Entanglement Revisited

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1 EJTP 6, No. 20 (2009) Electronic Journal of Theoretical Physics Entanglement Revisited Michail Zak Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109, USA Received 30 October 2008, Accepted 15 August 2008, Published 20 February 2009 Abstract: Quantum-classical hybrid that preserves the topology of the Schrödinger equation (in the Madelung form), but replaces the quantum potential with other, specially selected, function of probability density is introduced. Non-locality associated with a global geometrical constraint that leads to entanglement effect is demonstrated. Despite such a quantum-like characteristic, the hybrid can be of classical scale and all the measurements can be performed classically. This new emergence of entanglement shed light on the concept of non-locality in physics. Application of hybrid systems to instantaneous transmission of conditional information on remote distances is discussed. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Entanglement; Quantum Systems; Liouville Equation; Emergence; Madelung equations PACS (2008): Ud; Vf; a; Bg; Mn; Gg; Dv; w 1. Introduction Quantum entanglement is a phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. Different views of what is actually occurring in the process of quantum entanglement give rise to different interpretations of quantum mechanics. In this paper we will demonstrate that entanglement is not a prerogative of quantum systems: it occurs in other non-classical systems such as quantum-classical hybrids [1], and that will shed light on the concept of entanglement as a special type of global constraint imposed upon a broad class of dynamical systems. In order to do that, we will turn to quantum mechanics. Representing the Schrödinger Michail.Zak@jpl.nasa.gov

2 308 Electronic Journal of Theoretical Physics 6, No. 20 (2009) equation in the Madelung form, we observe the feedback from the Louville equation to the Hamilton-Jacobi equation in the form of the quantum potential. Preserving the same topology, we will replace the quantum potential by other type of feedbacks [2] and investigate the property of these hybrid systems. 2. Destabilizing Effect of Liouville Feedback We will start with derivation of an auxiliary result that illuminates departure from Newtonian dynamics. For mathematical clarity, we will consider here a one-dimensional motion of a unit mass under action of a force f depending upon the dimensionless velocity v and time t v = f(v, t), (1) If initial conditions are not deterministic, and their probability density is given in the form ρ 0 = ρ 0 (V ), where ρ 0, and ρdv =1 (2) while ρ is a single- valued function, then the evolution of this density is expressed by the corresponding Liouville equation t + (ρf) =0 (3) v The solution of this equation subject to initial conditions and normalization constraints (2) determines probability density as a function of V and t : ρ = ρ(v,t). In order to deal with the constraint (2), let us integrate Eq. (3) over the whole space assuming that ρ 0at V and f <.Then t ρdv =0, ρdv = const, (4) Hence, the constraint (3) is satisfied for t>0 if it is satisfied for t =0. Let us now specify the force f as a feedback from the Liouville equation f(v, t) =ϕ[ρ(v, t)] (5) and analyze the motion after substituting the force (5) into Eq.(1) v = ϕ[ρ(v, t)], (6) This is a fundamental step in our approach. Although the theory of ODE does not impose any restrictions upon the force as a function of space coordinates, the Newtonian physics does: equations of motion are never coupled with the corresponding Liouville equation. Moreover, it can be shown that such a coupling leads to non-newtonian properties of the

3 Electronic Journal of Theoretical Physics 6, No. 20 (2009) underlying model. Indeed, substituting the force f from Eq. (5) into Eq. (4), one arrives at the nonlinear equation for evolution of the probability density t + {ρϕ[ρ(v,t)]} =0 (7) V Let us now demonstrate the destabilizing effect of the feedback (5). For that purpose, it should be noted that the derivative / v must change its sign, at least once, within the interval <v<, in order to satisfy the normalization constraint (2). But since Sign v v = Signdϕ dρ Sign (8) v there will be regions of v where the motion is unstable, and this instability generates randomness with the probability distribution guided by the Liouville equation (8). It should be noticed that the condition (9) may lead to exponential or polynomial growth of v (in the last case the motion is called neutrally stable, however, as will be shown below, it causes the emergence of randomness as well if prior to the polynomial growth, the Lipchitz condition is violated). 3. Emergence of Randomness In order to illustrate mathematical aspects of the concepts of Liouville feedback, as well as associated with it instability and randomness let us take the feedback (6) in the form to obtain the following equation of motion f = σ 2 ln ρ (9) v v = σ 2 ln ρ, (10) v This equation should be complemented by the corresponding Liouville equation (in this particular case, the Liouville equation takes the form of the Fokker-Planck equation) t = 2 ρ σ2 (11) V 2 Here v stands for a particle velocity, and σ 2 is the constant diffusion coefficient. The solution of Eq. (11) subject to the sharp initial condition is ρ = 1 2σ πt exp( V 2 4σ 2 t ) (12) Substituting this solution into Eq. (10) at V = v one arrives at the differential equation with respect to v(t) v = v (13) 2t and therefore,

4 310 Electronic Journal of Theoretical Physics 6, No. 20 (2009) v = C t (14) where C is an arbitrary constant. Since v = 0 at t = 0 for any value of C, the solution (14) is consistent with the sharp initial condition for the solution (12) of the corresponding Liouvile equation (11). The solution (14) describes the simplest irreversible motion: it is characterized by the beginning of time where all the trajectories intersect (that results from the violation of Lipchitz condition at t =0, Fig.1), while the backward motion obtained by replacement of t with ( t) leads to imaginary values of velocities. One can notice that the probability density (13) possesses the same properties. For a fixed C, the solution (14) is unstable since d v dv = 1 2t > 0 (15) and therefore, an initial error always grows generating randomness. Initially, at t = 0, this growth is of infinite rate since the Lipchitz condition at this point is violated d v dv at t 0 (16) This type of instability has been introduced and analyzed in [3]. Fig. 1 Stochastic process and probability density Considering first Eq. (14) at fixed C as a sample of the underlying stochastic process (12), and then varying C(ω) (whereω is a variable running over different samples of the stochastic process, and ω V ), one arrives at the whole ensemble characterizing that process, (see Fig. 1). One can verify that, as follows from Eq. (12), [4], the expectation and the variance of this process are, respectively MV =0, DV =2σ 2 t (17) The same results follow from the ensemble (14) at C(ω). Indeed, the first equality in (17) results from symmetry of the ensemble with respect to v = 0; the second one follows from the fact that DV v 2 t (18)

5 Electronic Journal of Theoretical Physics 6, No. 20 (2009) It is interesting to notice that the stochastic process (14) is an alternative to the following Langevin equation, [4] v =Γ(t), MΓ=0, DΓ=σ (19) that corresponds to the same Fokker-Planck equation (11). Here Γ(t)is the Langevin (random) force with zero mean and constant variance σ. The results described in this sub-section can be generalized to n-dimensional case, [5]. 4. Emergence of Global Constraint In order to introduce and illuminate a fundamentally new non-newtonian phenomenon similar to quantum entanglement, let us assume that the function ϕ(ρ) ineq. (5) is invertible, i.e. ρ = ϕ 1 (f). Then as follows from Eq. (7) with reference to the normalization constraint (2) ϕ 1 [ v(ω, t)]dω = 1 (20) Other non-newtonian properties of solution to Eq. (7) such as shock waves in probability space have been studied in [2] and [5]. Similar shock wave effects have been investigated in [6]. Thus, the motions of the particles emerged from instability of Eq. (6) must satisfy the global kinematical constraint (20). It should be emphasized that the concept of a global constraint is one of the main attribute of Newtonian mechanics. It includes such idealizations as a rigid body, an incompressible fluid, an inextensible string and a membrane, a non-slip rolling of a rigid ball over a rigid body, etc. All of those idealizations introduce geometrical or kinematical restrictions to positions or velocities of particles and provides instantaneous speed of propagation of disturbances. However, the global constraint ρdv = 1 (21) is fundamentally different from those listed above for two reasons. Firstly, this constraint is not an idealization, and therefore, it cannot be removed by taking into account more subtle properties of matter such as elasticity, compressibility, etc. Secondly, it imposes restrictions not upon positions or velocities of particles, but upon the probabilities of their positions or velocities, and that is where the non-locality comes from. Continuing this brief review of global constraints, let us discuss the role of the reactions to these constraints, and in particular, let us find the analog of reactions of global constraints in quantum mechanics. One should recall that in an incompressible fluid, the reaction to the global constraint v 0 (expressing non-negative divergence of the velocity v) is a non-negative pressure p 0; in inextensible flexible (one- or twodimensional) bodies, the reaction of the global constraint g ij gij 0,i,j =1,2 (expressing

6 312 Electronic Journal of Theoretical Physics 6, No. 20 (2009) that the components of the metric tensor cannot exceed their initial values) is a nonnegative stress tensor σ ij 0, i, j=1,2. Turning to quantum mechanics and considering the uncertainty inequality ΔxΔp /2 (22) in which Δx and Δp are the standard deviation of coordinate and impulse, respectively as a global constraint, one arrives at the quantum potential 2 2 ρ 2m as a reaction of ρ this constraint in the Madelung equations t + ( ρ S) = 0 (23) m S t +( S)2 + V 2 2 ρ 2m ρ = 0 (24) Here ρ and S are the components of the wave function ϕ = ρe is/h,and is the Planck constant divided by 2π. But since Eq. (23) is actually the Liouville equation, the quantum potential represents a Liouville feedback similar to those introduced above via Eqs. (5) and (9). Due to this topological similarity with quantum mechanics, the models that belong to the same class as those represented by the system (7), (8) are expected to display properties that are closer to quantum rather than to classical ones. 5. Emergence of Entanglement Prior to introduction of the entanglement phenomenon, we will demonstrate additional non-classical effects displayed by the solutions to Eqs. (6) and (7) when ϕ(ρ) =ζρ, ζ = const. > 0 and therefore v = ζρ (25) t + ζ V (ρ2 ) = 0 (26) The solution of Eq. (26) subject to the initial conditions ρ 0 (V ) and the normalization constraint (2) is given in the following implicit form [7] ρ = ρ 0 (λ), V = λ + ρ 0 (λ)t (27) This solution subject to the initial conditions and the normalization constraint, describes propagation of initial distribution of the density ρ 0 (V ) with the speed V that is proportional to the values of this density, i.e. the higher values of ρ propagates faster than lower ones. As a result, any compressive part of the wave, where the propagation velocity is a decreasing function of V, ultimately breaks to give a triple-valued (but still continuous) solution for ρ(v,t). Eventually, this process leads to the formation of strong discontinuities that are related to propagating jumps of the probability density. In the theory of nonlinear waves, this phenomenon is known as the formation of a shock wave. Thus,

7 Electronic Journal of Theoretical Physics 6, No. 20 (2009) as follows from the solution (27), a single-valued continuous probability density spontaneously transforms into a triple-valued, and then, into discontinuous distribution. In aerodynamical application of Eq. (26), when ρ stands for the gas density, these phenomena are eliminated through the model correction: at the small neighborhood of shocks, the gas viscosity ν cannot be ignored, and the model must include the term describing dissipation of mechanical energy. The corrected model is represented by the Burgers equation t + V (ρ2 )=ν 2 ρ (28) V 2 As shown in [7], this equation has continuous single-valued solution (no matter how small is the viscosityν), and that provides a perfect explanation of abnormal behavior of the solution to Eq. (26). Similar correction can be applied to the case when ρ stands for the probability density if one includes Langevin forces Γ(t) into Eq. (25) v = ρ + νγ(t), < Γ(t) >= 0, < Γ(t)Γ(t ) >= 2δ(t t ) (29) Then the corresponding Fokker-Planck equation takes the form (28). It is reasonable to assume that small random forces of strength ν<<1 are always present, and that protects the mathematical model (25), (26) from singularities and multi-valuedness in the same way as it does in the case of aerodynamics. It is interesting to notice that Eq. (28) can be obtained from Eq. (25) in which random force is replaced by additional Liouville feedback v = ζρ ν ln ρ, ζ > 0, ν > 0, (30) V An interesting non-classical property of a solution of this equation is decrease of entropy. Indeed, H t = t ρ ln ρdv = = 1[ ρ 2 (ln ρ +1) ζ 1 ρ(ln ρ +1)dV = ζ 1 ζ V (ρ2 )ln(ρ +1)dV ρdv ]= 1 ζ < 0 (31) Obviously, presence of small diffusion, when ν<<1, does not change the inequality (31) during certain period of time. (However, eventually, for large times, diffusion takes over, and the inequality (31) is reversed). It is easily verifiable that the solution to Eq. (28) satisfies the constraint (2) if the corresponding initial condition does, [7]. Let us concentrate now on the solution of the system (30) and (28) remembering that it is a particular case of the system (6), (7) v = ζρ ν V ln ρ, ζ > 0, [ζ] = 1 sec, [ν] = 1 sec, (32) t + ζ V (ρ2 )=ν 2 ρ V 2, (33)

8 314 Electronic Journal of Theoretical Physics 6, No. 20 (2009) subject to a single-hump initial condition where A is the initial area of the hump, and ρ 0 (V )=Aδ(V ) at t =0, A = const. (34) v(t =0)=v 0 (35) The variable v in Eq. (32) is a dimensionless velocity v v/v 0, and the Reynolds number R = ζ A (36) 2ν We will be interested in the solution of the system (32), (33) for the case of large Reynolds number R, and ζ>>ν (37) In this case, Eq. ( 32) can be simplified by omitting the viscose term v = ζρ, ζ > 0 (38) However, omitting the last term in Eq. (33) would lead to qualitative changes outlined above, and in particular, it would prevent us to start with the sharp initial conditions (34). We will start with the solution to Eq. (33). It is different from the standard Burger s equation only by a physical interpretation of the variable ρ that is now a probability density (instead of density of a gas), but as shown in [7], the constraint (2) is satisfied automatically if it is satisfied for the initial condition (34). Thus, the solution to Eq.(33) subject to the conditions (2), (34) and (37) reads ρ = V ζt in 0 <V < 2Aζt and ρ = 0 outside (39) Fig. 2 The triangular shock wave of probability density and samples of associated stochastic process. The solution has a shock of density 2A [ρ] = ζt at V = 2Aζt (40)

9 Electronic Journal of Theoretical Physics 6, No. 20 (2009) Substituting the solution (39) into equation (38) one obtains v = v t in 0 <v< 2Aζt (41) and whence v =0 in v> 2Aζt (42) v = Ct in 0 <v< 2Aζt (43) v = C in v> 2Aζt (44) We will be interested here only in the region of Eqs. (41) and (43). Here C is an arbitrary constant. Since v =0att =0foranyvalueofC, the solution (43) is consistent with the sharp initial condition (34). For a fixed C, the solution (43) is unstable since d v dv = 1 2t > 0 (45) and therefore, an initial error always grows generating randomness whose probability is controlled by Eq.(33). Initially, at t = 0, this growth is of infinite rate since the Lipschitz condition at this point is violated d v dv at t 0 (46) Considering first Eq. (41) at fixed C as a sample of the underlying stochastic process (39), and then varying C(ω) (whereω is a variable running over different samples of the stochastic process, and ω V ), one arrives at the whole ensemble characterizing that process, (see Fig. 2). The same phenomenon has been observed in the solution to Eq. (13). However, here the stochastic process converges to the attractor represented by the curve (40) on the V t plane where the shock of the probability density occurs (see the red line in Fig. 2). The displacement x corresponding to the velocity v, strictly speaking, includes a diffusion term. However, because of vanishing viscosity (see Eq. (37), this term can be ignored, and therefore, samples with the same initial positions have the same velocities. Let us turn to physical interpretation of the solution. In terms of Eq.(38), the curve (40) is a superposition of different samples of the stochastic process that have the same position and velocity at the same instance of time. However, the accelerations of the superimposed samples, as well as the probability of their occurance, are different. If one introduces a continuous variable μ to distinguish the accelerations (assuming that this variable changes from zero to one, while μ V ), than, as follows from Eq. (38) with reference to Eq. (40) 1 2Aζ v(μ)dμ = ζ[ρ] = (47) t 0

10 316 Electronic Journal of Theoretical Physics 6, No. 20 (2009) This is a global kinematical constraint that bounds the accelerations of those samples of the stochastic process that are superimposed at the same point of the V t plane, (see the red curve in Fig. 2). In order to investigate the properties of the constrained samples (47), let us slightly modify the original system (32), (33) as v = ζρ ν V ln ρ, ζ > 0, if t T, and ζ<0 if t>t (48) t + ζ V (ρ2 )=ν 2 ρ (49) V 2 Now we will be interested in the case ζ<0. Starting from t>t, the triangular shock wave (see Fig. 2) will move backwards as a wave of expansion, and the shock will start dissipating. In order to avoid unnecessary mathematical details, we will concentrate attention only on the area around the shock itself disregarding behavior of the rest area of the triangle. Therefore we can drop viscose terms in both equations since formation of new shock waves will start up-stream away from the old shock, (see the grey area in Fig. 3). Hence, now we will deal with the system v = ζρζ < 0 (50) t + ζ V (ρ2 ) = 0 (51) The solution to these equations is, ρ = 2A ζt, and v = 2A ζt t forρ < V ζt, ρ = V ζt, and v = C(ω)t for 0 < V ζt < 2A ζt ρ =0, and v =0 for V ζt < 0 (52) The first and the last parts of the solution (52) describe the motion in front and in rear of the dispersing shock, (see the grey and the yellow areas, respectively, in Fig. 3). The middle part of the solution (52) describes the process of shock dispersion (see the red fan in Fig.3). The most important property of the motion in this area is the preservation of the global constraint (47): although samples of the stochastic process that occur inside of the fan are not superimposed any more, i.e. they have different locations and different velocities, nevertheless their accelerations are still bounded by the same constraint as those as they had during their superposition, and such a memory expresses qualitative effect of entanglement similar to those in quantum mechanics. However, there is a difference: in quantum mechanics, entanglement is referred to different particles, while in the system introduced above it is referred to different samples of the same particle. To illuminate this effect, assume that we observed some portion of entangled samples of a stochastic process. Then, based upon the global constraint (47), we can predict

11 Electronic Journal of Theoretical Physics 6, No. 20 (2009) Fig. 3 Dispersion of the shock, and associated entangled fan. properties of the rest, never observed, samples of the same stochastic process. It could be expected that extension of the model (48), (49) to multi-dimensional case v i = ζ i ρ ν V i ln ρ, (53) t + ζ i (ρ 2 )=ν 2 ρ, i=1,2,...n (54) V i V 2 would create entanglement of samples of different particles. 6. Discussion and Conclusion The formal mathematical difference between quantum and classical mechanics is better pronounced in the Madelung (rather than the Schrödinger) equation. Two factors contribute to this difference: the scale of the system introduced through the Planck constant and the topology of the Madelung equations (23), (24) that includes the feedback (in the form of the quantum potential) from the Liouville equation to the Hamilton-Jacobi equation. Ignoring the scale factor as well as the concrete form of the feedback, we concentrated upon preserving the topology while varying the types of the feedbacks. As a result, we arrived at a new class of dynamical systems: quantum-classical hybrids. A general approach to the choice of the feedback was introduced and discussed in [2]. More specific feedbacks linked to the behavioral models of Livings were presented in [5] via replacement of quantum potential with information potential. That replacement leads to

12 318 Electronic Journal of Theoretical Physics 6, No. 20 (2009) the capability of the model to evolve from disorder to order (compare to Eq. (31). The computational capabilities of the quantum-classical hybrids have been investigated in [1] where a special feedback computational potential- was selected The most effective way of implementation of quantum-classical hybrid is by means of analog devices such as VLSI chips used for neural net s analog simulations, [10]. Of special importance there is the square root circuit that was extended to the circuit for terminal repeller by Cetin,B, [11]. Thus, the objective of this paper is to demonstrate another fundamental property of quantum-classical hybrids: entanglement. So far this mysterious property that caused many discussions on the highest level of scientific community was considered as a prerogative of quantum mechanics. In this paper we demonstrate that a special form of Liouville feedback, Fig. 4, provides quantitatively similar entanglement effects: different samples of a stochastic process after being superimposed for a certain period of time attain a memory : their accelerations satisfy a global kinematical constraint (47). Fig. 4 Classical physics, Quantum physics, and quantum-classical hybrid. Acknowledgment The research described in this paper was performed at Jet Propulsion Laboratory California Institute of Technology under contract with National Aeronautics and Space Administration. References [1] Zak, M., Quantum-inspired maximizer, Journal of mathematical physics, (in press), 49, [2] Zak,M., Self-supervised dynamical systems, Chaos, Solitons & Fractals, 19, , [3] Zak, M., Terminal Attractors for Associative Memory in Neural Networks, Physics Letters A, Vol. 133, No. 1-2, pp [4] Risken,H., The Fokker-Planck Equation, Springer, N.Y [5] Zak, M., Physics of Life from First Principles, EJTP 4, No.16, December, 2007.

13 Electronic Journal of Theoretical Physics 6, No. 20 (2009) [6] Kulikov, M. Zak, Shock waves in a Bose-Einstein condensate Phys. Rev. A, Vol. 67, , 2003 [7] Whitham, G., Linear and nonlinear waves, Wily-Interscience Publ.;1974. [8] Zak, M., Entanglement-based communications, Chaos, Solitons & Fractals, 2000, 13, [9] Zak, M., Entanglement-based self-organization, Chaos, Solitons & Fractals, 14 (2002), [10] Mead, C, 1989, Analog VLSI and Neural Systems, Addison Wesley. [11] Cetin,B, 1994, TRUST: A new global optimization methodology, application to Artificial neural

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