# Quantum Computing Series, Part 6: Quantum Teleportation

Quantum teleportation is used to transmit information from one location to another. This quantum information, for example, can be the exact state of a particle such as an atom or a photon. The information transmits using the classical communication and previously shared quantum entanglement between the sending and receiving location. Because the transmission depends on classical communication, it cannot be used for faster-than-light transport or communication of classical bits. It has been proven possible to teleport one or more qubits of information between two (entangled) atoms; it has, however, not yet been achieved between molecules or anything larger.

It’s important to note that quantum information can be neither copied nor destroyed.

Teleportation can be applied not just to pure states but also to mixed states. The mixed states can be regarded as the state of a single subsystem of an entangled pair. Entanglement swapping is a simple example of quantum teleportation.

In a commonly used illustration, if Alice has a particle entangled with a particle owned by Bob, and Bob teleports it to Carol, then afterwards, Alice’s particle is entangled with Carol’s.

In other words, the state of Bob’s first particle can be teleported to Carol’s. Alice and Carol without even having any interaction with each other now have their particles entangled.

In quantum or classical information theory, it is best to work with the simplest possible unit of information, the two-state system. In classical information, this is a bit, represented using one or zero (or true or false). The quantum analog of a bit is a quantum bit, or qubit, a unit of quantum information. Qubits encode a type of information, called quantum information, which differs sharply from “classical” information. This difference is apparent, for example, in quantum information properties such as it being impossible to copy (the no-cloning theorem) and impossible to destroy (the no-deleting theorem).

#### Teleportation of Qubits

Quantum teleportation provides a mechanism to move a qubit from one location to another. And this is done without having to physically transport the underlying particle that a qubit is normally attached to. With the invention of the telegraph in 1830s, the information in the form of classical bits could be sent across great distances, even among the continents, at a high speed. Similarly, quantum teleportation holds the promise of one day moving the qubits likewise through open space.

The actual teleportation protocol requires the creation of an entangled quantum state or Bell state. In addition, it is also required that its two parts are shared between two locations. Essentially, before moving a qubit, a quantum channel between two sites must first be established. Further, teleportation also mandates establishing a classical-information link. This is required for transmission of two classical bits that accompany each qubit. This transmission is required to communicate the results of the measurements that must be done over ordinary classical communication channels. Such communication is similar to ordinary communications requiring media such as wires, radios, or lasers. Bell states are easily shared using photons from lasers. Therefore, teleportation could possibly be done through open space.

#### Teleportation of Quantum States

The quantum states of single atoms have successfully been teleported. An atom is made up of a number of different subatomic particles; these include primarily the electrons in the outer orbits and protons and neutrons in the nucleus, the qubits in the electronic state or electron shells that surround the nucleus of the atom, and the qubits in the nucleus itself. So far, physicists have only managed to teleport the qubits encoded in the electronic state of atoms. The teleportation of neither the nuclear state nor the nucleus itself has so far been possible. Therefore, teleporting an atom has still not been achieved; the only success is with the quantum state of an atom.

Performing such teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them. It is yet not clear how important teleporting the nuclear state is. An atom is affected by its nuclear state, for example, in hyperfine splitting. However, the usefulness of teleporting a nuclear state is debatable for any practical application.

#### Entanglement

An important aspect of quantum information theory is entanglement. It imposes statistical correlations between otherwise-distinct physical systems. Even when measurements are chosen and performed independently, these correlations hold true because of causal contact from one another. This is verified in Bell test experiments. When a measurement is done at a point in spacetime and an observation is made, it instantaneously affects the outcome in another region. This happens even before the light travels the distance between the two regions. This conclusion is incongruous with Special relativity. Such correlations, however, cannot be used to transmit information at a speed faster than that of light. In any condition, teleportation cannot be faster than the speed of light. This is true as a qubit cannot be reconstructed until the accompanying classical information arrives.

#### FTL Communication

Faster-than-light (FTL) or superluminal communication refers to the information transmission at a speed faster than light. Similarly, FTL travel means transfer of matter faster than the speed of light. The special theory of relativity states that a particle with rest mass and having subluminal velocity needs an infinite amount of energy to accelerate to the speed of light. Special relativity, however, does not prohibit the existence of particles that travel faster than light at all times (tachyons).

Apparent FTL phenomenon might be explained on the hypothesis of distorted spacetime regions permitting matter to travel faster than light. Per the current theories, matter still needs to travel subluminally (below the speed of light) with respect to the locally distorted spacetime region. However, general relativity still does not exclude the apparent FTL. Examples of apparent FTL proposals, even though their physical plausibility is uncertain, are the Alcubierre drive and the traversable wormhole.

Appropriately describing quantum teleportation requires a basic mathematical toolset. It is quite complex but completely comprehensible by anyone with a grasp of finite-dimensional linear algebra. Specifically, it heavily uses the theory of Hilbert spaces and projection matrices. A Hilbert space (a two-dimensional complex number-valued vector space) is used to describe a qubit; the formal manipulations do not make use of anything much more than that. To conclude, one does not require working knowledge of quantum mechanics to understand the mathematics underlying quantum teleportation. The knowledge of quantum mechanics, however, is indeed helpful in understanding the meaning of the equations used.

#### Superdense Coding

Quantum teleportation and superdense coding both deal with quantum entanglement but act in opposite ways. Quantum teleportation, as we know, sends one qubit with two classical bits. In contrast, using the technique of superdense coding, two bits of classical information can be sent using just one qubit. Thus, it is the inverse of quantum teleportation when it comes to entangled particles. Both superdense coding and quantum teleportation use entanglement between the sender and receiver in the form of Bell pairs (entanglement cannot be broken using local operations).

When the sender and receiver share a bell pair, two classical bits can be packed into one qubit.

#### An Illustration

Suppose Alice wants to use qubits (in place of classical bits) to send classical information to Bob. Alice goes ahead and encodes the classical information in a qubit and sends it over to Bob. Bob receives the qubit and then recovers the classical information via measurement. But how much classical information can a qubit carry? Since it is impossible to reliably distinguish non-orthogonal quantum states, the best that Alice can probably do is one classical bit of information per qubit. Holevo’s theorem discusses this bound on efficiency. Therefore, apparently, qubits offer no advantage over classical bits in carrying/transmitting information. However, with the additional assumption that Alice and Bob share an entangled state, two classical bits per qubit are possible. This doubling of efficiency is termed “superdense.” Also, it can be proved that a maximum of two bits of classical information can be sent (even while using entangled state) using one qubit.

The shared entangled state between Alice and Bob is vital to this procedure. Using local manipulation, a maximally entangled state can be transformed into another state.

#### Performing Unitary Operations

Now, Bob wants to find out which classical bits Alice wanted to send. For this, Bob performs the CNOT unitary operation. This is followed by unitary operation on the entangled qubit.

Let us say the resultant entangled qubit was B00. After applying the above unitary operations, the entangled qubit will become |00›

Let us say the resultant entangled qubit was B01. After applying the above unitary operations, the entangled qubit will become |01›

Let us say the resultant entangled qubit was B10. After applying the above unitary operations, the entangled qubit will become |10›

Let us say the resultant entangled qubit was B11. After applying the above unitary operations, the entangled qubit will become |11›

Alice performs one of the above four local operations and sends her qubit to Bob. Bob then decodes the desired message by performing a projective measurement in the Bell basis on the two particle system.

If Eve, a mischievous person, intercepts Alice’s qubit while it transmits to Bob, Eve can only obtain a part of an entangled state. Therefore, unless she can interact with Bob’s qubit, Eve cannot have any useful information.

#### General Dense Coding Scheme

General dense coding schemes can be formulated in the language used to describe quantum channels.

##### Quantum Channel

In quantum information theory, a quantum channel is a communication channel used to transmit both quantum and classical information. While the state of a qubit is an example of quantum information, any electronic information such as a spreadsheet, a web page, or a text document transmitted over the Internet is representative of classical information.

In a more technical context, quantum channels are defined as completely positive (CP) trace-preserving maps between spaces of operators. So, a quantum channel is essentially a quantum operation viewed not just as the reduced dynamics of a system but a pipeline for carrying quantum information. The term “quantum operation” can also be used to include trace-decreasing maps, while “quantum channel” is used specifically for trace-preserving maps.

In quantum teleportation, a sender wants to transmit to a distant receiver an arbitrary quantum state of a particle. Therefore, the teleportation process is a quantum channel. The apparatus to carry out the process itself requires a quantum channel for transmitting a particle to the receiver. Teleportation occurs by jointly measuring the sent particle and the remaining entangled particle. This measurement produces classical information. To complete the teleportation, the receiver must receive this information. Notably, even after the quantum channel has ceased to exist, the classical information can still be sent.

##### Illustrative Implementation of a Quantum Channel

Transmission of single photons over a fiber optic (or free-space) can be considered a simple implementation of a quantum channel. Single photons can easily travel up to 100 km in a standard fiber optics. The photon’s time-of-arrival (time-bin entanglement) or polarization are used as a basis to encode quantum information for purposes such as quantum cryptography. The channel is capable of transmitting the basis states (e.g. |0>, |1>) as well as superpositions of these states (e.g. |0>+|1>). While transmitting through the channel, state coherence is maintained. Contrast this with transmitting electrical pulses through wires (a classical channel), which can transmit only the classical information (e.g. 0s and 1s).

Importantly, using entanglement, both of the above bounds on capacities can be broken. The teleportation scheme, aided by entanglement, enables using a classical channel to transmit quantum information. Superdense coding can send two bits per qubit. It can, therefore, be concluded that entanglement plays a significant role in quantum communication.

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