Quantum teleportation

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In quantum information, quantum teleportation, or entanglement-assisted teleportation is a technique that transfers a quantum state to an arbitrarily distant location using a distributed entangled state and the transmission of some classical information. Quantum teleportation does not transport energy or matter, nor does it allow communication of information at superluminal speed.

1 Motivation
2 The result
- 2.1 Remarks
- 2.2 Alternative description
3 Entanglement swapping
4 N-state particles
5 General teleportation scheme
- 5.1 General description
- 5.2 Further details
6 References
7 External links

[edit] Motivation

This article will use standard nomenclature in quantum information: the two parties are Alice (A) and Bob (B), and a qubit is in general a superposition of quantum state labelled $|0\rangle$ and $|1\rangle$ . Equivalently, a qubit is a unit vector in two-dimensional Hilbert space.

Suppose Alice has a qubit in some arbitrary quantum state $|\psi\rangle$ . Assume that this quantum state is not known to Alice and she would like to send this state to Bob. Ostensibly, Alice has the following options:

1) She can attempt to physically transport the qubit to Bob.

2) She can broadcast this (quantum) information, and Bob can obtain the information via some suitable receiver.

3) She can perhaps measure the unknown qubit in her possession. The results of this measurement would be communicated to Bob, who then prepares a qubit in his possession accordingly, to obtain the desired state. (This hypothetical process is called classical teleportation.)

Option 1 is highly undesirable because quantum states are fragile and any perturbation en route would corrupt the state.

The unavailability of option 2 is the statement of the no-broadcast theorem, a consequence of the no cloning theorem.

Similarly, it has also been shown formally that classical teleportation is impossible; this is called the no teleportation theorem. This is another way to say that quantum information can not be measured reliably.

Thus, Alice seems to face an impossible problem. A solution was discovered by Bennet et al. (see reference below.) The parts of a maximally entangled two-qubit state are distributed to Alice and Bob. The protocol then involves Alice and Bob interacting locally with the qubit(s) in their possession and Alice sending two classical bits to Bob. In the end, the qubit in Bob's possession will be in the desired state.

[edit] The result

Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be written generally as: $|\psi_o\rangle = \alpha |0\rangle_o + \beta|1\rangle_o$ The subscript O refers to the original qubit to be teleported.

Our quantum teleportation scheme requires Alice and Bob to share a maximally entangled state beforehand, for instance the two-particle Bell state

$|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_{B} + |1\rangle_A \otimes |1\rangle_{B})$ ,

or one of the other Bell states. Alices takes one of the particles in the pair, and Bob keeps the other one. The subscripts A and B in the entangled state refer Alice's or Bob's particle.

So, Alice has two particles (O, the one she wants to teleport, and A, one of the entangled pair), and Bob has one particle, B. In the total system, the state of these three particles is given by

$|\psi_o\rangle \otimes |\Phi^+\rangle = (\alpha |0\rangle_o + \beta|1\rangle_o) \otimes \left(\frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_{B} + |1\rangle_A \otimes |1\rangle_{B})\right)$

Alice will then make a partial measurement in the Bell basis on the two qubits in her possession. To make the result of her measurement clear, we will rewrite the two qubits of Alice in the Bell basis via the following general identities (these can be easily verified):

$|0\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}} (|\Phi^+\rangle + |\Phi^-\rangle),$

$|0\rangle \otimes |1\rangle = \frac{1}{\sqrt{2}} (|\Psi^+\rangle + |\Psi^-\rangle),$

$|1\rangle \otimes |0\rangle = \frac{1}{\sqrt{2}} (|\Psi^+\rangle - |\Psi^-\rangle),$

and

$|1\rangle \otimes |1\rangle = \frac{1}{\sqrt{2}} (|\Phi^+\rangle - |\Phi^-\rangle).$

The three particle state shown above thus becomes:

Notice all we have done so far is a change of basis on Alice's part of the system. No operation has been performed and the three particles are still in the same state. The actual teleportation starts when Alice measures her two qubits in the Bell basis. Given the above expression, evidently the results of her (local) measurement is that the three-particle state would collapse to one of the following four states (with equal probability of obtaining each):

$|\Phi^+\rangle_A \otimes (\alpha |0\rangle + \beta|1\rangle)_B$
$|\Phi^-\rangle_A \otimes (\alpha |0\rangle - \beta|1\rangle)_B$
$|\Psi^+\rangle_A \otimes (\beta |0\rangle + \alpha|1\rangle)_B$
$|\Psi^-\rangle_A \otimes (-\beta |0\rangle + \alpha|1\rangle)_B$

Alice's two particles are now entangled to each other, in one of the four Bell states. The entanglement originally shared between Alice's and Bob's is now broken. Bob's particle takes on one of the four superposition states shown above. Note how Bob's qubit is now in a state that resembles the state to be teleported. The four possible states for Bob's qubit are unitary images of the state to be teleported.

The crucial step, the local measurement done by Alice on the Bell basis, is done. It is clear how to proceed further. Alice now has complete knowledge of the state of the three particles; the result of her Bell measurement tells her which of the four states the system is in. She simply has to send her results to Bob through a classical channel. Two classical bits can communicate which of the four results she obtained.

After Bob receives the message from Alice, he will know which of the four states his particle is in. Using this information, he performs a unitary operation on his particle to transform it to the desired state $\alpha |0\rangle + \beta|1\rangle$ :

If Alice indicates her result is $|\Phi^+\rangle$ , Bob knows his qubit is already in the desired state and does nothing. This amounts to the trivial unitary operation, the identity operator.

If the message indicates $|\Phi^-\rangle$ , Bob would send his qubit through the unitary gate given by the Pauli matrix

$\sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}$

to recover the state.

If Alice's message corresponds to $|\Psi^+\rangle$ , Bob applies the gate

$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$

to his qubit.

Finally, for the remaining case, the appropriate gate is given by

$\sigma_3 \sigma_1 = i \sigma_2 = \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}.$

Teleportation is therefore achieved.

Experimentally, the projective measurement done by Alice may be achieved via a series of laser pulses directed at the two particles.

[edit] Remarks

After this operation, Bob's qubit will take on the state $|\psi\rangle= \alpha |0\rangle + \beta|1\rangle$ , and Alice's qubit becomes (undefined) part of an entangled state. Teleportation does not result in the copying of qubits, and hence is consistent with the no cloning theorem.

There is no transfer of matter or energy involved. Alice's particle has not been physically moved to Bob; only its state has been transferred. The term "teleportation", coined by Bennett et al., reflects the indistinguishability of quantum mechanical particles.

The teleportation scheme combines the resources of two impossible procedures. If we remove the shared entangled state from Alice and Bob, the scheme becomes classical teleportation, which is impossible as mentioned before. On the other hand, if the classical channel is removed, then it becomes an attempt to achieve superluminal communication, again impossible (see no communication theorem).

For every qubit teleported, Alice needs to send Bob two classical bits of information. As we would expect, these two classical bits do not carry complete information about the qubit being teleported. If an eavesdropper intercepts the two bits, she may know exactly what Bob needs to do in order to recover the desired state. However, this information is useless if she can not interact with the entangled particle in Bob's possession.

[edit] Alternative description

In the literature, one might find alternative, but completely equivalent, descriptions of the teleportation protocol given above. Namely, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can also be implemented by quantum gates. Direct calculation shows that this gate is given by

$G = (H \otimes I) \; C_N$

, where H is the one qubit Walsh-Hadamard gate and $C N$ is the Controlled NOT gate.

[edit] Entanglement swapping

Entanglement can be applied not just to pure states, but also mixed states, or even the undefined state of an entangled particle. The so-called entanglement swapping is a simple and illustrative example.

If Alice has a particle which is entangled with a particle owned by Bob, and Bob teleports it to Carol, then afterwards, Alice's particle is entangled with Carol's.

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

                      ___
                     /   \
 Alice-:-:-:-:-:-Bob1 -:- Bob2-:-:-:-:-:-Carol
                     \___/

Now, if Bob performs a projective measurement on his two particles in the Bell state basis and communicates the results to Carol, as per the teleportation scheme described above, the state of Bob's first particle can be teleported to Carol's. Although Alice and Carol never interacted with each other, their particles are now entangled. This effect allows (at least in theory) to build a quantum repeater.

[edit] N-state particles

One can imagine how the teleportation scheme given above might be extended to N-state particles, i.e. particles whose states lie in the N dimensional Hilbert space. The combined system of the three particles now has a $N 3$ dimensional state space. To teleport, Alice makes a partial measurement on the two particles in her possession in some entangled basis on the $N 2$ dimensional subsystem. This measurement has $N 2$ equally probable outcomes, which are then communicated to Bob classically. Bob recovers the desired state by sending his particle through an appropriate unitary gate.

[edit] General teleportation scheme

[edit] General description

A general teleportation scheme can be described as follows. Three quantum systems are involved. System 1 is the (unknown) state ρ to be teleported by Alice. Systems 2 and 3 are in a maximally entangled state ω that are distributed to Alice and Bob, respectively. The total system is then in the state

$\rho \otimes \omega.$

A successful teleportation process is a LOCC quantum channel Φ that satisfies

$(\operatorname{Tr}_{12} \circ \Phi ) (\rho \otimes \omega) = \rho.$

where Tr₁₂ is the partial trace operation with respect systems 1 and 2, and $\circ$ denotes the composition of maps. This describes the channel in the Schrodinger picture.

Taking adjoint maps in the Heisenberg picture, the success condition becomes

$\langle \Phi(\rho \otimes \omega), I \otimes O \rangle = \langle \rho, O \rangle$

for all observable O on Bob's system. The tensor factor in $I \otimes O$ is $12 \otimes 3$ while that of $\rho \otimes \omega$ is $1 \otimes 23$ .

[edit] Further details

The proposed channel Φ can be described more explicitly. To begin teleportation, Alice performs a local measurement on the two subsystems (1 and 2) in her possession. Assume the local measurement have effects

${F_i} = {M_i ^2}.$

If the measurement registers the i-th outcome, the overall state collapses to

$(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I).$

The tensor factor in $(M_i \otimes I)$ is $12 \otimes 3$ while that of $\rho \otimes \omega$ is $1 \otimes 23$ . Bob then applies a corresponding local operation Ψ_i on system 3. On the combined system, this is described by

$(Id \otimes \Psi_i)(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I).$

where Id is the identity map on the composite system $1 \otimes 2$ .

Therefore the channel Φ is defined by

$\Phi (\rho \otimes \omega) = \sum_i (Id \otimes \Psi_i)(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I)$

Notice Φ satisfies the definition of LOCC. As stated above, the teleportation is said to be successful if, for all observable O on Bob's system, the equality

$\langle \Phi(\rho \otimes \omega), I \otimes O \rangle = \langle \rho, O \rangle$

holds. The left hand side of the equation is:

$\sum_i \langle (Id \otimes \Psi_i)(M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I), \; I \otimes O \rangle$

$= \sum_i \langle (M_i \otimes I)(\rho \otimes \omega)(M_i \otimes I), \; I \otimes \Psi_i ^*(O)\rangle$

where Ψ_i* is the adjoint of Ψ_i in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes

$\sum_i \operatorname{Tr} \; (\rho \otimes \omega)(F_i \otimes \Psi_i^*(O)).$

The success criterion for teleportation has the expression

$\sum_i \operatorname{Tr} \; (\rho \otimes \omega)(F_i \otimes \Psi_i ^*(O)) = \operatorname{Tr} \; \rho \cdot O.$

[edit] References

Theoretical proposal:
- C. H. Bennett, G. Brassard, C. Crépeau, 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) This is the seminal paper that laid out the entanglement protocol.
- G. Brassard, S Braunstein, R Cleve, Teleportation as a Quantum Computation, Physica D 120 43-47 (1998)
- G. Rigolin, Quantum Teleportation of an Arbitrary Two Qubit State and its Relation to Multipartite Entanglement, Phys. Rev. A 71 032303 (2005) (this document online)
- L. Vaidman, Teleportation of Quantum States, Phys. Rev. A, (1994)

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 and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 80, 6, 1121-1125 (1998)
- I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, N. Gisin, Long-Distance Teleportation of Qubits at Telecommunication Wavelengths, Nature, 421, 509 (2003)
- R. Ursin et.al., Quantum Teleportation Link across the Danube, Nature 430, 849 (2004)

First experiments with atoms:
- M. Riebe, H. Häffner, C. F. Roos, W. Hänsel, M. Ruth, J. Benhelm, G. P. T. Lancaster, T. W. Körber, 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