Teleportation

The idea of teleportation comes from science fiction and means  

At first sight it seems that this contradicts quantum mechanics. The Heisenberg uncertainty principle says that it is not possible to prepare a perfect copy of the object, because, in case of mechanical quantities with non-commuting operators (like positions and momenta), there is no way to have them measured exactly, in order to rebuild the system elsewhere with the same values of the quantities. 

The trick is, however, that the quantum teleportation we are going to describe, will not violate the Heisenberg principle, because the mechanical quantities needed will not be measured and the copy made based on their values. 

The teleportation protocol was proposed by Bennett and coworkers, and applied by the Anton Zeilinger group. The latter used the entangled states (EPR effect) of two photons described above.  

Assume that photon A (number 1) from the entangled state belongs to Alice, and photon B (number 2) to Bob. Alice and Bob introduce a common fixed coordinate system. Both photons have identical polarizations in this coordinate system, but neither Alice nor Bob know which. Alice may measure the polarization of her photon and send this information to Bob, who may prepare his photon in that state. This, however, does not amount to teleportation, because the original state could be a linear combination of the 0 (parallel) and 1) (perpendicular) states, and in such a case Alice s measurement would falsify the state due to wave function collapse (it would give either 0 or 1 ), cf. p. 23. 

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Besides quantum computations, entanglement has also been at the core of other active research such as quantum teleportation [32, 33], dense coding [34, 35], quantum communication [36], and quantum cryptography [37]. It is believed that the conceptual puzzles posed by entanglement have now become a physical source of novel ideas that might result in applications. 

D. Bouwmeester, A.K. Ekert, A. Zeilinger, The Physics of Quantum, Information Quantum Cryptography, Quantum Teleportation, Quantum, Computation (Springer, 2000)... 

Following is a highly idealized account of how one might teleport the quantum state of a boson. Consider a boson which has just two possible states, say, x and y—like a photon with its two polarizations. Suppose that we have boson 1 in an arbitrary state I (l) = a x ) b y ), where a and b are complex coefficients. We let this state become entangled with the two-boson state x2, which is the... 

The result is that the state T (l) = a xi) - - ji) of particle 1 has been teleported to an identical state I (3) = a x3) - - b yi) of particle 3. The original quantum state of the first particle is destroyed in the process. Note that the coefficients a and b, hence the precise quantum state being teleported, need never be known. 

Group theory aspects of information processing in quantum ensembles is discussed in the first paper of this chapter. It presents the basic concepts of the information processing and teleportation, based on the group theory approach. It also addresses the question whether the information processing can be carried out when the ensemble of qubits is not in a pure quantum state, being subject to thermalization prior to the quantum computation. 

So indeed, ever since the scheme was invented in 1997 there have been dozens of papers with a large impact (here is just a small selection) with algorithms performed including Grover [Jones 1998], Deutsch-Jozsa [Chuang 1998], Shor [Vandersypen 2001] recently and even teleportation [Nielsen 1998],... 

Finally, I would like to say a few words about entanglement in communication, as the last caveat to the Jozsa-Linden theorem. I will just use the example of quantum teleportation. 

In quantum teleportation, Alice is given a state pm = ip)(ip whose identity is unknown to her. She may do anything she wishes to this state and then she communicates with Bob via only a classical communication channel. Bob s aim is to create a state pout which best resembles the original state. 

In the absence of shared entanglement between Alice and Bob the mean fidelity of the output state F = (t/.jpout t/, ) is bounded. For example, for teleporting qubits, F < 2/3, whereas for the teleportation of coherent states in an infinite-dimensional Hilbert space it is bounded by F < 1/2 [Braunstein 2000 (b)]. 

Fidelities beating these bounds have been achieved experimentally when the sender and receiver share entanglement. So teleportation is a success and entanglement really matters. 

To close the loop, teleportation and indeed many other quantum communication protocols do not need a universal set of gates and do quite well with just those gates covered by the Gottesman-Knill theorem [Bartlett 2002], Although these protocols may be easily simulated, they have capabilities beyond classical communication channels - clearly simulation is not everything. 

Entanglement is a vital information resource employed in quantum teleportation, dense coding and quantum computation [Nielsen 2000], The fundamental role played by the entanglement in quantum information science was discussed in part I this part of the book is devoted to the generation and characterization of the entanglement of photons and their usage in quantum communication and computation protocols. 

To summarize, we have studied the interaction of two weak quantum fields with an optically dense medium of coherently driven four-level atoms in tripod configuration. We have presented a detailed semiclassical as well as quantum analysis of the system. The main conclusion that has emerged from this study is that optically dense vapors of tripod atoms are capable of realizing a novel regime of symmetric, extremely efficient nonlinear interaction of two multimode single-photon pulses, whereby the combined state of the system acquires a large conditional phase shift that can easily exceed 1r. Thus our scheme may pave the way to photon-based quantum information applications, such as deterministic all-optical quantum computation, dense coding and teleportation [Nielsen 2000]. We have also analyzed the behavior of the multimode coherent state and shown that the restriction on the classical correspondence of the coherent states severely limits their usefulness for QI applications. 

Approximate versions of the translational EPR state, wherein the -function correlations are replaced by finite-width (Gaussian) distributions, have been shown to characterize the quadratures of the two optical-field outputs of parametric down-conversion, or of a fiber interferometer with Kerr nonlinearity. Such states allow for various schemes of continuous-variable quantum information processing such as quantum teleportation [Braunstein 1998 (b) Furu-sawa 1998] or quantum cryptography [Silberhorn 2002], A similar state has also been predicted and realized using collective spins of large atomic samples [Polzik 1999 Julsgaard 2001]. It has been shown that if suitable interaction schemes can be realized, continuous-variable quantum states of the original EPR type could even serve for quantum computation. 

We plan to extend these experiments to a set-up with two identical cavities separated by several centimeters. In this way, we hope to study mesoscopic field superpositions having a non-local character. Tens of photons could be put in state superpositions involving the two cavities, revealing many situations unusual to a classical mind. Teleportation of the quantum state of a material particle at macroscopic distances could be achieved in this way, as well as other demonstrations of nonlocality [Milman 2004],... 

Quantum teleportation was first proposed in 1993 [Bennett 1993] and the year after for the special case of continuous variables [Vaidman 1994]. Teleportation is extremely important since direct transport of physical states is often hindered by exponential decoherence. With quantum teleportation the information is cleanly separated into a classical part, which can be transmitted over arbitrary distances, and a quantum mechanical part, which only needs to interact locally. 

A proposal for spin state teleportation was given in Ref. [Duan 2000 (d)]. Three atomic samples are needed as shown in Fig. 3 (a). Adjacent samples are oriented oppositely along the rc-axis so that both collective measurements on cells 1 and 2 and on cells 1 and 3 will be regular entangling interactions as... 

Figure 3. (a) Teleporting an unknown quantum state first cells 1 and 2 are entangled. Then... 

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