Since there are four possibilities, her choice of operation represents two bits of classical information. Note that transforming just one bit of an entangled pair means performing the identity transformation on the other bit. *Alice* then sends her qubit to *Bob* who must deduce which Bell basis state the qubits are in. *Bob* first applies a controlled-NOT to the two qubits of the entangled pair.

*Bob* then measures the second qubit. If the measurement returns |0〉, the encoded value was either 0 or 3; otherwise the value was either 1 or 2. *Bob* now applies *H* to the first bit and measures that bit (see Table below).

This allows him to distinguish between 0 and 3, and 1 and 2, as shown in the table below.

In principle, dense coding can permit secure communication: the qubit sent by Alice will only yield the two classical information bits to someone in possession of the entangled partner qubit. But more importantly, it shows why quantum entanglement is an information resource. It reveals a relationship between classical information, qubits, and the information content of quantum entanglement.

### 10.6 Teleportation

Teleportation is the ability to transmit the quantum state of a given particle using classical bits and to reconstruct that exact quantum state at the receiver. The no-cloning principle, however, requires that the quantum state of the given particle be necessarily destroyed. Instinctively, one perhaps realizes that teleportation may be realizable by manipulating a pair of entangled particles; if we could impose a specific quantum state on one member of an entangled pair of particles, then we would be instantly imposing a predetermined quantum state on the other member of the entangled pair. The teleportation algorithm is due to Charles Bennett and his team (1993)^{34}.

Teleportation of a laser beam consisting of millions of photons was achieved in 1998. In June 2002, an Australian team reported a more robust method of teleporting a laser beam. Teleportation of trapped ions was reported in 2004^{35}. In May 2010, a Chinese research group reported that they were able to “teleport” information 16 kilometers^{36}. In July 2017, a Chinese team reported “the first quantum teleportation of independent single-photon qubits from a ground observatory to a low Earth orbit satellite – through an up-link channel – with a distance up to 1400 km”^{37}. This experiment is an important step forward in establishing a global scale quantum internet in the future. In theory, there is no distance limit over which teleportation can be done. But since entanglement is a fragile thing, there are technological hurdles to be overcome.

To see how teleportation works, let *Alice* possess a qubit of unknown state |*Ø*〉 = a |0〉 + b |1〉. She wishes to send the state of this qubit to *Bob* through classical channels. In addition, *Alice* and *Bob* each possess one qubit of an entangled pair in the state

*Alice* applies the decoding step of dense coding to the qubit *∅* to be transmitted and her half of the entangled pair. The initial state is

of which *Alice* controls the first two qubits and *Bob* controls the last qubit. She now applies *C _{not}⊗ I* and

*H⊗ I⊗ I*to this state:

*Alice* measures in the Bell basis the first two qubit to get one of |00〉, |01〉, |10〉, or |11〉 with equal probability. That is, Alice’s measurements collapse the state onto one of four different possibilities, and yield two classical bits. *Alice* sends the result of her measurement as two classical bits to *Bob*. Depending on the result of the measurement, the quantum state of *Bob*’s qubit is projected to a(|0〉 + b|1〉, a(|1〉 + b|0〉, a(|0〉- b|1〉, or a(|1〉 -b|0〉 respectively^{38}. Note that *Alice*’s measurement has irretrievably altered the state of her original qubit *∅* , whose state she is trying to send to *Bob*. Thus, the no-cloning principle is not violated. Also, Bob’s particle has been put into a definite state.

When *Bob* receives the two classical bits from *Alice* he knows how the state of his half of the entangled pair compares to the original state of *Alice*’s qubit.

*Bob* can reconstruct the original state of *Alice*’s qubit *∅* by applying the appropriate decoder to his part of the entangled pair. Note that this is the encoding step of dense coding.

The interesting facts to note are as follows. First, the state that is transmitted is completely arbitrary (not chosen by Alice and unknown to her). Second, a message with only binary classical information, such as the result of the combined experiment made by Alice is definitely not sufficient information to reconstruct a quantum state; in fact, a quantum state depends on continuous parameters, while results of experiments correspond to discrete information only. Somehow, in the teleportation process, binary information has turned into continuous information! The latter, in classical information theory, would correspond to an infinite number of bits.

It also happens that Alice cannot determine the state of her particle with state *∅* by making a measurement and communicating the result to Bob because it is impossible to determine the unknown quantum state of a single particle (even if one accepts only an *a posteriori* determination of a perturbed state); one quantum measurement clearly does not provide sufficient information to reconstruct the whole state, and several measurements will not provide more information, since the first measurement has already collapsed the state of the particle. Note also that without the classical communication step, teleportation does not convey any information at all. The original qubit ends up in one of the computational basis states |0〉 or |1〉 depending on the measurement outcome.

Quantum teleportation can be used to move quantum states around, *e.g.,* to shunt information around inside a quantum computer or indeed between quantum computers. Quantum information can be transferred with perfect fidelity, but in the process the original must be destroyed. This might be especially useful if some qubit needs to be kept secret. Using quantum teleportation, a qubit could be passed around without ever being transmitted over an insecure channel. In addition, teleportation inside a quantum computer can be used as a security feature wherein only one version of sensitive data is ensured to exist at any one time in the machine. We need not worry about the original message being stolen after it has been teleported because it no longer exists at the source location. Furthermore, any eavesdropper would have to steal both the entangled particle and the classical particle in order to have any chance of capturing the information.

## 11. Conclusion of Part II

Thus far we have, by a variety of examples, shown the power of quantum computing. In Section 10 we concluded Part II by describing some of the prized algorithms.

## Supplementary material

Readers desirous of getting some hands-on experience with a few of the algorithms presented in Part II can visit https://acc.digital/Quantum_Algorithms/Quantum_Algorithms_June2018.zip . This material is provided by Vikram Menon using the IBM Q Experience for Researchers (see https://quantumexperience.ng.bluemix.net/qx/experience).

34 Bennett, et al (1993). See also: Rieffel & Polak (2000).

35 Riebe, et al (2004). Barrett, et al (2004).

36 Jin, et al (2010).

37 Ren, et al (2017). A report appears in: Emerging Technology from the arXiv. First Object Teleported from Earth to Orbit. MIT Technology Review, 10 July 2017, https://www.technologyreview.com/s/608252/first-object-teleported-from-earthto- orbit of the states a(|0〉 + b|1〉, a(|1〉 + b|0〉, a(|0〉- b|1〉, or a(|1〉 – b|0〉.

38 Note that due to the measurement Alice made, the measured qubits have collapsed. Hence only Bob’s qubit can be in one