# QUANTUM TELEPORTATION BREAKTHROUGH ANNOUNCED BY SCIENTISTS – Discovery opens up new ‘dimension’ in transmitting information @ Scientists have made a major breakthrough in quantum teleportation, successfully transferring something far more complex than ever before. & Facial recognition algorithms can be deployed to hunt for dark matter – The same type of neural networks that make facial recognition tools are now being used to look for and characterize dark matter. ´´Quantum teleportation is a process in which quantum information (e.g. the exact state of an atom or photon) can be transmitted (exactly, in principle) from one location to another, with the help of classical communication and previously shared quantum entanglement between the sending and receiving location. Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for faster-than-light transport or communication of classical bits.´´ @ Other News

https://en.wikipedia.org/wiki/Quantum_teleportation

https://www.technologyreview.com/s/614259/an-important-quantum-algorithm-may-actually-be-a-property-of-nature/

# QUANTUM TELEPORTATION BREAKTHROUGH ANNOUNCED BY SCIENTISTS

## Discovery opens up new ‘dimension’ in transmitting information

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The Independent Tech

Scientists have made a major breakthrough in quantum teleportation, successfully transferring something far more complex than ever before.

Quantum teleportation is a still mysterious phenomena that can see information flung across space.

The effect is not the same as the teleportation often seen in science fiction – where matter is moved through space – but instead is the transfer information about the quantum state of a particle. It could have huge and unimagined effects on the way we organise and transmit information, including the possibility of a vastly more complicated quantum internet.

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Until now, scientists have only been able to teleport quantum bits, or qubits. Those are the simplest possible piece of quantum information, where a particle can be in two states at once.

But a qutrit – which scientists have now successfully teleported – adds a whole extra dimension. If a qubit is polarised in two ways at once, a qutrit is polarised in three directions – a vastly more complicated unit, and a far greater challenge for the scientists involved.

It means, in effect, that scientists have more detailed information that can be teleported. A qubit could be 0 or 1; a qutrit can be 0, 1 or 2.

Sending a qutrit rather than a qubit means a vast increase in complexity, because of the nature of the process.

But it also means that the effect could be far more useful. The extra information that can be transferred could power technologies such as a possible quantum internet, allowing for instant and very secure communication.

“The higher the dimensions of your quantum system, the more secure you can ensure your communication is and the more information you can encode,” Ciarán Lee, from University College London, told New Scientist. “But going from a qubit to a qutrit is especially difficult — the tricks you use for qubits have to do with a nice symmetry that qutrits don’t have.”

The researchers report in their new paper, published in Physical Review Letters, that the demonstrations of their technique had a 75 per cent fidelity. That may appear low, but is far better than similar techniques that do not use quantum entanglement and is likely to improve in time.MORE ABOUTQUANTUM TELEPORTATION

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# An important quantum algorithm may actually be a property of nature

## Evidence that quantum searches are an ordinary feature of electron behavior may explain the genetic code, one of the greatest puzzles in biology.

by Emerging Technology from the arXivSep 12, 2019

Back in 1996, a quantum physicist at Bell Labs in New Jersey published a new recipe for searching through a database of N entries. Computer scientists have long known that this process takes around N steps because in the worst case, the last item on the list could be the one of interest.

However, this physicist, Lov Grover, showed how the strange rules of quantum mechanics allowed the search to be done in a number of steps equal to the square root of N.

That was a big deal. Searching databases is a foundational task in computer science, used for everything from finding telephone numbers to breaking cryptographic codes. So any speed-up is a significant advance.

Quantum mechanics provided an additional twist. At the time, Grover’s recipe was only the second quantum algorithm that had been proved faster than its classical counterpart. (The first was Peter Shor’s algorithm for factoring numbers, which he discovered in 1994.) Grover’s work was an important factor in preparing the way for the quantum computing revolution that is still ongoing today.

But despite the interest, implementing Grover’s algorithm has taken time because of the significant technical challenges involved. The first quantum computer capable of implementing it appeared in 1998, but the first scalable version didn’t appear until 2017, and even then it worked with only three qubits. So new ways to implement the algorithm are desperately needed.

Today Stéphane Guillet and colleagues at the University of Toulon in France say this may be easier than anybody expected. They say they have evidence that Grover’s search algorithm is a naturally occurring phenomenon. “We provide the first evidence that under certain conditions, electrons may naturally behave like a Grover search, looking for defects in a material,” they say.

That has obvious implications for quantum computing, but its real import may be much more profound. For some time, theorists have debated whether quantum search could explain one of the greatest mysteries about the origin of life. The idea that Grover searches occur in nature could finally solve the conundrum.

First some background. Because it is so fundamental, Grover’s search algorithm can be reformulated in a variety of ways. One of these is as a quantum walk across a surface—the way a quantum particle would move randomly from one point to another.

##### What is quantum communication?

###### Explainer: What is quantum communication?

Researchers and companies are creating ultra-secure communication networks that could form the basis of a quantum internet. This is how it works.

Clearly, this process is a kind of search of two-dimensional space. But because a quantum particle can explore many paths at the same time, it is much faster than a classical search.

The nature of the surface has an important influence on the search. For example, one type of surface consists of a square grid where the quantum particle has four possible moves at each vertex.

But there are many other possible grids; a triangular one, for example, where the quantum particle has three choices at each vertex. “The triangular grid is of particular interest because of its resemblance to several naturally occurring crystal-like materials,” say Guillet and co.

The team focused on simulating the way a Grover search works for electrons exploring triangular and square grids, but they also included other physically realistic effects, such as defects in the grid in the form of holes, and quantum properties such as interference effects.

The results are eye-opening. The question they ask is how quickly an electron can find the hole in a grid. And the team’s big breakthrough is to show that these simulations reproduce the way real electrons behave in real materials.

In other words, this is evidence that free electrons naturally implement the Grover search algorithm when moving across the surface of certain crystals.

That has immediate implications for quantum computing. “[This work] may be the path to a serious technological leap, whereby experimentalist would bypass the need for a full-fledged scalable and error-correcting Quantum Computer, and take the shortcut of looking for ‘natural occurrences’ of the Grover search instead,” say the team.

The work also has implications for our thinking about the genetic code and the origin of life. Every living creature on Earth uses the same code, in which DNA stores information using four nucleotide bases. The sequences of nucleotides encode information for constructing proteins from an alphabet of 20 amino acids.

But why these numbers—four and 20—and not some others? Back in 2000, just a few years after Grover published his work, Apoorva Patel at the Indian Institute of Science in Bangalore showed how Grover’s algorithm could explain these numbers.

Patel’s idea is related to the way DNA is assembled inside cells. In this situation, the molecular machinery inside a cell must search through the molecular soup of nucleotide bases to find the right one. If there are four choices, a classical search takes four steps on average. So the machinery would have to try four different bases during each assembly step.

But a quantum search using Grover’s algorithm is much quicker: Patel showed that when there are four choices, a quantum search can distinguish between four alternatives in a single step. Indeed, four is optimal number.

This thinking also explains why there are 20 amino acids. In DNA, each set of three nucleotides defines a single amino acid. So the sequence of triplets in DNA defines the sequence of amino acids in a protein.

But during protein assembly, each amino acid must be chosen from a soup of 20 different options. Grover’s algorithm explains these numbers: a three-step quantum search can find an object in a database containing up to 20 kinds of entry. Again, 20 is the optimal number.

In other words, if the search processes involved in assembling DNA and proteins is to be as efficient as possible, the number of bases should be four and the number of amino acids should to be 20—exactly as is found. The only caveat is that the searches must be quantum in nature.

When Patel published his idea, quantum physicists immediately pooh-poohed it. At the time, they were bogged down in their own attempts to control quantum processes, which they could do only by isolating quantum particles in extreme environments such as at temperatures close to absolute zero.

The obvious problem, they said, was that living things operate in a warm, messy environment in which quantum states would be immediately destroyed.

Biologists were equally dismissive, saying that quantum processes couldn’t possibly be at work inside living things.

Since then, an increasing body of evidence has emerged that quantum processes play an important role in a number of biological mechanisms. Photosynthesis, for example, is now thought to be an essentially quantum process.

The work of Guillet and co throws a new perspective on all this. It suggests that Grover’s algorithm is not only possible in certain materials; it seems to be a property of nature. And if that’s true, then the objections to Patel’s ideas start to crumble.

It may be that life is just an example of Grover’s quantum search at work, and that this algorithm is itself a fundamental property of nature. That’s a Big Idea if ever there was one.

Ref: arxiv.org/abs/1908.11213 : The Grover search as a naturally occurring phenomenonShareLinkAuthor

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The key to bigger quantum computers could be to build them like Legos

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## Facebook is making it harder to use ads to mess with the 2020 election

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#### Special issue: longevity

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ExpandClimate ChangeAug 19

## How nanoparticles that harvest light could curb climate emissions

### Syzygy Plasmonics has raised nearly $6 million to produce a cleaner form of hydrogen, using a novel type of photocatalyst. Humans and TechnologyAug 19 ## The anatomy of a sextortion spam campaign ### The latest genre of nuisance email tries to blackmail victims with threats to send embarrassing images or information to their contacts. A new analysis reveals just how much money this type of scam can generate. Read moreSilicon ValleyAug 19 ## China’s state media is paying Twitter to promote ads against Hong Kong protesters China’s official state news agency Xinhua is promoting tweets attacking the protesters and claiming they do not have wider support…. ExpandBlockchainAug 15 ## New money-laundering rules change everything for cryptocurrency exchanges ### Complying with regulators could mean the difference between going mainstream and remaining forever on the margins of the global financial system. 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It’ll certainly feel like longer.ComputingAug 14 ## Data leak exposes unchangeable biometric data of over 1 million people You can always change your password. Your fingerprints and face are another story…. ExpandSilicon ValleyAug 14 ## Facebook paid people to listen to voice recordings, too It’s the fifth big tech company forced to admit to the practice this year…. ExpandComputingAug 13 ## 16 million Americans will vote on hackable paperless machines ### Experts agree that paper ballots are needed, but eight American states will use completely paperless machines in the 2020 elections. BlockchainAug 13 ## North Korea is funding its weapons program with stolen cryptocurrency Hacking has become a central strategy for Kim Jong-un’s regime to generate revenue and soften the blow of economic sanctions. A new report from the United Nations, seen by the Associated Press,… ExpandTech PolicyAug 13 ## The tech employee backlash, Whole Foods edition The news: A group of anonymous Whole Foods employees is calling out Amazon (which bought the supermarket chain in 2017) for working with Immigration and Customs Enforcement (ICE). An open letter,… ## Having mastered Space Invaders, chess, and Go, AI tackles video soccer ### Google’s artificial-intelligence researchers have created a football simulator for training the next generation of machine-learning algorithms. ## Nvidia just made it easier to build smarter chatbots and slicker fake news ### Chip maker Nvidia is betting that AI’s language skills will advance rapidly—it’s releasing a powerful tool for putting together chatty programs. ## These companies claim to provide “fair-trade” data work. Do they? ## China has started a grand experiment in AI education. 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ExpandTech PolicyAug 6 ## CloudFlare dropping 8chan helps fight hate even if 8chan comes back ### CloudFlare has changed expectations of the moral obligations of technology companies. BiotechnologyAug 6 ## A fake eye that sheds fake tears could replace animal testing Is that supposed to be an eye? Yes, and the researchers who made this biomechanical version (which contains human cells) say it might eventually replace animals for testing…. ExpandSmart CitiesAug 6 ## A Japanese “flying car” has successfully made its first test flight Companies around the world are racing to be the first to launch self-flying vehicles…. ExpandTech PolicyAug 5 ## Democrats have told Google to make its contractors permanent employees The news: Ten Democratic senators—including presidential hopefuls Kamala Harris, Bernie Sanders, and Elizabeth Warren—have asked Google to turn its army of temporary workers into full-time employees…. 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What could go wrong? ## AI researchers need to stop hiding the climate toll of their work The Allen Institute for Artificial Intelligence (AI2) is proposing a new way to incentivize energy-efficient machine learning…. ## You’re not imagining it: always checking dating apps makes you feel worse study just out in the Journal of Social and Personal Relationships found that people who compulsively checked dating apps ended up feeling more lonely than before…. ExpandTech PolicyAug 1 ## How YouTubers plan to take on YouTube for better working conditions BlockchainAug 1 ## “Crypto rogue” nations want to use blockchains to undermine the US dollar ## Disgraced CRISPR scientist had plans to start a designer-baby business Read moreComputingAug 1 ## Business Lab podcast: Keren Elazari on the importance of hackers ### How thinking of helpful hackers as the immune systems of the internet can make your security stronger and better prepare and secure your digital presence. BiotechnologyAug 1 ## Scientists are making human-monkey hybrids in China ### Scientists may have taken a big—and controversial—leap by mixing human cells into monkey embryos. Read moreComputingAug 1 ## A new bill aims to protect US voters from the next Cambridge Analytica As the 2020 campaign season accelerates, a US lawmaker introduced a bill on Thursday that would regulate how political parties use voters’ data in federal elections…. ExpandBiotechnologyAug 1 ## The Impossible Burger is coming to a shop (and a Burger King) near you US regulators have approved a key ingredient in Impossible Foods’ plant-based burger patties, clearing the way for them to go on sale directly to consumers…. 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## Facial recognition algorithms can be deployed to hunt for dark matter

The same type of neural networks that make facial recognition tools are now being used to look for and characterize dark matter.

The dark stuff: We’ve never directly detected or measured dark matter, but we know it exists because we can see its influence on the universe. Many of the movements of different stars and galaxies cannot be explained by the gravity exerted by the matter we can see. Something else must be at play.

All matter, including dark matter, is capable of slightly bending light rays and causing “weak gravitational lensing” in the observations made by our telescopes. Scientists can use these distortions to map out the regions of the night sky where dark matter is most likely situated.

Looking for familiar faces: A team of researchers from ETH Zurich in Switzerland trained a neural network model (of the type often employed to analyze visual imagery) to look for subtle signs of weak gravitational lensing caused by dark matter. The model was trained by being fed simulated data that taught it what scientists typically look for when hunting for dark matter. The model ended up being 30% more accurate than human scientists at spotting and labeling potential signs of dark matter in images.

The machine-learning model was then used to analyze actual dark-matter maps created from the European Southern Observatory’s Kilo-Degree Survey (KiDS). Findings reported in Physical Review D this week show that the model was able to analyze the maps in finer detail than current methods, and modeled the likely location of dark matter more sharply.

The other dark mystery: Next, the team hopes to apply the model to other cosmological maps to learn more about the distribution and behavior of dark energy, a mysterious force that is driving the expansion of the universe.ShareLinkTaggedFace RecognitionAuthor

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# Quantum teleportation

Quantum teleportation is a process in which quantum information (e.g. the exact state of an atom or photon) can be transmitted (exactly, in principle) from one location to another, with the help of classical communication and previously shared quantum entanglement between the sending and receiving location. Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for faster-than-light transport or communication of classical bits. While it has proven possible to teleport one or more qubits of information between two (entangled) quanta,[1][2][3] this has not yet been achieved between anything larger than molecules.[4]

Although the name is inspired by the teleportation commonly used in fiction, quantum teleportation is limited to the transfer of information rather than matter itself. Quantum teleportation is not a form of transportation, but of communication: it provides a way of transporting a qubit from one location to another without having to move a physical particle along with it.

The term was coined by physicist Charles Bennett. The seminal paper[5] first expounding the idea of quantum teleportation was published by C. H. Bennett, G. BrassardC. CrépeauR. JozsaA. Peres, and W. K. Wootters in 1993.[6] Quantum teleportation was first realized in single photons,[7] later being demonstrated in various material systems such as atoms, ions, electrons and superconducting circuits. The latest reported record distance for quantum teleportation is 1,400 km (870 mi) by the group of Jian-Wei Pan using the Micius satellite for space-based quantum teleportation.[8][9][10]

## Non-technical summary

In matters relating to quantum or classical information theory, it is convenient to work with the simplest possible unit of information, the two-state system. In classical information, this is a bit, commonly represented using one or zero (or true or false). The quantum analog of a bit is a quantum bit, or qubit. Qubits encode a type of information, called quantum information, which differs sharply from “classical” information. For example, quantum information can be neither copied (the no-cloning theorem) nor destroyed (the no-deleting theorem).

Quantum teleportation provides a mechanism of moving a qubit from one location to another, without having to physically transport the underlying particle to which that qubit is normally attached. Much like the invention of the telegraph allowed classical bits to be transported at high speed across continents, quantum teleportation holds the promise that one day, qubits could be moved likewise.[11][12] As of 2015, the quantum states of single photons, photon modes, single atoms, atomic ensembles, defect centers in solids, single electrons, and superconducting circuits have been employed as information bearers.[13]

The movement of qubits does not require the movement of “things” any more than communication over the internet does: no quantum object needs to be transported, but it is necessary to communicate two classical bits per teleported qubit from the sender to the receiver. The actual teleportation protocol requires that an entangled quantum state or Bell state be created, and its two parts shared between two locations (the source and destination, or Alice and Bob). In essence, a certain kind of quantum channel between two sites must be established first, before a qubit can be moved. Teleportation also requires a classical information channel to be established, as two classical bits must be transmitted to accompany each qubit. The reason for this is that the results of the measurements must be communicated between the source and destination so as to reconstruct the qubit, or else the state of the destination qubit would not be known to the source, and any attempt to reconstruct the state would be random; this must be done over ordinary classical communication channels. The need for such classical channels may, at first, seem disappointing, and this explains why teleportation is limited to the speed of transfer of information, i.e., the speed of light. The main advantages is that Bell states can be shared using photons from lasers,[8] and so teleportation is achievable through open space, i.e., without the need to send information through cables or optical fibers.

The quantum states of single atoms have been teleported.[1][3][2] Quantum states can be encoded in various degrees of freedom of atoms. For example, qubits can be encoded in the degrees of freedom of electrons surrounding the atomic nucleus or in the degrees of freedom of the nucleus itself. It is inaccurate to say “an atom has been teleported”. It is the quantum state of an atom that is teleported. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them.[3] The importance of teleporting the nuclear state is unclear:[citation needed] the nuclear state does affect[how?] the atom, e.g. in hyperfine splitting, but whether such state would need to be teleported in some futuristic “practical” application is debatable.[according to whom?]

An important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems by creating or placing two or more separate particles into a single, shared quantum state. These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments. Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region, even though light hasn’t yet had time to travel the distance; a conclusion seemingly at odds with special relativity (EPR paradox). However such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the no-communication theorem. Thus, teleportation, as a whole, can never be superluminal, as a qubit cannot be reconstructed until the accompanying classical information arrives.

Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebraHilbert spaces and projection matrixes. A qubit is described using a two-dimensional complex number-valued vector space (a Hilbert space), which are the primary basis for the formal manipulations given below. A working knowledge of quantum mechanics is not absolutely required to understand the mathematics of quantum teleportation, although without such acquaintance, the deeper meaning of the equations may remain quite mysterious.

## Protocol

Diagram for quantum teleportation of a photon

The prerequisites for quantum teleportation are a qubit that is to be teleported, a conventional communication channel capable of transmitting two classical bits (i.e., one of four states), and means of generating an entangled EPR pair of qubits, transporting each of these to two different locations, A and B, performing a Bell measurement on one of the EPR pair qubits, and manipulating the quantum state of the other pair. The protocol is then as follows:

1. An EPR pair is generated, one qubit sent to location A, the other to B.
2. At location A, a Bell measurement of the EPR pair qubit and the qubit to be teleported (the quantum state {\displaystyle |\phi \rangle }) is performed, yielding one of four measurement outcomes, which can be encoded in two classical bits of information. Both qubits at location A are then discarded.
3. Using the classical channel, the two bits are sent from A to B. (This is the only potentially time-consuming step after step 1, due to speed-of-light considerations.)
4. As a result of the measurement performed at location A, the EPR pair qubit at location B is in one of four possible states. Of these four possible states, one is identical to the original quantum state {\displaystyle |\phi \rangle }, and the other three are closely related. Which of these four possibilities actually obtained, is encoded in the two classical bits. Knowing this, the EPR pair qubit at location B is modified in one of three ways, or not at all, to result in a qubit identical to {\displaystyle |\phi \rangle }, the qubit that was chosen for teleportation.

It is worth to notice that the above protocol assumes that the qubits are individually addressable, that means the qubits are distinguishable and physically labeled. However, there can be situations where two identical qubits are indistinguishable due to the spatial overlap of their wave functions. Under this condition, the qubits cannot be individually controlled or measured. Nevertheless, a teleportation protocol analogous to that described above can still be (conditionally) implemented by exploiting two independently-prepared qubits, with no need of an initial EPR pair. This can be made by addressing the internal degrees of freedom of the qubits (e.g., spins or polarizations) by spatially localized measurements performed in separated regions A and B shared by the wave functions of the two indistinguishable qubits.[14]

## Experimental results and records

Work in 1998 verified the initial predictions,[15] and the distance of teleportation was increased in August 2004 to 600 meters, using optical fiber.[16] Subsequently, the record distance for quantum teleportation has been gradually increased to 16 kilometres (9.9 mi),[17] then to 97 km (60 mi),[18] and is now 143 km (89 mi), set in open air experiments in the Canary Islands, done between the two astronomical observatories of the Instituto de Astrofísica de Canarias.[19] There has been a recent record set (as of September 2015) using superconducting nanowire detectors that reached the distance of 102 km (63 mi) over optical fiber.[20] For material systems, the record distance is 21 metres (69 ft).[21]

A variant of teleportation called “open-destination” teleportation, with receivers located at multiple locations, was demonstrated in 2004 using five-photon entanglement.[22] Teleportation of a composite state of two single qubits has also been realized.[23] In April 2011, experimenters reported that they had demonstrated teleportation of wave packets of light up to a bandwidth of 10 MHz while preserving strongly nonclassical superposition states.[24][25] In August 2013, the achievement of “fully deterministic” quantum teleportation, using a hybrid technique, was reported.[26] On 29 May 2014, scientists announced a reliable way of transferring data by quantum teleportation. Quantum teleportation of data had been done before but with highly unreliable methods.[27][28] On 26 February 2015, scientists at the University of Science and Technology of China in Hefei, led by Chao-yang Lu and Jian-Wei Pan carried out the first experiment teleporting multiple degrees of freedom of a quantum particle. They managed to teleport the quantum information from ensemble of rubidium atoms to another ensemble of rubidium atoms over a distance of 150 metres (490 ft) using entangled photons.[29][30][31] In 2016, researchers demonstrated quantum teleportation with two independent sources which are separated by 6.5 km (4.0 mi) in Hefei optical fiber network.[32] In September 2016, researchers at the University of Calgary demonstrated quantum teleportation over the Calgary metropolitan fiber network over a distance of 6.2 km (3.9 mi).[33]

Researchers have also successfully used quantum teleportation to transmit information between clouds of gas atoms, notable because the clouds of gas are macroscopic atomic ensembles.[34][35]

In 2018, physicists at Yale demonstrated a deterministic teleported CNOT operation between logically encoded qubits.[36]

## Formal presentation

There are a variety of ways in which the teleportation protocol can be written mathematically. Some are very compact but abstract, and some are verbose but straightforward and concrete. The presentation below is of the latter form: verbose, but has the benefit of showing each quantum state simply and directly. Later sections review more compact notations.

The teleportation protocol begins with a quantum state or qubit {\displaystyle |\psi \rangle }, in Alice’s possession, that she wants to convey to Bob. This qubit can be written generally, in bra–ket notation, as:

{\displaystyle |\psi \rangle _{C}=\alpha |0\rangle _{C}+\beta |1\rangle _{C}.}

The subscript C above is used only to distinguish this state from A and B, below.

Next, the protocol requires that Alice and Bob share a maximally entangled state. This state is fixed in advance, by mutual agreement between Alice and Bob, and can be any one of the four Bell states shown. It does not matter which one.

{\displaystyle |\Phi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})},
{\displaystyle |\Phi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B})},
{\displaystyle |\Psi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B})},
{\displaystyle |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B})}.

In the following, assume that Alice and Bob share the state {\displaystyle |\Phi ^{+}\rangle _{AB}.} Alice obtains one of the particles in the pair, with the other going to Bob. (This is implemented by preparing the particles together and shooting them to Alice and Bob from a common source.) The subscripts A and B in the entangled state refer to Alice’s or Bob’s particle.

At this point, 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

{\displaystyle |\psi \rangle _{C}\otimes |\Phi ^{+}\rangle _{AB}=(\alpha |0\rangle _{C}+\beta |1\rangle _{C})\otimes {\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}).}

Alice will then make a local measurement in the Bell basis (i.e. the four Bell states) on the two particles in her possession. To make the result of her measurement clear, it is best to write the state of Alice’s two qubits as superpositions of the Bell basis. This is done by using the following general identities, which are easily verified:

{\displaystyle |0\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle +|\Phi ^{-}\rangle ),}
{\displaystyle |0\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle +|\Psi ^{-}\rangle ),}
{\displaystyle |1\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle -|\Psi ^{-}\rangle ),}

and

{\displaystyle |1\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle -|\Phi ^{-}\rangle ).}

One applies these identities with A and C subscripts. The total three particle state, of AB and C together, thus becomes the following four-term superposition:

{\displaystyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\ =\\{\frac {1}{2}}{\Big \lbrack }\ &|\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 (\alpha |1\rangle _{B}-\beta |0\rangle _{B}){\Big \rbrack }.\\\end{aligned}}}[37]

The above is just 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 total state. The actual teleportation occurs when Alice measures her two qubits A,C, in the Bell basis

{\displaystyle |\Phi ^{+}\rangle _{AC},|\Phi ^{-}\rangle _{AC},|\Psi ^{+}\rangle _{AC},|\Psi ^{-}\rangle _{AC}}.

Experimentally, this measurement may be achieved via a series of laser pulses directed at the two particles[citation needed]. Given the above expression, evidently the result of Alice’s (local) measurement is that the three-particle state would collapse to one of the following four states (with equal probability of obtaining each):

• {\displaystyle |\Phi ^{+}\rangle _{AC}\otimes (\alpha |0\rangle _{B}+\beta |1\rangle _{B})}
• {\displaystyle |\Phi ^{-}\rangle _{AC}\otimes (\alpha |0\rangle _{B}-\beta |1\rangle _{B})}
• {\displaystyle |\Psi ^{+}\rangle _{AC}\otimes (\alpha |1\rangle _{B}+\beta |0\rangle _{B})}
• {\displaystyle |\Psi ^{-}\rangle _{AC}\otimes (\alpha |1\rangle _{B}-\beta |0\rangle _{B})}

Alice’s two particles are now entangled to each other, in one of the four Bell states, and the entanglement originally shared between Alice’s and Bob’s particles 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 result of Alice’s Bell measurement tells her which of the above four states the system is in. She can now send her result 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 {\displaystyle \alpha |0\rangle _{B}+\beta |1\rangle _{B}}:

• If Alice indicates her result is {\displaystyle |\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 {\displaystyle |\Phi ^{-}\rangle _{AC}}, Bob would send his qubit through the unitary quantum gate given by the Pauli matrix
{\displaystyle \sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}

to recover the state.

• If Alice’s message corresponds to {\displaystyle |\Psi ^{+}\rangle _{AC}}, Bob applies the gate
{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}}

to his qubit.

• Finally, for the remaining case, the appropriate gate is given by
{\displaystyle \sigma _{1}\sigma _{3}=-\sigma _{3}\sigma _{1}=-i\sigma _{2}={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}.}

Teleportation is thus achieved. The above-mentioned three gates correspond to rotations of π radians (180°) about appropriate axes (X, Y and Z) in the Bloch sphere picture of a qubit.

Some remarks:

• After this operation, Bob’s qubit will take on the state {\displaystyle |\psi \rangle _{B}=\alpha |0\rangle _{B}+\beta |1\rangle _{B}}, and Alice’s qubit becomes an (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.
• 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.

Alice’s state in qubit 2 is transferred to Bob’s qubit 0 using a priorly entangled pair of qubits between Alice and Bob, qubits 1 and 0.

## Alternative notations

Quantum teleportation in its diagrammatic form.,[38] employing Penrose graphical notation.[39] Formally, such a computation takes place in a dagger compact category. This results in the abstract description of quantum teleportation as employed in categorical quantum mechanics.

Quantum circuit representation of quantum teleportation

There are a variety of different notations in use that describe the teleportation protocol. One common one is by using the notation of quantum gates. In the above derivation, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can be written using quantum gates. Direct calculation shows that this gate is given by

{\displaystyle G=(H\otimes I)\;C_{N}}

where H is the one qubit Walsh-Hadamard gate and {\displaystyle C_{N}} is the Controlled NOT gate.

## Entanglement swapping

Teleportation can be applied not just to pure states, but also mixed states, that can be regarded as the state of a single subsystem of an entangled pair. 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 does 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.

A detailed diagrammatic derivation of entanglement swapping has been given by Bob Coecke,[40] presented in terms of categorical quantum mechanics.

## Generalizations of the Teleportation Protocol

The basic teleportation protocol for a qubit described above has been generalized in several directions, in particular regarding the dimension of the system teleported and the number of parties involved (either as sender, controller, or receiver).

#### d-dimensional systems

A generalization to {\displaystyle d}-level systems (so-called qudits) is straight forward and was already discussed in the original paper by Bennett et al.:[5] the maximally entangled state of two qubits has to be replaced by a maximally entangled state of two qudits and the Bell measurement by a measurement defined by a maximally entangled orthonormal basis. All possible such generalizations were discussed by Werner in 2001.[41] The generalization to infinite-dimensional so-called continuous-variable systems was proposed in [42] and led to the first teleportation experiment that worked unconditionally.[43]

#### Multipartite versions

The use of multipartite entangled states instead of a bipartite maximally entangled state allows for several new features: either the sender can teleport information to several receivers either sending the same state to all of them (which allows to reduce the amount of entanglement needed for the process) [44] or teleporting multipartite states [45] or sending a single state in such a way that the receiving parties need to cooperate to extract the information.[46] A different way of viewing the latter setting is that some of the parties can control whether the others can teleport.

## Logic gate teleportation

In general, mixed states ρ may be transported, and a linear transformation ω applied during teleportation, thus allowing data processing of quantum information. This is one of the foundational building blocks of quantum information processing. This is demonstrated below.

### 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

{\displaystyle \rho \otimes \omega .}

A successful teleportation process is a LOCC quantum channel Φ that satisfies

{\displaystyle (\operatorname {Tr} _{12}\circ \Phi )(\rho \otimes \omega )=\rho \,,}

where Tr12 is the partial trace operation with respect systems 1 and 2, and {\displaystyle \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

{\displaystyle \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 {\displaystyle I\otimes O} is {\displaystyle 12\otimes 3} while that of {\displaystyle \rho \otimes \omega } is {\displaystyle 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

{\displaystyle {F_{i}}={M_{i}^{2}}.}

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

{\displaystyle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}

The tensor factor in {\displaystyle (M_{i}\otimes I)} is {\displaystyle 12\otimes 3} while that of {\displaystyle \rho \otimes \omega } is {\displaystyle 1\otimes 23}. Bob then applies a corresponding local operation Ψi on system 3. On the combined system, this is described by

{\displaystyle (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 {\displaystyle 1\otimes 2}.

Therefore, the channel Φ is defined by

{\displaystyle \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

{\displaystyle \langle \Phi (\rho \otimes \omega ),I\otimes O\rangle =\langle \rho ,O\rangle }

holds. The left hand side of the equation is:

{\displaystyle \sum _{i}\langle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes O\rangle }
{\displaystyle =\sum _{i}\langle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes \Psi _{i}^{*}(O)\rangle }

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

{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O)).}

The success criterion for teleportation has the expression

{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O))=\operatorname {Tr} \;\rho \cdot O.}

## Local explanation of the phenomenon

local explanation of quantum teleportation is put forward by David Deutsch and Patrick Hayden, with respect to the many-worlds interpretation of quantum mechanics. Their paper asserts that the two bits that Alice sends Bob contain “locally inaccessible information” resulting in the teleportation of the quantum state. “The ability of quantum information to flow through a classical channel […], surviving decoherence, is […] the basis of quantum teleportation.”[47]

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