Quantum teleportation

Quantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer quantum information from one quantum system to another. It does not transport the system itself, nor does it allow communication of information at superluminal (faster than light) speed. Neither does it concern rearranging the particles of a macroscopic object to copy the form of another object. Its distinguishing feature is that it can transmit the information present in a quantum superposition, useful for quantum communication and computation.

More precisely, quantum teleportation is a quantum protocol by which a qubit a (the basic unit of quantum information) can be transmitted exactly (in principle) from one location to another. The prerequisites are a conventional communication channel capable of transmitting two classical bits (i.e. one of four states), and an entangled Bell pair (b,c) of qubits, with b at the origin and c at the destination. (So whereas b and c are intimately related, a is entirely independent of them other than being initially colocated with b.) The protocol has three steps: measure a and b jointly to yield two classical bits; transmit the two bits to the other end of the channel (the only potentially time-consuming step, due to speed-of-light considerations); and use the two bits to select one of four ways of recovering c. The upshot of this protocol is to permute the original arrangement ((a,b),c) to ((b′,c′),a), that is, a moves to where c was and the previously separated qubits of the Bell pair turn into a new Bell pair (b′,c′) at the origin.

It has been experimentally shown to work over distances of at least 16 kilometers.^[1]

1 Experiments
- 1.1 Example
2 Motivation
3 A summary
4 The result
- 4.1 Remarks
- 4.2 Alternative description
5 Entanglement swapping
6 N-state particles
7 General teleportation scheme
- 7.1 General description
- 7.2 Further details
8 See also
9 References
10 External links

Experiments

In 1997, quantum teleportation was first experimentally confirmed by a group in Innsbruck.^[2] Further work verified the results,^[3] and in August 2004 increased the distance of teleportation to 600 meters using optical fiber.^[4]

The longest distance yet claimed to be achieved for quantum teleportation is 16 km (10 miles) in May 2010 by Chinese scientists over free-space with an average of 89% accuracy.^[1] This distance is sufficient for use between ground stations and a satellite.

Example

For a primitive example. Before the information can be teleported the qubits of information as represented by the quantum states of atoms/ions and photons, electrons or other fundamental particles must be in the correct locations and properly entangled.

For example: If an atom's electron absorbs a photon of energy, occasionally that photon will radiate away as two photons of lower energy (corresponding to the drop in energy as the electron relaxes down to its lowest energy level). These two photons are created by the same event and are entangled. Their properties are interdependent.

The last preparation for "teleportation" is for the atom and one of the photons to stay with the sender of the example (commonly referred to as Alice). The other photon travels in a classical manner at the speed of light to the designated receiver (Bob). The equipment for teleportation to begin is now complete.

The information in this quantum system is therefore held by all three species; E.g. the energy is divided between the two photons b and c and some may still be held by the atom a. Only by probing all species will all the properties be known. The same goes for the spin of particles in similar experiments. It is therefore necessary to know what measurement will be performed on the photons and atom. Necessitating the classical communication -of what test to do-, between the holders of a,b Alice and the holder of c Bob. At the moment Alice determines the states of the atom and photon, the state of the other photon with Bob becomes fixed. The energy (or other information) is determined without uncertainty in this universe by the conservation of energy.

The information is only "teleported" when a joint quantum-mechanical measurement of both qubits has been performed that determines in which of the four Bell states the two qubits were in.

Motivation

Suppose Alice has a qubit in some arbitrary quantum state $|\psi\rangle$ . (A qubit may be represented as a superposition of states, labeled $|0\rangle$ and $|1\rangle$ .) 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:

She can attempt to physically transport the qubit to Bob.
She can broadcast this (quantum) information, and Bob can obtain the information via some suitable receiver.
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.

Option 2 is forbidden by the no-broadcast theorem.

Option 3 (classical teleportation) has also been formally shown to be impossible. (See the no teleportation theorem.) This is another way to say that quantum information cannot be measured reliably.

Thus, Alice seems to face an impossible problem. A solution was discovered by Bennett, et al. The components 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.

A summary

Assume that Alice and Bob share an entangled qubit AB. That is, Alice has one half, A, and Bob has the other half, B. Let C denote the qubit Alice wishes to transmit to Bob.

Alice applies a unitary operation on the qubits AC and measures the result to obtain two classical bits. In this process, the two qubits are destroyed. Bob's qubit, B, now contains information about C; however, the information is somewhat randomized. More specifically, Bob's qubit B is in one of four states uniformly chosen at random and Bob cannot obtain any information about C from his qubit.

Alice provides her two measured classical bits, which indicate which of the four states Bob possesses. Bob applies a unitary transformation which depends on the classical bits he obtains from Alice, transforming his qubit into an identical re-creation of the qubit C.

The result

Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be written generally, in bra-ket notation, as: $|\psi\rangle_C = \alpha |0\rangle_C + \beta|1\rangle_C.$

Our quantum teleportation scheme requires Alice and Bob to share a maximally entangled state beforehand, for instance one of the four Bell states

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

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

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

$|\Psi^-\rangle_{AB} = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_{B} - |1\rangle_A \otimes |0\rangle_{B})$ .

Alice takes one of the particles in the pair, and Bob keeps the other one. The subscripts A and B in the entangled state refer to Alice's or Bob's particle. We will assume that Alice and Bob share the entangled state $|\Phi^+\rangle_{AB}.$

So, Alice has two particles (C, 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

$|\Phi^+\rangle_{AB} \otimes |\psi\rangle_C = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)\otimes (\alpha |0\rangle_C + \beta|1\rangle_C).$

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 the following four-term superposition:

$\frac{1}{2} (\ |\Phi^+\rangle_{AC} \otimes (\alpha |0\rangle_B + \beta|1\rangle_B)\ +\ |\Phi^-\rangle_{AC} \otimes (\alpha |0\rangle_B - \beta|1\rangle_B)\ +\ |\Psi^+\rangle_{AC} \otimes (\beta |0\rangle_B + \alpha|1\rangle_B)\ +$

$\ + |\Psi^-\rangle_{AC} \otimes (-\beta |0\rangle_B + \alpha|1\rangle_B)\ ).$

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 result 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):

$|\Phi^+\rangle_{AC} \otimes (\alpha |0\rangle_B + \beta|1\rangle_B)$
$|\Phi^-\rangle_{AC} \otimes (\alpha |0\rangle_B - \beta|1\rangle_B)$
$|\Psi^+\rangle_{AC} \otimes (\beta |0\rangle_B + \alpha|1\rangle_B)$
$|\Psi^-\rangle_{AC} \otimes (-\beta |0\rangle_B + \alpha|1\rangle_B)$

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 $\alpha |0\rangle_B + \beta|1\rangle_B$ :

If Alice indicates her result is $|\Phi^+\rangle_{AC}$ , 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 $|\Phi^-\rangle_{AC}$ , 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 $|\Psi^+\rangle_{AC}$ , 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.

Remarks

After this operation, Bob's qubit will take on the state $|\psi\rangle_B= \alpha |0\rangle_B + \beta|1\rangle_B$ , 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, Brassard, Crépeau, Jozsa, Peres and Wootters, reflects the indistinguishability of quantum mechanical particles.

The teleportation scheme combines the resources of two separately 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^{[citation needed]} (see no communication theorem).

For every qubit teleported, Alice needs to send Bob two classical bits of information. 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 cannot interact with the entangled particle in Bob's possession.

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.

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.

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.

General teleportation scheme

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 Schrödinger 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 $1 \otimes 23$ .

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

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

References

General

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.
- L. Vaidman, Teleportation of Quantum States, Phys. Rev. A 49, 1473-1476 (1994)
- G Brassard, S Braunstein, R Cleve, Teleportation as a Quantum Computation, Physica D 120 43-47 (1998)
- A. Peres, "What is actually teleported?", IBM Journal of Research and Development, Vol. 48, Issue 1, (2004)(this document online)
- 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)
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)
- Y.-H. Kim, S.P. Kulik, and Y. Shih, Quantum teleportation of a polarization state with a complete bell state measurement, Phys. Rev. Lett. 86, 1370 (2001).
- 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:
- S. Olmschenk, D. N. Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, and C. Monroe, Quantum Teleportation between Distant Matter Qubits, Science 323, 486 (2009).
- 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 (2004).

Specific

^ ^a ^b Xian-Min Jin (16 May 2010). "Quantum teleportation achieved over 16 km". Nature. http://www.physorg.com/news193551675.html. Retrieved 2010-05-22.
^ Bouwmeester, D. et al. (1997). "Experimental quantum teleportation". Nature. pp. 575–579. http://www.nature.com/nature/journal/v390/n6660/full/390575a0.html. Retrieved 2010-05-22.
^ D. Boschi; S. Branca1, F. De Martini1, L. Hardy, and S. Popescu (1998). "Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels". Phys. Rev. Lett.. pp. 80, 1121. http://prl.aps.org/abstract/PRL/v80/i6/p1121_1. Retrieved 2010-05-22.
^ Rupert Ursin (August 2004). "Quantum teleportation across the Danube". Nature. http://www.nature.com/nature/journal/v430/n7002/full/430849a.html. Retrieved 2010-05-22.

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