# Quantum teleportation

Quantum teleportation is a technique discussed in quantum information science to transfer a quantum state to an arbitrarily distant location using an entangled state and the transmission of some classical information.

 Contents

## Background

In quantum mechanics, the state of a system is described by a mathematical entity called the wave function or state vector. (A system can mean basically any physical object, but think of particles such as photons, atoms, ions or molecules, maybe more than just one.)

According to Heisenberg's uncertainty principle, it is impossible even in principle to completely measure the state vector of an individual quantum system, as its different constituents ("amplitudes") are complementary: If you measure one quantity to precisely obtain its value, you necessarily disturb (change) some other quantity so that you cannot find out its value. Hence, it is impossible to get all the "quantum information", which completely describes a system, out of it.

This makes it impossible to create an exact duplicate of it. The no-cloning theorem asserts that this still holds if you try to find other techniques than measuring and recreating the system.

Surprisingly, you can however transfer the complete state of a system onto a similar system which is at some other place. The original system loses its state, of course, in order to follow the no-cloning postulate.

## Indistinguishability

Let's say that Alice has a rubidium atom (the element physicists in this field like to use for their experiments), which is in its ground state, and Bob also has such an atom, as well in its ground state. It is important to see that these two atoms are indistinguishable; that means that there really is no difference between them.

If Alice and Bob had, say, two glass balls, which exactly look alike, and they exchanged them, then something would change. If you had a powerful microscope, you could certainly find some difference between the two balls. For atoms of the same kind and in the same quantum state, however, there really is no difference at all. The physical situation with Alice having the first atom and Bob the second is exactly the same as vice versa.1 In a certain sense, it is even wrong to say that the two atoms have any individuality or identity. It would be more appropriate to say that the two locations in space both have the property that the fundamental quantum fields have those values which define the ground state of the rubidium atom.

## Quantum teleportation: the result

Now, imagine Alice's atom being in some complicated (excited) quantum state. Assume that we do not know this quantum state — and of course, we cannot find out by inspection (measurement). But what we can do is to teleport the quantum state to Bob's rubidium atom. After this operation, Bob's atom is exactly in the state that Alice's atom was before.

Now note that Bob's atom afterwards is indistinguishable from the Alice's atom before. In a way, the two are the same — because it does not make sense to claim that two atoms are different only because they are at different locations. If Alice had gone to Bob and given him the atom we would have exactly the same physical situation.

But Alice and Bob were not required to meet. They only needed to share entanglement.

To see what this means, let us abstract to qubits. The atoms are now in states of the form α |ground state> + β |first excited state> [itex] =: \alpha |0\rangle + \beta|1\rangle[itex] (using bra-ket notation).

## The method

As a prerequisite, Bob has produced two particles (or atoms, to stay with the concrete example) called I and II, which are maximally entangled, e.g. in the Bell state [itex]|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_I \otimes |0\rangle_{II} + (|1\rangle_I \otimes |1\rangle_{II})[itex].

This means that the particles I and II are neither in the state |0> nor in |1> but rather in both simultaneously, and if you measure one of them you will find it to be |0> or |1> each with probability 1/2. But the other is then always in the same state, and this connects them by some "spooky action at a distance", as Einstein has put it.

Bob passed particle I to Alice (by moving it there through a quantum channel) and kept particle II. From now on, Alice can send a quantum state to Bob, whenever she wants (or Bob to Alice).

If Alice now has a particle which she wants to teleport to Bob, she does a so-called Bell measurement on the particle to be sent and on particle I. The Bell measurement (for details see the article on it) projects Alice's two particle into one of the four Bell states. The two measured particles are afterwards in a known entangled state, specified by the measurement result, and the particle to be sent has lost its former state. That state, however, is not gone: it was teleported to Bob's particle II due to the previously existing entanglement.

One problem remains: The state of Bob's particle might be "rotated". This depends on the outcome of Alice's measurement, and Alice has to tell Bob this outcome by using some classical (i.e. ordinary) communication channel and Bob will then apply a unitary operation (a so-called Pauli operator) onto his particle.2 Afterwards, he really has the desired state.

## Experiments

The first experimental demonstration of entanglement was carried out at the University of Innsbruck in 1997, by the group of Anton Zeilinger. Their setup allowed to teleport the polarization quantum state of photons across an optical table. As source of entanglement they used parametric down-conversion. A slightly earlier experiment in Rome, by the group of DeMartini, showed the principle of teleportation, however it involved only two particles, instead of three: one carrying the state to be teleported, and two entangled ones, which provide the "quantum channel" and must be initially independent of the one carrying the state.

In 2004, another research group at the University of Innsbruck (Austria) and a group at NIST (Boulder, Colorado) demonstrated the teleportation of atoms: The electronic quantum state of a calcium (Innsbruck) or beryllium (Boulder) ion was teleported to another one, with all the ions being held in a linear ion trap.

## Star Trek beaming

That the term teleportation reminds of the beaming process in the Star Trek TV series might have been intentional. After all, the physicists coining the term might have thought, it is as if the particle would be brought there. But this analogue is prone to miss the point: Only the information about the quantum state is brought there, the particle to take up the state must already be present.

## Entanglement swapping

If a state being teleported is itself entangled with another state, the entanglement is teleported with it. To illustrate: If Alice has a particle which is entangled with a particle owned by Zeke, and she teleports it to Bob, then afterwards, Bob's particle is entangled with Zeke's.

A more symmetric way to describe the situation is the following: Alice has one particle, Bob two, and Charlie one. Alice's particle and Bob's first particle are entangled, and so are Bob's second and Charlie's particle:

 Alice -:-:-:-:- Bob1     Bob2 -:-:-:-:- Charlie


Now, Bob performs a Bell measurement on his two particles, which projects them into a Bell state, i.e. they are now entangled. But, more spectacularly, Alice's and Charlie's particles are now entangled as well, although the two never met:

 Alice           Bob1 -:- Bob2           Charlie
|                                         |
\-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-/


This effect allows (at least in theory) to build a quantum repeater (see there). Breaking news: [[1] (http://www.spacedaily.com/news/physics-04zi.html)] demonstrates one node of a quantum repeater.

## References

• theoretical proposal:
• C. H. Bennett, G. Brassard, C. Crpeau, R. Jozsa, A. Peres, & W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70 1895-1899 (1993) (this document online (http://www.research.ibm.com/quantuminfo/teleportation/teleportation.html))
• first experiments with photons:
• D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, & A. Zeilinger, Experimental quantum teleportation, Nature 390, 6660, 575-579 (1997).
• D. Boschi, S. Branca, F. De Martini, L. Hardy, & S. Popescu, Experimental realization of teleporting an unknown pure quantum state via dual classical an Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 80, 6, 1121-1125 (1998);
• first experiments with atoms:
• M. Riebe, H. Hffner, C. F. Roos, W. Hnsel, J. Benhelm, G. P. T. Lancaster, T. W. Krber, C. Becher, F. Schmidt-Kaler, D. F. V. James, R. Blatt: Deterministic quantum teleportation with atoms, Nature 429, 734 - 737 (2004)
• M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri & D. J. Wineland: Deterministic quantum teleportation of atomic qubits, Nature 429, 737

## External Websites

Quantum Teleportation at IBM (http://www.research.ibm.com/quantuminfo/teleportation/)

## Footnotes

1 with one technicality: If you swap two identical fermions their common wave function changes its sign. For this odd effect, see spin-statistics theorem.

2 As the measurement outcome will be one of the four Bell states, the message Alice has to send consists of 2 bits. Hence to teleport 1 qubit, we need 1 shared Bell pair (sometimes called 1 e-bit ("entangled bit")) and the transmission of 2 ordinary bits.

da:Kvanteteleportation

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