The Informationally-Complete Quantum Theory
Zeng-Bing Chen
Quantum mechanics is a cornerstone of our current understanding of nature and extremely successful in describing physics covering a huge range of scales. However, its interpretation remains controversial since the early days of quantum mechanics. What does a quantum state really mean? Is there any way out of the so-called quantum measurement problem? Here we present an informationally-complete quantum theory (ICQT) and the trinary property of nature to beat the above problems. We assume that a quantum system's state provides an informationally-complete description of the system in the trinary picture. We give a consistent formalism of quantum theory that makes the informational completeness explicitly and argue that conventional quantum mechanics is an approximation of the ICQT. We then show how our ICQT provides a coherent picture and fresh angle of some existing problems in physics. The computational content of our theory is uncovered by defining an informationally-complete quantum computer. \newline \noindent Subject Areas: Quantum Physics, Quantum Information
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\begin{document}
\title{\textbf{The Informationally-Complete Quantum Theory}}
\author{Zeng-Bing Chen}
\affiliation{Hefei National Laboratory for Physical Sciences at Microscale, Department of
Modern Physics, University of Science and Technology of China, Hefei, Anhui
230026, China}
\affiliation{The CAS Center for Excellence in QIQP and the Synergetic Innovation Center for
QIQP, University of Science and Technology of China, Hefei, Anhui 230026, China}
\date{\today }

\begin{abstract}
Quantum mechanics is a cornerstone of our current understanding of nature and
extremely successful in describing physics covering a huge range of scales.
However, its interpretation remains controversial since the
early days of quantum mechanics. What does a quantum state really
mean? Is there any way out of the so-called quantum measurement problem? Here
we present an informationally-complete quantum theory (ICQT) and the trinary
property of nature to beat the above problems. We assume that a quantum
system's state provides an informationally-complete description of the system
in the trinary picture. We give a consistent formalism of quantum theory that
makes the informational completeness explicitly and argue that conventional
quantum mechanics is an approximation of the ICQT. We then show how our ICQT
provides a coherent picture and fresh angle of some existing problems in
physics. The computational content of our theory is uncovered by defining an
informationally-complete quantum computer. \newline

\noindent Subject Areas: Quantum Physics, Quantum Information

\end{abstract}
\maketitle


\section{Introduction}

The unease of understanding quantum theory (QT) began at the very beginning of
its establishment. The famous Bohr-Einstein debate \cite{EPR,Bohr} inspired a
lively controversy on quantum foundations. QT is surely an empirically
successful theory, with huge applications ranging from subatomic world to
cosmology. However, why does it attract such a heated debate over its whole
history? The controversial issues on quantum foundations mainly focus on two
aspects: ($Q1$) What does a wave function (or a quantum state) really mean?
($Q2$) Is the so-called quantum measurement problem
\cite{WZ-book,Omnes,Zurek,Sun,Sun-rev,PhyRep} really a problem? The first
axiom of the standard QT states that a system's wave function provides a
complete description of the system. But accepting the wave function as QT's
central entity, what is the physical meaning of the wave function itself? In
this regard, there are two alternatives that the quantum state might be either
a state about an experimenter's knowledge or information about some aspect of
reality (an \textquotedblleft epistemic\textquotedblright\ viewpoint), or a
state of physical reality (an \textquotedblleft ontic\textquotedblright%
\ viewpoint). A recent result \cite{PBR} on this issue seems to support the
reality of quantum states, yet with ongoing controversy
\cite{PBR-comment,Aaronson}.

On the other hand, the quantum measurement problem is perhaps the most
controversial one on quantum foundations. According to the orthodox
interpretation (namely, the Copenhagen interpretation \cite{Omnes}) of QT, a
quantum system in a superposition of different states evolves
deterministically according to the Schr\"{o}dinger equation, but actual
measurements always collapse, in a truly random way, the system into a
definite state, with a probability determined by the probability amplitude according
to the Born rule. When, where, and how the quantum state really collapses are out of the reach of QT
as it is either \textquotedblleft uninteresting or unscientific to discuss
reality before measurement\textquotedblright\ \cite{Aaronson}.

Our classical world view implies that there exists a world that is objective
and independent of any observations. By sharp contrast, what is observed on a
quantum system is dependent upon the choice of experimental arrangements;
mutually exclusive (or complementary) properties cannot be measured accurately
at the same time, a fact known as the complementarity principle. In
particular, which type of measurements one would like to choose is totally a
\textit{free will} \cite{free-will} or a \textit{freedom of choice}
\cite{free-choi74,Bell-book,free-choiEX}. Such a freedom of choice underlies
the Pusey-Barrett-Rudolph theorem \cite{PBR} and the derivation of Bell's
inequalities \cite{free-choi74,Bell-book,free-choiEX,Bell}. However, one could
ask: What does a free will or a freedom of choice really mean and whose free
will or freedom of choice?

Thus, in the orthodox interpretation classical concepts are necessary for the
description of measurements (which type of measurements to choose and the
particular measurement results for chosen measurement) in QT, although the
measurement apparatus can indeed be described quantum mechanically, as done by
von Neumann \cite{vonN-book,Wigner}. Seen from its very structure, quantum
mechanics \textquotedblleft contains classical mechanics as a limiting case,
yet at the same time it requires this limiting case for its own
formulation\textquotedblright\ \cite{Landau}. In this sense current QT has a
classical-quantum \textit{hybrid} feature. At a cosmological scale, the
orthodox interpretation rules out the possibility of assigning a wave function
to the whole Universe, as no external observer could exist to measure the Universe.

Facing with the interpretational difficulties, various interpretations on QT
were proposed by many brilliant thoughts, such as the hidden-variable theory
\cite{Bell-book,Bohm} (initiated by the famous Einstein-Podolsky-Rosen paper
\cite{EPR} questioning the completeness of QT), many-worlds interpretation
\cite{Everett,Tipler}, the relational interpretation
\cite{Rovelli,Rovelli-book}, and the decoherence theory \cite{Zurek}, to
mention a few. Thus, ``questions concerning the
foundations of quantum mechanics have been picked over so thoroughly that
little meat is left'' \cite{Aaronson}. The discovery of
Bell's inequalities \cite{Bell} (recently questioned from the many-worlds
interpretation \cite{Tipler}) and the emerging field of quantum information
\cite{QuInf-book} might be among a few exceptions. The recent development of
quantum information science sparks the information-theoretical understanding
of quantum formalism \cite{Hardy,Fuchs,i-causality,Colbeck-Renner}.

Inspired by the classical-quantum hybrid feature of current QT and the
above-mentioned interpretational progresses, here we present an
informationally-complete quantum theory (ICQT) by removing any classical
concepts in our description of nature. The ICQT is based on the
\textit{informational completeness principle}: A quantum system's state
provides an informationally-complete description of the system. In other
words, quantum states represent an informationally-complete code of any
possible information that one might access. Current QT is \textit{not}
informationally-complete and thus suffers from interpretational difficulties.
After working out the informational completeness explicitly in our formalism,
we show that informationally-complete physical systems are characterized by
dual entanglement pattern, the emergent\textbf{ }dual Born rule and dual
dynamics. The computational content of our theory is uncovered by defining an
informationally-complete quantum computer with potential of outperforming
conventional quantum computers. Moreover, we consider the possible conceptual
applications of our theory, hoping to shed new light on some existing problems
in physics.

\section{An informationally complete description for finite-dimensional
systems}
The orthodox quantum measurement theory
\cite{WZ-book,Omnes,Zurek,Sun,Sun-rev,PhyRep} was proposed by von Neumann and
can be summarized as follows. For an unknown $d$-dimensional quantum state
$\left\vert \psi\right\rangle _{\mathcal{S}}$ of a quantum system
$\mathcal{S}$ to be measured, a measurement apparatus (\textquotedblleft a
pointer\textquotedblright) $\mathcal{A}$ is coupled to the system via a
unitary operator $\hat{U}_{\mathcal{SA}}(\hat{s},\hat{p})$. Here $\hat{s}$ is
system's observable whose eigenstate with respect to the eigenvalue $s_{j}$
reads $\left\vert j,\mathcal{S}\right\rangle $, namely, $\hat{s}\left\vert
j,\mathcal{S}\right\rangle =s_{j}\left\vert j,\mathcal{S}\right\rangle $
($j=1,2,...d$); $\hat{p}$ is the momentum operator which shifts pointer's
$\hat{q}$-reading ($[\hat{q},\hat{p}]=i$). Assuming that the pointer is
initialized in a \textquotedblleft ready\textquotedblright\ state $\left\vert
0,\mathcal{A}\right\rangle $ and expanding $\left\vert \psi,\mathcal{S}%
\right\rangle $ in terms of $\left\vert j,\mathcal{S}\right\rangle $ as
$\left\vert \psi,\mathcal{S}\right\rangle =\sum_{j}c_{j}\left\vert
j,\mathcal{S}\right\rangle $, then the system and the apparatus are mapped
into $\hat{U}_{\mathcal{SA}}(\hat{s},\hat{p})\left\vert \psi,\mathcal{S}%
\right\rangle \left\vert 0,\mathcal{A}\right\rangle =\sum_{j}c_{j}\left\vert
j,\mathcal{S}\right\rangle \left\vert q_{j},\mathcal{A}\right\rangle $. To
ideally measure $\hat{s}$, one has to assume that $\mathcal{A}$ must have at
least $d$ macroscopically distinguishable pointer\ positions (plus the ready
position corresponding to $\left\vert 0,\mathcal{A}\right\rangle $), and the
pointer state $\left\vert q_{j},\mathcal{A}\right\rangle $ and the measured
states $\left\vert j,\mathcal{S}\right\rangle $ have a one-to-one
correspondence. The above is the usual
pre-measurement progress. The orthodox interpretation of the measurement can
only predict the collapse of a definite state $\left\vert j,\mathcal{S}%
\right\rangle $ with a probability $\left\vert c_{j}\right\vert ^{2}$ given by
the probability amplitude $c_{j}$; the collapse occurs in a truly random way.
For latter convenience, we call ($\hat{s},\hat{p}$) as an observable pair. It
is interesting to note that a factorizable structure of the \textquotedblleft
measurement operation\textquotedblright\ $\hat{U}_{\mathcal{SA}}(\hat{s}%
,\hat{p})$ was discovered in the context of the dynamical approach to the
quantum measurement problem \cite{Sun,Sun-rev}.

To avoid the quantum measurement problem, here we take a key step by assuming
explicitly informational completeness, whose meaning will be clear below, in
our formalism of describing nature. To this end, starting from a separable
state $\left\vert \psi,\mathcal{S}\right\rangle \left\vert \phi,\mathcal{A}%
\right\rangle $, we introduce the third system, called the \textquotedblleft
programming system\textquotedblright\ ($\mathcal{P}$) hereafter. We assume
that $\mathcal{P}$ has $D_{\mathcal{P}}$ dimensions spanned by $D_{\mathcal{P}%
}$ orthogonal states, called programming states $\left\vert r,\mathcal{P}%
\right\rangle $ ($r=0,1,...D_{\mathcal{P}}-1$), where $D_{\mathcal{P}}$ is to
be determined by informational completeness. Let us define a unitary
programming operation
\begin{equation}
\hat{U}_{\mathcal{P(SA)}}=\sum_{r=0}^{D_{\mathcal{P}}-1}\left\vert
r,\mathcal{P}\right\rangle \left\langle r,\mathcal{P}\right\vert \hat
{U}_{\mathcal{SA}}(\hat{s}_{r},\hat{p}_{r}), \label{progU}%
\end{equation}
which means that if $\mathcal{P}$ is in $\left\vert r,\mathcal{P}\right\rangle
$, then do a unitary measurement operation $\hat{U}_{\mathcal{SA}}(\hat{s}%
_{r},\hat{p}_{r})$ on $\mathcal{SA}$. Now suppose that $\mathcal{P}$ is
prepared in an initial state $\left\vert \chi,\mathcal{P}\right\rangle
=\sum_{r}g_{r}\left\vert r,\mathcal{P}\right\rangle $. Then the state of the
whole system $\mathcal{PSA}$ after the programming operation reads
\begin{equation}
\left\vert \mathcal{P(SA)}\right\rangle =\sum_{r=0}^{D_{\mathcal{P}}-1}%
g_{r}\left\vert r,\mathcal{P}\right\rangle \left\vert r,\mathcal{SA}%
\right\rangle , \label{psa}%
\end{equation}
where $\left\vert r,\mathcal{SA}\right\rangle =\hat{U}_{\mathcal{SA}}(\hat
{s}_{r},\hat{p}_{r})\left\vert \psi,\mathcal{S}\right\rangle \left\vert
\phi,\mathcal{A}\right\rangle $ encodes the programmed entanglement between
$\mathcal{S}$ and $\mathcal{A}$. For a given $\left\vert r,\mathcal{P}%
\right\rangle $, the observable pair, denoted by ($\hat{s}_{r},\hat{p}_{r}$),
to be measured is determined by the Schmidt form of $\left\vert r,\mathcal{SA}%
\right\rangle $. Note that $\left\vert \mathcal{P(SA)}\right\rangle $ can also
be written in a Schmidt form with positive real coefficients \cite{pureEE1}.
Hereafter we suppose that the Schmidt decomposition of $\left\vert
\mathcal{P(SA)}\right\rangle $ has been done.

Now the key point of our formalism is to require that the programming system
$\mathcal{P}$ encodes all possible, namely, informationally complete,
measurement operations that are allowed to act upon the $\mathcal{SA}$-system.
To be \textquotedblleft informationally complete\textquotedblright, all
programmed measurement operations $\hat{U}_{\mathcal{SA}}(\hat{s}_{r},\hat
{p}_{r})$ can at least achieve the measurements of a complete set of operators
for $\mathcal{S}$; for the $d$-dimensional system, the complete set has
$d^{2}$\ operators \cite{ThewTomo}, i.e., the minimal $D_{\mathcal{P}}=d^{2}$.
Note that informationally complete set of operators or measurement is also
important for quantum state tomography \cite{ICO-Busch,ICM}.

Another trick in the above discussion is that, to enable the
informationally-complete programmed measurements, it seems that one needs
$D_{\mathcal{P}}$ different measurement apparatuses. Hereafter we take a step
further by dropping this specific measurement model by regarding the
$\mathcal{A}$-system as a single system with $D_{\mathcal{A}}$ ($\geq d$)
dimensions. In this case we have $D_{\mathcal{P}}\geq dD_{\mathcal{A}}$. The
step is necessary for seeking a model-independent and intrinsic description of
the whole system $\mathcal{PSA}$.

To have an easy understanding of our informationally-complete description of
physical systems, some remarks are necessary. First, we note that the third
system is also included in other interpretations of QT, such as the
many-worlds interpretation \cite{Everett,Tipler} and the relational
interpretation \cite{Rovelli,Rovelli-book}. However, the third system in our
formalism plays a role that is dramatically different from those
interpretations. Actually, imposing informational completeness into our
quantum description of nature distinguishes our theory from all previous
interpretations of QT. Second, the fact that $\left\vert r,\mathcal{SA}%
\right\rangle $, as entangled, can always be written in a Schmidt form implies
a symmetric role played by $\mathcal{S}$ and $\mathcal{A}$. Meanwhile, the
role of $\mathcal{P}$\ is dramatically different from that of either
$\mathcal{S}$ or $\mathcal{A}$. But $\mathcal{P}$ and the combined system $\mathcal{SA}$
play a symmetric role. We anticipate that such a feature could
have profound consequences, particularly for the internal consistency of the
theory. We will find that this is indeed the case when we consider the
dynamics within the ICQT.

\section{The emergent dual Born rule}
How to acquire information and which kind of information to acquire are two
questions of paramount importance. According to the ICQT, on one hand, the
only way to acquire information is to interact (i.e., entangle) the system
$\mathcal{S}$ and the apparatus $\mathcal{A}$ with each other; no interaction
leads to no entanglement and thus no information. This is in a similar spirit
as the relational interpretation \cite{Rovelli,Rovelli-book}, which treats the
quantum state as being observer-dependent, namely, the state is the relation
between the observer and the system. On the other hand, the programming system
$\mathcal{P}$, by interacting with $\mathcal{SA}$, dictates the way (actually,
the informational-complete way) on which kind of information to acquire about
the system $\mathcal{S}$. For instance, if the whole system is programmed to
measure $\hat{s}_{r}$, then $\mathcal{S}$ and $\mathcal{A}$ interact with each
other to induce the programmed measurement operations $\hat{U}_{\mathcal{SA}%
}(\hat{s}_{r},\hat{p}_{r})$. This process generates the entangled state
$\left\vert r,\mathcal{SA}\right\rangle $ with which $\mathcal{A}$
\textquotedblleft knows\textquotedblright, in a completely coherent way, all
information about $\mathcal{S}$ in the basis of $\hat{s}_{r}$; the amount of
entanglement contained in $\left\vert r,\mathcal{SA}\right\rangle $ quantifies
the amount of information acquired during this measurement. Also,
$\mathcal{P}$ \textquotedblleft knows\textquotedblright, also in a completely
coherent way, the information about which kind of information (here
$\left\vert r,\mathcal{SA}\right\rangle $) $\mathcal{A}$ has about
$\mathcal{S}$; the amount of the $\mathcal{P}$-$\mathcal{(SA)}$ entanglement
quantifies the amount of information on which kind of measurements to do. All
information is coherently and completely encoded there.

Note that information here is characterized by \textit{dual}
\textit{entanglement}---the $\mathcal{P}$-$\mathcal{(SA)}$ entanglement
(maximally, $\ln D_{\mathcal{P}}$) and the $\mathcal{S}$-$\mathcal{A}$
entanglement (maximally, $\ln D_{\mathcal{S}}$) contained in $\left\vert
r,\mathcal{SA}\right\rangle $. As both $\left\vert r,\mathcal{SA}\right\rangle
$ and $\left\vert \mathcal{P(SA)}\right\rangle $ are pure states, their
entanglement is uniquely quantified by the usual entanglement entropy
\cite{pureEE1,pureEE2}. This immediately identifies each of the squared
coefficients of their Schmidt decompositions as a probability to reconcile
with Shannon's definition of entropy. Put differently, in our
informationally-complete description of physical systems, entanglement is all
\textit{the} information; classical terms like probability arise in our
description because of either our reliance on classical concept of information
or an approximate and incomplete description to be shown below.

The status of quantum states in the informational completeness formalism thus
represents a complete reality of the whole system ($\mathcal{P}$,
$\mathcal{S}$, and $\mathcal{A}$). Such a reality picture is only possible by
taking into account the informational completeness explicitly in our
formalism. Quantum states do exist in a world that is informational and
objective. Whatever an observation might be, they always encode information
pertaining to that observation as programmed, without invoking observers or
having to appeal to any mysterious mechanisms to account for wave function
collapse; there is simply no wave function collapse. Here local quantum states
(i.e., states for each of $\mathcal{P}$, $\mathcal{S}$, and $\mathcal{A}$) are
all relative, but information encoded in dual entanglement is invariant under
the changes of local bases. If one likes, the choice of local bases can be
called a free will or freedom of choice, corresponding to certain
\textquotedblleft gauge\textquotedblright. Yet, all physical predictions of
the theory are encoded in dual entanglement and do not depend on the chosen gauge.

In certain sense, $\mathcal{P}$ and $\mathcal{A}$ act like a \textquotedblleft
quantum being\textquotedblright\ (\textquotedblleft qubeing\textquotedblright)
who holds coherently all the informationally-complete programmes on how to
entangle $\mathcal{S}$ and $\mathcal{A}$. In this way, the qubeing has all the
information about $\mathcal{S}$. However, our human beings, unlike the
qubeing, only have limited ability to acquire information, with limited
precisions, limited degrees of freedom, limited information detection and
storage, and so on; or simply we are so used to and familiar with classical
concepts on information and physical systems. For example, an experimenter,
Alice, would like to acquire information about $\left\vert \psi,\mathcal{S}%
\right\rangle $. First of all, she has to \textit{decide} which kind of
information she would like to know. After making a decision, she needs then to
\textit{observe} (that is, to interact with) the apparatus readily entangled
with $\mathcal{S}$. In principle, Alice's decisions and observations are all
physical processes which should be described quantum mechanically.
Nevertheless, Alice is macroscopic, and has so many quantum degrees of freedom
and limited ability (lack of full knowledge of the entire system). In this
case, she has to \textquotedblleft trace out\textquotedblright\ her quantum
degrees of freedom involved in her decisions (interaction with $\mathcal{P}$),
leading to a mixed state $\sum_{r}\left\vert g_{r}\right\vert ^{2}\left\vert
r,\mathcal{P}\right\rangle \left\vert r,\mathcal{SA}\right\rangle \left\langle
r,\mathcal{SA}\right\vert \left\langle r,\mathcal{P}\right\vert $. This state
allows a probability interpretation about Alice's freedom of choice: Each of
her decisions $\left\vert r,\mathcal{P}\right\rangle $ occurs with a
probability of $\left\vert g_{r}\right\vert ^{2}$. As far as a particular
choice $\left\vert r,\mathcal{P}\right\rangle $ has been made, again she has
to trace out her quantum degrees of freedom involved in her observation
(interaction with $\mathcal{A}$). This then leads to the usual Born rule about
$\left\vert \psi,\mathcal{S}\right\rangle $ for the given measurement. Thus, in the
ICQT, the Born rule is an emergent or derived rule determined by the dual entanglement
structure.

To summarize the above picture, the world view of the ICQT is fascinating. If
we regard the system $\mathcal{S}$ as an indivisible part of the qubeing
$\mathcal{PA}$, the whole system $\mathcal{PSA}$ then represents an
informationally complete and objective entity; it seems that the qubeing has
its own \textquotedblleft consciousness\textquotedblright, a kind of
miraculous quantum ability, to encode and access all its information in the
form of dual entanglement, in which the constituent parts of the qubeing are
mutually measured or defined. In other words, for the qubeing all information
(namely, all physical predictions) is encoded in dual entanglement via interaction
and not obtained via the usual quantum measurement with the unavoidable concept
of the wave function collapse.

The trinary picture (the division of $\mathcal{S}$, $\mathcal{A}$, and
$\mathcal{P}$) of physical systems arises here as a new feature of the ICQT,
as shown in Figure 1. To retain the informationally complete description
of nature, such a trinary picture seems to be unavoidable. The limitation of
informational contents in dual entanglement could be tentatively called
\textquotedblleft the trinity principle\textquotedblright, instead of the
conventional complementarity principle, to put the trinary property of
physical systems on the most fundamental ground.

The loss of the trinary picture of describing physical systems leads to the
\textit{emergent dual Born rule}, i.e., the probability description on which
kind of observables to measure and on which eigenvalue of the observable to
measure, due to, e.g., lack of full knowledge of the entire system in our
ICQT. The conventional von Neumann entropy quantifies this dual loss of
information. In other words, the conventional Born rule arises as a
consequence of the sacrifice of informationally-complete description in the
trinary picture; the sacrifice leads to a \textit{partial} reality of physical
system as described by conventional QT.

\begin{figure}
\centerline{\includegraphics[width=.35\textwidth]{taiji-icqc.eps}}
\caption{The trinary picture of the world. The division of system
$\mathcal{S}$, measurement apparatus $\mathcal{A}$, and programming system
$\mathcal{P}$ naturally arises in the informationally complete description of
physical systems. \textquotedblleft The Taiji pattern\textquotedblright\ shows
in an intuitive manner the $\mathcal{S}$-$\mathcal{A}$ interaction
(entanglement), while the green discs inside and outside the Taiji pattern
represent the programmed measurement operations $\hat{U}_{\mathcal{P(SA)}}$
between $\mathcal{P}$ and $\mathcal{SA}$. In ancient China, Taoists regarded
the Taiji pattern as a \textquotedblleft diagram of the
Universe\textquotedblright. The trinary picture of the world shown here is
ubiquitous in the sense that the world, at the most fundamental level, is
made up of a trinity: gravity ($\mathcal{P}$), elementary particles
($\mathcal{S}$) and their gauge fields ($\mathcal{A}$); the trinity should be
describable by the ICQT.}%
\label{TJ}%
\end{figure}


\section{The informationally-complete dynamics}

According to the above picture of nature, single free systems are simply
meaningless for acquiring information and thus in the \textquotedblleft
information-vacuum\textquotedblright\ state; a system, which does not give
information to (i.e., interact with) other systems in any way, simply does not
exist. The \textquotedblleft$\mathcal{S}$+$\mathcal{A}$\textquotedblright%
\ description in the usual QT is also inadequate because of its informational
incompleteness. Therefore, the dynamics of the ICQT has to obey the
informational completeness principle within the trinary picture and thus will
be dramatically different from the usual picture. Namely, the ICQT is
characterized by the indivisibility of its kinematics and dynamics to maintain
the informationally-complete trinary description.

Before considering the informationally-complete dynamics, let us introduce an
important conception of \textit{dual} \textit{measurability}: the
$\mathcal{P}$-$\mathcal{SA}$ measurability and the programmed $\mathcal{SA}%
\left\vert _{\mathcal{P}}\right.  $ measurability. The former means the
ability of measuring $\mathcal{P}$ with $\mathcal{SA}$ and vice versa; the
latter means the ability of measuring $\mathcal{A}$ with $\mathcal{S}$ and
vice versa, under a given programmed measurement operation of $\mathcal{P}$.
The $\mathcal{P}$-$\mathcal{SA}$ measurability (the programmed measurability
$\mathcal{SA}\left\vert _{\mathcal{P}}\right.  $) leads to $D_{\mathcal{P}%
}=D_{\mathcal{A}}D_{\mathcal{S}}$ ($D_{\mathcal{A}}=D_{\mathcal{S}}=D$) and
thus a symmetric role between $\mathcal{P}$ and $\mathcal{SA}$ ($\mathcal{S}$
and $\mathcal{A}$). Note that here measurability does not means certain von Neumann
measurement actually performed by an experimenter.

After the above preparation, now we give a definition of
informationally-complete physical systems: A physical system is said to be
informationally-complete if and only if (the use of \textquotedblleft if and
only if\textquotedblright\ will be explained below) it is consisted of
$\mathcal{S}$, $\mathcal{A}$ and $\mathcal{P}$ described as a trinity such
that the $\mathcal{P}$-$\mathcal{SA}$ measurability and the programmed
measurability $\mathcal{SA}\left\vert _{\mathcal{P}}\right.  $ are satisfied.
As a result of this definition, the $\mathcal{P}$-$\mathcal{SA}$ measurability
implies the existence of at most $D^{2}$ independent \textit{informationally
complete measurement (entanglement) operations} in the Hilbert spaces of both
$\mathcal{P}$ and $\mathcal{SA}$; these operations generate at most $D^{2}%
$-dimensional entangled states between $\mathcal{P}$ and $\mathcal{SA}$ and at
most $D^{2}$ entangled states between $\mathcal{S}$ and $\mathcal{A}$ such
that $\mathcal{P}$ and $\mathcal{SA}$ are mutually measuring and defining.
Meanwhile, under the given programming state of $\mathcal{P}$, the programmed
measurability $\mathcal{SA}\left\vert _{\mathcal{P}}\right.  $ implies the
existence of at most $D$-dimensional entangled states between $\mathcal{S}$ and
$\mathcal{A}$ such that $\mathcal{S}$ and $\mathcal{A}$ are mutually measuring
and defining, as programmed. For convenience, we also call the complete set of
states and operators defined in the Hilbert space of $\mathcal{P}$ or
$\mathcal{SA}$ are informationally complete. Accordingly, states and operators
for either $\mathcal{S}$ or $\mathcal{A}$ alone are \textit{informationally
incomplete}.

Let us suppose that the informationally-complete system $\mathcal{PSA}$ has a
general Hamiltonian $\hat{H}_{\mathcal{PSA}}$. We assume that the whole system
evolutes according to a Schr\"{o}dinger-like equation (we take $\hbar=1$),
namely, $i\frac{d}{dt}\left\vert \mathcal{PSA},t\right\rangle =\hat
{H}_{\mathcal{PSA}}\left\vert \mathcal{PSA},t\right\rangle $. In general,
$\hat{H}_{\mathcal{PSA}}=\hat{H}_{\mathcal{P}}+\hat{H}_{\mathcal{S}}+\hat
{H}_{\mathcal{A}}+\hat{H}_{\mathcal{PS}}+\hat{H}_{\mathcal{PA}}+\hat
{H}_{\mathcal{SA}}+\hat{H}_{\mathcal{P(SA)}}$, where the subscripts label the
corresponding systems. Now our problem is to determine how the informational
completeness principle constrains the form of $\hat{H}_{\mathcal{PSA}}$ and
thus the dynamics of the $\mathcal{PSA}$-system.

Note that we can choose an orthonormal basis (the \textquotedblleft
programming basis\textquotedblright) \{$\left\vert e_{n},\mathcal{P}%
\right\rangle ;n=0,1,..D_{\mathcal{P}}-1$\} to span the whole Hilbert space of
$\mathcal{P}$. We associate each programming state $\left\vert e_{n}%
,\mathcal{P}\right\rangle $ as an eigenstate of $\mathcal{P}$-system's
\textquotedblleft programming observable\textquotedblright%
\ $\hat{e}_{\mathcal{P}}$ with eigenvalue $e_{n}$.
It is easy to verify that the following Hamiltonian obeys the informational
completeness principle:
\begin{align}
\mathcal{\hat{I}}_{\mathcal{PSA}}  &  =\hat{H}_{\mathcal{P}}+\hat
{H}_{\mathcal{P(SA)}}=\hat{H}_{\mathcal{P}}(t)\nonumber\\
&  +\sum_{n=0}^{D_{\mathcal{P}}-1}\left\vert e_{n},\mathcal{P}\right\rangle
\left\langle e_{n},\mathcal{P}\right\vert \hat{H}_{\mathcal{SA}\left\vert
_{\mathcal{P}}\right.  }(e_{n},t), \label{IPSA}%
\end{align}
if we impose the $\mathcal{P}$-$\mathcal{SA}$ measurability condition as follows
\begin{equation}
\lbrack\hat{H}_{\mathcal{P(SA)}},\hat{H}_{\mathcal{P}}]=0, \label{pmc}%
\end{equation}
namely, $\hat{H}_{\mathcal{P(SA)}}$ and $\hat{H}_{\mathcal{P}}$\ are commutative.

In Eq.~(\ref{IPSA}) $\hat{H}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.}(e_{n}%
,t)=i\frac{d}{dt}\hat{U}_{\mathcal{SA}\left\vert _{\mathcal{P}%
}\right.}\left(e_{n},t\right) \cdot\hat{U}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.  }^{-1}\left(
e_{n},t\right)$ is an informationally-complete operator
set of $\mathcal{SA}$ and as such, $\hat{H}_{\mathcal{S}}+\hat{H}%
_{\mathcal{A}}+\hat{H}_{\mathcal{PS}}+\hat{H}_{\mathcal{PA}}+\hat
{H}_{\mathcal{SA}}$ has also been included therein, up to local unitary tranformations. $\mathcal{\hat{I}%
}_{\mathcal{PSA}}$ induces a dynamical evolution $\left\vert \mathcal{PSA}%
,t\right\rangle =\hat{U}_{\mathcal{PSA}}\left\vert \mathcal{PSA}%
,t=0\right\rangle $; the evolution operator $\hat{U}_{\mathcal{PSA}}$ always
has a factorizable structure
\begin{equation}
\hat{U}_{\mathcal{PSA}}(t)=\sum_{n=0}^{D_{\mathcal{P}}-1}\left\vert
e_{n},\mathcal{P}\right\rangle \left\langle e_{n},\mathcal{P}\right\vert
\hat{U}_{\mathcal{P}}(t)\hat{U}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.
}(e_{n},t), \label{factor}%
\end{equation}
as a result of the $\mathcal{P}$-$\mathcal{SA}$ measurability condition
(\ref{pmc}), such that $(\forall e_{n})$
\begin{align}
i\frac{d}{dt}\hat{U}_{\mathcal{P}}(t)  &  =\hat{H}_{\mathcal{P}}(t)\hat
{U}_{\mathcal{P}}(t),\nonumber\\
i\frac{d}{dt}\hat{U}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.  }%
(e_{n},t)  &  =\hat{H}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.  }%
(e_{n},t)\hat{U}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.  }(e_{n},t).
\label{dual-dyn}%
\end{align}
In this way, the dynamical evolutions of $\mathcal{P}$ and $\mathcal{SA}$ are
mutually defined, in accordance with the informational completeness principle.

The Hamiltonian $\mathcal{\hat{I}}_{\mathcal{PSA}}$ as given above respects
the $\mathcal{P}$-$\mathcal{SA}$ measurability. If we also require the
programmed measurability $\mathcal{SA}\left\vert _{\mathcal{P}}\right.  $
($\forall e_{n}$), the evolution governed by $\hat{H}_{\mathcal{SA}\left\vert
_{\mathcal{P}}\right.  }(e_{n},t)$ will similarly acquire the factorizable
structure as
\begin{align}
\hat{H}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.}(e_{n})  & =\sum
_{i=0}^{D}\left\vert \varepsilon_{i}(e_{n}),e_{n},\mathcal{S}\right\rangle
\left\langle \varepsilon_{i}(e_{n}),e_{n},\mathcal{S}\right\vert \nonumber\\
& \times\hat{H}_{\mathcal{A}}\left[  e_{n},\varepsilon_{i}(e_{n}),t\right]
+\hat{H}_{\mathcal{S}\left\vert _{\mathcal{P}}\right.  }(e_{n},t)\label{Hsap}%
\end{align}
with $\hat{H}_{\mathcal{A}}=i\frac{d}{dt}\hat{U}_{\mathcal{A}}\left[
e_{n},\varepsilon_{i}(e_{n}),t\right]\cdot\hat{U}_{\mathcal{A}}^{-1}\left[  e_{n}%
,\varepsilon_{i}(e_{n}),t\right]$. Here the orthonormal
basis for $\mathcal{S}\left\vert _{\mathcal{P}}\right.  $ is \{$\left\vert
\varepsilon_{i}(e_{n}),e_{n},\mathcal{S}\right\rangle $\}, where $\left\vert
\varepsilon_{i}(e_{n}),e_{n},\mathcal{S}\right\rangle $ is an eigenstate of an
observable $\hat{\varepsilon}_{\mathcal{S}\left\vert _{\mathcal{P}}\right.
}(e_{n})$ with eigenvalue $\varepsilon_{i}(e_{n})$ for given $e_{n}$.
Similarly to the $\mathcal{P}$-$\mathcal{SA}$ measurability condition
(\ref{pmc}), we need to impose the programmed measurability $\mathcal{SA}%
\left\vert _{\mathcal{P}}\right.  $ condition
\begin{equation}
\lbrack\hat{H}_{\mathcal{SA}\left\vert _{\mathcal{P}}\right.}(e_{n}),\hat{H}_{\mathcal{S}\left\vert _{\mathcal{P}}\right.  }(e_{n}%
,t)]=0\ \ \ \ (\forall e_{n}).\label{sapmc}%
\end{equation}
The $\mathcal{SA}\left\vert _{\mathcal{P}}\right.  $ dynamics is then similar
to the $\mathcal{P}$-$\mathcal{SA}$ dynamics considered above. Such a
\textit{dual dynamics} of the whole system $\mathcal{PSA}$ is an attribute of
the trinary description and quite distinct to the usual Schr\"{o}dinger evolution.

What is the physical significance of the programming basis \{$\left\vert e_{n},\mathcal{P}%
\right\rangle$\} and the associated observable $\hat{e}_{\mathcal{P}}$? Actually it is
physically transparent that \{$\left\vert e_{n},\mathcal{P}%
\right\rangle$\} can be identified with the Schmidt basis for the $\mathcal{P}$-$\mathcal{SA}$
decomposition, which is not affected by the local transformations generated by $\hat{H}_{\mathcal{P}}$. When applying the ICQT to quantum gravity \cite{icQFT}, the usual Schr\"{o}dinger equation
reduces to a constraint $\hat{H}_{\mathcal{PSA}}\left\vert \mathcal{PSA},t\right\rangle=0$, known as
the Wheeler-DeWitt equation. In this case, the programming observable $\hat{e}_{\mathcal{P}}$ then corresponds to a Dirac observable \cite{Rovelli-book,Thiemann}.

To end this section, it is important to note that the distinguished roles of
$\mathcal{P}$ and $\mathcal{SA}$ are relative. Depending on the specific form
of the trinary Hamiltonian, we could have another possibility that
$\mathcal{SA}$ can programme the evolution of $\mathcal{P}$. In this case, the
programming basis is chosen for $\mathcal{SA}$ and associated with an
observable being commutative with $\hat{H}_{\mathcal{SA}}$ such that the
$\mathcal{P}$-$\mathcal{SA}$ measurability and a similar dynamics as in
Eq.~(\ref{dual-dyn}) can still be obtained. Furthermore, the roles of
$\mathcal{P}$ and $\mathcal{SA}$ are actually symmetric due to a nice property
of the Schmidt decomposition \cite{Schmidt}, in which if an orthonormal basis
labelled by an index (e.g., $n$) is chosen for a system, then the orthonormal
basis for another system, by acting a unitary tranformation upon it, is
labelled by the same index. This property implies that both of the bases are
already the Schmidt bases, up to local unitary tranformations. A similar
consideration is applicable to the programmed measurability $\mathcal{SA}%
\left\vert _{\mathcal{P}}\right.  $, too.

\section{Relation with conventional quantum theory}

What is the relation between (\ref{dual-dyn}) and the usual Schr\"{o}dinger
equation? Here the dynamical evolution is such that systems $\mathcal{P}$ and
$\mathcal{SA}$ are mutually defined, and so systems $\mathcal{S}$ and
$\mathcal{A}$. In an informationally-complete quantum field theory, if we
regard $\mathcal{S}$ as particle fields and $\mathcal{A}$ as gauge fields, and
$\mathcal{SA}$ together as matter fields, then we immediately recognize that
system $\mathcal{P}$ must be the gravitational field (i.e., spacetime),
nothing else. If we think this way, an amazing picture (Figure 1) of our
world arises: The gravitational field and matter fields are mutually defined
and entangled---no matter, no gravity (spacetime) and vice versa, and for each
of their entangled patterns, particle fields and their gauge fields are
likewise mutually defined and entangled. If this is indeed what our nature
works to obey the informational completeness principle, the conventional QT
will be an approximation of our ICQT when nature's programming system, i.e.,
gravity, \textquotedblleft hides\textquotedblright\ its quantum effects. Under such
an approximation the ICQT reduces to conventional QT, characterized by the usual Schr\"{o}dinger equation and the probability description and as such, QT in its current form is thus 
informationally incomplete. This is in the exact sense that classical
Newtonian mechanics is an approximate theory of special relativity when a
physical system has a speed much less than the speed of light.

On the other hand, no matter how weak gravity is, it is forced to be there by the
informational completeness principle, to play a role for the internal
consistency of the theory. This unique role of gravity (or spacetime) in our theory is consistent
with the remarkable fact that only gravity is universally coupled to all other physical
fields (particle fields and gauge fields). Of course, our current quantum description is an
extremely good approximation. But for scales near the Planck one and for early
Universe, quantized spacetime acts as the programming system and the ICQT will be necessary. Thus, both facts (i.e., current QT works so well
and quantum gravity effects are so weak at normal scales) hide so deeply any new theoretical
architecture beyond current QT, like the ICQT. Even in the string theory, there is no change
of the underlying quantum formalism. By contrast, what we suggest within the ICQT is that at the level of quantized fields including quantized spacetime, everything is quantized and one does not have the usual separation of the quantum system and the observer. In this case, one has to give up the classical-quantum
hybrid feature of current QT. For this purpose, the most obvious way seems to be the elimination of the
measurement postulate in our fully quantum (namely, not classical-quantum hybrid) description of nature. As we hope to argue, giving up
the classical concepts associated with the measurement postulate in current QT does not
lead to any sacrifice of our predictive power as the complete information is encoded by the
dual entanglement structure.

If we take the above argument seriously, then \textit{the ICQT captures the
most remarkable trinity of nature}, namely, the division of nature by
particles, their gauge fields and gravity (spacetime), though the role of the
Higgs field needs a separate consideration. The previous two sections argued
the necessity of the informational completeness in the trinary description.
Here we see that it is also sufficient: We do not have to invoke more
programming systems to program $\mathcal{PSA}$ simply because we do not have
spacetime (gravity) out of spacetime (gravity)---Trinity is necessary and
sufficient. This eliminates the von Neumann chain in the usual quantum
measurement model.

Thus, the ICQT gives a strong motivation or reason for quantizing spacetime.
However, quantizing gravity as yet another field, as in concentional quantum
field theory, is not sufficient and does not automatically result in a correct
and consistent quantum theory of all known forces. Only when the informational
completeness in the trinary description is integrated into our quantum
formulation, can we have the desired theory of the Universe. Only after that,
can the relation between our theory and the conventional quantum theory be
rigorously discussed.

As we showed elsewhere \cite{icQFT}, following the above argument indeed leads
to a consistent framework of unifying spacetime (gravity) and matter, without the
fundamental inconsistencies \cite{Thiemann} between gravity and conventional
quantum field theory, implying the conceptual advantages of our theory. For
instance, with the theoretical input from loop quantum gravity predicting the
quantized geometry \cite{Rovelli-book,Thiemann,quanA1,quanA2,quanA3}, the
informationally complete quantum field theory naturally gives the holographic
principle \cite{tHooft,Susskind,holoRMP}. Such a strong limit on the allowed
states of the trinary system in any finite spacetime regime, as imposed by the
ICQT, paves the way to escape the infrared and ultraviolet singularities
(divergences) that occur in conventional quantum field theory.

The natural position of gravity in the ICQT cannot be accidental and may be a
strong evidence supporting our informationally-complete description of nature.
Thus, at the most fundamental level, our reasoning suggests once again the
trinary picture of our world consisting of particles, gauge fields and gravity
as a trinity. It is surprise to see that nature singles out gravity as a
programming field, which plays a role that is definitely different from matter
fields. This also indicates the distinct roles of matter-matter (particles and
their gauge fields) entanglement and spacetime-matter (i.e., gravity-matter)
entanglement and might explain the reason why quantizing gravity as usual
quantized fields suffers from well-known conceptual problems.

As an abstract mathematical structure, current QT is
content-irrelevant in the following sense. While it is believed to be
universally applicable to physical systems of any physical contents, ranging
from elementary particles and (super)strings to the whole Universe, what physical content that
it describes does not matter and the physical content never changes its very
structure. The situation for classical mechanics is quite similar in this
aspect. However, the ICQT changes this in a dramatic way in the sense that the
trinary picture of nature has to be integrated into a consistent formulation
to enable an informationally-complete description. The physical content that
the ICQT describes does matter as the states and their dynamical evolution of
the trinary system are constrained or structured into the dual forms specified
above. In particular, the inclusion of the programming system, identified with
gravity in the field-theoretical case, is very essential and necessary in our description.

\section{Informationally-complete quantum computation}

A new theory should make new predictions or/and motivate new applications. Of
course, previous interpretations of QT are very important for a better
understanding of the theory. However, no interpretations make new predictions
or/and motivate fundamentally new applications. Now we argue that our ICQT indeed motivates
new applications if we consider its computational power. Even though gravity
would play certain role in our future understanding of nature, artificial
informationally-complete quantum systems are realizable as a quite good approximation.

What is an informationally-complete quantum computer (ICQC)? We define the
ICQC as an artificial informationally-complete quantum systems, or a quantum
intelligent system (qubeing), which has an informationally-complete trinary
structure consisting of $\mathcal{S}$, $\mathcal{A}$, and $\mathcal{P}$. The
ICQC starts from an initial state $\left\vert \mathrm{IC}\text{\textrm{QC}%
}\right\rangle _{0}=\left\vert \psi,\mathcal{S}\right\rangle \left\vert
\phi,\mathcal{A}\right\rangle \left\vert \chi,\mathcal{P}\right\rangle $. As
usual, the $\mathcal{S}$ system has $n$ qubits, and thus dimensions of $2^{n}%
$. To be well defined, we also use qubits to make up the $\mathcal{A}$ system
and the $\mathcal{P}$ system; $\mathcal{A}$ ($\mathcal{P}$) has
$n_{\mathcal{A}}$ ($n_{\mathcal{P}}$) qubits and dimensions of
$2^{n_{\mathcal{A}}}$ ($2^{n_{\mathcal{P}}}$). To satisfy the informational
completeness principle, we have $n_{\mathcal{A}}=n$\ and $n_{\mathcal{P}}=2n$.
Our ICQC then works by applying certain patterns of universal quantum logic
gates (single-qubit and two-qubit ones), determined by quantum algorithm
pertaining to the question under study. Generally speaking, as an artificially
controllable quantum system the patterns of gates are allowed to exhaust all
unitary operations on the whole $\mathcal{PAS}$ system, which we denote
collectively by $\hat{V}_{\mathcal{PAS}}=\hat{U}(\mathcal{P},\mathcal{A}%
,\mathcal{S},\mathcal{AS},\mathcal{PA},\mathcal{PS},\mathcal{PAS})$. At the
end of running the ICQT, we perform the programmed measurement operation
$\hat{U}_{\mathcal{P(SA)}}$ on $\mathcal{PAS}$. The resulting final state of
the ICQC reads $\left\vert \mathrm{IC}\text{\textrm{QC}}\right\rangle =\hat
{U}_{\mathcal{P(SA)}}\hat{V}_{\mathcal{PAS}}\left\vert \mathrm{IC}%
\text{\textrm{QC}}\right\rangle _{0}$. Here, $\hat{U}_{\mathcal{P(SA)}}=\hat{U}_{\mathcal{P}}\sum_{p=0}^{4^{n}-1}\left\vert
p,\mathcal{P}\right\rangle \left\langle p,\mathcal{P}\right\vert \hat
{U}(p,\mathcal{A},\mathcal{S})$
and all $\hat{U}(p,\mathcal{A},\mathcal{S})$ span a complete operator set for
$\mathcal{SA}$.

Is the ICQC defined above a usual quantum computer merely with more
($n+n_{\mathcal{A}}+n_{\mathcal{P}}=4n$) qubits, but without the
informational-completeness and trinary structure? The answer is definitely
\textquotedblleft no\textquotedblright\ because of the conceptual
difference between the two quantum computing devices. To see this, we prepare
each qubit of $\mathcal{S}$ in the initial
state $\left\vert +,\mathcal{S}\right\rangle =\frac{1}{\sqrt{2}}(\left\vert
0,\mathcal{S}\right\rangle +\left\vert 1,\mathcal{S}\right\rangle )$ such that
$\left\vert \psi,\mathcal{S}\right\rangle $ is in a superposition of all
$2^{n}$ bit-values with equal probability amplitude: $\left\vert
\psi,\mathcal{S}\right\rangle =\frac{1}{\sqrt{2^{n}}}\sum_{x=0}^{2^{n}%
-1}\left\vert x,\mathcal{S}\right\rangle $. The initial states of
$\mathcal{A}$ and $\mathcal{P}$ are likewise prepared: $\left\vert
\phi,\mathcal{A}\right\rangle =\frac{1}{\sqrt{2^{n}}}\sum_{x=0}^{2^{n}%
-1}\left\vert y,\mathcal{A}\right\rangle $ and $\left\vert \chi,\mathcal{P}%
\right\rangle =\frac{1}{\sqrt{4^{n}}}\sum_{x=0}^{4^{n}-1}\left\vert
z,\mathcal{P}\right\rangle $. Such a coherent superposition of conventional
quantum computer's initial states is believed to be the very reason for the
speedup of quantum algorithms \cite{QuInf-book,SciAm}. Now let us make a further
simplification by doing nothing anymore on $\mathcal{A}$ and $\mathcal{S}$,
namely, the ICQC only acts $\hat{U}_{\mathcal{P(SA)}}$ on the initial state
as
\begin{equation}
\left\vert \mathrm{ICQC},\mathcal{PSA}\right\rangle =\hat{U}_{\mathcal{P(SA)}%
}\left\vert \psi,\mathcal{S}\right\rangle \left\vert \phi,\mathcal{A}%
\right\rangle \left\vert \chi,\mathcal{P}\right\rangle . \label{icqcS}%
\end{equation}


For such a simplified ICQC, $\hat{U}_{\mathcal{P(SA)}}$ can encode all
possible (i.e., informationally complete) programmed measurement operations
upon $\mathcal{A}$ and $\mathcal{S}$. These operations are actually all
allowed quantum algorithms and their outputs on $n$-qubit state $\left\vert
\psi,\mathcal{S}\right\rangle $, in the terminology of conventional quantum
computing. Then we immediately see that in the ICQC, one has \textit{dual
parallelism}: Parallelism in initial states as usual \textit{and} parallelism
of programmed operations (algorithms and outputs). In other words, \textit{a
single ICQC with }$4n$\textit{ qubits could compute in parallel all algorithms
of usual quantum computers with }$n$\textit{ qubits}. Due to this particular
dual parallelism enabled by the ICQC, it is reasonable to expect much higher
computational power with the ICQC.

Actually, the ICQC is, by definition, the most powerful computational machine
on qubit systems in the sense of informational completeness; otherwise it is
informationally incomplete. Finding algorithms on the ICQC to explicitly
demonstrate the computational power of the ICQC is surely a future interesting
problem. Also, computational complexity and error-tolerance in the ICQC
framework are two important issues. If nature does use the informational
completeness as a guiding principle, it computes the world we currently know;
such a world could be simulated and thus comprehensible by the ICQC (i.e.,
\textquotedblleft qubitization\textquotedblright\ within an
informationally-complete trinary description) in principle.

\section{Other conceptual applications}

Below we give, only very briefly, a few conceptual applications of the
informational completeness principle and the ICQT, hoping to shed new light on
some long-standing open questions in physics.

An important question is how to understand the occurrence of the classical
world surrounding us, including the second law of thermodynamics and the arrow
of time, in our new framework characterized by the informational completeness
principle and the trinary picture of nature. Though we cannot present
quantitative analysis of the problem here, a qualitative and conceptual answer
to the problem is quite transparent: For informationally complete quantum
systems, always-on interactions lead to the increase of entanglement for
$\mathcal{S}$-$\mathcal{A}$ and $\mathcal{P}$-$\mathcal{SA}$ systems; the
Universe as a whole has an increasing entanglement, a kind of
\textit{entanglement arrow of time}. It is easy to verify the entanglement
creation by considering the $\mathcal{PSA}$ evolution governed by
$\mathcal{\hat{I}}_{\mathcal{PSA}}$ from a separable state. At a
thermodynamic/macroscopic scale, tracing out thermodynamically/macroscopically
irrelevant degrees of freedom, only as an approximate description of the
underlying informationally complete physics, leads to the second law of
thermodynamics, the arrow of time, and ultimately, the classical world.

We note related analysis on the role of entanglement in the thermodynamic
arrow of time in the framework of conventional
\cite{arrow-Maccone,arrow-Terry} or time-neutral formulation \cite{Hartle} of
quantum mechanics. As gravity arguably plays an essential role in our
informationally complete description of nature, it is intriguing to see that
gravity plays some role in the occurrence of the second law of thermodynamics
and the arrow of time, as hinted in the study of black-hole thermodynamics
\cite{Bekenstein,BekensteinPRD,Hawking,Thiemann}. In the Di\'{o}si-Penrose
model \cite{Diosi,Penrose}, gravity was argued to play certain role for the
wave function collapse.

Now let us briefly consider the potential conceptual applications to cosmology.
Obviously, the conceptual difficulty of applying usual QT to the whole
Universe disappears in our ICQT. Actually, the ICQT is interpretation-free and
does not need an observer as the observer is a part of the Universe; the
description of the Universe by the ICQT would give us all information as it
could be. The contituent parts in trinity are mutually defining and
measuring in a specific dual entanglement structure, eliminating any
subjective aspects regarding the current interpretations of quantum
states---The existence of the Universe does not rely on the existence of
potential observers observing the Universe. Their entanglement in the dual
form encodes, without relying on any external observers, all physical
information and can give all physical predictions of the theory.

There is no reason why we cannot describe our human beings
as an informationally complete (classical, but ultimately, quantum) system via
a trinary description. In this way, some aspects of human beings could be
comprehensible purely from the informational and physical point of view. For instance,
if we define Alice's body and all of her sense organs as $\mathcal{A}$
and her outside world as $\mathcal{S}$, then Alice knows her world or gets known by her world
via interaction (i.e., information exchange) between $\mathcal{S}$ and $\mathcal{A}$.
Now an intriguing problem arises here: What is the programming system $\mathcal{P}$ in this context? A straightforward way is simply to define $\mathcal{P}$ (or, the correlations between $\mathcal{P}$ and $\mathcal{SA}$) as the mind (consciousness). By analogy to the above quantum trinary description, the mind $\mathcal{P}$ and $\mathcal{SA}$ are mutually defined in an
informationally-complete sense. This prescription thus provides an interesting possibility of understanding
the most mysterious part (namely, consciousness) of human beings from an informational and physical perspective. Particularly, Alice's brain undertakes only \emph{partial} (though the most important) functioning of her mind according to the above definition; the remaining functioning of her mind is distributed nonlocally in such a way that enables programming the interaction between $\mathcal{S}$ and $\mathcal{A}$. Note that, in an informationally-complete field theory \cite{icQFT}, the programming system is the quantized spacetime. If we take the above analogy more seriously, a very strange conclusion seems to be unavoidable: Alice's mind $\mathcal{P}$ should be ultimately explainable by the spacetime $\mathcal{P}$, namely, the mind is certain spacetime code of Alice's $\mathcal{SA}$. The reason behind the conjecture is the strong belief that the informational completeness should underlie the world, ranging from the elementary trinity (elementary fermions, their gauge fields, and spacetime) to our human beings and the whole Universe---actually, everything in the world is built from the informationally-complete elementary trinity.

In certain sense, it seems that human beings work as a quantum-decohering ICQC. Similarly to the fact that an ICQC is conceptually different from a normal quantum computer, a classical computing device with an integrated trinary structure similar to the ICQC should be quite different from the normal Turing machine and could be capable of
simulating human-like intelligence better. Does this mean certain ``consciousness computing'', or
``intelligence computing''? Further consideration in the context will be given in future.

Therefore, it could well be that the informational completeness is of significance in a
broader sense and should be a basic requirement for any physical systems,
classical or quantum. It is in this sense that the informational completeness deserves
to be named as a principle. It is a missed principle in our current
understanding of nature and a rule behind the comprehensibility of the
world---The informationally complete world is comprehensible by
informationally complete human beings.

\section{Conclussions and outlook}

In the present work, we have presented an interpretation-free QT under the
assumption that quantum states of physical systems represent an
informationally-complete code of any possible information that one might
access. To make the informational completeness explicitly in our formalism,
the trinary picture of describing physical systems seems to be necessary.
Physical systems in trinity evaluate and are entangled both in a dual form;
quantum entanglement plays a central role in the ICQT---Our world is
information given in terms of entanglement at the most fundamental level. So
\textit{the ICQT modifies two postulates (on quantum states and on dynamics) of
current quantum mechanics in a fundamental way and eliminates the measurement
postulate from our description}. We give various evidences and conceptual
applications of the ICQT, to argue that the ICQT, naturally identifying
gravity as nature's programming system, might be a candidate theory capable of
unifying matter and gravity (spacetime) in an informationally complete quantum
framework; for further development on our theory in the context of quantum
gravity, see Ref.~\cite{icQFT}. In this sense, the conventional QT will be an
approximation of our ICQT when quantum effect of gravity is ignored or
unimportant. Such an approximation leads to the approximate Schr\"{o}dinger
equation and the probability description of current QT. This is in the exact
sense that classical Newtonian mechanics is an approximate description of
relativistic systems. The ICQT motivates an interesting application to
informationally-complete quantum computing.

As we argued above, current quantum mechanics is \textit{not}
informationally-complete and thus suffers from interpretational difficulties.
The explicit demand of informational completeness not only removes the
conceptual problem of our current understanding of quantum mechanics, but also
leads to a profound constraint on formulating quantum theory. Thus, the ICQT
should not be understood simply as another interpretation of current QT; rather, it,
by giving up the classical concept of probability associated with the measurement
postulate, generalizes current quantum formalism---the physical prediction (outcomes
of an observable and the corresponding probabilities) of a quantum measurement in conventional
QT is now entailed by entanglement; no entanglement implies no information and thus no
prediction. As we noted previously, adding informational completeness requirement into our
current quantum formalism leads to serious consequences: Informational
completeness not only restricts the way on how to describe physical systems,
but also the way how they interact/entangle with each other. This will thus
give a very strong constraint on what physical progresses could have happened
or be allowed to happen.

On one hand, the ICQT provides a coherent conceptual picture of, or sheds new
light on, understanding some problems or phenomena in current physics,
including the intrinsic trinity of particles, gauge fields and gravity, the
occurrence of the classical world, the arrow of time, and the holographic
principle. On the other hand, some other problems, such as the complementarity
principle, quantum nonlocality \cite{Tipler} and quantum communication, should
be reconsidered from the viewpoint of the ICQT. All current quantum
communication protocols \cite{QuInf-book,BB84,teleport-PRL} have to make use
of classical concepts on information. It is thus very interesting to see how
to do communication in the ICQT and, particularly, to see whether or not it is
possible to achieve unconditionally secure communication.

According to the ICQT, the world underlying us is all about information
(entanglement); it is informationally complete, deterministic, self-defining,
and thus objective. Such a world view (\textquotedblleft quantum
determinism\textquotedblright) is of course quite different from what we learn
from current quantum mechanics, but in some sense, returns to Einstein's world
view and not surprisingly, represents an embodiment of Wheeler's thesis known
as \textquotedblleft it from bit\textquotedblright\ \cite{Wheeler}. Such a
viewpoint calls for a reconsideration of our current understanding on physical
reality, information, spacetime (gravity) and matter, as well as their links.
Let us cite the famous Einstein-Podolsky-Rosen paper \cite{EPR} here:
\textquotedblleft\textit{While we have thus shown the wave function does not
provide a complete description of the physical reality, we left open the
question of whether or not such a description exists. We believe, however,
that such a theory is possible}.\textquotedblright\ It is too early to judge
whether or not our ICQT completes current quantum mechanics in the
Einstein-Podolsky-Rosen sense as experiments will be the ultimate judgement.
But if nature does work like a description provided by the ICQT, nature will
be very funny and more importantly, nature does be comprehensible.

\ \ \newline\textbf{Acknowledgements}\newline I am grateful to Chang-Pu Sun for bringing Ref.~\cite{Tipler} into my attention, and to Xian-Hui Chen, Dong-Lai Feng and Yao Fu for enjoyable
discussions. This work was supported by the National Natural Science
Foundation of China (Grant No. 61125502) and the CAS.


\begin{thebibliography}{99}                                                                                               %


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\bibitem {Bohr}N. Bohr, \textit{Can quantum-mechanical description of physical
reality be considered complete?} Phys. Rev. \textbf{48}, 696-702 (1935).

\bibitem {WZ-book}J.A. Wheeler and W.H. Zurek, eds., \textit{Quantum Theory
and Measurement} (Princeton Univ. Press, Princeton, New Jersey, 1983).

\bibitem {Omnes}R. Omn\`{e}s, \textit{The Interpretation of Quantum Mechanics}
(Princeton Univ. Press, Princeton, New Jersey, 1994).

\bibitem {Zurek}W.H. Zurek, \textit{Decoherence, einselection, and the quantum
origins of the classical}, Rev. Mod. Phys. \textbf{75}, 715-775 (2003).

\bibitem {Sun}C.-P. Sun, \textit{Quantum dynamical model for wave-function
reducation in classical and macroscopic limits}, Phys. Rev. A \textbf{48},
898-906 (1993).

\bibitem {Sun-rev}C.-P. Sun, X.-X. Yi, and X.-J. Liu, \textit{Quantum
dynamical approach of wavefunction collapse in measurement progress and its
application to quantum Zeno effect}, Fortschr. Phys. \textbf{43}, 585-612 (1995).

\bibitem {PhyRep}A. Allahverdyan, R. Balian, and T.M. Niewenhuizen,
\textit{Understanding quantum measurement from the solution of dynamical
models}, Phys. Rep. \textbf{525}, 1-166 (2013).

\bibitem {PBR}M.F. Pusey, J. Barrett, and T. Rudolph, \textit{On the reality
of the quantum state}, Nature Phys. \textbf{8}, 475-478 (2012).

\bibitem {PBR-comment}G.C. Ghirardi and R. Romano, \textit{On the completeness
of quantum mechanics and the interpretation of the state vector}, J. Phys.:
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\bibitem {Aaronson}S. Aaronson, \textit{Quantum mechanics: Get real}, Nature
Phys. \textbf{8}, 443-444 (2012).

\bibitem {free-will}J. Conway and S. Kochen, \textit{The free will theorem},
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\bibitem {Bell-book}J.S. Bell, \textit{Speakable and Unspeakable in Quantum
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\bibitem {free-choi74}J.F. Clauser and M.A. Horne, \textit{Experimental
consequences of objective local theories}, Phys. Rev. D \textbf{10}, 526-535 (1974).

\bibitem {free-choiEX}T. Scheidl, R. Ursin, J. Kofler, S. Ramelow, X.-S. Ma,
T. Herbst, L. Ratschbacher, A. Fedrizzi, N. Langford, T. Jennewein, and A.
Zeilinger, \textit{Violation of local realism with freedom of choice}, Proc.
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\bibitem {vonN-book}J. von Neumann, \textit{Mathematical Foundations of
Quantum Mechanics,} Vol. 2 (Princeton Univ. Press, Princeton, New Jersey, 1996).

\bibitem {Wigner}E.P. Wigner, \textit{The problem of measurement}, Am. J.
Phys. \textbf{31}, 6-15 (1963).

\bibitem {Landau}L.D. Landau and E.M. Lifshitz, \textit{Quantum Mechanics:
Non-Relativistic Theory}, 3rd Ed. (Butterworth-Heinemann, Oxford, 1981).

\bibitem {Bohm}D. Bohm, \textit{A suggested interpretation of the quantum
theory in terms of `hidden' variables. I}, Phys. Rev. \textbf{85}, 166-179 (1952).

\bibitem {Everett}H. Everett III, \textit{\textquotedblleft Relative
state\textquotedblright\ formulation of quantum mechanics}, Rev. Mod. Phys.
\textbf{29}, 454-462 (1957).

\bibitem {Tipler}F.J. Tipler, \textit{Quantum nonlocality does not exist},
Proc. Natl. Acad. Sci. (USA) \textbf{111}, 11281-11286 (2014).

\bibitem {Rovelli}C. Rovelli, \textit{Relational quantum mechanics}, Int. J.
Theor. Phys. \textbf{35}, 1637-1678 (1996).

\bibitem {Rovelli-book}C. Rovelli, \textit{Quantum Gravity} (Cambridge Univ.
Press, Cambridge, 2004).

\bibitem {QuInf-book}M.A. Nielsen and I.L. Chuang, \textit{Quantum Computation
and Quantum Information} (Cambridge Univ. Press, Cambridge, 2000).

\bibitem {Hardy}L. Hardy, \textit{Quantum theory from five reasonable axioms},
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%BeginExpansion
$<$%
%EndExpansion
http://arXiv.org/quant-ph/0101012v4 (2001).

\bibitem {Fuchs}C.A. Fuchs, \textit{Quantum mechanics as quantum information
(and only a little more)}, Prepint at
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%BeginExpansion
$<$%
%EndExpansion
http://arXiv.org/quant-ph/0205039%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
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\bibitem {i-causality}M. Paw\l owski, T. Paterek, D. Kaszlikowski, V. Scarani,
A. Winter, and M. \.{Z}ukowski, \textit{Information causality as a physical
principle}, Nature \textbf{461}, 1101-1104 (2009).

\bibitem {Colbeck-Renner}R. Colbeck and R. Renner, \textit{No extension of
quantum theory can have improved predictive power}, Nature Commun. \textbf{2},
411 (2011).

\bibitem {pureEE1}C.H. Bennett, H. Bernstein, S. Popescu, and B. Schumacher,
\textit{Concentrating partial entanglement by local operations}, Phys. Rev. A
\textbf{53}, 2046-2062 (1996).

\bibitem {ThewTomo}R.T. Thew, K. Nemoto, A.G. White, and W.J. Munro,
\textit{Qudit quantum-state tomography}, Phys. Rev. A \textbf{66}, 012303 (2002).

\bibitem {ICO-Busch}P. Busch, \textit{Informationally complete set of physical
quantities}, Int. J. Theor. Phys. \textbf{30}, 1217-1227 (1991).

\bibitem {ICM}J.M. Renes, R. Blume-Kohout, A.J. Scott, and C.M. Caves,
\textit{Symmetric informationally complete quantum measurements}, J. Math.
Phys. \textbf{45}, 2171-2180 (2004).

\bibitem {pureEE2}S. Popescu and D. Rohrlich, \textit{Thermodynamics and the
measure of entanglement}, Phys. Rev. A \textbf{56}, R3319-R3321 (1997).

\bibitem {icQFT}Z.-B. Chen, \textit{An informationally-complete unification of
quantum spacetime and matter}, arXiv:1412.3662.

\bibitem {Thiemann}T. Thiemann, \textit{Lectures on loop quantum gravity},
\textit{Lecture Notes in Physics} \textbf{631}, 41-135 (2003).

\bibitem {Schmidt}J. Audretsch, \textit{Entangled Systems: New Directions in
Quantum Physics} (Wiley-VCH Verlag, Weinheim, 2007) pp.~149-150.

\bibitem {quanA1}C. Rovelli and L. Smolin, \textit{Discreteness of area and
volume in quantum gravity}, Nucl. Phys. B \textbf{442}, 593-619 (1995).

\bibitem {quanA2}C. Rovelli, \textit{Black hole entropy from loop quantum
gravity}, Phys. Rev. Lett. \textbf{77}, 3288-3291 (1996).

\bibitem {quanA3}A. Ashtekar, J. Baez, A. Corichi, and A. Krasnov,
\textit{Quantum geometry and black hole entropy}, Phys. Rev. Lett.
\textbf{80}, 904-907 (1998).

\bibitem {tHooft}G. 't Hooft, \textit{Dimensional reduction in quantum
gravity}, Prepint at
%TCIMACRO{\TEXTsymbol{<}}%
%BeginExpansion
$<$%
%EndExpansion
http://arXiv.org/gr-qc/9310026%
%TCIMACRO{\TEXTsymbol{>} }%
%BeginExpansion
$>$
%EndExpansion
(1993).

\bibitem {Susskind}L. Susskind, \textit{The world as a hologram}, J. Math.
Phys. \textbf{36}, 6377-6396 (1995).

\bibitem {holoRMP}R. Bousso, \textit{The holographic principle}, Rev. Mod.
Phys. \textbf{74}, 825-874 (2002).

\bibitem {SciAm}S. Aaronson, \textit{The limits of quantum computers}, Sci.
Am. \textbf{298}, 62-69 (March, 2012).

\bibitem {arrow-Maccone}L. Maccone, \textit{Quantum solution to the
arrow-of-time dilemma}, Phys. Rev. Lett. \textbf{103}, 080401 (2009).

\bibitem {arrow-Terry}D. Jennings and T. Rudolph, \textit{Entanglement and the
thermodynamic arrow of time}, Phys. Rev. E \textbf{81}, 061130 (2010).

\bibitem {Hartle}J.B. Hartle, \textit{The quantum mechanical arrows of time},
in \textit{Fundamental Aspects of Quantum Theory: A Two-Time Winner}, edited
by D. Struppa and J. Tollaksen (Springer, Milan, 2013).

\bibitem {Bekenstein}J.D. Bekenstein, \textit{Black holes and the second law},
Nuovo Cimento Lett. \textbf{4}, 737-740 (1972).

\bibitem {BekensteinPRD}J.D. Bekenstein, \textit{Black holes and entropy},
Phys. Rev. D \textbf{7}, 2333-2346 (1973).

\bibitem {Hawking}S.W. Hawking, \textit{Black hole explosions}, Nature
\textbf{248}, 30-31 (1974).

\bibitem {Diosi}L. Di\'{o}si, \textit{A universal master equation for the gravitational violation of quantum mechanics}, Phys. Lett. A \textbf{120}, 377-381 (1987).

\bibitem {Penrose}R. Penrose, \textit{On Gravity's role in Quantum State Reduction}, Gen. Rel. Grav. \textbf{28}, 581-600 (1996).

\bibitem {BB84}C.H. Bennett and G. Brassard, \textit{Quantum cryptography:
Public key distribution and coin tossing}, in \textit{Proceedings of the IEEE
International Conference on Computers, Systems and Singal Proceeding,
Bangalore, India} (IEEE, New York, 1984), p.175.

\bibitem {teleport-PRL}C.H. Bennett, G. Brassard, C. Cr\'{e}peau, R. Jozsa, A.
Peres, and W.K. Wootters, \textit{Teleporting an unknown quantum state via
dual classical and Einstein-Podolsky-Rosen channels}, Phys. Rev. Lett.
\textbf{73}, 3081-3084 (1993).

\bibitem {Wheeler}J.A. Wheeler, \textit{Recent thinking about the nature of
the physical world: It from bit}, Ann. N.Y. Acad. Sci. \textbf{655}, 349-364 (1992).

\ \
\end{thebibliography}


\end{document}

1412.1079.extracted_bib
{EPRA. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777-780 (1935).
1412.1079.extracted_bib
{BohrN. Bohr, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696-702 (1935).
1412.1079.extracted_bib
{WZ-bookJ.A. Wheeler and W.H. Zurek, eds., Quantum Theory and Measurement (Princeton Univ. Press, Princeton, New Jersey, 1983).
1412.1079.extracted_bib
{OmnesR. Omn\`{es, The Interpretation of Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey, 1994).
1412.1079.extracted_bib
{ZurekW.H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715-775 (2003).
1412.1079.extracted_bib
{SunC.-P. Sun, Quantum dynamical model for wave-function reducation in classical and macroscopic limits, Phys. Rev. A 48, 898-906 (1993).
1412.1079.extracted_bib
{Sun-revC.-P. Sun, X.-X. Yi, and X.-J. Liu, Quantum dynamical approach of wavefunction collapse in measurement progress and its application to quantum Zeno effect, Fortschr. Phys. 43, 585-612 (1995).
1412.1079.extracted_bib
{PhyRepA. Allahverdyan, R. Balian, and T.M. Niewenhuizen, Understanding quantum measurement from the solution of dynamical models, Phys. Rep. 525, 1-166 (2013).
1412.1079.extracted_bib
{PBRM.F. Pusey, J. Barrett, and T. Rudolph, On the reality of the quantum state, Nature Phys. 8, 475-478 (2012).
1412.1079.extracted_bib
{PBR-commentG.C. Ghirardi and R. Romano, On the completeness of quantum mechanics and the interpretation of the state vector, J. Phys.: Conf. Series 442, 012002 (2013).
1412.1079.extracted_bib
{AaronsonS. Aaronson, Quantum mechanics: Get real, Nature Phys. 8, 443-444 (2012).
1412.1079.extracted_bib
{free-willJ. Conway and S. Kochen, The free will theorem, Found. Phys. 36, 1441-1473 (2006).
1412.1079.extracted_bib
{Bell-bookJ.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987).
1412.1079.extracted_bib
{free-choi74J.F. Clauser and M.A. Horne, Experimental consequences of objective local theories, Phys. Rev. D 10, 526-535 (1974).
1412.1079.extracted_bib
{free-choiEXT. Scheidl, R. Ursin, J. Kofler, S. Ramelow, X.-S. Ma, T. Herbst, L. Ratschbacher, A. Fedrizzi, N. Langford, T. Jennewein, and A. Zeilinger, Violation of local realism with freedom of choice, Proc. Natl. Acad. Sci. (USA) 107, 19708-19713 (2010).
1412.1079.extracted_bib
{BellJ.S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics (Long Island City, N.Y.) 1, 195-200 (1964).
1412.1079.extracted_bib
{vonN-bookJ. von Neumann, Mathematical Foundations of Quantum Mechanics, Vol. 2 (Princeton Univ. Press, Princeton, New Jersey, 1996).
1412.1079.extracted_bib
{WignerE.P. Wigner, The problem of measurement, Am. J. Phys. 31, 6-15 (1963).
1412.1079.extracted_bib
{LandauL.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd Ed. (Butterworth-Heinemann, Oxford, 1981).
1412.1079.extracted_bib
{BohmD. Bohm, A suggested interpretation of the quantum theory in terms of `hidden' variables. I, Phys. Rev. 85, 166-179 (1952).
1412.1079.extracted_bib
{EverettH. Everett III, \textquotedblleft Relative state\textquotedblright\ formulation of quantum mechanics, Rev. Mod. Phys. 29, 454-462 (1957).
1412.1079.extracted_bib
{TiplerF.J. Tipler, Quantum nonlocality does not exist, Proc. Natl. Acad. Sci. (USA) 111, 11281-11286 (2014).
1412.1079.extracted_bib
{RovelliC. Rovelli, Relational quantum mechanics, Int. J. Theor. Phys. 35, 1637-1678 (1996).
1412.1079.extracted_bib
{Rovelli-bookC. Rovelli, Quantum Gravity (Cambridge Univ. Press, Cambridge, 2004).
1412.1079.extracted_bib
{QuInf-bookM.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).
1412.1079.extracted_bib
{HardyL. Hardy, Quantum theory from five reasonable axioms, Prepint at %TCIMACRO{\TEXTsymbol{<% %BeginExpansion $<$% %EndExpansion http://arXiv.org/quant-ph/0101012v4 (2001).
1412.1079.extracted_bib
{FuchsC.A. Fuchs, Quantum mechanics as quantum information (and only a little more), Prepint at %TCIMACRO{\TEXTsymbol{<% %BeginExpansion $<$% %EndExpansion http://arXiv.org/quant-ph/0205039% %TCIMACRO{\TEXTsymbol{> % %BeginExpansion $>$ %EndExpansion (2002).
1412.1079.extracted_bib
{i-causalityM. Paw\l owski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. \.{Zukowski, Information causality as a physical principle, Nature 461, 1101-1104 (2009).
1412.1079.extracted_bib
{Colbeck-RennerR. Colbeck and R. Renner, No extension of quantum theory can have improved predictive power, Nature Commun. 2, 411 (2011).
1412.1079.extracted_bib
{pureEE1C.H. Bennett, H. Bernstein, S. Popescu, and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046-2062 (1996).
1412.1079.extracted_bib
{ThewTomoR.T. Thew, K. Nemoto, A.G. White, and W.J. Munro, Qudit quantum-state tomography, Phys. Rev. A 66, 012303 (2002).
1412.1079.extracted_bib
{ICO-BuschP. Busch, Informationally complete set of physical quantities, Int. J. Theor. Phys. 30, 1217-1227 (1991).
1412.1079.extracted_bib
{ICMJ.M. Renes, R. Blume-Kohout, A.J. Scott, and C.M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171-2180 (2004).
1412.1079.extracted_bib
{pureEE2S. Popescu and D. Rohrlich, Thermodynamics and the measure of entanglement, Phys. Rev. A 56, R3319-R3321 (1997).
1412.1079.extracted_bib
{icQFTZ.-B. Chen, An informationally-complete unification of quantum spacetime and matter, arXiv:1412.3662.
1412.1079.extracted_bib
{ThiemannT. Thiemann, Lectures on loop quantum gravity, Lecture Notes in Physics 631, 41-135 (2003).
1412.1079.extracted_bib
{SchmidtJ. Audretsch, Entangled Systems: New Directions in Quantum Physics (Wiley-VCH Verlag, Weinheim, 2007) pp.~149-150.
1412.1079.extracted_bib
{quanA1C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442, 593-619 (1995).
1412.1079.extracted_bib
{quanA2C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77, 3288-3291 (1996).
1412.1079.extracted_bib
{quanA3A. Ashtekar, J. Baez, A. Corichi, and A. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80, 904-907 (1998).
1412.1079.extracted_bib
{tHooftG. 't Hooft, Dimensional reduction in quantum gravity, Prepint at %TCIMACRO{\TEXTsymbol{<% %BeginExpansion $<$% %EndExpansion http://arXiv.org/gr-qc/9310026% %TCIMACRO{\TEXTsymbol{> % %BeginExpansion $>$ %EndExpansion (1993).
1412.1079.extracted_bib
{SusskindL. Susskind, The world as a hologram, J. Math. Phys. 36, 6377-6396 (1995).
1412.1079.extracted_bib
{holoRMPR. Bousso, The holographic principle, Rev. Mod. Phys. 74, 825-874 (2002).
1412.1079.extracted_bib
{SciAmS. Aaronson, The limits of quantum computers, Sci. Am. 298, 62-69 (March, 2012).
1412.1079.extracted_bib
{arrow-MacconeL. Maccone, Quantum solution to the arrow-of-time dilemma, Phys. Rev. Lett. 103, 080401 (2009).
1412.1079.extracted_bib
{arrow-TerryD. Jennings and T. Rudolph, Entanglement and the thermodynamic arrow of time, Phys. Rev. E 81, 061130 (2010).
1412.1079.extracted_bib
{HartleJ.B. Hartle, The quantum mechanical arrows of time, in Fundamental Aspects of Quantum Theory: A Two-Time Winner, edited by D. Struppa and J. Tollaksen (Springer, Milan, 2013).
1412.1079.extracted_bib
{BekensteinJ.D. Bekenstein, Black holes and the second law, Nuovo Cimento Lett. 4, 737-740 (1972).
1412.1079.extracted_bib
{BekensteinPRDJ.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333-2346 (1973).
1412.1079.extracted_bib
{HawkingS.W. Hawking, Black hole explosions, Nature 248, 30-31 (1974).
1412.1079.extracted_bib
{DiosiL. Di\'{osi, A universal master equation for the gravitational violation of quantum mechanics, Phys. Lett. A 120, 377-381 (1987).
1412.1079.extracted_bib
{PenroseR. Penrose, On Gravity's role in Quantum State Reduction, Gen. Rel. Grav. 28, 581-600 (1996).
1412.1079.extracted_bib
{BB84C.H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in Proceedings of the IEEE International Conference on Computers, Systems and Singal Proceeding, Bangalore, India (IEEE, New York, 1984), p.175.
1412.1079.extracted_bib
{teleport-PRLC.H. Bennett, G. Brassard, C. Cr\'{epeau, R. Jozsa, A. Peres, and W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 73, 3081-3084 (1993).
1412.1079.extracted_bib
{WheelerJ.A. Wheeler, Recent thinking about the nature of the physical world: It from bit, Ann. N.Y. Acad. Sci. 655, 349-364 (1992). \ \