(0)\right|^{2}+\frac{1}{2}C_{+}\left|\alpha_{1}(0)\alpha_{2}(0)\right|\cos\left[\left(\omega_{1}-\omega_{2}\right)t-\varphi_{1}+\varphi_{2}\right]$
$\displaystyle\quad-\frac{1}{2}C_{-}\left|\alpha_{1}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{1}-\omega_{3}\right)t-\varphi_{1}+\varphi_{3}\right]$
$\displaystyle\quad-\frac{1}{2}\left|\alpha_{1}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-\omega_{4}\right)t-\varphi_{1}+\varphi_{4}\right]+\frac{1}{4}C_{+}^{2}\left|\alpha_{2}(0)\right|^{2}$
$\displaystyle\quad-\frac{1}{2}C_{+}C_{-}\left|\alpha_{2}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]$
$\displaystyle\quad-\frac{1}{2}C_{+}\left|\alpha_{2}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{2}-\omega_{4}\right)t-\varphi_{2}+\varphi_{4}\right]+\frac{1}{4}C_{-}^{2}\left|\alpha_{3}(0)\right|^{2}$
$\displaystyle\quad+\frac{1}{2}C_{-}\left|\alpha_{3}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{3}-\omega_{4}\right)t-\varphi_{3}+\varphi_{4}\right]+\frac{1}{4}\left|\alpha_{4}(0)\right|^{2}\vspace{-6pt}$
(122) $\displaystyle\textrm{Prob}\left(\downarrow_{x}\uparrow_{x}\right)$
$\displaystyle=\frac{1}{4}\left|\alpha_{1}(0)\right|^{2}-\frac{1}{2}C_{+}\left|\alpha_{1}(0)\alpha_{2}(0)\right|\cos\left[\left(\omega_{1}-\omega_{2}\right)t-\varphi_{1}+\varphi_{2}\right]$
$\displaystyle\quad+\frac{1}{2}C_{-}\left|\alpha_{1}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{1}-\omega_{3}\right)t-\varphi_{1}+\varphi_{3}\right]$
$\displaystyle\quad-\frac{1}{2}\left|\alpha_{1}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-\omega_{4}\right)t-\varphi_{1}+\varphi_{4}\right]+\frac{1}{4}C_{+}^{2}\left|\alpha_{2}(0)\right|^{2}$
$\displaystyle\quad-\frac{1}{2}C_{+}C_{-}\left|\alpha_{2}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]$
$\displaystyle\quad+\frac{1}{2}C_{+}\left|\alpha_{2}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{2}-\omega_{4}\right)t-\varphi_{2}+\varphi_{4}\right]+\frac{1}{4}C_{-}^{2}\left|\alpha_{3}(0)\right|^{2}$
$\displaystyle\quad-\frac{1}{2}C_{-}\left|\alpha_{3}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{3}-\omega_{4}\right)t-\varphi_{3}+\varphi_{4}\right]+\frac{1}{4}\left|\alpha_{4}(0)\right|^{2}\vspace{-6pt}$
(123) $\displaystyle\textrm{Prob}\left(\downarrow_{x}\downarrow_{x}\right)$
$\displaystyle=\frac{1}{4}\left|\alpha_{1}(0)\right|^{2}-\frac{1}{2}C_{-}\left|\alpha_{1}(0)\alpha_{2}(0)\right|\cos\left[\left(\omega_{1}-\omega_{2}\right)t-\varphi_{1}+\varphi_{2}\right]$
$\displaystyle\quad-\frac{1}{2}C_{+}\left|\alpha_{1}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{1}-\omega_{3}\right)t-\varphi_{1}+\varphi_{3}\right]$
$\displaystyle\quad+\frac{1}{2}\left|\alpha_{1}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{1}-\omega_{4}\right)t-\varphi_{1}+\varphi_{4}\right]+\frac{1}{4}C_{-}^{2}\left|\alpha_{2}(0)\right|^{2}$
$\displaystyle\quad+\frac{1}{2}C_{+}C_{-}\left|\alpha_{2}(0)\alpha_{3}(0)\right|\cos\left[\left(\omega_{2}-\omega_{3}\right)t-\varphi_{2}+\varphi_{3}\right]$
$\displaystyle\quad-\frac{1}{2}C_{-}\left|\alpha_{2}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{2}-\omega_{4}\right)t-\varphi_{2}+\varphi_{4}\right]+\frac{1}{4}C_{+}^{2}\left|\alpha_{3}(0)\right|^{2}$
$\displaystyle\quad-\frac{1}{2}C_{+}\left|\alpha_{3}(0)\alpha_{4}(0)\right|\cos\left[\left(\omega_{3}-\omega_{4}\right)t-\varphi_{3}+\varphi_{4}\right]+\frac{1}{4}\left|\alpha_{4}(0)\right|^{2}.$
(124)

Computer simulations confirm again that in the presence of non-zero
interaction Hamiltonian the quantum dynamics of the composite state vector
$|\Psi(t)\rangle$ is able to undergo cycles of quantum entanglement and
disentanglement depending on the choice of the initial state vector
$|\Psi(0)\rangle$ (Figure 4). When, the initial state is
$|\Psi(0)\rangle=|\uparrow_{x}\uparrow_{x}\rangle$ (Figure 4A) or
$|\Psi(0)\rangle=|\downarrow_{x}\downarrow_{x}\rangle$ (Figure 4D), the
quantum dynamics leaves the state $|\Psi(t)\rangle$ separable at all times, as
indicated by the zero normalized entanglement number,
$e(\Psi)/e(\Psi)_{\max}=0$. However, in these cases each qubit manifests its
own quantum interference pattern, which is independent on the interference
pattern manifested by the other qubit. This, reveals the importance of the
internal Hamiltonians to generate quantum interference patterns only within
the reduced subspaces of the component subsystems. When, the initial state is
$|\Psi(0)\rangle=|\uparrow_{x}\downarrow_{x}\rangle$ (Figure 4B) or
$|\Psi(0)\rangle=|\downarrow_{x}\uparrow_{x}\rangle$ (Figure 4C), the quantum
dynamics of the state $|\Psi(t)\rangle$ undergoes cycles of quantum
entanglement and disentanglement. In the presence of quantum entanglement, the
quantum interference patterns manifested by the two qubits become correlated,
which can be appreciated by comparison with the case when the interaction
Hamiltonian is turned off.

Figure 4:  Expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$ corresponding to
probabilities of obtaining the given measurement outcomes for the $x$-spin
components of the two qubits. The initial state $|\Psi(0)\rangle$ at $t=0$ is
$|\uparrow_{x}\uparrow_{x}\rangle$ in panel (A),
$|\uparrow_{x}\downarrow_{x}\rangle$ in panel (B),
$|\downarrow_{x}\uparrow_{x}\rangle$ in panel (C) and
$|\downarrow_{x}\downarrow_{x}\rangle$ in panel (D). The internal Hamiltonians
were modeled with $\Omega_{1}=\Omega_{2}=0.3$ rad/ps. The interaction
Hamiltonian was non-zero with $\omega_{s}=0.3$ rad/ps. The amount of quantum
entanglement at each moment of time was measured using the normalized
entanglement number $e(\Psi)/e(\Psi)_{\max}$.

In the absence of interaction Hamiltonian (Figure 5), the quantum dynamics can
no longer generate quantum entanglement from initially separable composite
quantum state $|\Psi(0)\rangle$. For all four initial quantum states, the
quantum dynamics leaves the state $|\Psi(t)\rangle$ separable at all times.
When the initial state is $|\Psi(0)\rangle=|\uparrow_{x}\uparrow_{x}\rangle$
(Figure 5A) or $|\Psi(0)\rangle=|\downarrow_{x}\downarrow_{x}\rangle$ (Figure
5D), the expectation values for the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$ remain the same
as in the corresponding cases with non-zero interaction Hamiltonian (Figure
4A,D). However, when the initial state is
$|\Psi(0)\rangle=|\uparrow_{x}\downarrow_{x}\rangle$ (Figure 5B) or
$|\Psi(0)\rangle=|\downarrow_{x}\uparrow_{x}\rangle$ (Figure 5C), the
expectation values for the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$ differ from those
shown in the corresponding cases with non-zero interaction Hamiltonian (Figure
4B,C). The explanation of these findings is based on the presence or absence
of non-zero off-diagonal elements (coherences) in the interaction Hamiltonian
when expressed in the spin $xx$ basis, analogously to the explanation that was
already given for the spin $zz$ observables. In fact, the matrix form (109) of
the interaction Hamiltonian is preserved exactly the same during the
conversion from spin $zz$ basis to spin $xx$ basis. Noteworthy, the internal
Hamiltonians $\hat{H}_{A}$ and $\hat{H}_{B}$ do change their matrix form
during the conversion from spin $zz$ basis to spin $xx$ basis, which in turn
explains why the expectation values of spin $xx$ or spin $zz$ observables for
initial states where both spins start aligned in the same direction, either
display quantum interference patterns (Figure 5A,D) or stay constant (Figure
3A,D), respectively.

Figure 5:  Expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$ corresponding to
probabilities of obtaining the given measurement outcomes for the $x$-spin
components of the two qubits. The initial state $|\Psi(0)\rangle$ at $t=0$ is
$|\uparrow_{x}\uparrow_{x}\rangle$ in panel (A),
$|\uparrow_{x}\downarrow_{x}\rangle$ in panel (B),
$|\downarrow_{x}\uparrow_{x}\rangle$ in panel (C) and
$|\downarrow_{x}\downarrow_{x}\rangle$ in panel (D). The internal Hamiltonians
were modeled with $\Omega_{1}=\Omega_{2}=0.3$ rad/ps. The interaction
Hamiltonian was zero with $\omega_{s}=0$ rad/ps. The amount of quantum
entanglement at each moment of time was measured using the normalized
entanglement number $e(\Psi)/e(\Psi)_{\max}$.

##  8 Quantum Dynamics of Initial Quantum Entangled States

The importance of non-zero interaction Hamiltonian for entangling two qubits
starting from an initially separable quantum state, goes in the other
direction as well. In other words, non-zero interaction Hamiltonian is
required to disentangle two qubits starting from an initially entangled
quantum state. For illustration, consider the following four maximally
entangled initial states $|\Psi(0)\rangle$ given by

$\displaystyle\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle+|\downarrow_{z}\downarrow_{x}\rangle\right)$
$\displaystyle=\frac{1}{2}\left(|E_{1}\rangle+C_{-}|E_{2}\rangle+C_{+}|E_{3}\rangle-|E_{4}\rangle\right)$
(125)
$\displaystyle\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle+|\downarrow_{z}\uparrow_{x}\rangle\right)$
$\displaystyle=\frac{1}{2}\left(|E_{1}\rangle+C_{+}|E_{2}\rangle-
C_{-}|E_{3}\rangle+|E_{4}\rangle\right)$ (126)
$\displaystyle\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle-|\downarrow_{z}\downarrow_{x}\rangle\right)$
$\displaystyle=\frac{1}{2}\left(|E_{1}\rangle-
C_{+}|E_{2}\rangle+C_{-}|E_{3}\rangle+|E_{4}\rangle\right)$ (127)
$\displaystyle\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle-|\downarrow_{z}\uparrow_{x}\rangle\right)$
$\displaystyle=\frac{1}{2}\left(|E_{1}\rangle-C_{-}|E_{2}\rangle-
C_{+}|E_{3}\rangle-|E_{4}\rangle\right)$ (128)

and suppose that we measure the expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$. When the initial
state $|\Psi(0)\rangle$ is
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle+|\downarrow_{z}\downarrow_{x}\rangle\right)$
(Figure 6A) or
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle-|\downarrow_{z}\uparrow_{x}\rangle\right)$
(Figure 6D), the state $|\Psi(t)\rangle$ remains maximally entangled at all
times even though the spin $xx$ observables undergo dynamics that manifests
quantum interference patterns. However, when the initial state
$|\Psi(0)\rangle$ is
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle+|\downarrow_{z}\uparrow_{x}\rangle\right)$
(Figure 6B) or
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle-|\downarrow_{z}\downarrow_{x}\rangle\right)$
(Figure 6C), the state $|\Psi(t)\rangle$ undergoes cycles of disentanglement
and entanglement. The time points at which the composite quantum state
$|\Psi(t)\rangle$ becomes separable coincide with unit quantum probability
creating a peak at one of the two states $|\uparrow_{x}\downarrow_{x}\rangle$
or $|\downarrow_{x}\uparrow_{x}\rangle$, while all other orthogonal basis
states remain empty with zero quantum probability.

Figure 6: Expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$ corresponding to
probabilities of obtaining the given measurement outcomes for the $x$-spin
components of the two qubits. The initial state $|\Psi(0)\rangle$ at $t=0$ is
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle+|\downarrow_{z}\downarrow_{x}\rangle\right)$
in panel (A),
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle+|\downarrow_{z}\uparrow_{x}\rangle\right)$
in panel (B),
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle-|\downarrow_{z}\downarrow_{x}\rangle\right)$
in panel (C) and
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle-|\downarrow_{z}\uparrow_{x}\rangle\right)$
in panel (D). The internal Hamiltonians were modeled with
$\Omega_{1}=\Omega_{2}=0.3$ rad/ps. The interaction Hamiltonian was non-zero
with $\omega_{s}=0.3$ rad/ps. The amount of quantum entanglement at each
moment of time was measured using the normalized entanglement number
$e(\Psi)/e(\Psi)_{\max}$.

In the case when the interaction Hamiltonian is turned off, all four maximally
entangled initial quantum states
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle+|\downarrow_{z}\downarrow_{x}\rangle\right)$
(Figure 7A),
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle+|\downarrow_{z}\uparrow_{x}\rangle\right)$
(Figure 7B),
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle-|\downarrow_{z}\downarrow_{x}\rangle\right)$
(Figure 7C) and
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle-|\downarrow_{z}\uparrow_{x}\rangle\right)$
(Figure 7D) undergo quantum dynamics that leaves the composite state
$|\Psi(t)\rangle$ maximally entangled at all times. Now, we are ready to prove
a general theorem according to which quantum dynamics resulting only from
internal Hamiltonians is unable to change the amount of quantum entanglement
that is already possessed by the initial quantum state.

Figure 7: Expectation values of the projectors
$\hat{\mathcal{P}}\left(\uparrow_{x}\uparrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\uparrow_{x}\downarrow_{x}\right)$,
$\hat{\mathcal{P}}\left(\downarrow_{x}\uparrow_{x}\right)$ and
$\hat{\mathcal{P}}\left(\downarrow_{x}\downarrow_{x}\right)$ corresponding to
probabilities of obtaining the given measurement outcomes for the $x$-spin
components of the two qubits. The initial state $|\Psi(0)\rangle$ at $t=0$ is
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle+|\downarrow_{z}\downarrow_{x}\rangle\right)$
in panel (A),
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle+|\downarrow_{z}\uparrow_{x}\rangle\right)$
in panel (B),
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\uparrow_{x}\rangle-|\downarrow_{z}\downarrow_{x}\rangle\right)$
in panel (C) and
$\frac{1}{\sqrt{2}}\left(|\uparrow_{z}\downarrow_{x}\rangle-|\downarrow_{z}\uparrow_{x}\rangle\right)$
in panel (D). The internal Hamiltonians were modeled with
$\Omega_{1}=\Omega_{2}=0.3$ rad/ps. The interaction Hamiltonian was zero with
$\omega_{s}=0$ rad/ps. The amount of quantum entanglement at each moment of
time was measured using the normalized entanglement number
$e(\Psi)/e(\Psi)_{\max}$.

Unitary quantum dynamics resulting from the Schrödinger equation that is only
due to internal Hamiltonians is unable to change the amount of quantum
entanglement that is already present in the initial composite quantum state
$|\Psi(0)\rangle$.

###### Proof.

Without loss of generality, suppose that we have a composite system of two
qubits governed by the Hamiltonian

$\hat{H}=\hat{H}_{A}\otimes\hat{I}_{B}+\hat{I}_{A}\otimes\hat{H}_{B}.$ (129)

Express the initial quantum state in Schmidt basis

$|\Psi(0)\rangle=\lambda_{1}|\uparrow_{A}\rangle|\uparrow_{B}\rangle+\lambda_{2}|\downarrow_{A}\rangle|\downarrow_{B}\rangle,$
(130)

where the basis vectors are orthogonal

$\langle\downarrow_{A}|\uparrow_{A}\rangle=\langle\downarrow_{B}|\uparrow_{B}\rangle=0.$
(131)

Solving the Schrödinger equation gives the composite quantum state at any time
point $t$ using the matrix exponential of the Hamiltonian

$|\Psi(t)\rangle=e^{-\frac{\imath}{\hbar}\left(\hat{H}_{A}\otimes\hat{I}_{B}+\hat{I}_{A}\otimes\hat{H}_{B}\right)t}|\Psi(0)\rangle.$
(132)

Using the commutativity of $\hat{H}_{A}\otimes\hat{I}_{B}$ and
$\hat{I}_{A}\otimes\hat{H}_{B}$, namely,
$\left[\hat{H}_{A}\otimes\hat{I}_{B},\hat{I}_{A}\otimes\hat{H}_{B}\right]=0$,
we can apply a special case of the Baker–Campbell–Hausdorff formula Campbell
(1897); Baker (1905); Strichartz (1987); Nishimura (2013)

$e^{-\frac{\imath}{\hbar}\left(\hat{H}_{A}\otimes\hat{I}_{B}+\hat{I}_{A}\otimes\hat{H}_{B}+\frac{1}{2}\left[\hat{H}_{A}\otimes\hat{I}_{B},\hat{I}_{A}\otimes\hat{H}_{B}\right]\right)t}=e^{-\frac{\imath}{\hbar}\left(\hat{H}_{A}\otimes\hat{I}_{B}\right)t}e^{-\frac{\imath}{\hbar}\left(\hat{I}_{A}\otimes\hat{H}_{B}\right)t}.$
(133)

Using the power series definition of the matrix exponential, we have

$|\Psi(t)\rangle=\sum_{k=0}^{\infty}\frac{1}{k!}\left[-\frac{\imath}{\hbar}\left(\hat{H}_{A}\otimes\hat{I}_{B}\right)t\right]^{k}\sum_{n=0}^{\infty}\frac{1}{n!}\left[-\frac{\imath}{\hbar}\left(\hat{I}_{A}\otimes\hat{H}_{B}\right)t\right]^{n}|\Psi(0)\rangle.$
(134)

Taking into account that powers of the identity operator are also identity, we
obtain

$|\Psi(t)\rangle=\left(e^{-\frac{\imath}{\hbar}\hat{H}_{A}t}\otimes\hat{I}_{B}\right)\left(\hat{I}_{A}\otimes
e^{-\frac{\imath}{\hbar}\hat{H}_{B}t}\right)|\Psi(0)\rangle=e^{-\frac{\imath}{\hbar}\hat{H}_{A}t}\otimes
e^{-\frac{\imath}{\hbar}\hat{H}_{B}t}|\Psi(0)\rangle.$ (135)

Further substitution of the initial quantum state in Schmidt basis followed by
distribution of the time evolution operators gives

$|\Psi(t)\rangle=\lambda_{1}e^{-\frac{\imath}{\hbar}\hat{H}_{A}t}|\uparrow_{A}\rangle\otimes
e^{-\frac{\imath}{\hbar}\hat{H}_{B}t}|\uparrow_{B}\rangle+\lambda_{2}e^{-\frac{\imath}{\hbar}\hat{H}_{A}t}|\downarrow_{A}\rangle\otimes
e^{-\frac{\imath}{\hbar}\hat{H}_{B}t}|\downarrow_{B}\rangle.$ (136)

Because both internal time evolution operators
$e^{-\frac{\imath}{\hbar}\hat{H}_{A}t}$ and
$e^{-\frac{\imath}{\hbar}\hat{H}_{B}t}$ are unitary, they preserve inner
products

$\displaystyle\langle\downarrow_{A}(t)|\uparrow_{A}(t)\rangle$
$\displaystyle=\langle\downarrow_{A}|e^{\frac{\imath}{\hbar}\hat{H}_{A}t}e^{-\frac{\imath}{\hbar}\hat{H}_{A}t}|\uparrow_{A}\rangle=\langle\downarrow_{A}|1|\uparrow_{A}\rangle=0$
(137) $\displaystyle\langle\downarrow_{B}(t)|\uparrow_{B}(t)\rangle$
$\displaystyle=\langle\downarrow_{B}|e^{\frac{\imath}{\hbar}\hat{H}_{B}t}e^{-\frac{\imath}{\hbar}\hat{H}_{B}t}|\uparrow_{B}\rangle=\langle\downarrow_{B}|1|\uparrow_{B}\rangle=0.$
(138)

This implies that the time evolved composite state decomposed in Schmidt basis
is

$|\Psi(t)\rangle=\lambda_{1}|\uparrow_{A}(t)\rangle|\uparrow_{B}(t)\rangle+\lambda_{2}|\downarrow_{A}(t)\rangle|\downarrow_{B}(t)\rangle,$
(139)

which has exactly the same Schmidt coefficients as $|\Psi(0)\rangle$. The
amount of quantum entanglement measured by the entanglement number $e(\Psi)$
is only a function of the two Schmidt coefficients $\lambda_{1}$ and
$\lambda_{2}$. Therefore, during quantum dynamics that is only due to internal
Hamiltonians, the entanglement number stays constant,
$e\left[\Psi(0)\right]=e\left[\Psi(t)\right]$. Note that we did not resort
anywhere in the proof to the fact that we have only two component systems, or
that the individual Hilbert spaces have only two complex dimensions. Hence,
the presented proof straightforwardly generalizes to multipartite systems with
an arbitrary number of dimensions of the individual Hilbert spaces. ∎

##  9 Quantum Coherence Cannot Bind Conscious Experiences

Having explored the precise meaning of various quantum information-theoretic
concepts, we are ready to apply quantum information theory to a specific
problem of consciousness, namely, the problem of binding of conscious
experiences. (Single mind) Awake healthy humans experience an integrated
mental picture consisting of visual, auditory, gustatory, olfactory, and
sensorimotor perceptions. Being a single mind can be understood
introspectively as the total content of all conscious experiences that you
have at a single time instant. Single minds are conscious Georgiev (2017).

(Collection of minds) Being within a collection of minds can be understood
through contemplation of what it is like to be a participant in a conversation
with another human. When you talk with a friend, you may only guess what it is
like to be inside your friend’s mind, but you do not have direct access to
your friend’s conscious experiences. Thus, you and your friend are two
separate minds. Taken together, you and your friend, form a collection of
minds. The collection of conscious minds is not conscious because it does not
possess a single mind Georgiev (2017). Some quantum mind advocates use the
phrase _quantum coherence_ as a magical buzzword that presumably explains the
physical mechanism that binds your conscious experience, e.g., your visual
experiences with your auditory experiences, into a single integrated conscious
mind that is you. Unfortunately, this cannot be the case due to the intimate
relationship between subsystem coherence and separability of the composite
system as discussed in Sections 5 and 6. The latter important point could be
better appreciated by considering the following illuminating example. (Alice
and Bob have two separate minds) Take two people, Alice and Bob, each of which
has an individual conscious mind. Let Alice have a mind defined by the pure
(quantum coherent) state $|\Psi_{A}\rangle$ and let Bob have a mind defined by
the pure (quantum coherent) state $|\Psi_{B}\rangle$. The quantum mechanical
axioms for composition then imply that the composite system is also in a pure
(quantum coherent) state
$|\Psi_{AB}\rangle=|\Psi_{A}\rangle\otimes|\Psi_{B}\rangle$. Obviously, if all
pure (quantum coherent) states had a conscious mind, then $|\Psi_{AB}\rangle$
would be paradoxically a global conscious mind as well Georgiev (2017). In
order to ban paradoxical occurrence of minds within other larger minds,
quantum theory needs to postulate that factorizable non-entangled quantum
states correspond to a collection of minds like Alice and Bob Georgiev (2020a,
b). Quantum purity (colloquially referred to as “quantum coherence” in
decoherence theory) is related to separability, non-interaction and lack of
binding, rather than binding of conscious experiences.

###### Proof.

Arrive at contradiction (i.e., occurrence of minds within other minds) by
assuming the opposite (i.e., that quantum purity binds conscious experiences).
In mathematical logic, arriving at contradiction implies that the assumed
premise is false. ∎

Extremely cold temperatures in quantum technologies attempt to reduce the
quantum interactions in order to ensure that the initial composite state of
qubits is separable in the computational basis, and stays unaltered when no
computational quantum gates are performed Yang et al. (2020). In other words,
in order for human programmer to be able to extract any useful information
from the quantum computation, the quantum computer should be isolated from
interaction with its physical environment. However, we as conscious minds do
not exist in a void isolated from interaction from the rest of the universe.
On the contrary, we constantly receive sensory information from the
surrounding world. What is more, we enjoy so greatly being aware of the
surrounding world that _total sensory deprivation_ of healthy human subjects
in a dark room is experienced as a mental torture Roth et al. (1987);
Rasmussen and Lunde (1989); Reyes (2007); Raz (2013). Furthermore, there is no
external programmer who is supposed to extract useful information by observing
the quantum state of our brain. Both of these points show that even the
motivation behind quantum mind models insisting on preserving quantum
coherence through putative isolation from the environment is faulty. Quantum
dynamics of brain molecules occurs at picosecond timescale and rapid quantum
entanglement due to quantum interactions could lead to extensive correlations
between different quantum brain observables. Because composite quantum
entangled states are much more complex than separable states, it is quite
natural to expect that quantum entanglement due to quantum interactions is
directly related to the complexity of conscious experiences, whereas quantum
separability due to non-interaction of individually quantum coherent
components is related to disbinding or splitting of conscious experiences into
a collection of simpler, elementary minds Georgiev (2020b). (Kolmogorov
complexity) The complexity $K$ of description of a given string of symbols $S$
is the minimal number of classical information bits needed to describe the
string $S$ in some fixed universal description language Kolmogorov (1968);
Shen et al. (2017). (Condensation reduces complexity) Modeling consciousness
in terms of _condensation_ of multiple components in the same quantum state
$|\psi\rangle\otimes|\psi\rangle\otimes\ldots\otimes|\psi\rangle$ inside some
kind of room temperature superconductor (as suggested in Marshall (1989)) goes
against the desirable link between complexity and richness of conscious
experiences. Complexity of quantum states requires high level of entanglement
as opposed to separability Susskind (2020). Consider a simple system of 3
qutrits whose Hilbert space $\mathcal{H}$ is $3\times 3\times 3=27$
dimensional. To specify a general quantum entangled state of these 3 qutrits,
we will need 27 different complex quantum probability amplitudes

$|\Psi\rangle=\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\ \vdots\\\
a_{27}\end{array}\right).$ (140)

If we suppose that for the encoding of each $a_{i}$ (e.g., as a root of some
polynomial equation) it takes $K$ bits of information, for a general quantum
entangled state we will need $27K$ bits.

If the state is factorizable, however, in the form

$|\Psi\rangle=\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\right)\otimes\left(\begin{array}[]{c}a_{4}\\\ a_{5}\\\
a_{6}\end{array}\right)\otimes\left(\begin{array}[]{c}a_{7}\\\ a_{8}\\\
a_{9}\end{array}\right),$ (141)

then we will need only $3+3+3=9$ different complex quantum probability
amplitudes to fully specify the separable tensor product state. Thus, for a
general separable quantum state of 3 qutrits we will need $9K$ bits.

Finally, if all of the 3 qutrits are condensed in the same state

$|\Psi\rangle=\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\right)\otimes\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\right)\otimes\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\right),$ (142)

then we will need only $3$ different complex quantum probability amplitudes to
fully specify the condensed state. The resulting complexity is only $3K$ bits.
The problem with condensation is obvious, namely, the state becomes less and
less complex, therefore conscious experiences will become poorer and poorer as
the condensation progresses. In fact, the previous criticism with regard to
quantum coherence and quantum separability also applies to condensation. In
particular, quantum systems that are in a tensor product state are independent
on each other and it makes no sense to claim that they have a single mind, on
the contrary, there is a good reason to conclude that such quantum systems
correspond to a collection of separate, independent minds.

##  10 Conclusions

Application of quantum information theory to studying neurophysiological
processes is able to provide novel insights that are hard to anticipate from
classical principles alone Georgiev (2017). To amend the current lack of
concise but rigorous introductions to quantum physics, we commenced this work
with a brief description of the Hilbert space formalism of quantum mechanics
and explained its physical meaning using a minimal quantum toy model. Then,
employing the tools of quantum mechanics, we have determined the appropriate
quantum dynamical timescale of cognitive processes and pinpointed the physical
mechanism that binds conscious experiences within a single mind.

Our first main result is that the dynamic timescale of quantum processes,
which supports consciousness, has to be constrained by the Planck–Einstein
relation between energy and frequency. Because the typical energies driving
biological processes are greater than the energy of the thermal noise, the
resulting dynamics is fixed to picosecond timescale or faster, which is in the
realm of quantum chemistry Lowe and Peterson (2005). This requires rethinking
of the role of classical neuronal electric spiking at millisecond timescale
Rieke et al. (1999), not as generating elementary conscious events, but as a
form of short term memory comprised of billions of almost identical picosecond
conscious events Georgiev (2017). Of course, the electric spike eventually
fades away as the neuron repolarizes, which means that the ongoing stream of
consciousness will forget the experiences that have triggered the spike unless
another biological form of storing long term memories is used, such as
changing the electric excitability of the neuronal membrane Zhang and Linden
(2003), strengthening of the activity of the stimulated synapses Citri and
Malenka (2007) or morphological reorganization of the axo-dendritic neuronal
trees Chklovskii (2004).

Our second main result is that the quantum interactions due to non-zero
interaction Hamiltonian in the Schrödinger equation are responsible for
dynamic changes in the amount of quantum entanglement, which leads to
decoherence of the individual quantum subsystems along with their interaction.
We also proved a rigorous quantum theorem according to which unitary quantum
dynamics that is only due to internal Hamiltonians is unable to change the
amount of quantum entanglement already present in the initial composite
quantum state. This implies that turning off the interaction Hamiltonian will
indeed preserve quantum coherence of individual component subsystems starting
from an initially separable quantum state, however, the quantum probabilities
for different physical observables measured on one of the subsystems will
remain independent from those measured on the other subsystem, which in turn
precludes any role of quantum coherence in cognitive binding of conscious
experiences. In fact, decomposition of a composite quantum state into a tensor
product of pure (quantum coherent) states is an indicator of splitting of
different conscious experiences into a collection of individual conscious
minds, whereas inseparability of a composite quantum entangled state is a
possible indicator of binding of different conscious experiences into a single
integrated conscious mind Georgiev (2017).

Studying the Schrödinger equation may appear intimidating, but it is a
profitable investment that pays off with a high interest. The mathematical
constraints that enter into the composition of a physically admissible
Hamiltonian, together with the axioms that relate quantum states with the
expectation values of quantum observables, are sufficient for the derivation
of general quantum information-theoretic no-go theorems that hold for all
physical processes including those that support consciousness Georgiev (2013,
2017). These quantum theorems then could serve as theoretical tools to
differentiate between plausible and implausible physical solutions of open
problems in cognitive neuroscience.

This research received no external funding. Not applicable.

Not applicable.

The morphology file of digitally reconstructed pyramidal neuron (NMO_09565)
from layer 5 of rat motor cortex is publicly available from NeuroMorpho.Org
inventory (http://NeuroMorpho.Org; accessed on 19 April 2021).

The author declares no conflict of interest.

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