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Gravity as Quantum Entanglement Force

We conjecture that the total quantum entanglement of matter and vacuum in the universe tends to increase with time, like entropy, and that an effective force is associated with this tendency. We also suggest that gravity and dark energy are types of quantum entanglement forces, similar to Verlinde's entropic force, and give holographic dark energy with an equation of state comparable to current observational data. This connection between quantum entanglement and gravity could give some new insights into the origins of gravity, dark energy, and the arrow of time.


I. INTRODUCTION
Recently proposed Verlinde's idea [1] linking gravitational force to entropic force attracted much attention [2][3][4][5][6][7][8][9]. He derived the Newton's equation and the Einstein equation from this relation. Padmanabhan also proposed a similar idea [10]. However, Verlinde's proposal is based on some rather unusual assumptions such as the proportionality of the holographic entropy on the distance, and the holographic principle holding on equipotential surfaces.
In this paper, we conjecture that in general quantum entanglement of matter or the vacuum in the universe increases like the entropy and there is a new kind of force ('quantum entanglement force', henceforth) associated with this tendency similar to the entropic force. (This force is different from the 'entanglement force' of polymer science.) From this perspective it is also suggested that gravity and dark energy are types of the quantum entanglement force associated with the increase of the entanglement, similar to Verlinde's entropic force linked to the second law of thermodynamics. Our model relies on the well established quantum entanglement theory and uses less assumptions.
In a series of works [11][12][13][14][15] we have investigated the quantum informational nature of gravity, using especially the quantum entanglement and the Landauer's principle. Using the concepts, we suggested that dark energy is related to the quantum entanglement of the vacuum fluctuation [11] at a cosmic horizon [16,17], and derived the first law of black hole thermodynamics from the second law of thermodynamics [12]. Recently, we also suggested [14,15] that classical Einstein gravity can be derived by considering quantum entanglement entropy and an information erasure at Rindler horizons and Jacobson's idea linking the Einstein equation to thermodynamics. All our results imply that gravity has something to do with quantum information, especially quantum entanglement.
In Sec. II we discuss the relation between entanglement and the holographic principle. In Sec. III we introduce the concept of quantum entanglement force and suggest that gravity is a quantum entanglement force. In Sec. VI the predictions of our dark energy theory are compared with the recent observational data. Section V contains discussions.

II. ENTANGLEMENT AND HOLOGRAPHY
In quantum information science, quantum entanglement is a central concept and a precious resource allowing various quantum information processing such as quantum key distribution. The entanglement is a quantum nonlocal correlation which can not be prepared by local operations and classical communication. For pure states the entanglement entropy S Ent is a good measure of entanglement. For a bipartite system AB described by a full density matrix ρ AB , S Ent is the von Neumann entropy S Ent = −T r(ρ A lnρ A ) for a reduced density matrix ρ A ≡ T r B ρ AB obtained by partial tracing part B.
As a typical example in quantum field theory, consider a massless scalar field φ in a flat spacetime with Hamiltonian [18] where π(x) is the momentum of the field. For a spherical region as shown in Fig. 1, one can expand the field with spherical harmonics on a discrete radial coordinate with a UV-cutoff to obtain an effective Hamiltonian for discretized field oscillators φ lmj [18], which contains terms like φ lmj * φ lm(j+1) , where lmj are angular and radial indices. These terms represent a nearest neighbor interaction along the r direction. One can find similar terms generating entanglement between two regions for more general spacetime and fields. The entanglement of the generic quantum field vacuum was also shown by using the Reeh-Schlieder theorem [19,20]. In general, the vacuum entanglement entropy of a spherical region with a radius r with quantum fields can be expressed in the form where β is an O(1) constant that depends on the nature of the field and b is the UV-cutoff. By performing numerical calculations on a sphere lattice Srednicki obtained a value β = 0.3 for the massless filed. A similar value was obtained for the entanglement entropy for a massless scalar field in the Friedmann universe in Ref. [21,22]. More generally we may add up contributions from other fields with β = β j [21]. If there are N j spin degrees of freedom of the j-th field, this implies M r m FIG. 1. The space around a massive object with mass M can be divided into two subspaces, the inside and the outside of an imaginary spherical surface with a radius r. The surface Σ has the entanglement entropy Sent ∝ r 2 and entanglement energy Eent ≡ Σ TentdSent. If there is a test particle with mass m, it feels an effective attractive force in the direction of increase of entanglement.
where the Planck length l P = G/c 3 . If we choose b = 1/M P , where M P is the reduced Planck mass, then It will be very important to obtain the value of α by explicit calculation in the future. It was conjectured that the Bekenstein-Hawking Entropy saturates the information bound that a region of space with a surface area A can contain [23]. If S ent saturates this bound, i.e., S ent = S BH , then α = π. Why are we considering the quantum entanglement as an essential concept for gravity? First, there are interesting similarities between the holographic entropy and the entanglement entropy of a given surface. Both are proportional to its area in general and related to quantum nonlocality. Second, when there is a gravitational force, there is always a Rindler horizon for some observers, which acts as information barrier for the observers. This can lead to ignorance of information beyond the horizons, and the lost information can be described by the entanglement entropy [14]. The spacetime should bend itself so that the increase of the entanglement entropy compensate the lost information of matter. In [14] we suggested that the Einstein equation is just an equation describing this relation. Third, if we use the entanglement entropy of quantum fields instead of thermal entropy of the holographic screen, we can understand the microstates of the screen and explicitly calculate, in principle, relevant physical quantities using the quantum field theory in the curved spacetime. The microstates can be thought of as just quantum fields on the surface or its discretized oscillators. Finally, identifying the holographic entropy as the entanglement entropy could explain why the derivations of the Einstein equation is involved with entropy, the Planck constant and, hence, quantum mechanics. All these facts indicate that quantum mechanics and gravity has an intrinsic connection, and the holographic principle itself has something to do with quantum entanglement.

III. ENTANGLEMENT ENERGY AND ENTANGLEMENT FORCE
Separable (i.e., not entangled) states are fragile in the sense that the states can be easily entangled with environments surrounding the states. A well known example is the Schrödinger cat paradox. In the paradox, no matter how well we seperate a box which contains the cat from its environment, one can not fully block the information leakage of the cat toward the environment outside the box. Thus, even if we carefully prepared the superposition of the cat's state |dead cat + |live cat , the state easily gets entangled with the environment to be |dead cat |env 0 + |live cat |env 1 , where |env 0 and |env 1 represent the corresponding states of the environment. This entanglement between the cat and the environment induces 'decoherence' of the cat's effective density matrix when we trace out the environment states.
Although this process is reversible in principle, practically the reverse process is hardly observable in the macroscopic scale. Ironically, this is one of the reasons why it is so hard to observe controllable quantum entanglement in a laboratory and to build a practical quantum computer using the states. The quantum system of interest gets so easily entangled with its environment uncontrollably and loses coherence within the system. Decoherence is also related to the emergence of the classical world from the quantum world [24].
It might seem strange that the entanglement in the universe is increasing in general, considering that quantum evolution of a density matrix is described by using a unitary matrix U as ρ → U † ρU , which is reversible. This paradoxical situation is very similar to the case with the entropy; Even though the Schrödinger equation and the Einstein equation is time reversible, we see many time irreversible phenomena in the macroscopic world, and total entropy always does not decrease. One way of handling this 'Loschmidt's paradox' is assuming that the early universe had a very small entropy due to, for example, inflation. Similarly we can assume that the early universe began with very small entanglement too. Thus, we can expect that the universe has a strong tendency to increase entanglement among its constituents (matter, quantum field, spacetime) as well as entropy. This might give us some new insights on the issues of arrow of time. The direction of the time (i.e., arrow of time) seems to be the direction in which the total entanglement of the universe increases. That is, for a sufficiently large macroscopic system and its environment. In [13] we argued that the time evolution of the universe is related to the expansion of the cosmic event horizon and its entanglement entropy. Then, what is the relation between gravity and the entanglement? In [11], it was pointed out that a cosmic horizon has a kind of thermal energy called entanglement energy related to S ent and suggested that it is the origin of dark energy. (Here, we explicitly denoted the Boltzman constant k B for clearness.) This energy can be interpreted as effective energy obtained by tracing out the Hilbert space describing the outside of the horizon. It is also the energy of the vacuum fluctuation around the horizon. In [14] we pointed out that this energy is very similar to the equipartition energy of the horizon [1,25]. If this energy is a function of parameters r i , one can define a generalized force which is similar to entropic force. We call this force a 'quantum entanglement force'. Now, we conjecture that quantum entanglement of matter and vacuum in the universe tend to increase over time like entropy, and that gravity is a kind of this quantum entanglement force, similar to Verlinde's entropic force.
From now on, we reinterpret the Verlinde's theory in terms of our entanglement theory. To do this we consider a situation in Fig. 1. First, one can integrate Eq. (7) on the isothermal spherical surface Σ with radius r surrounding a mass M where we have used the Unruh temperature for T ent . In [14] we identified T U as the Rindler horizon temperature observed by a test particle under the influence of the mass M . For the holographic condition E = M c 2 to hold on the surface, α should be 2π, which exceeds the Bekenstein bound. This discrepancy could be removed by using the relation E = 2k B T S by Padmanabhan [26], which seems to be valid when there is an active gravitational mass. In that case we recover α = π. Following the Verlinde's approach, we express the tendency of maximizing entanglement by the condition where e V (r) denotes the gravitational redshift. Then, this equation leads to where we have used Eq. (7). In the weak gravity limit GM ≪ r , V ≃ GM/r, e V ≃ 1, and which is just the Newton's gravity as Verlinde showed with the thermal entropy instead of S ent . Since we used the gravitational redshift for the derivation, the appearance of the Newton's gravity is not so surprising. What we want to see here is the relation between gravity felt by the test particle and quantum entanglement of the whole system. The test particle moves in a way that the total entanglement of the system maximizes.

IV. ENTANGLEMENT AND DARK ENERGY
In [11], we suggested that a cosmic causal horizon with a radius R h ∼ O(H −1 ) has a kind of thermal energy E h ∼ T h S h ∝ R h , and this energy is the dark energy. Here H is the Hubble parameter, T h is the horizon temperature, and S h is its entropy. To be specific, we considered the entanglement energy E ent associated with the cosmic event horizon for E h . (Similar suggestions based on the Verlinde's idea [27][28][29][30] appeared recently.) In this chapter we will redo the calculation in [11] except that we integrate Eq. (7) on the horizon surface instead of the radial direction. We will see that this gives a factor 2 smaller E ent than that in [11].
As before, by integrating dE ent on the surface of the event horizon we obtain where we chose T ent = c/2πk B R h , the Hawking-Gibbons temperature of the horizon. Using Eq. (8) we find that this dark energy corresponds to a constant quantum entanglement force which makes the cosmological horizon expand. (A similar value for the entropic force was obtained independently by Easson et al [30] using a surface action.) Thus, we can say that dark energy is an effective force of the universe associated with the increase of the quantum entanglement of the universe, or the radius of the cosmic causal horizon. Now, the entanglement energy density of the event horizon is given by which has the form of the holographic dark energy [31]. From the above equation we immediately obtain a formula for the constant If S ent saturates the Bekenstein bound, then α = π, hence d = 1. The constant d determining the equation of state w Λ of the dark energy and the final fate of the universe, had been constrained only by observations before our work in [11]. Theoretically, the value d = 1 is favored, because it reproduces the de Sitter universe when the dark energy dominates.
One can compare predictions of our theory directly with current observational data. The equation of state for holographic dark energy is given by [31,32] and its change rate at the present is [31,33] where z is the redshift parameter, Ω 0 Λ is the present value of the density parameter for the dark energy, w Λ (z) ≃ w 0 + w 1 (1 − R) and R is the scale factor of the universe. For Ω 0 Λ = 0.73 and d = 1 these equations give w 0 = −0.903 and w 1 = 0.208. These theoretical values for ω Λ are comparable to the current observational data. Although the cosmological constant is most favored by observations, there is still a large range of values allowed by the data for the time dependent w Λ . For example, the combination of WMAP(Wilkinson Microwave Anisotropy Probe) 7-year data, the baryon acoustic oscillation, Type Ia supernovae, and the Hubble constant yields ω 0 = −0.93 ± 0.13 and w 1 = −0.41 +0.72 −0.71 at the 68% CL [34]. (Note that the observational uncertainty for w 1 is still large, and w 1 was 0.11 ± 0.7 in WMAP-5 year data [35].) Alternatively, we can rely on the quantum field theory to avoid the use of the Bekenstein bound. If we use the approximation β j ≃ 0.3 for all j and use j N j = 118 of the standard model of particle physics in Eq. (4), we obtain d = 0.67, w 0 ≃ −1.18 and w 1 ≃ 0.407. For the minimal supersymmetric standard model, j N j = 244, d = 0.962, ω 0 ≃ −0.925 and ω 1 ≃ 0.22. Thus, our theory with quantum field theory is still in good agreement with the observational data, and favors supersymmetric theories than non-supersymmetric ones.

V. DISCUSSIONS
In this work, we have tried to reconcile Verlinde's theory with our theory based on quantum entanglement. There are many similarities between two theories. If we identify the horizon entropy as the entanglement entropy and the equipartition energy as the entanglement energy, we can give the theory a better ground.
We conjectured that quantum entanglement of matter and fields in the universe tend to increase over time, and there is an effective force associated with this tendency. This force might be very general in the nature and we expect someday one may measure this force during a quantum information experiments using quantum optics or solid state quantum devices.
It is also suggested that dark energy and, more fundamentally, gravity itself are quantum entanglement forces, similar to Verlinde's entropic force. If the entanglement entropy of the universe saturates the Bekenstein bound, this gives the holographic dark energy with equation of state consistent with current observational data. Our quantum informational interpretation on gravity may provide some new insights on the nature of gravity, dark energy, and arrow of time.