aa r X i v : . [ phy s i c s . g e n - ph ] M a r AdS/CFT correspondence: the fountain of quantumyouth
Ovidiu Racorean
General Direction of Information TechnologyBanul Antonache str. 52-60, sc.C, ap.19, Bucharest, RomaniaE-mail: [email protected]
Abstract.
We argue, in the context of AdS/CFT correspondence, that the structure of thegeometry dual to two entangled CFTs is a time non-orientable spacetime. Further, weelevate this argument to whatever entangled quantum systems. Accordingly, we shouldexpect that entangled quantum systems (particles in subsidiary) to not experience theflow of time. As a result, the lifetime of entangled particles should be considerablylonger than that of their unentangled counterparts.
Essay written for the Gravity Research Foundation2020 Awards for Essays on GravitationMarch 13, 2020uantum entanglement is one of the most counter-intuitive properties of quantumsystems. Einstein, Podolsky and Rosen used entanglement to argue that quantummechanics predicts a breakdown in locality [1]. In recent years, the concept ofentanglement is applying into the understanding of phenomena related the to highenergy and gravity physics. This leads to important discoveries about the connectionof entanglement and geometry. It has been proposed in [2], [3], that spacetimeconnectedness in AdS is related to quantum entanglement in the dual field theory.Moreover, this occurrence was elevated in [4], [5], [6] to a general principle: whenevertwo quantum systems are entangled, there should be some version of a physical spacetimeconnection between them.In this essay we will argue that if this scenario is correct, than the spacetimeconnecting the two quantum systems should be time non-orientable. This statement isbased on another interesting insight into the phenomenology of quantum entanglementthat comes from emergence of the thermodynamic arrow of time [7], [8], [9], in thepresence of initial high-correlations between the quantum systems. On this line of work,we previously argued [10], in the context of AdS/CFT correspondence, that the globalspacetime dual to quantum entanglement in the field theory must be time non-orientable.The argument of time non-orientability of the global spacetime dual to anyentangled quantum systems has one notable consequence: the entangled pair is notexperiencing the flow of time. As a result, the lifetime of quantum entangled particlesshould be considerably longer than their unentangled counterparts.To motivate this result, let us start considering two non-interacting copies of CFTon sphere S d noted left ( L ) and right ( R ), such that we can decompose Hilbert space H LR of the composite system as H LR = H L ⊗ H R . We assume that the two subsystemsare initially uncorrelated to begin with, such that the joint state of the system is theproduct state, ρ LR = ρ L ⊗ ρ R , with ρ L , ρ R and ρ LR the density matrix of the left,right ant the composite system, respectively, corresponding to the thermal states of thefield theory. From this initial product state the correlations between the two quantumsystems can only increase.We employ here as a natural measure of the correlations between the two subsystemsthe mutual information: I ( ρ LR ) = S ( ρ L ) + S ( ρ R ) − S ( ρ LR ) , (1)where S ( ρ L ), S ( ρ R ) and S ( ρ LR ) are the entropies of the left, right and the compositesystem, respectively, defined as the von Neumann entropy, S ( ρ ) = − T r ( ρlogρ ).Since we are in a low-entropy environment, I ( ρ LR ) = 0, which reduces the Eq. (1)to: S ( ρ LR ) = S ( ρ L ) + S ( ρ R ) . (2)As a result, the mutual information and consequently the entropy of the compositesystem can only increase. To see this, let us now consider entangling some of thedegrees of freedom of the individual components, evolving in this way the entropy2f the joint state from the initial, S i ( ρ LR ) = S i ( ρ L ) + S i ( ρ R ), to the final entropy S f ( ρ LR ) ≤ S f ( ρ L ) + S f ( ρ R ). The initial and final states of the composite system, arerelated unitarily, thus S f ( ρ LR ) = S i ( ρ LR ), such that we have∆ S ( ρ L ) + ∆ S ( ρ R ) ≥ . (3)This result is consistent with the second law of thermodynamics which states thatthe entropy of an isolated system can only increase. In this case, in accordance withthe formulation of the second law, the flow of time is directed toward the standardthermodynamic arrow.On the gravity side of the AdS/CFT correspondence, the interpretation of thisinitially uncorrelated state is straightforward [3]. The two separate physical systemsdetermined by the density matrix ρ L and ρ R , correspond in the dual geometricdescription, to a global time-oriented spacetime having two separate asymptoticallyAdS spacetime.We consider now that the two copies of the field theory are high-correlated, tobegin with. In this scenario the joint state of the two subsystems can be represented,as ρ LR = | Ψ β i h Ψ β | , with | Ψ β i defined as the thermofield double state, | Ψ β i = 1 p Z β X n e − βEn | E n i L | E n i R , (4)where | E n i L ( | E n i R ) is the n-th energy eigenvector for the subsystem L ( R ) , β isthe inverse temperature and Z β ( − is a normalization constant. We may remark herethat the two subsystems are initially entangled in a pure state. Note however that theindividual states of the two subsystems are the thermal states.In stark contrast to initially uncorrelated case, we can emphasize here that theinitial pure state of the two copies of the field theory ensures that ρ L and ρ R areisospectral so that S ( ρ L ) = S ( ρ R ). Let us now start from the thermofield doublestate and gradually disentangle the degrees of freedom. The initial entanglement ofthe composite system forces the individual entropies S ( ρ L ) and S ( ρ R ) to move in thesame direction, such that, ∆ S ( ρ L ) = ∆ S ( ρ R ), at all times. In addition, we can say thatthe initial pure state, implies that the individual entropies can only decrease, such that∆ S ( ρ L ) = ∆ S ( ρ R ) ≤ S ( ρ L ) + ∆ S ( ρ R ) ≤ , (5)a result which suggests that ,from the perspective of the second law ofthermodynamics, both orientations of the thermodynamic arrow are allowed, such thatthere is no opportunity for the dominance of one direction of time over the other [7],[8], [9].The initial high correlations of the two CFTs ensure that in the dual spacetimethe arrow of time may be oriented normal, from the past to the future or in reverse,from the future to the past. In this context, since there is no preferred orientation of3ime on the spacetime, we may conjecture that the gravity dual is a time non-orientablespacetime, [10].Let us now consider a scenario in which we start with the two copies of the fieldtheory entangled in the thermofield double state and gradually disentangle the degreesof freedom to zero. It has been pointed out in [3], that as the entanglement between Land R CFTs decreases to zero, the mutual information also goes to zero, I ( ρ L R ) = 0. Asa result, we remain in the end with two completely separate physical systems which donot interact, such that the joint state of the system is the product state ρ LR = ρ L ⊗ ρ R ,as in the case of initially uncorrelated CFTs.From the gravity dual perspective, the above scenario is translated in the followingstatement: starting with a global time non-orientable spacetime and decreaseing tozero the entanglement between the dual copies of CFT, the resulting spacetime is timeorientable. Roughly speaking, disentangling the degrees of freedom between two copiesof field theory implies, on the geometry side, a transition from a time non-orientablespacetime to a spacetime having a definite orientation of time, thus a time-orientablespacetime.Let us now assume in the spirit of [4], [5], [6] that the structure of global spacetimeplay a role in the non-gravitational systems. Thus even the singlet state of particlespair spins should be related by some version of a physical spacetime connection betweenthem. As we have argued in the discussion above the global spacetime the entangledpair lives in must be a time non-orientable spacetime. This statement as abstract as itmight be does have remarkable consequences.As a result of the absence of a preferred orientation of time on time non-orientablespacetime manifolds, the entangled pair ensemble does not evolve in time. In otherwords, the entangled pair is not experiencing the flow of time. We can think of someproperties devolving from here, like a possible resolution of the non-locality of entangledquantum states. Here we like to draw the reader attention to another particularityimplied by the time non-orientability of the spacetime dual to entangle d quantumsystems. The absence of the flow of time is reflected in the lifetime of the entangled pair.Accordingly, when particles are considered, the lifetime of quantum entangled particlesshould be considerably longer than the lifetime of their unentangled counterparts.To see this more clearly, let us destroy (by some measurements) the entanglementbetween the pair of particles. From the point of view of the geometry, there is a transitionto a time orientable spacetime, in such a way that the unentangled particles live nowon a time-orientable spacetime. The, now, unentangled pair will experience the flow oftime and consequently all phenomenon associated to motion.We conclude the essay by pointing out that measuring the lifetime of entangledparticle compared to the lifetime of the same unentangled particles could shed light onsome theoretical aspects of AdS/CFT duality.4 eferences [1] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical RealityBe Considered Complete?, Phys. Rev. 47, 777 (1935).[2] J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021.[3] M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42, 2323(2010) [arXiv:1005.3035 [hep-th].[4] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, arXiv:1306.0533 [hep-th].[5] K. Jensen and A. Karch, Holographic Dual of an Einstein-Podolsky-Rosen Pair has a Wormhole,Phys. Rev. Lett. 111, 211602 (2013).[6] J. Sonner, Holographic Schwinger Effect and the Geometry of Entanglement, Phys. Rev. Lett. 111,211603, 2013;[7] M. H. Partovi, Entanglement versus stosszahlansatz: disappearance of the thermodynamic arrowin a highcorrelation environment. Phys. Rev. E 77, 021110 (2008).[8] L. Maccone, Quantum solution to the arrow-of-time dilemma. Phys. Rev. Lett. 103, 080401 (2009).[9] K. Micadei, J. P. S. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, T. B.Batalho, R. M. Serra, and E. Lutz, Reversing the thermodynamic arrow of time using quantumcorrelations, arXiv:1711.03323v1 (2017).[10] O. Racorean, Quantum entanglement, two-sided spacetimes and the thermodynamic arrow of time,arXiv:1904.04012, (2019).[1] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical RealityBe Considered Complete?, Phys. Rev. 47, 777 (1935).[2] J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021.[3] M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42, 2323(2010) [arXiv:1005.3035 [hep-th].[4] J. Maldacena and L. Susskind, Cool horizons for entangled black holes, arXiv:1306.0533 [hep-th].[5] K. Jensen and A. Karch, Holographic Dual of an Einstein-Podolsky-Rosen Pair has a Wormhole,Phys. Rev. Lett. 111, 211602 (2013).[6] J. Sonner, Holographic Schwinger Effect and the Geometry of Entanglement, Phys. Rev. Lett. 111,211603, 2013;[7] M. H. Partovi, Entanglement versus stosszahlansatz: disappearance of the thermodynamic arrowin a highcorrelation environment. Phys. Rev. E 77, 021110 (2008).[8] L. Maccone, Quantum solution to the arrow-of-time dilemma. Phys. Rev. Lett. 103, 080401 (2009).[9] K. Micadei, J. P. S. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, T. B.Batalho, R. M. Serra, and E. Lutz, Reversing the thermodynamic arrow of time using quantumcorrelations, arXiv:1711.03323v1 (2017).[10] O. Racorean, Quantum entanglement, two-sided spacetimes and the thermodynamic arrow of time,arXiv:1904.04012, (2019).