aa r X i v : . [ qu a n t - ph ] J a n The Informationally-Complete Quantum Theory
Zeng-Bing Chen
National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China
Quantum mechanics is a cornerstone of our current understanding of nature and extremelysuccessful in describing physics covering a huge range of scales. However, its interpretation remainscontroversial since the early days of quantum mechanics. What does a quantum state reallymean? Is there any way out of the so-called quantum measurement problem? Here we presentan informationally-complete quantum theory (ICQT) and the trinary property of nature to beat theabove problems. We assume that a quantum system’s state provides an informationally-completedescription of the system in the trinary picture. We give a consistent formalism of quantumtheory that makes the informational completeness explicitly and argue that conventional quantummechanics is an approximation of the ICQT. We then show how our ICQT provides a coherentpicture and fresh angle of some existing problems in physics. The computational content of ourtheory is uncovered by defining an informationally-complete quantum computer.
I. INTRODUCTION
The unease of understanding quantum theory (QT)began at the very beginning of its establishment. Thefamous Bohr-Einstein debate [1, 2] inspired a livelycontroversy on quantum foundations. QT is surelyan empirically successful theory, with huge applicationsranging from subatomic world to cosmology. However,why does it attract such a heated debate over itswhole history? The controversial issues on quantumfoundations mainly focus on two aspects: ( Q
1) Whatdoes a wave function (or a quantum state) really mean?( Q
2) Is the so-called quantum measurement problem [3–8] really a problem? The first axiom of the standardQT states that a system’s wave function provides acomplete description of the system. But accepting thewave function as QT’s central entity, what is the physicalmeaning of the wave function itself? In this regard,there are two alternatives that the quantum state mightbe either a state about an experimenter’s knowledge orinformation about some aspect of reality (an “epistemic”viewpoint), or a state of physical reality (an “ontic”viewpoint). A recent result [9] on this issue seems tosupport the reality of quantum states, yet with ongoingcontroversy [10, 11].On the other hand, the quantum measurement prob-lem is perhaps the most controversial one on quantumfoundations. According to the orthodox interpretation(namely, the Copenhagen interpretation [4]) of QT, aquantum system in a superposition of different statesevolves deterministically according to the Schr¨odingerequation, but actual measurements always collapse, in atruly random way, the system into a definite state, witha probability determined by the probability amplitudeaccording to the Born rule. When, where, and how thequantum state really collapses are out of the reach of QTas it is either “uninteresting or unscientific to discussreality before measurement” [11].Our classical world view implies that there exists aworld that is objective and independent of any observa-tions. By sharp contrast, what is observed on a quantum system is dependent upon the choice of experimentalarrangements; mutually exclusive (or complementary)properties cannot be measured accurately at the sametime, a fact known as the complementarity principle. Inparticular, which type of measurements one would liketo choose is totally a free will [12] or a freedom of choice [13–15]. Such a freedom of choice underlies the Pusey-Barrett-Rudolph theorem [9] and the derivation of Bell’sinequalities [13–16]. However, one could ask: What doesa free will or a freedom of choice really mean and whosefree will or freedom of choice?Thus, in the orthodox interpretation classical conceptsare necessary for the description of measurements (whichtype of measurements to choose and the particularmeasurement results for chosen measurement) in QT,although the measurement apparatus can indeed be de-scribed quantum mechanically, as done by von Neumann[17, 18]. Seen from its very structure, quantum mechanics“contains classical mechanics as a limiting case, yetat the same time it requires this limiting case for itsown formulation” [19]. In this sense current QT hasa classical-quantum hybrid feature. At a cosmologicalscale, the orthodox interpretation rules out the possibilityof assigning a wave function to the whole Universe, as noexternal observer could exist to measure the Universe.Facing with the interpretational difficulties, variousinterpretations on QT were proposed by many brilliantthoughts, such as the hidden-variable theory [13, 20](initiated by the famous Einstein-Podolsky-Rosen paper[1] questioning the completeness of QT), many-worldsinterpretation [21, 22], the relational interpretation [23,24], and the decoherence theory [5], to mention a few.Thus, “questions concerning the foundations of quantummechanics have been picked over so thoroughly that littlemeat is left” [11]. The discovery of Bell’s inequalities [16](recently questioned from the many-worlds interpretation[22]) and the emerging field of quantum information[25] might be among a few exceptions. The recentdevelopment of quantum information science sparksthe information-theoretical understanding of quantumformalism [26–29].Inspired by the classical-quantum hybrid feature ofcurrent QT and the above-mentioned interpretationalprogresses, here we present an informationally-completequantum theory (ICQT) by removing any classicalconcepts in our description of nature. The ICQT is basedon the informational completeness principle : A quantumsystem’s state provides an informationally-completedescription of the system. In other words, quantumstates represent an informationally-complete code of anypossible information that one might access. CurrentQT is not informationally-complete and thus suffersfrom interpretational difficulties. After working out theinformational completeness explicitly in our formalism,we show that informationally-complete physical systemsare characterized by dual entanglement pattern, theemergent dual Born rule and dual dynamics. Thecomputational content of our theory is uncovered bydefining an informationally-complete quantum computerwith potential of outperforming conventional quantumcomputers. Moreover, we consider the possible concep-tual applications of our theory, hoping to shed new lighton some existing problems in physics.
II. AN INFORMATIONALLY COMPLETEDESCRIPTION FOR FINITE-DIMENSIONALSYSTEMS
The orthodox quantum measurement theory [3–8] wasproposed by von Neumann and can be summarizedas follows. For an unknown d -dimensional quantumstate | ψ i S of a quantum system S to be measured,a measurement apparatus (“a pointer”) A is coupledto the system via a unitary operator ˆ U SA (ˆ s, ˆ p ). Hereˆ s is system’s observable whose eigenstate with respectto the eigenvalue s j reads | j, Si , namely, ˆ s | j, Si = s j | j, Si ( j = 1 , , ...d ); ˆ p is the momentum operatorwhich shifts pointer’s ˆ q -reading ([ˆ q, ˆ p ] = i ). Assumingthat the pointer is initialized in a “ready” state | , Ai and expanding | ψ, Si in terms of | j, Si as | ψ, Si = P j c j | j, Si , then the system and the apparatus aremapped intoˆ U SA (ˆ s, ˆ p ) | ψ, Si | , Ai = X j c j | j, Si | q j , Ai . (1)To ideally measure ˆ s , one has to assume that A musthave at least d macroscopically distinguishable pointerpositions (plus the ready position corresponding to | , Ai ), and the pointer state | q j , Ai and the measuredstates | j, Si have an one-to-one correspondence. Theabove is the usual pre-measurement progress. Theorthodox interpretation of the measurement can onlypredict the collapse of a definite state | j, Si with aprobability | c j | given by the probability amplitude c j ;the collapse occurs in a truly random way. For latterconvenience, we call (ˆ s, ˆ p ) as an observable pair. It isinteresting to note that a factorizable structure of the“measurement operation” ˆ U SA (ˆ s, ˆ p ) was discovered in the context of the dynamical approach to the quantummeasurement problem [6, 7].To avoid the quantum measurement problem, herewe take a key step by assuming explicitly informationalcompleteness, whose meaning will be clear below, inour formalism of describing nature. For measuringinformation on S , one could of course choose variousbases, namely, entangle S and A in different bases. Allinformation, including the basis information, must beencoded by certain quantum system in the ICQT to avoidany classical terms or concepts . To this end, startingfrom a separable state | ψ, Si | φ, Ai , we introduce thethird system, called the “programming system” ( P )hereafter to encode the basis information of S and A .We assume that P has D P dimensions spanned by D P orthogonal states, called programming states | r, Pi ( r = 0 , , ...D P − D P is to be determinedby informational completeness. Let us define a unitaryprogramming operationˆ U P ( SA ) = D P − X r =0 | r, Pi h r, P| ˆ U SA (ˆ s r , ˆ p r ) , (2)which means that if P is in | r, Pi , then do a unitarymeasurement operation ˆ U SA (ˆ s r , ˆ p r ) on SA . Nowsuppose that P is prepared in an initial state | χ, Pi = P r g r | r, Pi . Then the state of the whole system PSA after the programming operation reads |P ( SA ) i = D P − X r =0 g r | r, Pi | r, SAi , (3)where | r, SAi = ˆ U SA (ˆ s r , ˆ p r ) | ψ, Si | φ, Ai encodes theprogrammed entanglement, if any, between S and A . Fora given | r, Pi , the pair observables [denoted by (ˆ s r , ˆ p r )]and their information to be measured is determined bythe Schmidt form of | r, SAi . Note that |P ( SA ) i canalso be written in a Schmidt form with positive realcoefficients [30]. Hereafter we suppose that the Schmidtdecomposition of |P ( SA ) i has been done such that g r >
0. Now the key point of our formalism is to requirethat the programming system P encodes all possible,namely, informationally complete, measurement oper-ations that are allowed to act upon the SA -system.To be “informationally complete”, all programmedmeasurement operations ˆ U SA (ˆ s r , ˆ p r ) can at least achievethe measurements of a complete set of operators for S ; for the d -dimensional system, the complete sethas d operators [31], i.e., the minimal D P = d .Note that informationally complete set of operatorsor measurement is also important for quantum statetomography [32, 33].Another trick in the above discussion is that,to enable the informationally complete programmedmeasurements, it seems that one needs D P differentmeasurement apparatuses. Hereafter we take a stepfurther by dropping this specific measurement modelby regarding the A -system as a single system (notnecessarily having ˆ p and ˆ q as in a specific model thatwe considered above) with D A ( ≥ d ) dimensions and assuch, the standard pre-measurement process describedabove is simply to entangle S and A (see the next Sectionfor further discussion). In this case we have D P ≥ dD A .The step is necessary for seeking a model-independentand intrinsic description of the whole system PSA .To have an easy understanding of our informationally-complete description of physical systems, some remarksare necessary. First, we note that the third system isalso included in other interpretations of QT, such asthe many-worlds interpretation [21, 22], the relationalinterpretation [23, 24], and the “objective quantummeasurement” [34]. However, the third system in ourformalism plays a role that is dramatically different fromthose interpretations. Actually, imposing informationalcompleteness into our quantum description of naturedistinguishes our theory from all previous interpretationsof QT. Second, the fact that | r, SAi , as entangled, canalways be written in a Schmidt form implies a symmetricrole played by S and A . Meanwhile, the role of P isdramatically different from that of either S or A . But P and the combined system SA play a symmetric role.We anticipate that such a feature could have profoundconsequences, particularly for the internal consistency ofthe theory. We will find that this is indeed the case whenwe consider the dynamics within the ICQT. III. ENTANGLEMENT IS MEASUREMENT:THE EMERGENT DUAL BORN RULE
How to acquire information and which kind ofinformation to acquire are two questions of paramountimportance. According to the ICQT, on one hand,the only way to acquire information is to interact (i.e.,entangle) the system S and the apparatus A with eachother; no interaction leads to no entanglement and thusno information. This is in a similar spirit as the relationalinterpretation [23, 24], which treats the quantum stateas being observer-dependent, namely, the state is therelation between the observer and the system. On theother hand, the programming system P , by interactingwith SA , dictates the way (actually, the informational-complete way) on which kind of information to acquireabout the system S . For instance, if the whole systemis programmed to measure ˆ s r , then S and A interactwith each other to induce the programmed measurementoperations ˆ U SA (ˆ s r , ˆ p r ). This process generates theentangled state | r, SAi with which A “knows”, in acompletely coherent way, all information about S in thebasis of ˆ s r ; the amount of entanglement contained in | r, SAi quantifies the amount of information acquiredduring this measurement. Also, P “knows”, againin a completely coherent way, the information aboutwhich kind of information (here | r, SAi ) A has about S ; the amount of the P -( SA ) entanglement quantifies the amount of information on which kind of measurements todo. All information is coherently and completely encodedthere by certain quantum system.Thus, for any given system S in our description one hasto ask two questions: How S gets entangled with anothermeasurement system A and how many independentways can it be entangled with A ? The answer to thelatter question is completely contained in the entangledstate |P ( SA ) i , while the answer to the former is theprogrammed entanglement | r, SAi —the two questionsare answered by entanglement at two different levels,called dual entanglement .Now let us state a key point in our ICQT. Namely, entanglement, necessary and sufficient for acquiringinformation, is the measurement and the physical pre-dictions of the theory as any possible information iscompletely encoded in the particular dual entanglementstructure of the whole system. To see this, let us notethat the reduced density operators for P and SA read ρ P = tr SA [ |P ( SA ) i hP ( SA ) | ] = D P − X r =0 g r | r, Pi h r, P| ,ρ SA = tr P [ |P ( SA ) i hP ( SA ) | ] = D P − X r =0 g r | r, SAi h r, SA| , (4)implying that all information about P ( SA ) is completelycontained in the set { g r , | r, Pi} ( { g r , | r, SAi} ), thephysical predictions of the theory. Yet, all thesephysical predictions are already encoded completely inthe P -( SA ) entanglement. In other words, the P -( SA ) entanglement is sufficient to predict { g r , | r, Pi} and { g r , | r, SAi} , a task that we could expect for a mea-surement. Similar analysis applies to the programmedentanglement | r, SAi as well.As both | r, SAi and |P ( SA ) i are pure states, theirentanglement is uniquely quantified by the usual entan-glement entropy [30, 35]; the P -( SA ) entanglement ismaximally ln D P and the S - A entanglement containedin | r, SAi is maximally ln D S . This immediatelyidentifies each of the squared coefficients of theirSchmidt decompositions as a probability to reconcile withShannon’s definition of entropy. Put differently, in ourinformationally-complete description of physical systems,entanglement does be all the information; classicalterms like probability arise in our description becauseof either our reliance on classical concept of informationor certain approximate and incomplete description tobe shown below. By regarding entanglement directlyas measurement of complete information, one can avoidthe classical-quantum hybrid feature of current quantumtheory or any classical concepts having to use therein.The status of quantum states (more precisely, dualentanglement) in the informational completeness for-malism thus represents a complete reality of the wholesystem ( P , S , and A , the trinity). Such a realitypicture (“quantum reality”) is only possible by takinginto account the informational completeness explicitlyin our formalism. Quantum states do exist in a worldthat is informational and objective. Whatever anobservation might be, informationally-complete statesalways encode information pertaining to that observationas programmed, without invoking observers or havingto appeal to any mysterious mechanisms to account forwave function collapse; there is simply no wave functioncollapse. Here local quantum states (i.e., states for eachof P , S , and A ) are all relative, but information encodedin dual entanglement is invariant under the changes oflocal bases, a basic property of entanglement. If onelikes, the choice of local bases can be called a free willor freedom of choice, corresponding to certain “gauge”.Yet, all physical predictions of the theory are encodedin dual entanglement and do not depend on the chosengauge.In certain sense, P and A act like a “quantum being”(“qubeing”) who holds coherently all the informationally-complete programmes on how to entangle S and A .In this way, the qubeing has all the information about S . However, our human beings, unlike the qubeing,only have limited ability to acquire information, withlimited precisions, limited degrees of freedom, limitedinformation detection and storage, and so on; or simplywe are so used to and familiar with classical conceptson information and physical systems . For example, anexperimenter, Alice, together with her apparatus, wouldlike to acquire information about | ψ, Si . First of all, shehas to decide which kind of information she would liketo know. After making a decision, she needs then to observe (that is, to interact with) her apparatus readilyentangled with S . In principle, Alice’s decisions andobservations are all physical processes which should bedescribed quantum mechanically. Nevertheless, Aliceand her apparatus are macroscopic and have so manyquantum degrees of freedom and limited ability (lack offull knowledge of the entire system). In this case, shehas to “trace out” those quantum degrees of freedominvolved in her decisions (interaction with P ), leading toa mixed state P r | g r | | r, Pi | r, SAi h r, SA| h r, P| . Thisstate allows a probability interpretation about Alice’sfreedom of choice: Each of her decisions | r, Pi occurswith a probability of | g r | . As far as a particular choice | r, Pi has been made, again she has to trace out herquantum degrees of freedom involved in her observation(interaction with A ). This then leads to the usual Bornrule about | ψ, Si for the given measurement. Thus, in theICQT, the Born rule, also in dual form, is an emergentor derived rule determined by the dual entanglementstructure.To summarize the above picture, the world view ofthe ICQT is fascinating. If we regard the system S asan indivisible part of the qubeing PA , the whole system PSA then represents an informationally complete andobjective entity; it seems that the qubeing has its own“consciousness”, a kind of miraculous quantum ability,to encode and access all its information in the form of dual entanglement, in which the constituent partsof the qubeing are mutually measured or defined. Inother words, for the qubeing all information (namely,all physical predictions) is encoded in dual entanglementvia interaction and not obtained via the usual quantummeasurement with the unavoidable concept of the wavefunction collapse.The trinary picture (the division of S , A , and P )of physical systems arises here as a new feature of theICQT, as shown in Fig. 1. To retain the informationallycomplete description of nature, such a trinary pictureseems to be unavoidable. The limitation of informationalcontents in dual entanglement could be tentativelycalled “the trinity principle”, instead of the conventionalcomplementarity principle, to put the trinary property ofphysical systems on the most fundamental ground.The loss of the trinary picture of describing physicalsystems leads to the emergent dual Born rule , i.e., theprobability description on which kind of observables tomeasure and then on which eigenvalue of the observableto measure, due to, e.g., lack of full knowledge ofthe entire system in our ICQT. The conventional vonNeumann entropy quantifies this dual loss of information.In other words, the conventional Born rule arises as aconsequence of the sacrifice of informationally-completedescription in the trinary picture; the sacrifice leadsto a partial reality of physical system as described byconventional QT. IV. THE INFORMATIONALLY-COMPLETEDYNAMICS
According to the above picture of nature, singlefree systems are simply meaningless for acquiringinformation; a system, which does not give information to(i.e., interact with) other systems in any way, simply doesnot exist. The “ S + A ” description in the usual QT is alsoinadequate because of its informational incompleteness.Therefore, the dynamics of the ICQT will be dramaticallydifferent from the usual picture as it requires interacting P + SA so as to obey the informational completenessprinciple within the trinary picture. Without specifyingthe P + SA dynamics to maintain the informationally-complete trinary description, it is meaningless or infor-mationally incomplete to specify local states of singlesystems in P + SA . Namely, the ICQT is characterized bythe indivisibility of its kinematics and dynamics . Below,we give some key features of the informationally-completedynamics.Before considering the informationally complete dy-namics, let us introduce an important concept of dual measurability : the P - SA measurability and theprogrammed SA | P measurability. The former means theability of measuring P with SA and vice versa; the lattermeans the ability of measuring A with S and vice versa,under a given programmed measurement operation of P .The P - SA measurability (the programmed measurability P A S U P ( ) SA U P ( ) SA ^^ FIG. 1: The trinary picture of the world. The divisionof system S , measurement apparatus A , and programmingsystem P naturally arises in the informationally completedescription of physical systems. “The Taiji pattern” showsin an intuitive manner the S - A interaction (entanglement),while the green discs inside and outside the Taiji patternrepresent the programmed measurement operations ˆ U P ( SA ) between P and SA . In ancient China, Taoists regarded theTaiji pattern as a “diagram of the Universe”. The trinarypicture of the world shown here is ubiquitous in the sensethat the world, at the most fundamental level, is made upof a trinity: gravity (i.e., spacetime, P ), elementary matterfermions ( S ) and their gauge fields ( A ); the trinity should bedescribable by the ICQT. SA | P ) leads to D P = D A D S ( D A = D S = D ) andthus a symmetric role between P and SA ( S and A ).Note that here measurability does not means certainvon Neumann measurement actually performed by anexperimenter in the usual sense. As we pointed outabove, in the ICQT entanglement is the measurement.Thus, dual measurability is simply another side of dualentanglement.After the above preparation, now we give a definitionof informationally-complete physical systems: A physicalsystem is said to be informationally-complete if and onlyif (the use of “if and only if” will be explained below)it is consisted of S , A and P described as a trinitysuch that the P - SA measurability and the programmedmeasurability SA | P are satisfied. As a result of thisdefinition, the P - SA measurability implies the existenceof at most D independent informationally completemeasurement (entanglement) operations in the Hilbertspaces of both P and SA ; these operations generate atmost D -dimensional entangled states between P and SA and at most D entangled states between S and A suchthat P and SA are mutually measuring and defining.Meanwhile, under the given programming state of P , theprogrammed measurability SA | P implies the existenceof at most D -dimensional entangled states between S and A such that S and A are mutually measuring anddefining, as programmed. For convenience, we also call the complete set of states and operators defined in theHilbert space of P or SA are informationally complete.Accordingly, states and operators for either S or A aloneare informationally incomplete. Thus, in the ICQT therole of observables defined for either S or A is quitedifferent from the role of observables defined for P .Let us suppose that the informationally-completesystem PSA has a general Hamiltonian ˆ H PSA . Weassume that the whole system evolutes according toa Schr¨odinger-like equation (we take ~ = 1), namely, i ddt |P ( SA ) , t i = ˆ H PSA |PSA , t i . In general, ˆ H PSA =ˆ H P + ˆ H S + ˆ H A + ˆ H PS + ˆ H PA + ˆ H SA + ˆ H P ( SA ) , wherethe subscripts label the corresponding systems. Nowour problem is to determine how the informationalcompleteness principle constrains the form of ˆ H PSA andthus the dynamics of the
PSA -system.Note that we can choose an orthonormal basis (the“programming basis”) {| e n , Pi ; n = 0 , , ..D P − } tospan the whole Hilbert space of P . We associate eachprogramming state | e n , Pi as an eigenstate of P -system’s“programming observable” ˆ e P with eigenvalue e n . It iseasy to verify that the following Hamiltonian obeys theinformational completeness principle:ˆ I PSA = ˆ H P + ˆ H P ( SA ) = ˆ H P + D P − X n =0 | e n , Pi h e n , P| ˆ H SA| P ( e n , t ) , (5)if we impose the P - SA measurability condition as follows[ ˆ H P ( SA ) , ˆ e P ] = [ ˆ H P , ˆ e P ] = 0 , (6)namely, ˆ e P commutes with both ˆ H P ( SA ) and ˆ H P . If onetakes ˆ e P = ˆ H P , Eq. (6) reads simply as[ ˆ H P ( SA ) , ˆ H P ] = 0 . (7)In Eq. (5) ˆ H SA| P ( e n , t ) = i ddt ˆ U SA| P ( e n , t ) · ˆ U − SA| P ( e n , t ) is an informationally complete operator setof SA . Meanwhile, ˆ I PSA takes a form like ˆ I PSA = ˆ H P +ˆ H PS + ˆ H PA + ˆ H P ( SA ) such that P universally coupleswith S and A . For this purpose, it is advantageousto single out the “empty” (or “absent”) states | , Pi , | , Ai , and | , Si from the state spaces of the trinity.For instance, | , Si means the absence of S . In thisway, e.g., ˆ H PS = P D P − n =0 | e n , Pi h e n , P| ⊗ | , Ai h , A| ⊗ ˆ H S| P ( e n , t ), where | e , Pi ≡ | , Pi . Thus, by introducingthese absent states as legal states for the trinity, ˆ I PSA can be written in a compact form as in Eq. (5). Notethat in ˆ I PSA , Hamiltonians ˆ H S , ˆ H A , and ˆ H SA donot appear in ˆ I PSA . These “local” Hamiltonians ( ˆ H S ,ˆ H A , and ˆ H SA ) decoupled from P induce local unitarytransformations upon S or/and A , corresponding tocertain gauge, which can be gauged out in physicalpredictions encoded in dual entanglement.ˆ I PSA induces an evolution |P ( SA ) , t i =ˆ U PSA |P , t = 0 i |SA , t = 0 i ; the evolution operatorˆ U PSA always has a factorizable structureˆ U PSA ( t ) = D P − X n =0 | e n , Pi h e n , P| ˆ U P ( t ) ˆ U SA| P ( e n , t ) , (8)as a result of the P - SA measurability condition (6), suchthat ( ∀ e n ) i ddt ˆ U P ( t ) = ˆ H P ˆ U P ( t ) ,i ddt ˆ U SA| P ( e n , t ) = ˆ H SA| P ( e n , t ) ˆ U SA| P ( e n , t ) . (9)In this way, the dynamical evolutions of P and SA aremutually defined, in accordance with the informationalcompleteness principle.The Hamiltonian ˆ I PSA as given above respects the P - SA measurability. If we also require the programmedmeasurability SA | P ( ∀ e n ), the evolution governed byˆ H SA| P ( e n , t ) depends on which system ( S or A ) definesthe programming observable. For example, if one can findthe programming observable ˆ ε S| P for S , the programmedevolution for SA will similarly acquire the factorizablestructure asˆ H SA| P ( e n ) = D X i =0 | ε i ( e n ) , e n , Si h ε i ( e n ) , e n , S|× ˆ H A [ e n , ε i ( e n ) , t ] + ˆ H S| P ( e n , t ) (10)with ˆ H A = i ddt ˆ U A [ e n , ε i ( e n ) , t ] · ˆ U − A [ e n , ε i ( e n ) , t ]. Herethe orthonormal basis for S | P is {| ε i ( e n ) , e n , Si} , where | ε i ( e n ) , e n , Si is an eigenstate of ˆ ε S| P ( e n ) with eigenvalue ε i ( e n ) for given e n . Similarly to the P - SA measurabilitycondition (6), we need to impose the programmedmeasurability SA | P condition ( ∀ e n )[ ˆ H SA| P ( e n ) , ˆ ε S| P ( e n )] = [ˆ ε S| P ( e n ) , ˆ H S| P ( e n , t )] = 0 . (11)The SA | P dynamics is then similar to the P - SA dynamics considered above. Such a dual dynamics ofthe whole system PSA is an attribute of the trinarydescription and quite distinct to the usual Schr¨odingerevolution.What is the physical significance of the programmingbasis {| e n , Pi} and the associated observable ˆ e P ?Actually it is physically transparent that {| e n , Pi} as thephysical predictions can be identified with the Schmidtbasis for the P - SA decomposition, which is not affectedby the local transformations generated by ˆ H P . Amore interesting possibility is to interpret ˆ e P as aquantum nondemolition observable [38, 39]; note thatthe pre-measurement involves a nondemolition couplingbetween S and A [34]. Then the P - SA measurabilitycondition (6) is a (sufficient) condition for a quantumnondemolition measurement of ˆ e P . In the context of quantum nondemolition observable, a more generalmeasurability condition could be imposed. For instance,the P - SA measurability condition in Eq. (6) might bereplaced by[ ˆ U P ( t ) , ˆ e P ] |P ( SA ) , t i = [ ˆ U PSA ( t ) , ˆ e P ] |P ( SA ) , t i = 0 . (12)As we require that the evolution |P ( SA ) , t i =ˆ U PSA |P , t = 0 i |SA , t = 0 i results in a state already inthe Schmidt form |P ( SA ) , t i = D P − X r =0 g r | e n , Pi | e n , SAi , | e n , SAi ≡ ˆ U SA| P ( e n , t ) |SA , t = 0 i , (13)it is easy to prove that {| e n , SAi} forms an orthonormalbasis and ˆ U P ( t ) |P , t = 0 i = D P − X r =0 g r | e n , Pi . (14)Equation (14) looks as if the single system P brings theproterties { g r , | e n , Pi} of the whole P - SA system. Thisis of course not true as the Schmidt basis is the jointproperties of the whole system. The specific form ofˆ U P ( t ) |P , t = 0 i stems from the specific choice of ˆ e P inthe Schmidt basis of |P ( SA ) , t i . Actually, in this casewe can choose ˆ e P = ρ P .When applying the ICQT to quantum gravity coupledwith matter quantum fields [36], the usual Schr¨odingerequation becomes a constraint ˆ H PSA |P ( SA ) , t i = 0,known as the Wheeler-DeWitt equation. In this case, theprogramming observable ˆ e P then corresponds to a Diracobservable [24, 37]. In this field-theoretical case, thesignificance and necessity of the informationally completedynamics in trinity is physically more transparent . Forinstance, if P is the quantized gravitational field, then thelocal Hamiltonians ˆ H S , ˆ H A , and ˆ H SA are simply ruledout as the gravitational field universally couples with anyform of matter.To end this section, it is important to note that thedistinguished roles of P and SA are relative. Dependingon the specific form of the trinary Hamiltonian, we couldhave another possibility that SA can programme theevolution of P . In this case, the programming basisis chosen for SA and associated with an observablebeing commutative with ˆ H SA such that the P - SA measurability and a similar dynamics as in Eq. (9) canstill be obtained. Furthermore, the roles of P and SA are actually symmetric due to a nice property of theSchmidt decomposition [40], in which if an orthonormalbasis labelled by an index (e.g., n ) is chosen for a system,then the orthonormal basis for another system, by actinga unitary transformation upon it, is labelled by thesame index. This property implies that both of thebases are already the Schmidt bases, up to local unitarytransformations. A similar consideration is applicable tothe programmed measurability SA | P , too. V. RELATION WITH CONVENTIONALQUANTUM THEORY
What is the relation between the ICQT and the usualQT? Before answering this question, first of all we haveto ask ourselves: Why are we bothered to revise theconventional QT into the current formulation of such astrange appearance? Here we must introduce at the verybeginning the interacting/entangling P + SA trinity witha dynamical evolution (being always unitary) determinedby the informational completeness principle such that thephysical properties of, e.g., P and SA are coherently andcompletely stored in the P - SA entanglement and canonly be predicted conditionally on each other. Anyway,in non-relativistic quantum mechanics there seems tobe no physical motivation to introduce P . This is insharp contrast to traditional QT, where the “ S + A ”description is sufficient and isolated, single systems(free particles, free quantum fields, and so on) canhave certain physical properties which can be accessedby a mysterious and non-unitary measurement process.However, one of the most important lessons learned fromgeneral relativity is that spacetime is dynamical and thesame thing as gravity. Therefore, there are, even inprinciple, no perfectly isolated systems as they must livein and couple with dynamical spacetime. If we quantizeevery thing of nature, even gravity (spacetime), whichmechanism could trigger a non-unitary measurementprocess? Of course, one could simply ignore, as anapproximation, the dynamical and quantum nature ofspacetime as the common wisdom does. Then, whysuch an ignorance could be a safe approximation withoutcausing any internal inconsistency or incompletenessof traditional QT? In any case, spacetime is such anelementary physical entity. Anyone who does not shuteyes to these problems, among other interpretationaldifficulties, has to conclude that traditional QT mustbe incomplete if its consistency is trustworthy; theinformational completeness principle is a possible remedyto complete current quantum formalism, as we suggest.Ultimately, we should describe nature with quantumfield theory. An informationally-complete quantum fieldtheory [36] can indeed be formulated; the ICQT de-veloped here for finite-dimensional quantum mechanicalsystems is thus the conceptual preparation and themathematical formulation for that purpose. Therein,if we regard S as particle (i.e., matter fermion) fieldsand A as their gauge fields, and SA together as matterfields, then we immediately recognize that system P must be the gravitational field (i.e., spacetime), nothingelse, as only gravitational field, while self-interacting,universally interacts with all other fields. Recall thatin our P + SA trinity, P must universally couple with S and A . If we think this way, an amazing picture(Fig. 1) of our world arises: The gravitational field andmatter fields are mutually defined and entangled—nomatter, no gravity (spacetime) and vice versa, and foreach of their entangled patterns, matter fermion fields and their gauge fields are likewise mutually defined andentangled. If this is indeed what our nature works to obeythe informational completeness principle guaranteeingthe completeness of the theory from the outset, theconventional QT will be an approximation of our ICQTwhen we ignore quantum effects of nature’s programmingsystem, i.e., gravity. Under such an approximationthe ICQT reduces to conventional QT, characterizedby the usual Schr¨odinger equation and the probabilitydescription, and as such, QT in its current form is thusinformationally incomplete. This is in the exact sensethat classical Newtonian mechanics is an approximatetheory of special relativity when a physical system has aspeed much less than the speed of light.On the other hand, no matter how weak gravityis, it is forced to be there by the informationalcompleteness principle, to play a role for completing aconsistent quantum theory. This unique role of gravity(or spacetime) in our theory is consistent with theremarkable fact that only gravity is universally coupledto all other physical fields (particle fields and gaugefields). Of course, our current quantum descriptionis an extremely good approximation. But for scalesnear the Planck one and for early Universe, quantizedspacetime acts as the programming system and the ICQTwill be necessary. Thus, both facts (i.e., current QTworks so well and quantum gravity effects are so weakat normal scales) hide so deeply any new theoreticalarchitecture beyond current QT, like the ICQT. Even inthe string theory, there is no change of the underlyingquantum formalism. By contrast, what we suggestwithin the ICQT is that at the level of quantized fieldsincluding quantized spacetime, everything is quantizedand one does not have the usual separation of quantumsystems and observers. In this case, one has to giveup the classical-quantum hybrid feature of current QT.For this purpose, the most obvious way seems to bethe elimination of the measurement postulate in ourfully quantum (namely, not classical-quantum hybrid)description of nature. As we hope to argue, giving upthe classical concepts associated with the measurementpostulate in current QT does not lead to any sacrificeof our predictive power as the complete information isencoded by the dual entanglement structure.If we take the above argument seriously, then the ICQTcaptures the most remarkable trinity of nature , namely,the division of nature by matter fermions, their gaugefields, and gravity (spacetime), though the role of theHiggs field needs a separate consideration. The previoustwo sections argued the necessity of the informationalcompleteness in the trinary description. Here we seethat it is also sufficient: We do not have to invoke moreprogramming systems to program PSA simply because we do not have spacetime (gravity) out of spacetime(gravity)—Trinity is necessary and sufficient . Thiseliminates the von Neumann chain in the usual quantummeasurement model.One of the most challenging problems in currentphysics is how to put QT and general gravity intoa single, consistent theory. To achieve this, it isencouraging to have a quantum formalism like theICQT, in which gravity must be quantized and plays anessential role. As we showed elsewhere [36], following theabove arguments indeed leads to a consistent quantumframework of unifying spacetime (gravity) and matter,without the fundamental inconsistencies [37] betweengravity and conventional quantum field theory, implyingthe conceptual advantages of our theory. For instance,with the theoretical input from loop quantum gravitypredicting the quantized geometry [24, 37, 41–43], theinformationally complete quantum field theory naturallygives the holographic principle [44–46]. Such a stronglimit on the allowed states of the trinary system inany finite spacetime regime, as imposed by the ICQT,paves the way to escape the infrared and ultravioletsingularities (divergences) that occur in conventionalquantum field theory.Thus, the ICQT gives a strong motivation or reasonfor quantizing spacetime/gravity; there is no trinity ifthere is no quantized gravity. The natural positionof gravity in the ICQT cannot be accidental and maybe a strong evidence supporting our informationallycomplete description of nature. It is surprise to seethat nature singles out gravity as a programming field,which plays a role that is definitely different from matterfields. However, quantizing gravity as yet anotherfield, as in conventional quantum field theory, is notsufficient and does not automatically result in a correctand consistent quantum theory of all known forces.Only when the informational completeness in the trinarydescription is integrated into our quantum formulation,can we have the desired theory of the Universe. Thedistinct roles of matter-matter (particles and their gaugefields) entanglement and spacetime-matter (i.e., gravity-matter) entanglement indicate the reason why quantizinggravity as usual quantized fields suffers from well-knownconceptual problems.As an abstract mathematical structure, current QTis content-irrelevant in the following sense. While it isbelieved to be universally applicable to physical systemsof any physical contents, ranging from elementaryparticles and (super)strings to the whole Universe, whatphysical content that it describes does not matter andthe physical content never changes its very structure.The situation for classical mechanics is quite similarin this aspect. However, the ICQT changes this in adramatic way in the sense that the trinary picture ofnature has to be integrated into a consistent formulationto enable an informationally-complete description. Thephysical content that the ICQT describes does matter asthe states and their dynamical evolution of the trinarysystem are constrained or structured into the dual formsspecified above. In particular, the inclusion of theprogramming system, identified with gravity in the field-theoretical case, is very essential and necessary in ourdescription.
VI. INFORMATIONALLY-COMPLETEQUANTUM COMPUTATION
A new theory should make new predictions or/andmotivate new applications. Of course, previous in-terpretations of QT are very important for a betterunderstanding of the theory. However, no interpretationsmake new predictions or/and motivate fundamentallynew applications. Now we argue that our ICQTindeed motivates new applications if we consider itscomputational power. Even though gravity would playcertain role in our future understanding of nature,artificial informationally-complete quantum systems arerealizable as a quite good approximation.What is an informationally complete quantum com-puter (ICQC)? We define the ICQC as an artifi-cial informationally complete quantum systems, or aquantum intelligent system (qubeing), which has aninformationally complete trinary structure consistingof S , A , and P . The ICQC starts from an initialstate | ICQC i = | ψ, Si | φ, Ai | χ, Pi . As usual, the S system has n qubits, and thus dimensions of 2 n . Tobe well defined, we also use qubits to make up the A system and the P system; A ( P ) has n A ( n P )qubits and dimensions of 2 n A (2 n P ). To satisfy theinformational completeness principle, we have n A = n and n P = 2 n . Our ICQC then works by applying certainpatterns of universal quantum logic gates (single-qubitand two-qubit ones), determined by quantum algorithmpertaining to the question under study. Generallyspeaking, as an artificially controllable quantum systemthe patterns of gates are allowed to exhaust all unitaryoperations on the whole PAS system, which we denotecollectively by ˆ V PAS = ˆ U ( P , A , S , AS , PA , PS , PAS ).At the end of running the ICQC, we perform theprogrammed measurement operation ˆ U P ( SA ) on PAS .The resulting final state of the ICQC reads | ICQC i =ˆ U P ( SA ) ˆ V PAS | ICQC i . Here,ˆ U P ( SA ) = n − X p =0 | p, Pi h p, P| ˆ U P ˆ U ( p, A , S ) , (15)where the pair observables defined by ˆ U ( p, A , S ) span acomplete operator set for SA .Is the ICQC defined above a usual quantum computermerely with more ( n + n A + n P = 4 n ) qubits, but withoutthe informational-completeness and trinary structure?The answer is definitely “no” because of the conceptualdifference between the two quantum computing devices.To see this, we prepare each qubit of S in the initialstate | + , Si = √ ( | , Si + | , Si ) such that | ψ, Si is in asuperposition of all 2 n bit-values with equal probabilityamplitude: | ψ, Si = √ n P n − x =0 | x, Si . The initialstates of A and P are likewise prepared: | φ, Ai = √ n P n − x =0 | y, Ai and | χ, Pi = √ n P n − x =0 | z, Pi . Sucha coherent superposition of conventional quantum com-puter’s initial states is believed to be the very reason forthe speedup of quantum algorithms [25, 47]. Now let usmake a further simplification by doing nothing anymoreon A and S , namely, the ICQC only acts ˆ U P ( SA ) on theinitial state as | ICQC , P ( SA ) i = ˆ U P ( SA ) | ψ, Si | φ, Ai | χ, Pi . (16)For such a simplified ICQC, ˆ U P ( SA ) can encode allpossible (i.e., informationally complete) programmedmeasurement operations upon A and S . Theseoperations are actually all allowed quantum algorithmsand their outputs on n -qubit state | ψ, Si , in theterminology of conventional quantum computing. Thenwe immediately see that in the ICQC, one has dualparallelism : Parallelism in initial states as usual and parallelism of programmed operations (algorithms andoutputs). In other words, a single ICQC with n qubitscould compute in parallel all algorithms of usual quantumcomputers with n qubits . Due to this particular dualparallelism enabled by the ICQC, it is reasonable toexpect much higher computational power with the ICQC.Actually, the ICQC is, by definition, the mostpowerful computational machine on qubit systems inthe sense of informational completeness; otherwise itis informationally incomplete. Finding algorithms onthe ICQC to explicitly demonstrate the computationalpower of the ICQC is surely a future interesting problem.Also, computational complexity and error-tolerance inthe ICQC framework are two important issues. If naturedoes use the informational completeness as a guidingprinciple, it computes the world we currently know; sucha world could be simulated and thus comprehensible bythe ICQC (i.e., “qubitization” within an informationally-complete trinary description) in principle. VII. OTHER CONCEPTUAL APPLICATIONS
Below we give, only very briefly, a few conceptualapplications of the informational completeness principleand the ICQT, hoping to shed new light on some long-standing open questions in physics.An important question is how to understand the oc-currence of the classical world surrounding us, includingthe second law of thermodynamics and the arrow of time,in our new framework characterized by the informationalcompleteness principle and the trinary picture of nature.Though we cannot present quantitative analysis of theproblem here, a qualitative and conceptual answer tothe problem is quite transparent: For informationallycomplete quantum systems, interactions lead to P - SA entanglement and the programmed S - A entanglement;the Universe as a whole has an increasing entanglement,a kind of entanglement arrow of time (see also Ref.[36]). It is easy to verify the entanglement creation byconsidering the PSA evolution governed by ˆ I PSA froma separable state. At a thermodynamic/macroscopicscale, tracing out thermodynamically/macroscopically irrelevant degrees of freedom, only as an approximatedescription of the underlying informationally completephysics, leads to the second law of thermodynamics, thearrow of time, and ultimately, the classical world.We note related analysis on the role of entanglementin the thermodynamic arrow of time in the frameworkof conventional [48, 49] or time-neutral formulation [50]of quantum mechanics. As gravity arguably plays anessential role in our informationally complete descriptionof nature, it is intriguing to see that gravity playssome role in the occurrence of the second law ofthermodynamics and the arrow of time, as hinted in thestudy of black-hole thermodynamics [37, 51–53]. In theDi´osi-Penrose model [54, 55], gravity was argued to playcertain role for the wave function collapse.Now let us briefly consider the potential conceptualapplications to cosmology. Obviously, the conceptualdifficulty of applying usual QT to the whole Uni-verse disappears in our ICQT. Actually, the ICQT isinterpretation-free and does not need an observer asthe observer is a part of the Universe; the descriptionof the Universe by the ICQT would give us allinformation as it could be. The constituent parts intrinity are mutually defining and measuring in a specificdual entanglement structure, eliminating any subjectiveaspects regarding the current interpretations of quantumstates—The existence of the Universe does not rely on theexistence of potential observers observing the Universe.Entanglement in the dual form encodes, without relyingon any external observers, all physical information andcan give all physical predictions of the theory.There is no reason why we cannot describe our humanbeings as an informationally complete (classical, butultimately, quantum) system via a trinary description.In this way, some aspects of human beings couldbe comprehensible purely from the informational andphysical point of view. For instance, if we defineAlice’s body and all of her sense organs as A and heroutside world as S , then Alice knows her world or getsknown by her world via interaction (i.e., informationexchange) between S and A . Now an intriguing problemarises here: What is the programming system P inthis context? A straightforward way is simply todefine P (or, the correlations between P and SA ) asthe mind (consciousness). By analogy to the abovequantum trinary description, the mind P and SA aremutually defined in an informationally-complete sense.This prescription thus provides an interesting possibilityof understanding the most mysterious part (namely,consciousness) of human beings from an informationaland physical perspective. Particularly, Alice’s brainundertakes only partial (though the most important)functioning of her mind according to the above definition;the remaining functioning of her mind is distributednonlocally in such a way that enables programmingthe interaction between S and A . Note that, in aninformationally-complete field theory [36], the program-ming system is the quantized spacetime. If we take the0above analogy more seriously, a very strange conclusionseems to be unavoidable: Alice’s mind P should beultimately explainable by the spacetime P , namely, themind is certain (nonlocal) spacetime code of Alice’s SA .The reason behind the conjecture is the strong beliefthat the informational completeness should underlie theworld, ranging from the elementary trinity (elementaryfermions, their gauge fields, and spacetime) to our humanbeings and the whole Universe—actually, everything inthe world is built from the informationally-completeelementary trinity; the matter-spacetime trinity is anindivisible single entity.In certain sense, it seems that human beings workas a quantum-decohering ICQC. Similarly to the factthat an ICQC is conceptually different from a normalquantum computer, a classical computing device with anintegrated trinary structure similar to the ICQC shouldbe quite different from the normal Turing machine andcould be capable of simulating human-like intelligencebetter. Does this mean certain “consciousness comput-ing”, or “intelligence computing”? Further considerationin the context will be given in future.Therefore, it could well be that the informationalcompleteness is of significance in a broader senseand should be a basic requirement for any physicalsystems, classical or quantum. It is in this sensethat the informational completeness deserves to benamed as a principle. It is a missed principle inour current understanding of nature and a rule behindthe comprehensibility of the world—The informationallycomplete world is comprehensible by informationallycomplete human beings. VIII. CONCLUSSIONS AND OUTLOOK
In the present work, we have presented aninterpretation-free QT under the assumption thatquantum states of physical systems represent aninformationally-complete code of any possible informa-tion that one might access. To make the informationalcompleteness explicitly in our formalism, the trinarypicture of describing physical systems seems to benecessary. Physical systems in trinity evaluate and areentangled both in a dual form; quantum entanglementplays a central role in the ICQT—Our world isinformation given in terms of entanglement at the mostfundamental level. So the ICQT modifies two postulates(on quantum states and on dynamics) of current quantummechanics in a fundamental way and eliminates themeasurement postulate from our description; as a resultof the modifications, the observables can be either infor-mationally complete (for P or SA ) or informationallyincomplete (for S or A ) . We give various evidencesand conceptual applications of the ICQT, to arguethat the ICQT, naturally identifying gravity as nature’sprogramming system in the field-theoretic case, might bea candidate theory capable of unifying matter and gravity (spacetime) in an informationally complete quantumframework; for further development on our theory inthe context of quantum gravity coupled with matter,see Ref. [36]. In this sense, the conventional QT will bean approximation of our ICQT when quantum effect ofgravity is ignored. Such an approximation leads to theapproximate Schr¨odinger equation and the probabilitydescription of current QT. This is in the exact sensethat classical Newtonian mechanics is an approximatedescription of relativistic systems. The ICQT motivatesan interesting application to informationally-completequantum computing.As we argued above, current quantum mechanics is not informationally-complete because of its classical-quantum hybrid feature and thus, suffers from interpreta-tional difficulties. The explicit demand of informationalcompleteness not only removes the conceptual problemof our current understanding of quantum mechanics,but also leads to a profound constraint on formulatingquantum theory. Thus, the ICQT should not beunderstood simply as another interpretation of currentQT; rather, it, by giving up the classical concept ofprobability associated with the measurement postulate,generalizes current quantum formalism—the physicalprediction (outcomes of an observable and the cor-responding probabilities) of a quantum measurementin conventional QT is now entailed by entanglement;no entanglement implies no information and thus noprediction. As we noted previously, adding informationalcompleteness requirement into our current quantumformalism leads to serious consequences: Informationalcompleteness not only restricts the way on how todescribe physical systems, but also the way how theyinteract/entangle with each other [56]. This will thusgive a very strong constraint on what physical processescould have happened or be allowed to happen.On one hand, the ICQT provides a coherent conceptualpicture of, or sheds new light on, understanding someproblems or phenomena in current physics, includingthe intrinsic trinity of matter fermions, gauge fieldsand gravity, the occurrence of the classical world, thearrow of time, and the holographic principle. Onthe other hand, some other problems, such as thecomplementarity principle, quantum nonlocality [22]and quantum communication, should be reconsideredfrom the viewpoint of the ICQT. All current quantumcommunication protocols [25, 57, 58] have to make useof classical concepts on information. It is thus veryinteresting to see how to do communication in the ICQTand, particularly, to see whether or not it is possible toachieve unconditionally secure communication.According to the ICQT, the world underlying us is allabout information (entanglement); it is informationallycomplete, deterministic, self-defining, and thus objective.Such a world view (“quantum determinism”) is of coursequite different from what we learn from current quantummechanics, but in some sense, returns to Einstein’s worldview and not surprisingly, represents an embodiment1of Wheeler’s thesis known as “it from bit” [59]. Sucha viewpoint calls for a reconsideration of our currentunderstanding on physical reality, information, spacetime(gravity), and matter, as well as their links. Let uscite the famous Einstein-Podolsky-Rosen paper [1] here:“ While we have thus shown the wave function does notprovide a complete description of the physical reality, weleft open the question of whether or not such a descriptionexists. We believe, however, that such a theory ispossible .” It is too early to judge whether or not our ICQTcompletes current quantum mechanics in the Einstein-Podolsky-Rosen sense cited above, as experiments willbe the ultimate judgement. But if nature does worklike a description provided by the ICQT, nature willbe very funny and more importantly, nature does be comprehensible via a self-defining structure. Einsteinmight be very happy to see that two of his importanttheoretical achievements, namely, general relativity(after being quantized in modern language) and theconcept of quantum entanglement (discovered by him,together with Podolsky and Rosen), are very essentialfor our information-complete quantum description.
Acknowledgements
I am grateful to Chang-Pu Sun for bringing Refs. [22, 34]into my attention, and to Xian-Hui Chen, Dong-Lai Fengand Yao Fu for enjoyable discussions. I also acknowledgesUniversity of Science and Technology of China, where thework was initiated. [1] A. Einstein, B. Podolsky, and N. Rosen,
Can quantum-mechanical description of physical reality be consideredcomplete?
Phys. Rev. , 777-780 (1935).[2] N. Bohr, Can quantum-mechanical description of physicalreality be considered complete?
Phys. Rev. , 696-702(1935).[3] J.A. Wheeler and W.H. Zurek, eds., Quantum Theoryand Measurement (Princeton Univ. Press, Princeton,New Jersey, 1983).[4] R. Omn`es,
The Interpretation of Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey, 1994).[5] W.H. Zurek,
Decoherence, einselection, and the quantumorigins of the classical , Rev. Mod. Phys. , 715-775(2003).[6] C.-P. Sun, Quantum dynamical model for wave-functionreducation in classical and macroscopic limits , Phys. Rev.A , 898-906 (1993).[7] C.-P. Sun, X.-X. Yi, and X.-J. Liu, Quantum dynam-ical approach of wavefunction collapse in measurementprogress and its application to quantum Zeno effect ,Fortschr. Phys. , 585-612 (1995).[8] A. Allahverdyan, R. Balian, and T.M. Niewenhuizen, Understanding quantum measurement from the solutionof dynamical models , Phys. Rep. , 1-166 (2013).[9] M.F. Pusey, J. Barrett, and T. Rudolph,
On the realityof the quantum state , Nature Phys. , 475-478 (2012).[10] G.C. Ghirardi and R. Romano, On the completeness ofquantum mechanics and the interpretation of the statevector , J. Phys.: Conf. Series , 012002 (2013).[11] S. Aaronson,
Quantum mechanics: Get real , NaturePhys. , 443-444 (2012).[12] J. Conway and S. Kochen, The free will theorem , Found.Phys. , 1441-1473 (2006).[13] J.S. Bell, Speakable and Unspeakable in Quantum Me-chanics (Cambridge University Press, Cambridge, 1987).[14] J.F. Clauser and M.A. Horne,
Experimental consequencesof objective local theories , Phys. Rev. D , 526-535(1974).[15] T. 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 realismwith freedom of choice , Proc. Natl. Acad. Sci. (USA) ,19708-19713 (2010). [16] J.S. Bell,
On the Einstein-Podolsky-Rosen paradox ,Physics (Long Island City, N.Y.) , 195-200 (1964).[17] J. von Neumann, Mathematical Foundations of QuantumMechanics,
Vol. 2 (Princeton Univ. Press, Princeton,New Jersey, 1996).[18] E.P. Wigner,
The problem of measurement , Am. J. Phys. , 6-15 (1963).[19] L.D. Landau and E.M. Lifshitz, Quantum Mechan-ics: Non-Relativistic Theory , 3rd Ed. (Butterworth-Heinemann, Oxford, 1981).[20] D. Bohm,
A suggested interpretation of the quantumtheory in terms of “hidden” variables. I , Phys. Rev. ,166-179 (1952).[21] H. Everett III, “Relative state” formulation of quantummechanics , Rev. Mod. Phys. , 454-462 (1957).[22] F.J. Tipler, Quantum nonlocality does not exist , Proc.Natl. Acad. Sci. (USA) , 11281-11286 (2014).[23] C. Rovelli,
Relational quantum mechanics , Int. J. Theor.Phys. , 1637-1678 (1996).[24] C. Rovelli, Quantum Gravity (Cambridge Univ. Press,Cambridge, 2004).[25] M.A. Nielsen and I.L. Chuang,
Quantum Computationand Quantum Information (Cambridge Univ. Press,Cambridge, 2000).[26] L. Hardy,
Quantum theory from five reasonable axioms ,Prepint at < http://arXiv.org/quant-ph/0101012v4(2001).[27] C.A. Fuchs, Quantum mechanics as quantum in-formation (and only a little more) , Prepint at < http://arXiv.org/quant-ph/0205039 > (2002).[28] M. Paw lowski, T. Paterek, D. Kaszlikowski, V. Scarani,A. Winter, and M. ˙Zukowski, Information causality as aphysical principle , Nature , 1101-1104 (2009).[29] R. Colbeck and R. Renner,
No extension of quantumtheory can have improved predictive power , NatureCommun. , 411 (2011).[30] C.H. Bennett, H. Bernstein, S. Popescu, and B.Schumacher, Concentrating partial entanglement by localoperations , Phys. Rev. A , 2046-2062 (1996).[31] R.T. Thew, K. Nemoto, A.G. White, and W.J. Munro, Qudit quantum-state tomography , Phys. Rev. A ,012303 (2002).[32] P. Busch, Informationally complete set of physical quantities , Int. J. Theor. Phys. , 1217-1227 (1991).[33] J.M. Renes, R. Blume-Kohout, A.J. Scott, and C.M.Caves, Symmetric informationally complete quantummeasurements , J. Math. Phys. , 2171-2180 (2004).[34] S.-W. Li, C.Y. Cai, X.F. Liu, and C.P. Sun, Objectivityin quantum measurements , Found. Phys. , 654-667(2018).[35] S. Popescu and D. Rohrlich, Thermodynamics and themeasure of entanglement , Phys. Rev. A , R3319-R3321(1997).[36] Z.-B. Chen, An informationally-complete unification ofquantum spacetime and matter , arXiv:1412.3662.[37] T. Thiemann,
Lectures on loop quantum gravity , LectureNotes in Physics , 41-135 (2003).[38] V.B. Braginsky, Y.I. Vorontsov, and K.S. Thorne,
Quantum nondemolition measurements , Science ,547-557 (1980).[39] V.B. Braginsky and F.Ya. Khalili,
Quantum nondemo-lition measurements: the route from toys to tools , Rev.Mod. Phys. , 1-11 (1996).[40] J. Audretsch, Entangled Systems: New Directions inQuantum Physics (Wiley-VCH Verlag, Weinheim, 2007)pp. 149-150.[41] C. Rovelli and L. Smolin,
Discreteness of area and volumein quantum gravity , Nucl. Phys. B , 593-619 (1995).[42] C. Rovelli,
Black hole entropy from loop quantum gravity ,Phys. Rev. Lett. , 3288-3291 (1996).[43] A. Ashtekar, J. Baez, A. Corichi, and A. Krasnov, Quantum geometry and black hole entropy , Phys. Rev.Lett. , 904-907 (1998).[44] G. ’t Hooft, Dimensional reduction in quantum gravity ,Prepint at < http://arXiv.org/gr-qc/9310026 > (1993).[45] L. Susskind, The world as a hologram , J. Math. Phys. , 6377-6396 (1995).[46] R. Bousso, The holographic principle , Rev. Mod. Phys. , 825-874 (2002).[47] S. Aaronson, The limits of quantum computers , Sci. Am. , 62-69 (March, 2012).[48] L. Maccone,
Quantum solution to the arrow-of-timedilemma , Phys. Rev. Lett. , 080401 (2009).[49] D. Jennings and T. Rudolph,
Entanglement and thethermodynamic arrow of time , Phys. Rev. E , 061130(2010).[50] J.B. Hartle, The quantum mechanical arrows of time , in
Fundamental Aspects of Quantum Theory: A Two-TimeWinner , edited by D. Struppa and J. Tollaksen (Springer,Milan, 2013).[51] J.D. Bekenstein,
Black holes and the second law , NuovoCimento Lett. , 737-740 (1972).[52] J.D. Bekenstein, Black holes and entropy , Phys. Rev. D , 2333-2346 (1973).[53] S.W. Hawking, Black hole explosions , Nature , 30-31(1974).[54] L. Di´osi,
A universal master equation for the gravitationalviolation of quantum mechanics , Phys. Lett. A , 377-381 (1987).[55] R. Penrose,
On Gravity’s role in Quantum StateReduction , Gen. Rel. Grav. , 581-600 (1996).[56] Z.-B. Chen, Synopsis of a Unified Theory for All Forcesand Matter , arXiv:1611.02662.[57] C.H. Bennett and G. Brassard,
Quantum cryptography:Public key distribution and coin tossing , in
Proceedingsof the IEEE International Conference on Computers,Systems and Singal Proceeding, Bangalore, India (IEEE,New York, 1984), p.175.[58] C.H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A.Peres, and W.K. Wootters,
Teleporting an unknownquantum state via dual classical and Einstein-Podolsky-Rosen channels , Phys. Rev. Lett. , 3081-3084 (1993).[59] J.A. Wheeler, Recent thinking about the nature of thephysical world: It from bit , Ann. N.Y. Acad. Sci.655