# Practical Teleportation The Art Of Space Time Transposition

Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a widely recognized concept in philosophy and fiction, particularly science fiction. The idea of a time machine was popularized by H. G. Wells' 1895 novel The Time Machine.[1]

## Practical Teleportation The Art Of Space Time Transposition

It is uncertain if time travel to the past is physically possible, and such travel, if at all feasible, may give rise to questions of causality. Forward time travel, outside the usual sense of the perception of time, is an extensively observed phenomenon and well-understood within the framework of special relativity and general relativity. However, making one body advance or delay more than a few milliseconds compared to another body is not feasible with current technology. As for backward time travel, it is possible to find solutions in general relativity that allow for it, such as a rotating black hole. Traveling to an arbitrary point in spacetime has very limited support in theoretical physics, and is usually connected only with quantum mechanics or wormholes.

Quantum-mechanical phenomena such as quantum teleportation, the EPR paradox, or quantum entanglement might appear to create a mechanism that allows for faster-than-light (FTL) communication or time travel, and in fact some interpretations of quantum mechanics such as the Bohm interpretation presume that some information is being exchanged between particles instantaneously in order to maintain correlations between particles.[45] This effect was referred to as "spooky action at a distance" by Einstein.

Many have argued that the absence of time travelers from the future demonstrates that such technology will never be developed, suggesting that it is impossible. This is analogous to the Fermi paradox related to the absence of evidence of extraterrestrial life. As the absence of extraterrestrial visitors does not categorically prove they do not exist, so the absence of time travelers fails to prove time travel is physically impossible; it might be that time travel is physically possible but is never developed or is cautiously used. Carl Sagan once suggested the possibility that time travelers could be here but are disguising their existence or are not recognized as time travelers.[30] Some versions of general relativity suggest that time travel might only be possible in a region of spacetime that is warped a certain way,[clarification needed] and hence time travelers would not be able to travel back to earlier regions in spacetime, before this region existed. Stephen Hawking stated that this would explain why the world has not already been overrun by "tourists from the future".[51]

Philosophers have discussed the nature of time since at least the time of ancient Greece; for example, Parmenides presented the view that time is an illusion. Centuries later, Isaac Newton supported the idea of absolute time, while his contemporary Gottfried Wilhelm Leibniz maintained that time is only a relation between events and it cannot be expressed independently. The latter approach eventually gave rise to the spacetime of relativity.[69]

The Novikov self-consistency principle, named after Igor Dmitrievich Novikov, states that any actions taken by a time traveler or by an object that travels back in time were part of history all along, and therefore it is impossible for the time traveler to "change" history in any way. The time traveler's actions may be the cause of events in their own past though, which leads to the potential for circular causation, sometimes called a predestination paradox,[83] ontological paradox,[84] or bootstrap paradox.[84][85] The term bootstrap paradox was popularized by Robert A. Heinlein's story "By His Bootstraps".[86] The Novikov self-consistency principle proposes that the local laws of physics in a region of spacetime containing time travelers cannot be any different from the local laws of physics in any other region of spacetime.[87]

We can identify \(L^2=\alpha '\sqrt2\, g_YM^2 N\,\). After this, we can easily see from (1) that the spacetime described by (1) is nothing but \(AdS_5\times S^5.\) The supergravity limit necessarily implies,

Although this conjecture has not been proven yet, it passes several essential checks, such as matching the spectrum of chiral operators and correlation functions. One obtains a precise dictionary between field theory correlators and correlators of fields living inside the AdS spacetime [1,2,3, 117,118,119].Footnote 2 Holography is being used to study hydrodynamic transport coefficients, phase transitions in condensed matter systems, some aspects of QCD, open quantum systems, quantum chaos, black hole information paradox etc [13, 40, 120,121,122,123,124,125,126,127,128].Footnote 3

\(U=0\) and \(V=0\) are the two horizons. From (9), it is evident that the metric is well defined even when either \(U=0\) or \(V=0.\) While doing the coordinate transformation, we implicitly assumed that \(r > r_+,\) thereby making U negative and V positive. Similarly, for the region \(r U will be positive and V will be negative. Then we again end up with the same form of the metric as shown in (9). Finally, the Penrose diagram for the spacetime looks like as shown in Fig. 2.Footnote 5 The spacetime now has four regions, as shown in the Fig. 2. The two singularities occur at \(U\, V=1\,\) \( (r=0)\), and the \(U\, V=-1\,\,\) \((r=\infty )\) corresponds to the two asymptotic AdS boundaries. Combining all four regions, we can now interpret the full two-sided Euclidean BTZ space as a wormhole connecting the two asymptotically-AdS spaces. The wormhole is non-traversable in the sense that no signal can be sent from the region- L to the region R as shown in the Fig. 2, but two people, Alice and Bob, will be able to jump from these two sides and reach the middle point (the bifurcation point where \(U=0\) and \(V=0\) line intersect as shown in the Fig. 2) and exchange notes. Although we have used mainly the BTZ metric, all these analyses can be extended to higher dimensions.

AdS/CFT is a two-way street, we briefly now discuss the dual of this geometry. Within the context of holography, each geometry corresponds to a certain state of the dual field theory. From the boundary point of view, the CFT lives on a space described by two coordinates, both of which are periodic. The space looks like a product of two spheres: \(S_\beta ^1 \times S^d-1.\) \(S_\beta \) is coming from the \(\tau \) coordinate, and \(S^d-1\) is coming from the rest of the angular coordinates. For (eternal) BTZ, we have \(d=2\), and for a constant time slice, the boundary will be the sum of two disconnected spheres \( S^1+ S^1\). Then the Euclidean time direction then connects these two spheres. Then following [77], we can write down the dual state as,

where \( E_i\rangle \) denotes the energy eigenstate of the CFT placed on the sphere, L and R indicate the two asymptotic regions, the sum over i goes over all the eigenstatesFootnote 6 and \(Z(\beta )\) is the thermal partition function for one copy of the CFT. The star denotes the CPT conjugation. From (10) it is evident that this state is an entangled state defined on a Hilbert space of the form \(\mathcal H=\mathcal H_L\otimes \mathcal H_R.\) In general finite dimensional quantum systems, TFD is a useful way to purify a given thermal state, we discuss this aspect in the next Sect. 3.

In a, Alice sends a signal at time \(-t,\) (shown by yellow dashes) then she measures a part of the Hawking radiation and exchanges information with Bob at time \(t=0\) (shown with gray line). This helps Bob send a negative energy shock (shown with solid black). Because of this, the signal reaches Bob at time t due to a Shapiro time advance. This is the essence of quantum teleportation [84, 141, 142]. In b, following [84], the same scenario is depicted in terms of operators, the message \(\varPhi _L(t_L)\) sent by Alice from the left boundary at time \(t_L\), experiences the negative energy shock generated due to the double trace coupling \(\mathcal O_L \mathcal O_R\) at \(t=0\), and finally reaches to Bob \(\varPhi _R(t_R)\) at the right boundary at time \(t_R.\) This diagram is motivated from [83, 84]

So far, in the present section, we have discussed the teleportation through a wormhole from the point of view of the bulk gravity. The coupling and the teleportation in the gravity have a straightforward representation in the boundary theory described by a TFD, wherein coupling the two Hamiltonians, information is teleported from one Hilbert space to another. In quantum simulators, which can realize very general states and engineer interesting evolutions, one can ask the question of the generality of such a gravity-inspired teleportation scheme. We review some recent developments in understanding the underlying mechanism of teleportation and their applicability in general many-body models in Sect. 4. In the next section, Sect. 3, we first set up some useful notations and summarize important results on quantum information scrambling, which makes the basis for the following sections.

An example of a pure state in the doubled Hilbert state is the EPR state. In its most simple form it can be understood as the product of N Bell pairs, \( \mathrm EPR \rangle =( \varPhi ^+ \rangle )^\otimes N\), where \( \varPhi ^+ \rangle =( 00 \rangle + 11 \rangle )/\sqrt2\) is a maximally entangled state between a pair consisting of one qubit from each Hilbert space, here (0, 1) are the computational basis or the qubit basis. This definition can be rewritten using the basis elements of each Hilbert space as