Beam me up Scotty: But only for quantum particles

Scientists have managed to perform quantum teleportation - another step in the dream of realizing a quantum computer and perhaps also for the launch of Scotty...

In 1993, a group of six scientists including Charles Bennett and Asher Peres published a paper titled, "Teleportation of an unknown quantum state using classical dual channels and APR". The article was considered a revolution because Began the theoretical research in quantum teleportation. This is a term that originates from science fiction and means: to cause an object to disappear in one place while an exact copy of it appears in another.

In the years that followed, other scientists demonstrated experimental teleportation in a variety of systems: single photons, trapped ions, and so on. It is believed that quantum teleportation has a useful future in information processing, quantum communication and maybe it will even help in the development of a future quantum computer. But science fiction fans dream of other applications for teleportation and will surely be disappointed to hear that people and other macroscopic bodies will not be able to be launched in the near future. While quantum teleportation transfers a quantum state from one particle to another, it does not transfer mass. In addition, the original state of the quantum particle is destroyed during the teleportation. Of course, this is because of the non-replication theorem, which does not allow the creation of a replication of a given quantum system and also because of Heisenberg's uncertainty principle, as will be explained later. Finally, launching via teleportation has a finite speed limit: it is possible to launch via teleportation at a speed that does not exceed the speed of light or the speed of light, but not at a speed higher than it - and this is in accordance with Einstein's special theory of relativity.

Bennett, Peres and the other researchers believed that correlations between pairs of quantum particles in the Einstein-Podolsky-Rosen (APR) experiment, i.e. quantum entanglement, could be used to transmit information. After all, Einstein himself feared that the APR experiment could transmit information "telepathically" (that is, at a speed higher than that of light). But we know that immediate information transfer through an APR experiment is impossible. Bennett thought that perhaps the APR experiment could be used to transmit information through "teleportation".

An entangled state of two particles describes a single quantum system in a state where the particles lose their self-identity in a certain sense. The entangled state contains no information about the individual particles, but only indicates that the two particles will be in opposite states. It means that a measurement performed on one particle causes the other particle to be in the opposite state. There is no limit to the distance the particles can be separated from each other. They can be as vast a distance apart as we like; But the equations of quantum mechanics predict that when a measurement is made in one particle it causes the other particle to be in the opposite state. This strange effect caused Einstein to call the APR experiment "action at a distance of ghosts".

At first it was thought that quantum teleportation could not be realized: if we want to accurately clone or duplicate an object when it is sent to another place, we will have to duplicate the object before it is sent. That is, we will have to measure the exact position and momentum of each and every atom in the body; And only then launch it to the desired destination and finally rebuild it on the basis of instructions received through a classical and not quantum channel. But the first step of rigorously simultaneously measuring the position and momentum of each atom grossly violates Heisenberg's uncertainty principle.

Heisenberg's uncertainty principle forbids us to perform an exact measurement on a quantum system and obtain complete information about its quantum state. According to the uncertainty principle, the more precisely we measure the quantum particle, the more we disturb it by the measurement process. This is until we reach a certain point where we have already completely disturbed the original quantum state of the system and we still haven't extracted enough information from it to make an exact replication of it.

In 1993 Bennett, Peres and the other scientists found a way around this limitation on the ability to make copies of the particle's quantum state.

Let's say that Alice is given a quantum system, a photon, and the photon is prepared in a quantum state that is unknown to Alice. Alice aims to convey to Bob enough information about the quantum system so that he can create a copy of it. If Alice had knowledge of the quantum state of the photon she would have had enough information to convey to Bob. But she has no way of knowing him. If Alice were to measure the quantum state of her original particle it would cause some information to be lost. In this situation, Bob will not be able to restore the state of the particle.

Therefore, the most efficient way for Alice to provide Bob with information about her particle A is to send Bob the particle A itself. But if she cannot send the original particle, she can make the particle interact with another particle B in a known state (ancilla). After the interaction, particle B remains in the unknown state of A, while Alice's original particle A remains in the known state. This way the particle B contains complete information about the particle A of Alice. Alice sends Bob the state of particle B rather than the state of the original particle A and Bob repeats her actions to make a copy of her unknown state of particle A. For the quantum nonreplication theorem to hold, the original unknown state of Alice's particle A is destroyed in the process: this is the removal of the unknown state from Alice's hands and its appearance in Bob's time later. This is the basis of the teleportation process. There is a transfer of quantum information from one system to another but not quantum replication. The teleportation process does not happen instantaneously, because it requires sending a classical signal from Alice to Bob for it to restore the copy from the unknown state of Alice's A particle.

In more detail, the standard teleportation process consists of a classical part and a non-classical part; That is, the information about the state of Alice's original particle is sent in two parts in a classical channel and a quantum channel, and after Bob receives both pieces of information, he restores the quantum state of Alice's original particle:

1. The non-classical channel: in the beginning, two particle states 2 and 3 are prepared in quantum entanglement mode, an APR pair (the ancilla). Alice aims to transfer to Bob an unknown state of a particle, which we will call particle 1. Therefore, we have a system that includes the pair of APR particles 2 and 3 and the particle with the unknown state to Alice of particle 1, which Alice wants to teleport. At this stage, the APR pair still do not contain any information about the original state of particle 1. If we measure particle 2 or particle 3, we will not receive any information about particle 1. Alice receives one APR particle (particle 2), while Bob receives a particle The second APR (particle 3). Only in the next step do you quantumly interweave the states of the APR particle system with the original state of Alice's particle 1, which you want to launch.
2. In order to link the state that you want to launch in the teleportation of particle 1 to the interlaced state, the state of particle 2 in Alice's possession (the Ancilla), Alice performs a Bell state measurement on these two particles. This measurement process ends with it destroying the unknown original state of particle 1 and obtaining one combined state of particles 1 and 2 out of four possible interlaced combined Bell states. All four possible measurement results have the same probability.
3. The classic channel: as you remember Alice's particle 2 is in quantum entanglement with Bob's particle 3. When Alice performs a measurement on her particle 1 and particle 2, Bob's particle 3 is affected by Alice's measurement. When Alice's particles 1 and 2 are in one of the Bell states, then particle 3 immediately also settles into one of four states: the four possible states that Bob's particle 3 can be in are a combination of the total state of particles 1, 2, and 3 before the measurement is made by Ellis and of the measurement result performed by Ellis on particles 1 and 2. Each of these four states for Bob's particle 3 contains the original state of particle 1 that Alice sends. In the first situation Bob does not need to do anything to get a copy of the state of particle 1 that Alice sent him. In the other three there is some action that Bob would have to perform to convert his particle 3 into a copy of the original state of particle 1.
4. To complete the teleportation process, Alice needs to transmit the results of her measurement to Bob in a classical channel so that Bob knows which of the four states his particle is in and he can apply the appropriate action to get a copy of the original state of Alice's particle 1. Bob can implement the correct action only if he receives the result of Bell state measurement performed by Alice. This information is necessary to complete the teleportation process and is transmitted through a classic communication channel. Therefore the maximum speed for quantum teleportation is the speed of light. The non-classical information transfer between the APR particles also does not occur immediately, because there is no transfer of signals between the APR particles, particle 2 to particle 3.

While after completing the teleportation Bob is left with an exact copy of the original state of particle 1, Alice is left with particles 1 and 2 with quantum states that are unremarkable to the original state of particle 1. Therefore particle 3 is not a duplicate of particle 1, but can be seen Teleported particle 1.
Now let's think about the teleportation process in terms of quantum bits (qubits). Alice (transmitter) and Bob (receiver) share two halves of a certain entangled state (APR state) of two qubits. Alice wants to transfer the state of a qubit in her possession to Bob. It prepares a single state of a qubit. The state of this qubit in Alice's possession is not important. Alice measures (an irreversible operation that destroys quantum information and replaces it with classical information) the qubit in her possession and the entangled half state (the second qubit) and she will transmit on a classical channel to Bob two classical bits about the result of this measurement. This measurement leaves the third qubit, which is in Bob's hands, in a state that is composed of the original state of Alice's qubit and also some action. There are two options for this action: either there is no action if Alice's two classic bits are 00. Or there are three options for action if they are not 00. To complete the teleportation move Alice transmits the two classic bits to Bob on a classic channel (transferring a state from one place to another). Bob uses them to perform a recovery operation and thus he knows which option to choose and he recovers at his exit a copy of the qubit that Alice sent at the entrance.

Why is the classical information necessary? Let's say Bob is impatient and doesn't want to wait for Alice's measurement. He decides to guess Alice's classic measurement before she arrives. Bob reconstructs the unknown original state of particle 1 as a random shuffle of the four states in which his particle 3 ends up. A complex state is obtained that does not give any information about the unknown pure state of Alice's particle 1. It must be so, because the quantum correlation or entanglement between particle 2 at the entrance and particle 3 that is guessed at the exit can be used to send faster-than-light signals. Only after Alice performs the measurement is the teleportation complete and the message from Alice to Bob is transmitted.

During the teleportation, the APR particles 2 and 3 initially interacted and then they separated and a state of quantum entanglement was created: particle 2 reached its final destination at Alice and particle 3 at its final destination at Bob. This means that objects can only be teleported between Alice and Bob or between other destinations where the particles are and not to unknown destinations that we have not visited. Very important information for Star Trek travelers.

Researchers began to think about the inclusion of quantum teleportation; On operations in qubits, Bell measurements and entangled quantum states - all of which are within the reach of contemporary technology - to build a universal theoretical model for a quantum computer. The standard teleportation course leads to a limited efficiency of such a model for a quantum computer (25%). Therefore, this efficiency needs to be improved to 100%, meaning that the reliability of the transmitted qubit will be 1. Because of the need for quantum error corrections, it is very difficult to develop efficient quantum components this way. In 2008 two researchers, Tohiya Hiroshima and Satoshi Ishizaka proposed a move called "port-based teleportation" (PBT). The goal of PBT is that the actions that Bob performs following the classical information he receives from Alice will be simpler.

The PBT protocol also requires a common quantum entanglement for Alice and Bob. Both researchers formulated a deterministic and probabilistic PBT course. We will focus on the deterministic course. In the deterministic process there are the following steps:

1. Alice and Bob both share 2N entangled states of qubits (ports): Bob has half of the qubits, meaning he has N qubits B that match the output ports and Alice also has half of the qubits, she has N qubits A that match the input ports.
2. Alice wants to send Bob an unknown state of qubit C. Alice performs a joint measurement for A and C when there are N possible measurement results on her qubits A and C. Let's say that Alice gets some one measurement result out of all these N possible results, measurement result i.
3. Alice tells Bob the result i using a classic channel. Bob gets the teleported state by choosing one of the N ports in his output that matches Alice's measurement result and destroying all the other ports (qubits): that is, Bob destroys the entire entanglement except for one port. Bob always infers that the selected port has the teleported state, so he does not need to perform any operation (using Alice's measurement) to extract a copy of the original state of Alice's C qubit.

This move therefore provides a deterministic and reliable quantum processor with probability 1. This is provided that we have infinite ports, that is, when N tends to infinity. If N is finite then this teleportation move is approximate.

But in 1997 two researchers (Nielsen and Chang) proved a no-go theorem (a type of theorem in physics that states that some situation in quantum theory or some protocol cannot exist: for example hidden variable theories that try to explain the probability of quantum theory): There cannot be a processor universal quantum that can be programmed and is deterministic; And if we provide it with deterministic instructions then it is inevitable that the result itself will contain noise. That is, a programmable quantum processor can only exist if it operates probabilistically.

In December 2012, Sergi Strelchak of the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, who led research together with Jonathan Oppenheim of Cambridge and University College London and Michael Horodky of the University of Gdansk, showed that there are problems with the PBT protocol: the protocol requires an enormous amount of Alice interweaving to launch Lviv one lonely situation.
In this respect, if you want to realize the PBT teleportation in a practical way, it will be very difficult to do so. Therefore, the amount of interweaving in Alice must be reduced and teleported to a sequence of quantum states. This way we will get a more efficient program and efficient non-local quantum computing.
There is another problem: if one wants to launch quantum states one after the other in this mode, Bob erases most of the entanglement, and as the entanglement thins out or distorts, errors are introduced in the interlacing, when the error increases as more states are launched - thus it is not possible to launch quantum states one after the other.

The idea of ​​the researchers from England and Magdansk is to introduce a cycle course to reduce the amount of entanglement in PBT type teleportation and to launch a large amount of quantum states one after the other or simultaneously. The researchers combine two types of teleportation protocols: 1) Alice and Bob use a finite set of states. Alice thus performs the teleportation launch of the state, when Bob has to perform a correction (action) to receive the launched state. 2) Alice and Bob use an infinite stack of states and Alice launches the state, when Bob needs no correction on his side.

To reduce the amount of entanglement, the researchers propose two protocols that recycle the entanglement in Alice's half-entangled state, which consists of N ports (qubits): in the first protocol, Alice teleports qubits one after the other using an entangled state shared by her and Bob, which, as mentioned, consists of 2N qubits. Alice thus applies the PBT teleportation move many times. But instead of getting rid of the tangled state at the end of the process, Alice and Bob keep it and Alice recycles the tangled state for reuse. In the second protocol, Alice teleports to her states at once - not one after the other - so she randomly associates each of the states with one of the ports. Here, too, the half entangled state diminishes in direct proportion to the number of states Alice sends, when there is a limit to the number of qubits she can send and she must recycle the entangled state.

For the abstract of the scientific article