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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 (faster than light) speed, but is useful to quantum communication and computation. Image File history File links Broom_icon. ... In quantum mechanics, quantum information is physical information that is held in the state of a quantum system. ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... It has been suggested that Quantum coherence be merged into this article or section. ... Physical information refers generally to the information that is contained in a physical system. ... This article is about matter in physics and chemistry. ... Faster-than-light (also superluminal or FTL) communications and travel are staples of the science fiction genre. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ...

Motivation

The two parties are Alice (A) and Bob (B), and a qubit is, in general, a superposition of quantum state labelled and . Equivalently, a qubit is a unit vector in two-dimensional Hilbert space. The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as cryptography and physics. ... A qubit representation by a Bloch sphere. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... Quite literally, quantum state describes the state of a quantum system. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

Suppose Alice has a qubit in some arbitrary quantum state . 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. The no-broadcast theorem is a result in quantum information theory. ...

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 cannot be measured reliably. In quantum information theory, the no teleportation theorem states that quantum information cannot be measured with complete accuracy. ...

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.

The result

Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be written generally as: .

Our quantum teleportation scheme requires Alice and Bob to share a maximally entangled state beforehand, for instance one of the four Bell states It has been suggested that Quantum coherence be merged into this article or section. ... The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. ...

,

,

,

.

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 .

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

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):

and

The three particle state shown above thus becomes the following four-term superposition:

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): Collapse is a puzzle game published in 1999 by the software company GameHouse. ...

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. A qubit representation by a Bloch sphere. ...

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 :

• If Alice indicates her result is , 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 , Bob would send his qubit through the unitary gate given by the Pauli matrix

to recover the state. The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. ...

• If Alice's message corresponds to , Bob applies the gate

to his qubit.

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

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 , 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 (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.

The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unknown quantum state. ... In quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. ...

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 A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits. ...

where H is the one qubit Walsh-Hadamard gate and C_N is the Controlled NOT gate. The Hadamard transform (Hadamard transformation, also known as Walsh-Hadamard transformation) is an example of a generalized class of Fourier transforms. ... The Controlled NOT gate is a Universal gate, an essential component in the construction of a quantum computer. ...

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. A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ...

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 N³ dimensional state space. To teleport, Alice makes a partial measurement on the two particles in her possession in some entangled basis on the N² dimensional subsystem. This measurement has N² 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

A successful teleportation process is a LOCC quantum channel Φ that satisfies LOCC, or Local Operations and Classical Communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is communicated classically to another part where usually another local operation is performed. ... A quantum channel is a communication channel which can transmit quantum information, as opposed to a classical information channel which is a communication channel transmitting only classical information. ...

where Tr₁₂ is the partial trace operation with respect systems 1 and 2, and denotes the composition of maps. This describes the channel in the Schrodinger picture. In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...

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

for all observable O on Bob's system. The tensor factor in is while that of is .

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

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

The tensor factor in is while that of is . Bob then applies a corresponding local operation Ψ_i on system 3. On the combined system, this is described by

where Id is the identity map on the composite system .

Therefore the channel Φ is defined by

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 LOCC, or Local Operations and Classical Communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is communicated classically to another part where usually another local operation is performed. ...

holds. The left hand side of the equation is:

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

The success criterion for teleportation has the expression

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.
  • L. Vaidman, Teleportation of Quantum States, Phys. Rev. A, (1994)
  • 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)

• 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 (2004).

Asher Peres (born 1934 and died January 1, 2005) was an Israeli physicist, considered a pioneer in quantum information theory. ... Physical Review is one of the oldest and most-respected scientific journals publishing research on all aspects of physics. ... Physical Review is one of the oldest and most-respected scientific journals publishing research on all aspects of physics. ... Anton Zeilinger Anton Zeilinger (born on 20 May 1945 in Ried im Innkreis, Austria) is a professor of physics at the University of Vienna, previously University of Innsbruck. ... Nature is a prominent scientific journal, first published on 4 November 1869. ... Physical Review is one of the oldest and most-respected scientific journals publishing research on all aspects of physics. ...

External links

• signandsight.com:"Spooky action and beyond" - Interview with Prof. Dr. Anton Zeilinger about quantum teleportation. Date: 2006-02-16
• Quantum Teleportation at IBM
• Physicists Succeed In Transferring Information Between Matter And Light
• Quantum telecloning: Captain Kirk's clone and the eavesdropper