Entanglement in Livings

Home , Madelung equations

Journal of Quantum Information Science, 2012, 2, 66-77 http://dx.doi.org/10.4236/jqis.2012.23012 Published Online September 2012 (http://www.SciRP.org/journal/jqis)

Michail Zak Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA Email: [email protected]

Received May 5, 2012; revised May 28, 2012; accepted June 28, 2012

ABSTRACT This paper discusses quantum-inspired models of Livings from the viewpoint of information processing. The model of Livings consists of motor dynamics simulating actual behavior of the object, and mental dynamics representing evolution of the corresponding knowledge-base and incorporating it in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics The paper concentrates on discovery of a new type of entanglement that correlates the probabilities of actions of Livings rather than the actions themselves.

Keywords: Entanglement; Information; Motor/Mental Dynamics; Probability; Feedback

1. Introduction matical formalism is quantum-inspired: it is based upon coupling the classical dynamical system representing the This paper is based upon models of Livings introduced in motor dynamics with the corresponding Liouville equa- our earlier publications [1,2] and motivated by an attempt tion describing the evolution of initial uncertainties in to interpret special properties of probability density dis- terms of the probability density and representing the tribution when conditional densities are incompatible [3], mental dynamics. (Compare with the Madelung equation and for that reason, the joint density does not exists. We that couples the Hamilton-Jacobi and Liouville equations will start with a brief description of mathematical models via the quantum potential).The coupling is implemented of Livings. by the information-based supervising forces that can be associated with the self-awareness. These forces funda- 1.1. Dynamical Model of Livings mentally change the pattern of the probability evolution, In this paper, the underlying dynamical model that cap- and therefore, leading to a major departure of the behav- tures behavior of Livings is based upon extension of the ior of living systems from the patterns of both Newtonian First Principles of classical physics to include phenome- and statistical mechanics. Further extension, analysis, nological behavior of living systems, i.e. to develop a interpretation, and application of this approach to com- new mathematical formalism within the framework of plexity in Livings and emergent intelligence have been classical dynamics that would allow one to capture the addressed in the papers referenced above. specific properties of natural or artificial living systems In the next introductory sub-sections we will briefly such as formation of the collective mind based upon ab- review these models without going into mathematical stract images of the selves and non-selves, exploitation of details. Instead we will illustrate their performance by the this collective mind for communications and predictions Figure 1. of future expected characteristics of evolution, as well as The model is represented by a system of nonlinear for making decisions and implementing the correspond- ODE and a nonlinear parabolic PDE coupled in a mas- ing corrections if the expected scenario is different from ter-slave fashion. The coupling is implemented by a the originally planned one. The approach is based upon feedback that includes the first gradient of the probability our previous publications (see References) that postulate density, and that converts the first order PDE (the Liou- that even a primitive living species possesses additional ville equation) to the second order PDE (the Fokker- non-Newtonian properties which are not included in the Planck equation). Its solution, in addition to positive dif- laws of Newtonian or statistical mechanics. These prop- fusion, can display negative diffusion as well, and that is erties follow from a privileged ability of living systems the major departure from the classical Fokker-Planck to possess a self-image (a concept introduced in psy- equation. The nonlinearity is generated by a feedback chology) and to interact with it. The proposed mathe- from the PDE to the ODE. As a result of the nonlinearity,

Figure 1. Classical physics, quantum physics, and physics life. the solutions to PDE can have attractors (static, periodic, results in quantum superposition of two or more values or chaotic) in probability space. The multi-attractor limit of the quantity. If the quantity is measured, the state will sets allow one to introduce an extension of neural nets be randomly collapsed onto one of the values in the su- that can converge to a prescribed type of a stochastic perposition (with a probability proportional to the square process in the same way in which a regular neural net of the amplitude of that eigenstate in the linear combina- converges to a prescribed deterministic attractor. The tion). Let us compare the behavior of the model of Liv- solution to ODE represents another major departure from ings from that viewpoint, Figure 2. classical ODE: due to violation of Lipchitz conditions at As follows from Figure 2, all the particular solutions states where the probability density has a sharp value, the intersect at the same point v = 0 at t = 0, and that leads to solution loses its uniqueness and becomes random. non-uniqueness of the solution due to violation of the However, this randomness is controlled by the PDE in Lipcshitz condition. Therefore, the same initial condition such a way that each random sample occurs with the v = 0 at t = 0 yields infinite number of different solutions corresponding probability (see Figure 1). forming a family; each solution of this family appears The model represents a fundamental departure from with a certain probability guided by the corresponding both Newtonian and statistical mechanics. In particular, Fokker-Planck equation. For instance, in case of the so- negative diffusion cannot occur in isolated systems lution plotted in Figure 2, the “winner” solution is without help of the Maxwell sorting demon that is strictly v  0 since it passes through the maxima of the prob- forbidden in statistical mechanics. The only conclusion ability density. However, with lower probabilities, other to be made is that the model is non-Newtonian, although solutions of the same family can appear as well. Obvi- it is fully consistent with the theory of differential equa- ously, this is a non-classical effect. Qualitatively, this tions and stochastic processes. Strictly speaking, it is a property is similar to those of quantum mechanics: the matter of definition weather the model represents an iso- system keeps all the solutions simultaneously and dis- lated or an open system since the additional energy ap- plays each of them “by a chance”, while that chance is plied via the information potential is generated by the controlled by the evolution of probability density. It system “itself” out of components of the probability den- should be emphasized that the choice of displaying a sity. In terms of a topology of its dynamical structure, the certain solution is made by the Livings model only once, proposed model links to quantum mechanics: if the information potential is replaced by the quantum potential, the model turns into the Madelung equations that are equivalent to the Schrödinger equation. The system of ODE describes a mechanical motion of the system driven by information forces. Due to specific properties of these forces, this motion acquires characteristics similar to those of quantum mechanics. These properties are discussed below. The most important property is Superposi- tion. In quantum mechanics, any observable quantity corresponds to an eigenstate of a Hermitian linear opera- Figure 2. Switching between superposition and classical tor. The linear combination of two or more eigenstates states.

Copyright © 2012 SciRes. JQIS 68 M. ZAK and in particular, at the instant of time when the feedback thermodynamics while the living systems are not. In or- is removed and the dynamical system becomes a Newto- der to further illustrate the connection between the life/ nian’s one. Therefore, the removal of the feedback can be non-life discrimination and the second law of thermody- associated with a quantum measurement. Modified ver- namics, consider a small physical particle in a state of sions of such quantum properties as uncertainty and en- random migration due to thermal energy, and compare its tanglement are also described in the referenced papers. diffusion i.e. physical random walk, with a biological The model illuminates the “border line” between liv- random walk performed by a bacterium. The fundamen- ing and non-living systems. The model introduces a bio- tal difference between these two types of motions (that logical particle that, in addition to Newtonian properties, may be indistinguishable in physical space) can be de- possesses the ability to process information. The prob- tected in probability space: the probability density evolu- ability density can be associated with the self-image of tion of the physical particle is always linear and it has the biological particle as a member of the class to which only one attractor: a stationary stochastic process where this particle belongs, while its ability to convert the den- the motion is trapped. On the contrary, a typical prob- sity into the information force—with the self-awareness ability density evolution of a biological particle is non- (both these concepts are adopted from psychology). Con- linear: it can have many different attractors, but eventu- tinuing this line of associations, the equation of motion ally each attractor can be departed from without any can be identified with a motor dynamics, while the evo- “help” from outside. lution of density—with a mental dynamics. Actually the That is how H. Berg [7], describes the random walk of mental dynamics plays the role of the Maxwell sorting an E. coli bacterium: “If a cell can diffuse this well by demon: it rearranges the probability distribution by cre- working at the limit imposed by rotational Brownian ating the information potential and converting it into a movement, why does it bother to tumble? The answer is force that is applied to the particle. One should notice that the tumble provides the cell with a mechanism for that mental dynamics describes evolution of the whole biasing its random walk. When it swims in a spatial gra- class of state variables (differed from each other only by dient of a chemical attractant or repellent and it happens initial conditions), and that can be associated with the to run in a favorable direction, the probability of tum- ability to generalize that is a privilege of living systems. bling is reduced. As a result, favorable runs are extended, Continuing our biologically inspired interpretation, it and the cell diffuses with drift.” Berg argues that the cell should be recalled that the second law of thermodynam- analyzes its sensory cue and generates the bias internally, ics states that the entropy of an isolated system can only by changing the way in which it rotates its flagella. This increase. This law has a clear probabilistic interpretation: description demonstrates that actually a bacterium inter- increase of entropy corresponds to the passage of the acts with the medium, i.e. it is not isolated, and that rec- system from less probable to more probable states, while onciles its behavior with the second law of thermody- the highest probability of the most disordered state (that namics. However, since these interactions are beyond the is the state with the highest entropy) follows from a sim- dynamical world, they are incorporated into the proposed ple combinatorial analysis. However, this statement is model via the self-supervised forces that result from the correct only if there is no Maxwell’ sorting demon, i.e. interactions of a biological particle with “itself”, and that nobody inside the system is rearranging the probability formally “violates” the second law of thermodynamics. distributions. But this is precisely what the Liouville Thus, this model offers a unified description of the pro- feedback is doing: it takes the probability density  gressive evolution of living systems. More sophisticated from the mental dynamics, creates functions of this den- effects that shed light on relationships between Livings sity, converts them into a force and applies this force to the equation of motor dynamics. As already mentioned above, because of that property of the model, the evolution of the probability density becomes nonlinear, and the entropy may decrease “against the second law of thermodynamics”, Figure 3. Obviously the last statement should not be taken literary; indeed, the proposed model captures only those aspects of the living systems that are associated with their behavior, and in particular, with their motor-mental dynamics, since other properties are beyond the dynamical formalism. Therefore, such physiological processes that are needed for the metabo- lism are not included into the model. That is why this model is in a formal disagreement with the second law of Figure 3. Deviation from thermodynamics.

Copyright © 2012 SciRes. JQIS M. ZAK 69 and the second law of thermodynamics is considered in version of the system (1) and (2) [4].  v 2 ln , (3) 1.2. Analytical Formulation v  2 The proposed model that describes mechanical behavior  2 ,dV  1 (4) of a Living can be presented in the following compressed t V 2   invariant form where v stands for the velocity, and  2 is the constant v  v ln , (1) diffusion coefficient. 2  , (2) Remark 1. Here and below we make distinction be- V tween the random variable v(t) and its values V in prob- where ν is velocity vector, ρ is probability density, ζ is ability space. universal constant, and α (D, w) is a tensor co-axial with The solution of Equation (4) subject to the sharp initial the tensor of the variances D that may depend upon these condition variances. Equation (1) represents the second Newton’s 2 law in which the physical forces are replaced by infor- 1 V  exp2  (5) mation forces via the gradient of the information poten- 2 πt 4 t tial  ln  , while the constant ζ connects the infor- describes diffusion of the probability density. Substitut- mation and inertial forces formally replacing the Planck ing this solution into Equation (3) at V = v one arrives at constant in the Madelung equations of quantum mechan- the differential equation with respect to v (t) ics. Equation (2) represents the continuity of the prob- v ability density (the Liouville equation), and unlike the v  (6) classical case, it is non-linear because of dependence of 2t the tensor α upon the components of the variance D. This and, therefore, model is equipped by a set of parameters w that control the properties of the solutions discussed above. The only vCt (7) realistic way to reconstruct these parameters for an object where C is an arbitrary constant. Since v = 0 at t = 0 for to be discovered is to solve the inverse problem: given any value of C, the solution (7) is consistent with the time series of sensor data describing dynamics of an un- sharp initial condition for the solution (5) of the corre- known object, find the parameters of the underlying dy- sponding Liouville Equation (4). namical model of this object within the formalism of The solution (7) describes the simplest irreversible Equations (1) and (2). As soon as such a model is recon- motion: it is characterized by the “beginning of time” structed, one can predict future object behavior by run- where all the trajectories intersect (that results from the ning the model ahead of actual time as well as analyze a violation of the Lipcsitz condition at t = 0, Figure 5), hypothetical (never observed) object behavior by appro- while the backward motion obtained by replacement of t priate changes of the model parameters. But the most with (–t) leads to imaginary values of velocities. One can important novelty of the proposed approach is the capa- notice that the probability density (5) possesses the same bility to detect Life that occurs if, at least, some of properties. Further analysis of the solution (7) demon- “non-Newtonian” parameters are present. The method- strates that it is unstable since ology of such an inverse problem is illustrated in Figure d1v 4.   0 (8) Our further analysis will be based upon the simplest d2vt

Figure 4. Data-driven model discovery.

Its solution depends upon the roots of the characteris- tic equation

22  20bbbb12 12 11 22  (18) Since both the roots are real in our case, let us assume for concreteness that they are of the same sign, for in-

stance, 12 1,  1 . Then the solution to Equation (17) is represented by the family of straight lines

vCvC21,const. (19) Figure 5. Stochastic process and probability density. Substituting this solution into Equation (54) yields

11 bCbbCb dv 2211 12 11 12   at t  0 (9) v12 Ct, v CCt (20) dv Thus, the solutions to Equations (10) and (11) are rep- 1.3. Example of Entanglement in Livings resented by two-parametrical families of random samples, as expected, while the randomness enters through the In order to introduce entanglement in a Living system, time-independent parameters C and C that can take any we will start with Equations (11) and (12) and generalize real numbers. Let us now find such a combination of the them to the two-dimensional case variables that is deterministic. Obviously, such a combi-  nation should not include the random parameters C or C . va111ln  a 12 ln , (10) It easily verifiable that vv12

ddbCb11 12  lnvv12 ln (21) va221ln  a 22 ln , (11) ddtt 2 t vv12 and therefore,  22  2 aaaa112  12 21 22 , (12) dd tVVV 12V 2 lnvv12 ln 1 (22) ddtt As in the one-dimensional case, this system describes Thus, the ratio (22) is deterministic although both the diffusion without a drift numerator and denominator are random. This is a fun- The solution to Equation (12) has a closed form damental non-classical effect representing a global con- 11 straint. Indeed, in theory of stochastic processes, two  expbVV , i 1,2. (13) ij i j random functions are considered statistically equal if 2det atˆ 4t ij they have the same statistical invariants, but their point- Here to-point equalities are not required (although it can hap- pen with a vanishingly small probability). As demon- 1  ˆˆ ˆ baaaaaij ij 11 11,, 22 22 strated above, the diversion of determinism into ran- (14) domness via instability (due to a Liouville feedback), and aaaaaabbˆ ˆˆˆ,,  , 12 21 12 21 ij ji ij ji then conversion of randomness to partial determinism (or Substituting the solution (13) into Equations (19) and coordinated randomness) via entanglement is the funda- (11), one obtains mental non-classical paradigm that may lead to instantaneous transmission of conditional information on remote bv11 1 bv 12 2 v1  (15) distance that has been discussed in [2]. 2t

bv21 1 bv 22 2 2. Entanglement vb2 , ijb ijaˆ ij (16) 2t Prior to deriving a fundamentally new type of entangle- Eliminating t from these equations, one arrives at an ment in living, we will discuss a concept of global con- ODE in the configuration space straints in Physics and its application to more general dvbvbv interpretation of entanglement. 2211222,0vatv 0, (17) Quantum entanglement is a phenomenon in which the dvbvbv 21 1111122 quantum states of two or more objects have to be This is a classical singular point treated in text books described with reference to each other, even though the on ODE. individual objects may be spatially separated. Qualitat-

Copyright © 2012 SciRes. JQIS M. ZAK 71 ively similar effect has been demonstrated in living is another type of the global constraint in physics: the systems: as follows from the example considered in the normalization constraint (see Equation (4)). This con- previous section, the diversion of determinism into ran- straint is fundamentally different from those listed above domness via instability (due to a Liouville feedback), and for two reasons. Firstly, it is not an idealization, and then conversion of randomness to partial determinism (or therefore, it cannot be removed by taking into account coordinated randomness) via entanglement is the funda- more subtle properties of matter such as elasticity, com- mental non-classical paradigm that may lead to instanta- pressibility, discrete structure, etc. Secondly, it imposes neous transmission of conditional information on remote restrictions not upon positions or velocities of particles, distance [2]. In this connection, it is reasonable to make but upon the probabilities of their positions or velocities, some comments about non-locality in physics. As al- and that is where the entanglement comes from. Indeed, ready mentioned above, the living systems and quantum if the Liouville equation is coupled with equations of systems have similar topology (see Figure 1). In this motion as in quantum mechanics, the normalization con- section, we will analyze the similarities in terms of en- dition imposes a global constraint upon the state vari- tanglement that follow from this topology. ables, and that is the origin of quantum entanglement. In quantum physics, the reactions of the normalization con- 2.1. Criteria for Non-Local Interactions straints can be associated with the energy eigenvalues Based upon analysis of all the known interactions in the that play the role of the Lagrange multipliers in the con- Universe and defining them as local, one can formulate ditional extremum formulation of the Schrödinger equa- the following criteria of non-local interactions: they are tion [5]. In living systems, the Liouville equation is also not mediated by another entity, such as a particle or field; coupled with equations of motion (although the feedback their actions are not limited by the speed of light; the is different). And that is why the origin of entanglement strength of the interactions does not drop off with dis- in living systems is the same as in quantum mechanics. tance. All of these criteria lead us to the concept of the global constraint as a starting point. 2.3. Speed of Action Propagation Further illumination of the concept of quantum entan- 2.2. Global Constraints in Physics glement follows from comparison of quantum and New- It should be recalled that the concept of a global con- tonian systems. Such a comparison is convenient to per- straint is one of the main attribute of Newtonian me- form in terms of the Madelung version of the Schrödinger chanics. It includes such idealizations as a rigid body, an equation: incompressible fluid, an inextensible string and a mem-    S 0 (23) brane, a non-slip rolling of a rigid ball over a rigid body, tm etc. All of those idealizations introduce geometrical or 22 kinematical restrictions to positions or velocities of parti- S 2    SV 0 (24) cles and provide “instantaneous” speed of propagation of t 2m  disturbances. Let us discuss the role of the reactions of these constraints. One should recall that in an income- Here  and S are the components of the wave func- iS / pressible fluid, the reaction of the global constraint tion  e , and is the Planck constant divided v 0 (expressing non-negative divergence of the by 2π . The Newtonian mechanics (  0 ), in terms of velocity v) is a non-negative pressure p  0 ; in inexten- the S and  as state variables, is of a hyperbolic type, sible flexible (one- or two-dimensional) bodies, the reac- and therefore, any discontinuity propagates with the fi- 0 tion of the global constraint gij g ij , i, j = 1, 2 (ex- nite speed S/m, i.e. the Newtonian systems do not have pressing that the components of the metric tensor cannot non-localities. But the quantum mechanics (  0 ) is of a exceed their initial values) is a non-negative stress tensor parabolic type. This means that any disturbance of S or

 ij  0 , i, j = 1, 2. It should be noticed that all the known  in one point of space instantaneously transmitted to forces in physics (the gravitational, the electromagnetic, the whole space, and this is the mathematical origin of the strong and the weak nuclear forces) are local. How- non-locality. But is this a unique property of quantum ever, the reactions of the global constraints listed above evolution? Obviously, it is not. Any parabolic equation do not belong to any of these local forces, and therefore, (such as Navier-Stokes equations or Fokker-Planck equa- they are non-local. Although these reactions are being tion) has exactly the same non-local properties. However, successfully applied for engineering approximations of the difference between the quantum and classical non- theoretical physics, one cannot relate them to the origin localities is in their physical interpretation. Indeed, the of entanglement since they are result of idealization that Navier-Stokes equations are derived from simple laws of ignores the discrete nature of the matter. However, there Newtonian mechanics, and that is why a physical inter-

Copyright © 2012 SciRes. JQIS 72 M. ZAK pretation of non-locality is very simple: If a fluid is in- do not discriminate between stable and unstable motions. compressible, then the pressure plays the role of a reac- But unstable motions cannot be realized and observed, tion to the geometrical constraint v 0 , and it is and therefore, a special mathematical analysis must be transmitted instantaneously from one point to the whole added to found out the existence and observability of the space (the Pascal law). One can argue that the incom- motion under consideration. However, then another ques- pressible fluid is an idealization, and that is true. How- tion can be raised: why turbulence as a postinstability ever, it does not change our point: Such a model has a lot version of an underlying laminar flow can be observed of engineering applications, and its non-locality is well and measured? In order to answer this question, we have understood. The situation is different in quantum me- to notice that the concept of stability is an attribute of chanics since the Schrodinger equation has never been mathematics rather than physics, and in mathematical derived from Newtonian mechanics: It has been postu- formalism, stability must be referred to the correspond- lated. In addition to that, the solutions of the Schrodinger ing class of functions. For example: a laminar motion equation are random, while the origin of the randomness with sub-critical Reynolds number is stable in the class does not follow from the Schrodinger formalism. That is of deterministic functions. Similarly, a turbulent motion why the physical origin of the same mathematical phe- is stable in the class of random functions. Thus the same nomenon cannot be reduced to simpler concepts such as physical phenomenon can be unstable in one class of “forces”: It should be accepted as an attribute of the functions, but stable in another, enlarged class of func- Schrodinger equation. tions. Let us turn now to the living systems. The formal dif- Thus, we are ready now to the following conclusion: ference between them and quantum systems is in a feed- any stochastic process in Newtonian dynamics describes back from the Liouville equation to equations of motion: the physical phenomenon that is unstable in the class of the gradient of the quantum potential is replaced by the the deterministic functions. information forces, while the equations of motion are This elegant union of physics and mathematics has written in the form of the second Newton’s law rather been disturbed by the discovery of quantum mechanics than in the Hamilton-Jacoby form. In this paper, we have that complicated the situation: Quantum physicists claim considered information forces that turn the corresponding that quantum randomness is the “true” randomness unlike Liouville equation into the Fokker-Planck equation the “deterministic” randomness of chaos and turbulence. which is parabolic, and therefore, all the changes of the Richard Feynman in his “Lectures on Physics” stated that probability density propagates instantaneously as in the randomness in quantum mechanics in postulated, and quantum systems. Thus, both quantum systems and liv- that closes any discussions about its origin. However, ing systems possess the same non-locality: instantaneous recent result disproved existence of the “true” random- propagation of changes in the probability density, and ness. Indeed, as shown in [2], the origin of randomness in this is due to similar topology of their dynamical struc- quantum mechanics can be traced down to instability ture, and in particular, due to a feedback from the Liou- generated by quantum potential at the point of departure ville equation. from a deterministic state if for dynamical analysis one transfer from the Shrodinger to the Madelung equation. 2.4. Origin of Randomness in Physics (For details see [2]). As demonstrated there, the instability triggered by failure of the Lipchitz condition splits the Since entanglement in quantum systems as well as in solution into a continuous set of random samples repre- living systems is exposed via instantaneous propagation senting a “bridge” to quantum world. Hence, now we can of changes in the probability density, it is relevant to ask state that any stochastic process in physics describes the what is the origin of randomness in physics. The concept physical phenomenon that is unstable in the class of the of randomness has a long history. Its philosophical as- deterministic functions. Actually this statement can be pects first were raised by Aristotle, while the math- used as a definition of randomness in physics. Finally ematic- cal foundations were introduced and discussed one may ask why the instability discussed above has not much later by Henry Poincare who wrote: “A very slight been detected in the Schrödinger equation. The answer is cause, which escapes us, determines a considerable effect simple if one recalls that stability analysis is based upon which we cannot help seeing, and then we say this effect a departure from the basic state into a perturbed state, is due to chance.” Actually Poincare suggested that the and such departure requires an expansion of the basic origin of randomness in physics is the dynamical insta- space. However, Schrödinger and Madelung equations in bility, and this viewpoint has been corroborated by the- the expanded spaces are not necessarily equivalent any ory of turbulence and chaos. However, the theory of dy- more, and that confirm the fact that stability is not an namical stability developed by Poincare and Lyapunov invariant of a dynamical system: it can depend upon the revealed the main flaw of physics: its fundamental laws definition of a distance between the basic and perturbed

Copyright © 2012 SciRes. JQIS M. ZAK 73 motions, and these definitions are different for Hilbert ii XtXt,, i  X i X i  ,1,2 i . (33) and physical spaces. for tt  read 3. Partial Entanglement in Livings 1 11XX| 2 2 In this section we introduce a new, more sophisticated 4πgXtt11 2   entanglement that does not exists in quantum mechanics, (34)  2 but can be found in Livings. This finding is based upon XX11  exp( 2 existence of incompatible stochastic processes that are 4g11Xtt 2   considered below. 1 221XX  2 3.1. Incompatible Stochastic Processes 4πgXtt22 1   (35) Classical probability theory defines conditional proba- 2  XX   bility densities based upon the existence of a joint prob-  exp  22  4g 2 Xtt  ability density. However, one can construct correlated  22 1  stochastic processes that are represented only by condi- As shown in [3], a joint density for the conditional tional densities since a joint probability density does not densities (34) and (35) exists only in special cases of the exist. For that purpose, consider two coupled Langevin diffusion coefficients g and g when the conditional equations [6]. 11 22 probabilities are compatible. These conditions are x11121 gxLt (25)   XX| ink ,ln 11 2 0 (36) x  gxLt (26) 12 22212  XX12 2 X 2| X 1 where the Langevin forces Lt1   and Lt2  satisfy the Indeed conditions  XX12,, 1 XX 12  X2d Ltiiii 0, LtLt   2 gi  t t  (27)  (37) Then the joint probability density  X , X de-  12  XX X , d , scribing uncertainties in values of the random variables 212  1  x1 and x2 evolves according to the following Fokker- whence Planck equation  XX  22 112  22 ln  ln X1 , d gX11 222 g 22 X 1 (28) 221XX  t X12X (38)  Let us now modify Equations (25) and (26) as follow-  ln  ,X 2 d ing  2* and that leads to Equation (36). x11121 gxLt (29) Thus, the existence of the join density  X12, X for 2* the conditional densities  X X and  X X x22212 gxLt  (30) 112 221  requires that where x* and x* are fixed values of x and x that 1 2 1 2 2 play role of parameters in Equations (29) and (30), re- 2 XX XX   11  2 2 spectively. Now the uncertainties of x and x are char- 1 2 220 (39) XX1244gX11 2 gX 22 1 acterized by conditional probability densities 11 X | X 2   and 22X | X1 while each of these densities is gov- erned by its own Fokker-Planck equation Obviously the identity (39) holds only for specially 2 2  selected functions g X and g X , and there- 1 gX 1 (31) 11 2  22 1 11 2 2 fore, existence of the joint density is an exception rather t X 1 than a rule. 2 22 2  gX22 1 (32) tX2 3.2. Partial Entanglement The solutions of these equations subject to sharp initial In order to prove existence of a new form of entangle- conditions ment, let us modify the system Equations (10)-(12) as

Copyright © 2012 SciRes. JQIS 74 M. ZAK following: the solutions (44) and (45) into Equations (40) and (42),  respectively. Then with reference to Equation (6), one vav1112  ln  112 vv (40) obtains v1 v1 2 v1  (48) 112VV 112VV 2t  aV11 2 2 (41) t V 1 v2 v2  (49)  2t vav2221  ln  22 vv1 (42) v1 and therefore 2 vCt (50) 221VV 221VV 11  aV (43) t 22 1 V 2 2 vCt22 (51) Since here we do not postulate existence of a joint It should be recalled that according to the terminology density, the system is written in terms of conditional introduced in Section I, the system (40)-(41) and the sys- densities, while Equations (41) and (43) are similar to tem (42)-(43) can be considered as dynamical models for Equations (31) and (32). The solutions of these PDE can interaction of two communicating agents where Equa- be written in the form similar to the solutions (34) and tions (40) and (42) describes their motor dynamics, and (35) Equations (41) and (43)—mental dynamics, respectively. 1 Also it should be reminded that the solutions (50) and  VV  112 (51) are represented by one-parametrical families of 4πaV11 2 tt  (44) random samples, as in Equation (7), while the random 2 ness enters through the time-independent parameters C VV11  1  exp and C that can take any real numbers. As follows from 4aV tt   2 11 2 Figure 2, all the particular solutions (50) and (51) inter-

1 sect at the same point v1,2  0 at t = 0, and that leads to 221VV  non-uniqueness of the solution due to violation of the 4πaVtt  22 1  Lipcshitz condition. Therefore, the same initial condition (45) 2 v  0 at t = 0 yields infinite number of different solu- VV  1,2  exp 22 tions forming a family; each solution of this family ap- 4aVtt  11 1 pears with a certain probability guided by the corresponding Fokker-Planck Equations (41) and (43), respec- As noticed in the previous sub-section, the existence of tively. Similar scenario was described in the Introduction the joint density  VV12,  for the conditional densities  VV and  VV requires that of this paper. But what unusual in the system (40)-(43) is 112 221 correlations: although Equations (41) and (43) are corre- 2 lated, and therefore, mental dynamics are entangled, 2 VV VV   11  2 2    0 (46) Equations (40) and (42) are not correlated (since they can VV1244 a 112 V a 221 V be presented in the form of independent Equations (48) and (49), respectively), and therefore, the motor dynam- In this case, the joint density exists (although its find- ics are not entangled. This means that in the course of ing is not trivial [3]), and the system (40)-(43) can be communications, each agent “selects” a certain pattern of reduced to a system similar to (10)-(12). But here we will behavior from the family of solutions (50) and (51) re- be interested in case when the joint density does not exist. spectively, and these patterns are independent; but the probabilities of these “selections” are entangled via It is much easier to find such functions aV11 2, aV 22 1  for which the identity (46) does not hold, and we assume Equations (41) and (43). Such sophisticated correlations that cannot be found in physical world, and they obviously represent a “human touch”. Unlike the entanglement in 2  2 VV VV system with joint density (such as that in Equations   11  2 2    0 (47) (10)-(12)) here the agents do not share any deterministic VV44 a V a V 12 112 221invariants (compare to Equation (22)). Instead the agents can communicate via “best guesses” based upon known In this case the system (40)-(43) cannot be simplified. probability densities distributions. In order to analyze this system in details, let substitute In order to quantify the amount of uncertainty due to

Copyright © 2012 SciRes. JQIS M. ZAK 75 incompatibility of the conditional probability densities tailed analysis. Omitting such an analysis, let us start (44) and (45), let us introduce a concept of complex with a trivial case when probability [3]. b = 0 (60)

f VV12,, aVV 12 ibVV 12, (52) In this case the system (59) reduces to the following

 two integral equations with respect to one unknown aVV ,  f11()VaVVVibVV (, 12 )d 2 (, 12 )dV 2 12   (53) aVV12, aV11 ( ) ibV 11 ( ) 112VV,,    aVV12,d V 2  f22(VaVVVibVV ) (, 12 )d 1 (, 12 )dV 1 (61)  (54)   aVV12, 212VV,   aV22 ( ) ibV 22 ( )  aVV12,d V 2 Following the formalism of conditional probabilities,  the conditional density will be defined as This system is overdetermined unless the compatibility f (,VV ) aVV (, ) ibVV (, ) f 12 12 12 conditions (36) are satisfied. 1|2 fV() aV () ibV () As known from classical mechanics, the incompatibil- 22 22 22 (55) aa bb a b ab ity conditions are usually associated with a fundamen- 22i 2 2 tally new concept or a physical phenomenon. For in- ab22 ab 22 22 22 stance, incompatibility of velocities in fluid (caused by f (,VV ) aVV (, ) ibVV (, ) non-existence of velocity potential) introduces vorticity f 12 12 12 2|1 in rotational flows, and incompatibility in strains de- fV11() aV 11 () ibV 11 () (56) scribes continua with dislocations. In order to interpret aa bb a b ab 11 1 1 the incompatibility (36), let us return to the system (59). 22i 2 2 ab11 ab 2 2 Discretizing the functions in Equations (59) and replac- with the normalization constraint ing the integrals by the corresponding sums, one reduces Equations (59) to a system of n algebraic equations with  12 22 respect to n unknowns. This means that the system is abdd VV12 1 (57)   closed, and cases when a solution does not exist are ex- ceptions rather than a rule. Therefore, in most cases, for This constraint can be enforced by introducing a nor- any arbitrarily chosen conditional densities, for instant, malizing multiplier in Equation (52) which will not affect for those given by Equations (44) and (45), the system the conditional densities (55) and (56). (59) defines the complex joint density in the form (52). Clearly Now we are ready to discuss a physical meaning of the  12 imaginary component of the complex probability density. aab22, and aVdd V  1 (58)   12 Firstly, as follows from comparison of Equations (59)   and (61), the imaginary part of the probability density Now our problem can be reformulated in the following appears as a response to incompatibility of the condi- manner: given two conditional probability densities (44) tional probabilities, and therefore, it can be considered as and (45), and considering them as real parts of (unknown) a “compensation” for the incompatibility. Secondly, as complex densities (55) and (56), find the corresponding follows from the inequalities (58), the imaginary part complex joint density (52), and therefore, all the mar- consumes a portion of the “probability mass” increasing ginal (53) and (54), as well as the imaginary parts of the thereby the degree of uncertainty in the real part of the conditional densities. In this case one arrives at two cou- complex probability density. Hence the imaginary part of pled integral equations with respect to two unknowns the probability density can be defined as a measure of the aVV12,  and bVV12, (while the formulations of uncertainty “inflicted” by the incompatibility into the real aVV112, , aV212,V , bVV112,  and bVV212, follow part of this density. from Equations (53) and (54)). These equations are In order to avoid solving the system of integral equa- aa bb aa bb tions (59), we can reformulate the problem in an inverse VV,,,22 VV 11,(59) fashion by assuming that the complex joint density is 112 ab22212 ab22 22 22 given. Then the real parts of the conditional probabilities The system (59) is nonlinear, and very little can be that drive Equations (40) and (41) can be found from said about general property of its solution without de- simple formulas (55) and (56).

Let us illustrate this new paradigm, and consider two 4. Conclusions players (for instance, in a poker-like game), assuming In this paper, a novel approach to the concept of entan- that each player knows his own state as well as the com- glement and its application to communications in Livings plex joint probability density, is introduced and discussed. The paper combines several departures from classical methods in physics and in  VV12,, aVV 12 ibVV 12, (62) probability theory. But he does not know the state of his adversary. Firstly, it introduces a non-linear version of the Liou- Without going in details of the game, let us draw a ville equation that is coupled with the equation of motion sketch of a common-sense-based logic that could be ap- (in Newtonian dynamics they are uncoupled). This new plied by the players. Since the player 1 knows the joint dynamical architecture grew up from quantum physics probability density (62) and he also knows the value of (in the Madelung version) when the quantum potential his own state variable was replaced by information forces. The advantage of this replacement for modeling communications between Vvattt* (63) 11 0 intelligent agents representing living systems is addressed and discussed. He can find the state variable of the player 2 at tt 0 as a function of the real part of the probability density at Secondly, it exploits a paradigm coming from incom- the fixed value (63) of its own state variable patible conditional probabilities that leads to non-existence of a joint probability (in classical probability theory, * VFa21  1  at Vvtt11, 0 (64) existence of a joint probability is postulated). That led to discovery of a new type of entanglement that correlates where V is the value of the state variable of the second 21 not actions of Livings, but rather the probability of these player in view of the first player. Now the question is: actions. what is the best guess of the player 1 about the next Thirdly, it introduces a concept of imaginary probabil- move of the player 2 at tt ? The simple logic sug- 0 ity as a measure of uncertainty generated by incompati- gests that it is the move that maximizes the real part of bility of conditional probabilities. the joint density, i.e. the value Vv * is to be 21 21  All of these departures actually extend and comple- found from the following condition ment the classical methods making them especially suc- cessful in analysis of communications in Living repre- F aV v* SupF F a (65) 1 2(1) 2(1) 1 1  sented by new mathematical formalism. Obviously the player 2 has the same dilemma, and his Thus this paper discusses quantum-inspired models of choice could be Livings from the viewpoint of information processing. The model of Livings consists of motor dynamics simu- *  lating actual behavior of the object, and mental dynamics F2aV 2(1) v 2(1) SupF 2  a (66)  representing evolution of the corresponding knowl- However both players rely only upon the real part of edge-base and incorporating it in the form of information the complex joint density instead of a real joint density flows into the motor dynamics. Due to feedback from (that does not exist in this case). But as follows from the mental dynamics, the motor dynamics attains quantum- inequalities (58), the values of density of the real part are like properties: its trajectory splits into a family of dif- lowered due to loss of the probability mass, and this in- ferent trajectories, and each of those trajectories can be creases the amount of uncertainty in player’s predictions. chosen with the probability prescribed by the mental dy- In order to minimize that limitation, the players can in- namics The paper concentrates on discovery of a new voke the imaginary part of the joint density that gives type of entanglement that correlates not actions of Liv- them qualitative information about the amount of uncer- ings, but rather the probability of these actions. tainty at the selected maxima. This information could give a reason to reconsider the previous decision and REFERENCES move away from the maxima at which uncertainty of the [1] M. Zak, “Physics of Life from First Principles,” Elec- density is large. tronic Journal of Theoretical Physics, Vol. 4, No. 16(II) Thus the game starts with a significant amount of un- 2007, pp. 11-96. certainties that will grow exponentially with next moves. [2] M. Zak, “From Quantum Computing to Intelligence,” Such subtle and sophisticated relationship is typical for Nova Science Publishers, New York, 2011. communications between humans, and the proposed [3] M. Zak, “Incompatible Stochastic Processes and Complex model captures it via partial entanglement introduced Probabilities,” Physics Letters A, Vol. 238, No. 1, 1998, above. pp. 1-7. doi:10.1016/S0375-9601(97)00763-9

[4] M. Zak, “Entanglement of Memories and Reality in [6] H. Risken, “The Fokker-Planck Equation,” Springer-Ver- Models of Livings,” Electronic Journal of Theoretical lag, New York, 1989. Physics, 2012 (in press). [7] H. Berg, “Random Walk in Biology,” Prinston University [5] L. Landau and E. Lifshitz, “Quantum Mechanics,” Nauka, Press, Princeton, 1983. Moscow, 1997.