Prime Numbers, Encryption and the Linux Factor Command
Have you ever needed to print the prime factors of a number on the Linux command line? Me neither. However, a tool does exist for it. Enter the factor command.
The factor command is part of the GNU Core Utilities package, therefore it is available on almost any Linux system. This little beauty has the singular purpose of producing the prime factors of any number. To me, this is pretty neat. To anyone interested in learning cryptography or number theory, this may be a useful, if not fun, little utility.
Prime Numbers and Prime Factors
Prime numbers have long been subjects of great interest to mathematicians, especially in the field of combinatorics. They are interesting because they are whole numbers that are only divisible by themselves and one. For example, the only way to multiply two whole numbers together to produce '5' is '5 x 1=5', whereas '6' factors by '3x2=6' as well as by one and itself.
Six is a composite number, numbers that are not prime numbers.
According to Wolfram-Alpha, a Mersenne prime is a prime that fits the formula M = 2^n - 1. Or, one subtracted from any power of 2. They were named for Martin Mersenne, a 17th-century French monk that studied the numbers. There exists an infinite number of Mersenne prime numbers.
Prime numbers can get large in and of themselves: the current, largest prime number has 24,862,048 digits!
Multiplying prime numbers together, even large ones is a straightforward task. The product of two prime numbers is called a semi-prime. A cheap desk calculator can do this with ease. Plenty of people can count by prime numbers and multiply big numbers without paper using a variety of techniques.
Prime factors are a different problem altogether. Any composite number can be made up of many different combinations of prime numbers.
Simply put, prime factors are the prime numbers that can be factored from any number, other than 1 and itself.
Factoring is the process of breaking down a number into the two numbers originally multiplied together. For example, 9 is the product of the prime number 3 and itself. This seems simple with very small numbers like 9, especially if you've had experience factoring hundreds of polynomials in high school. As with polynomials, prime factors become infinitely more complex the larger the numbers involved.
Big Prime Number, Big Problem
Multiplying big prime numbers, while still relatively easy, results in even bigger non-prime numbers. The number 330 has prime factors of 2, 3, 5, and 11. The larger your numbers get the more possible factorizations. Now, go through one by one and multiply each of those prime numbers together in different combinations until you get 330. Not impossible, but certainly more difficult than 9.
Factoring very large numbers, like the Mersenne prime numbers can take powerful computers years to complete.
Any method used to multiply and divide large numbers is useless when factoring prime numbers. Prime factors can only be found through trial and error. To find the correct pairing of primes to factor requires testing every prime against each other from 2 up to the nth prime, except for one and the very large number in question.
Linux and its Factor command use an algorithm called Pollard-Brent Rho to derive prime factors for relatively small numbers. The algorithm is quite powerful and can calculate the eighth Fermat number, 2^256+1 in approximately 20 seconds, depending upon hardware constraints. Distributions not using the GNU MP will have reduced capability with this command.
Prime Factors and Encryption
Cryptography is an essential aspect of modern society. Computers transfer data to other computers all over the world every nanosecond of every day. Threats to the security and accuracy of information pathways challenge hardware and infrastructure improvements as fast as they are conceived.
Globally, industries rely on encryption protocols to protect themselves and, sometimes only tangentially, consumers from identity theft, fraud, and violations of privacy. Prime factors are very useful in creating encryption keys to secure information over digital transmission media.
Because all composite numbers are made up of prime numbers, any composite number can be used as the public key, that allows messages to be encrypted. Only those in possession of the secret key can decrypt the message into plain text.
So any public key will be a very large composite number. The secret key will be the very large prime numbers, otherwise known as the prime factors of the composite number.
Prime factor cryptography guarantees security and privacy by creating a factorization problem that even supercomputers, let alone the most advanced consumer electronics, would be hard-pressed to solve within a century. Because of this, some law enforcement entities seek to restrict cryptography and prime factor usage to prevent freedom fighters and terrorists alike from obtaining secure means of communication.
It is fairly common to hear or read the phrase 1024-bit encryption. This describes the number used for the public key. The public key number used will be an integer with more than 2^1023 digits but less than 2^1024 digits. The secret key would be the two primes that produce this integer.
While modern supercomputers cannot crack this encryption in any reasonable amount of time, quantum computing will eventually render this method useless.
Linux Factor Command
As nifty as the Factor command may be, it is not useful in modern cryptography. Since Factor cannot find the prime factors of large numbers within any reasonable amount of time, it would be to simplistic for modern cryptography. But it is useful for learning cryptography basics or simply enjoying the elegance of numbers.
Factor Command Syntax and Options (or lack thereof)
The factor command has no functional options. The only options that exist are --help and --version. It simply takes a argument or list of arguments in the form of an integer number. It will also accept an integer from standard input (STDIN).
[[email protected] ~]$ factor 11 11: 11 [[email protected] ~]$ factor 77 77: 7 11 [[email protected] ~]$ factor 34578 11 77 34578: 2 3 3 17 113 11: 11 77: 7 11
The Linux factor command is a cool bit of computing history and it's interesting that it has remained a part of Unix since 1979. In 1986, Paul Rubin wrote a free software version of factor for the GNU project.
Some UNIX/Linux variants consider factor a game rather than a utility.The current GNU documentation categorizes Factor as a numerical operation, which makes more sense in my opinion. It finds use with number theorists, number enthusiasts, and when you require simple derivation of prime factors.
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Researchers at Google AI Quantum have announced a successful experiment in which for the first time a quantum computer has performed a task that ordinary computers based on integrated circuits are incapable of doing in a reasonable amount of time. This technical milestone paves the way for far-reaching advances in physics, chemistry, astronomy, materials science, machine learning and a host of other fields.
The results were produced using Google’s quantum computer, dubbed Sycamore. It is the product of a collaboration between 75 scientists led by Frank Arute at Google, NASA, Oak Ridge National Laboratory and more than a dozen other facilities in Germany and the United States. They compared how fast their machine and the world’s most powerful supercomputer, Summit, could produce a random number from a specially designed circuit one million times.
The experiment was then repeated multiple times on increasingly complex algorithms until they could show that while a quantum computer generated a result, a classical computer could not. During their final experiment, Sycamore produced its one million random numbers in 200 seconds. Summit was estimated to need 10,000 years to perform the same calculations.
This exponential increase in computing speed is the first documented instance of so-called quantum supremacy. The term was popularized by John Preskill in 2011 to describe the set of problems that are shown to be intractable for even the best modern computers but that should be relatively straightforward for the quantum computers being developed, thus providing a measure to determine if a given quantum computer had in fact surpassed the computational ability of conventional electronics.
Quantum supremacy also defines certain engineering milestones. While quantum computers have always held the promise of being able to do exponentially more processes per second than conventional machines, they have proven exponentially more difficult to build and maintain. It was not at all clear that quantum computers would in practice ever surpass supercomputers. Nonetheless, Google’s research indicates that there is at least one case where quantum computers are supreme, and suggests that there are many others.
The end goal, however, is not just to produce random numbers. An off-the-shelf laptop can produce a million random numbers in seconds if the algorithms used to produce them are not purposefully made complicated, as were the test cases for Sycamore and Summit. Rather, quantum computers have in theory the capability of solving in minutes problems that even the best supercomputers would likely not solve in the lifespan of our solar system. Two of these include simulating the motion of atomic and subatomic particles and factoring integers of several hundred digits.
To solve them, one must go beyond familiar binary models of computation which are used in today’s personal computers, tablets and phones. These devices store and process information in their memory using distinct physical states, usually some sort of switch being turned off or on, and the data they contain is often described as a sequence of the symbols 0 and 1. One unit of information, a bit, consists of either a 0 or 1 and the number of bits, usually discussed as bytes (where one byte equals eight bits), is the measure of the size of a computer’s memory.
This method of storing and retrieving information takes a small but finite amount of time, an amount which is not noticeable for a single calculation yet can grow large very quickly. High-end modern laptops can perform tens of billions of operations per second while the Summit supercomputer is capable of 148 million billion operations per second. And yet, while Summit could multiply two 300-digit numbers almost instantaneously, it would take the supercomputer—using its most advanced algorithms—billions of years to factor the product. A quantum computer is hypothesized to be able to perform the same operation in minutes.
The original rationale for quantum computers was not to factor large numbers, a key part in certain types of encryption, but to directly simulate rather than approximate quantum mechanics. This field of physics, the study of the motion of matter at its smallest scales, is inherently probabilistic. The position and momentum of a particle are not, as in our everyday life, described as a pair of numbers but as two sets of well-defined probabilities. In the early 1980s, Soviet mathematician Yuri Manin and American physicists Paul Benioff and Richard Feynman realized that if a machine could be devised to perform operations using this property of matter, it would be able to calculate the motion of matter exactly as it occurs in nature.
Instead of switches, Manin, Benioff and Feynman proposed to store information in a fundamental particle such as a photon, the basic unit of light. The value of the “qubit” is stored within the inherent rotation of the photon, which is either positive or negative. The difference between a bit and a qubit, and this is key, is that a qubit initially has both the positive and negative values. Only when the photon interacts with some external particle or wave will it fall into a single state, and it will do so following the probabilistic laws of quantum mechanics. This is known as “state superposition.”
In addition to superposition, quantum computing also takes advantage of a second property of fundamental particles known as “entanglement.” It is possible to take two (or more) particles and force them to interact in such a way that even though separated, each particle acts as part of the same system. What results from this is the ability to act on a single entangled particle, which instantaneously acts on all others within the entangled system.
The combination of state superposition and entanglement is what make quantum computers so much more powerful than classical computers. A computer with 266 bits can store or process 266 pieces of information at a time. A quantum computer with 266 qubits can store or process 2^266 (10^80, a one followed by eighty zeros) pieces of information at a time, a number equivalent to the number of atoms in the observable universe.
Yet qubits are incredibly difficult to operate on. The particles that are storing information react with their surroundings, either nearby matter or the so-called vacuum of spacetime, which is not “nothing” but in fact a constant creation and annihilation of particles. This can cause unknown but definite interactions—called quantum decoherence—with one particle which translates to each other particle with which it is entangled, forcing researchers to reset the entire system. Each particle serving as a qubit must be isolated as much as possible from these unwanted connections, typically by physically isolating them and cooling their surroundings to temperatures close absolute zero.
While it is impossible to suppress all quantum decoherence, for that would involve stopping the motion of matter, an impossibility, a great deal of research from groups around the world has gone into eliminating most of the extraneous motion. This effort is what has allowed Arute’s team to successfully align and operate Sycamore, which consists of 53 working qubits, outperforming the world’s most powerful supercomputer, which consists of many trillions of bits.
This technology is expected to herald advances in a variety of fields. Quantum computers, when they are more capable of surpassing supercomputers in all problems, not just one, will be able to more quickly and accurately find exoplanets, determine the properties of new materials, study the outcome of chemical reactions, and produce more advanced forms of artificial intelligence. They are at the same time a striking confirmation of humanity’s ability to understand and master nature.
Quantum computers under capitalism, however, have the capacity for reinforcing oppression. Standard encryption schemes will be broken in minutes or seconds, giving nations or corporations the ability to spy on their rivals and the working class, as well as infiltrate, control and destroy the electronic systems of whole countries. Employees at their workplace can be tracked with even greater efficiency and forced to work longer and harder. Immigrants can be hunted down with facial recognition and other forms of tracking with increased ease. And the algorithms used by Google, Facebook and other tech companies in conjunction with the US military and intelligence agencies will have an unparalleled ability to censor the internet, particularly left-wing, anti-capitalist and socialist publications.
While Google’s Sycamore quantum computer is nowhere near capable of such feats, the social and political consequences of a private company or a capitalist government having control of such a machine must be understood. At the same time, this must galvanize struggle against capitalism and for the establishment of a society where such vast and fundamental advances can be changed from tools of violence and repression to instruments for securing a prosperous and fulfilling life for all people.


Quantum computers are not like classical computers. I don't mean that in the sense that quantum computers perform calculations in a different manner, or that they might be faster, or more clever. No, I mean that quantum computers come with a whole set of issues (read: headache-inducing problems) that normal computers don't.
To reduce these problems, researchers have taken to hiding quantum information, albeit not very successfully. It turns out that using more than one type of qubit offers a bit more camouflage to quantum information.
Quantum hide and seek
Before we get to the latest results, let me paint a picture of pain for you. In a quantum computer, calculations are achieved by manipulating the value of a target qubit—the quantum computing equivalent of a bit—in a way that depends on the value of other qubits. The problem is doing this cleanly.
In most cases, any qubits in a system are identical, so if I have a tool that can change one qubit, the same tool will change the neighboring qubits. These tools are unavoidably blunt, so modifying one qubit has a good chance of changing its neighbors.
Let's look at a specific example: a quantum computer that consists of a string of ions sitting in a trap (an ion is an atom with a missing electron). The ions influence each other by the way they collectively rock back and forth in the trap.
This collective motion is used to couple qubits together, but it is very easy to disrupt. Imagine that I want to set the qubit state of the central ion. To do that, I have to shine a laser onto it; it will (eventually) absorb a photon, changing its state. But nothing says that it will absorb the first photon that hits it. A photon that is not absorbed will be scattered, like a pinball off a bumper. That recoil changes the motion of the ion in the trap, disrupting the collective motion of all ions. This reduces the effectiveness of (and eventually kills off) the collective behavior that's needed for quantum computation.
But wait—it gets worse. The scattered photon can hit a neighboring qubit and be absorbed. If that happens, you have introduced an error in your computation. You may have intended to set the state of qubit No.3, but you have also changed the state of qubit No.2 as well.
To solve this problem, a group of researchers has shown how to use a quantum bystander to maintain the state of the qubits for much longer. Instead of using a string of identical ions, the researchers use two different ions. Beryllium ions are used for computation, and, in between each beryllium ion, they place a calcium ion. This protects quantum information in several ways.
The photons scattered from the beryllium ions can't easily reach other beryllium ions because the calcium ion is in the way. The calcium ion requires an entirely different color of light, so the scattered light from the beryllium ion doesn't change the quantum state of the calcium ion, while light scattered from the calcium ion doesn't affect the beryllium ion.
Yet these neighbors are not completely isolated from each other. The qubits are still coupled through the motion of the ions in the trap. Here, the calcium ion also plays a role. When the ions absorb or scatter light, they get a kick that makes their motion in the trap more vigorous. This motion needs to be controlled so that the links between qubits remain under control. To do this, the researchers can slow the calcium ions down (using lasers, naturally). By slowing the calcium ion down, the researchers suck energy out of all the trapped ions, bringing them back under control.
But what is really cool is how the researchers brought it all together in a three-qubit demonstration system (two beryllium ions and one calcium ion). The researchers put the calcium ion in a known quantum state, then perform a set of operations on all three qubits. Imperfections mean that there will eventually be some difference between the intended quantum state (i.e., the quantum information) and the target quantum state. This difference will grow with time thanks to the ions all having a slightly different environment.
This difference is revealed (at least partially) by measuring the state of the calcium ion. This can be done without destroying the quantum state of the beryllium ions.
In response to the measured state of the calcium ion, the trap and the state of the beryllium ions are carefully adjusted. Then the calcium ion is cooled and its state is reset. From there, the entire operation of coupling the calcium ions with the beryllium ions can be repeated.
The researchers compared the reliability of their qubit state (including entangled states) with and without the trick of adjusting the trap and the state of the beryllium ions. Without adjustment, the quantum information stored in the beryllium ions quickly decays away. However, with these careful corrections, the researchers were able to perform 50 operations on the beryllium ions without losing the quantum state.
The researchers' control system is not perfect—the information still decays away, but the decay rate is a good 20 times slower than it would be if they were just using two beryllium ions.
The best bit, though, is that there is nothing stopping the researchers scaling up to more ions. Three qubits is puny compared to other quantum computers. But hitting nine-plus qubits should be possible, which is about state of the art for ion-based quantum computers. Furthermore, the cooling and control should allow for scaling to even larger numbers of qubits. It's all pretty exciting.


From Santa Barbara, California, to Hefei, China, scientists are developing a new type of computer that will make today’s machines look like toys.
Harnessing the mysterious power of quantum mechanics, the technology will do in minutes what even supercomputers have been unable to do for thousands of years. In the fall of 2019, Google unveiled an experimental quantum computer that showed it was possible.Two years later, a laboratory in China did a lot of the same.
But quantum computing won’t reach its potential without the help of another technological breakthrough. Call it the “quantum internet” – a network of computers that can send quantum information between remote machines.
At Delft University of Technology in the Netherlands, a team of physicists has taken a major step toward the computer network of the future, using a technique called quantum teleportation to send data across three physical locations. Previously, only two could do this.
New experiments show that scientists can scale quantum networks across a growing number of sites. “We are now building small quantum networks in the laboratory,” said Delft physicist Ronald Hanson, who led the team. “But our idea is to eventually build a quantum internet.”
Their study was published this week A paper published in the scientific journal Nature, demonstrating the power of phenomena that Albert Einstein once thought impossible. Quantum teleportation – he called it “horror from afar“—information can be transferred between locations without actually moving the physical matter that holds it.
This technology could profoundly change the way data is transferred from one place to another. It draws on more than a century of research involving quantum mechanics, a field of physics that dominates the subatomic realm and behaves differently than anything we experience in our daily lives. Quantum teleportation not only moves data between quantum computers, but also in a way that no one can intercept.
“Not only does this mean that a quantum computer can solve your problem, but it doesn’t know what the problem is,” says Tracy Eleanor Northup, a researcher at the Institute of Experimental Physics at the University of Innsbruck, who is also exploring quantum teleportation. “It doesn’t work that way today. Google knows what you’re running on its servers.”
If certain objects are very small (like electrons or particles of light) or very cold (like exotic metals cooled to almost absolute zero or minus 460 degrees Fahrenheit), quantum computers take advantage of the strange way they behave. In these cases, a single object can behave like two separate objects at the same time.
Traditional computers perform computations by manipulating “bits” of information, each bit containing either a 1 or a 0. By exploiting the strange behavior of quantum mechanics, a qubit or qubit can store a combination of 1s and 0s – a little like how a spinning coin holds the tantalizing possibility that when it ends up flat on a table it will Appears head or tail.
This means that two qubits can hold four values at the same time, three qubits can hold eight, four can hold 16, and so on. As the number of qubits grows, the capabilities of quantum computers will increase exponentially.
Researchers believe these devices could one day accelerate the development of new drugs, advance advances in artificial intelligence, and quickly crack the encryption that protects computers vital to national security. Globally, governments, academic labs, start-ups and tech giants are spending billions to explore the technology.
In 2019, Google announced that its machine had achieved what scientists call “quantum supremacy,” meaning it can perform experimental tasks that conventional computers can’t. But most experts believe it will be at least a few more years — at least — before quantum computers can actually do something useful that you can’t do with another machine.
Part of the challenge is that if you read from it, the qubit breaks or “falls out” — it becomes a normal bit that can only hold 0 or 1, but not both. But by stringing together many qubits and developing ways to prevent decoherence, scientists hope to make machines that are both powerful and practical.
Ultimately, ideally, these will be joined to networks that can send information between nodes, allowing them to be used anywhere, just as cloud computing services from companies like Google and Amazon make processing power widely available today.
But this also has its own problems. Due in part to decoherence, quantum information cannot simply be copied and sent over traditional networks. Quantum teleportation offers another option.
While it can’t move objects from one place to another, it can move information using a quantum property called “entanglement”: a change in the state of one quantum system instantly affects the state of another, distant quantum system.
“After entanglement, you can no longer describe these states individually,” Dr. Northup said. “Fundamentally, it’s a system now.”
These entangled systems could be electrons, particles of light, or other objects. In the Netherlands, Dr Hansen and his team used so-called nitrogen vacancy centers – tiny spaces in synthetic diamonds where electrons can be trapped.
The team constructed three quantum systems, named Alice, Bob, and Charlie, and connected them in a straight line with multiple strands of optical fiber. Scientists can then entangle these systems by sending individual photons — particles of light — between them.
First, the researchers entangled two electrons—one belonging to Alice and the other to Bob. In effect, the electrons are given the same spin and thus bind or entangle together in a common quantum state, each storing the same information: a specific combination of 1s and 0s.
The researchers could then transfer this quantum state to another qubit within Bob’s synthetic diamond, the carbon nucleus. Doing so freed Bob’s electron, which the researchers could then entangle with another electron belonging to Charlie.
By performing specific quantum operations on Bob’s two qubits (the electron and the carbon core), the researchers were able to bond the two entanglements together: Alice plus Bob bond to Bob plus Charlie.
Result: Alice is entangled with Charlie, which allows data to travel across all three nodes.
When data is transmitted in this way, there is no need to actually transmit the distance between nodes, and there is no loss. “Information can be fed into one side of the connection and then appear on the other side,” Dr. Hansen said.
Information also cannot be intercepted. A future quantum internet powered by quantum teleportation could provide a theoretically unbreakable new type of encryption.
In the new experiment, the network nodes were not far apart—only about 60 feet apart. But previous experiments have shown that quantum systems can be entangled over longer distances.
The hope is that, after several years of research, quantum teleportation will be able to span miles. “We’re now trying to do this outside the lab,” said Dr. Hansen.


By Amar Shah
When the mathematical rules for quantum mechanical theory were first created, Niels Böhr and Werner Heisenberg proposed a way to interpret these rules and explain their physical implications: this became known as the Copenhagen interpretation of quantum mechanics. The idea of superposition is instrumental in this: that until the property of a particle is measured, it can be thought of as in two different states at the same time. The most famous illustration of this is Schrödinger’s cat. If you leave a cat in a box, after a period of time you no longer know whether the cat is dead or alive. Thus, the cat is in a superposition of being dead and alive. If you open the box to find a dead cat, then sometime while the cat was in the box it went from alive to dead.
Böhr also created a model for the movement of electrons rotating around an atom’s nucleus like planets rotate around the sun. In the atomic case, there are specific energy levels that electrons can have. This can be thought of as specific orbitals that electrons follow around the nucleus of an atom(represented by integers n = 1, 2, 3, …).
Böhr was one of the first to speculate about the changes in these energy levels. He hypothesized that changes in an electron’s orbit are like changes in the “aliveness” of Schrödinger’s cat. He called this change in an electron’s orbit a quantum jump and predicted that they occur with estimable probabilities, but are random and instantaneous unless you are continuously monitoring them. However, Zlatko Minev’s new experiment observes a quantum jump between different energy levels of an artificial three energy level atom and concludes that it is “continuous, coherent, and deterministic.” Not only is Minev’s team able to predict when the jump is about to occur, but the jump itself is not an instantaneous event as Böhr predicted, but a continuous change in the energy level.
Minev creates an artificial atom with three energy levels: Ground, Bright, and Dark. These energy levels can be thought of as similar to the energy levels or orbitals of an electron. When electrons jump to the lower energy level, the system emits a photon. Minev exploits this in order to make his measurements for quantum jumps between the Ground and Bright Levels. The excitation (increase in the particle’s energy level) to the Bright level is recorded by a photodetector that measures photons emitted. Each time a photon is detected, it registers as a click, which alerts the experimenter that a quantum jump from Ground to Bright has occurred. If there are not many clicks for a period of time, one can infer through the process of elimination that quantum jumps from Ground to Dark are occurring. While the photodetector generally has a poor collection efficiency and oftentimes misses photons or “clicks,” Minev’s experimental set-up minimizes the error in photodetection.
Researchers run experiments to take note of when such “clicks” may stop. Even though they do not directly measure the change from Ground to Dark, researchers use this to detect an advance warning signal for the quantum jump. Researchers test a different version of the experiment in which they wait for the clicks to stop and subsequently suspend all system drives. Doing so freezes the evolution of the system causing all changes in energy level to stop. From here, they are able to reverse the trajectory of a quantum jump mid-flight. This means that the energy level will return to the Ground state.
These results have large consequences for many fields, namely quantum computing. Quantum computers use artificial atoms, called qubits, that are useful for storing quantum information. Sometimes, there are quantum jumps in the qubits which may cause errors in the calculations of quantum computers. Having an advanced warning of these jumps can help researchers mitigate these errors. Beyond that, these results could cause a large shift in how people think about quantum mechanics. Quantum jumps are not always completely random and spontaneous, but can be predictable and even reversible.
Bigblueboo. “Loop Physics GIF by Bigblueboo.” GIPHY, GIPHY, 19 Jan. 2020, giphy.com/gifs/bigblueboo-physics-atom-bohr-ToMjGplMhvFmZ6GdWCI.
Faye, Jan, “Copenhagen Interpretation of Quantum Mechanics”, The Stanford Encyclopedia of Philosophy (Winter 2019 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2019/entries/qm-copenhagen/>.
Minev, Z. K. et al. “To Catch and Reverse a Quantum Jump Mid-Flight.” Nature 570.7760 (2019): 200–204. Crossref. Web.
Gleiser, Marcelo. “The Idea That Changed The World: 100 Years Of Quantum Jumps.” NPR, NPR, 14 Aug. 2013, www.npr.org/sections/13.7/2013/08/14/211650524/the-idea-that-changed-the-world-100-years-of-quantum-jumps.
Pitkanen, M. “Copenhagen Interpretation Dead: Long Live ZEO Based Quantum Measurement Theory!” Research Gate, www.researchgate.net/publication/335882247_Copenhagen_interpretation_dead_long_live_ZEO_based_quantum_measurement_theory
Yale University. “Physicists can predict the jumps of Schrödinger’s cat (and finally save it).” ScienceDaily. ScienceDaily, 3 June 2019. <www.sciencedaily.com/releases/2019/06/190603124621.htm>.


Morphing DNA makes motorBy Kimberly Patch, Technology Research News
DNA molecules are prime candidates for helping humans make microscopic machines because they have a long history of assembling things on the molecular scale. Every one of a human's 75 to100 trillion cells exists because a DNA molecule automatically unzipped, created the duplicate a cell needs to divide, then folded itself neatly back up again.
Researchers at New York University have taken a significant step forward in being able to instruct artificial DNA molecules to move in specific ways with a method that allows certain portions of DNA to bind to each other, and then release. This reversible binding method allows for control of the shape of a DNA molecule, or machine.
The researchers demonstrated the mechanism by making a four-step rotary motor out of DNA.
The motor is a four-stranded DNA molecule that, prompted by separate strands of DNA, will go through a mechanical cycle over and over again. Because the process is a reversible cycle, there are no waste products.
The four-stranded DNA molecule is essentially a pair of double helixes of DNA connected at several points along their lengths.
When the researchers add molecules of control DNA to a solution full of the motor molecules, the short, single-stranded control molecules join with the larger molecules and rearrange them by connecting two of the double strands in one place and cutting them in another. The researchers then remove the control strands using fuel strands of DNA, which are also short single-stranded lengths of DNA. This leaves the motor molecule in a different physical shape than when it started -- the end of one double strand of the DNA is rotated 180 degrees relative to the strands next to it.
The process can be reversed by adding a different type of control strand to the solution, and that control strand can also be removed by a different type of fuel strand after it changes the molecule back. "The system can be cycled numerous times... and there are no breakdown products," said Nadrian Seeman, a chemistry professor at New York University.
The process can be adapted to many different sequences of DNA, said Seeman. "Many different species of this device can be made by changing the sequences in the region where the... strands bind," he said.
This means a wide range of similar rotary devices can be created by changing the fuel strands and the places where they bind, he said. Ten different molecules can result in 1,024 different structures, for instance.
The researchers are currently working on a method to insert the DNA devices into molecular lattices, said Seeman. This would enable still more structures. An array of four by four molecules, for instance, could produce 65,536 different shapes. "This may enable us to build nanofabrication facilities to produce new molecular species," he said.
The range of motion the molecular motors can produce ranges from .04 to 4 nanometers, but the researchers have produced motions as large as 35 nanometers using arrays, according to Seeman. A nanometer is one millionth of a millimeter. On this scale, an E. coli bacterium is a relative giant, with a girth of 1 micron, or 1,000 nanometers. A line of ten carbon atoms measures about one nanometer.
The research is "great stuff," said Erik Winfree, an assistant professor of computer science and computation and neural systems at the California Institute of Technology. The method is a step forward in terms of DNA mechanics, he said. "It expands our toolbox for designing molecular machines."
The research is ultimately aimed at making nanorobotics practical, according to Seeman. "It could be used to configure a molecular pegboard or control molecular assemblers. The ability to achieve many different shapes means that you can create many different patterns; different patterns in a timed sequence are the essence of a machine or robot," he said.
Molecular machines could be used to assemble drugs molecule-by-molecule, and molecular robots may eventually work inside the human body.
It will be about a decade before the method can be used to make practical devices, said Seeman.
Seeman's research colleagues were Hao Yan, Xiaoping Zhang and Zhiyong Shen. They published the research in the January 3, 2002 issue of Nature. The research was funded by the National Science Foundation (NSF), Office of Naval Research (ONR), the National Institutes of Health (NIH) and the Defense Advanced Research Projects Agency (DARPA).
Timeline: 10 years
TRN Categories: Biological, Chemical, DNA and Molecular Computing; Nanotechnology
Story Type: News
Related Elements: Technical paper, "A Robust DNA Mechanical Device Controlled by Hybridization Topology," Nature, January 3, 2002.
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In a recent experiment, scientists were able to observe quasiparticles propagating across a string of ions, creating waves of quantum entanglement in their wake. Experiments like this one, which study systems with multiple quantum bodies, are crucial to learning about the behavior of quasiparticles and their interactions with more traditional particles.
It’s tempting to think that quasiparticles are not particles at all. Quasiparticles are “objects” that emerge within a complex system, such as a solid object. The collective behavior of the particles in the solid can create the impression of a new particle. The impression—or quasiparticle—moves through the solid as if it were a real particle moving through empty space, and it behaves according to the same rules.
Nevertheless, within their system, quasiparticles can have real effects on their environment. Most recently, scientists were able to track the propagation of quasiparticles called magnons through a collection of atoms. Now, scientists have been able to watch as that propagation changed the behavior of these atoms. And in the process, the quasiparticles reached speeds where a conventional model, which we use to understand time, breaks down.
To make these observations, the researchers lined up seven ions and targeted the fourth ion, exactly in the middle of the line, with a laser. The laser changes the ion’s quantum spin direction.
Changing the spin of the fourth (middle) ion sends out quasiparticles in both directions, much in the same way that a pebble, dropped into a pond, sends out a ripple in all directions.
In this case, the "quasiparticle" was essentially a wave of altered spin states. Before beginning the experiment, all ions had the same spin direction. But once the first ion’s spin had been reversed, it quickly changed the spins of the two ions that flanked it, starting a chain reaction—a wave, or quasiparticle, moving in each direction. The quasiparticles generated are called magnons.
As the two magnons moved away from the middle of the line, entanglement moved with them. That is, as the magnon moving to the right passed over ion 5, and as the one moving to the left passed over ion 3, ions 3 and 5 became entangled with each other.
The scientists were able to measure how the entanglement changed with time as the two magnons propagated away from each other. Their results agreed very closely with prediction—pairs of ions were briefly measured to be entangled as the pair of magnons moved over them, and then ceased to be entangled once the magnon was gone.
The experiment also had a second layer. The scientists were able to “tune” the range of interactions between the ions in the system. In other words, they could adjust how far one ion’s influence on its neighbors reaches. In the first part of the experiment, each ion’s spin essentially only influenced its immediate neighbors’. In the second, the researchers were able to adjust it so that the ions’ spin can jump over adjacent ions, changing the spins of more distant ones.
The resulting collective behavior of the ions still produced quasiparticles, but quasiparticles of a different sort, moving at a different speed. As they tuned the system to three different interaction ranges, the quasiparticles became faster and faster, ultimately approaching infinite speed.
Actual infinite speed is not possible, even in a quantum system, due to a speed limit called the Lieb-Robinson bound. The actual top speed of a quasiparticle may vary depending on the system it inhabits, but it is always finite. However, according to the researchers, the Lieb-Robinson bounds are “trivial” in certain circumstances, such as in their tuned system, meaning that there’s essentially no restriction on the speeds of the quasiparticles under certain circumstances.
The unbounded speed also breaks down conventional notions of time. A relativistic model called the light-cone is often used to understand time. Light-cones are graphs of the furthest light beams that can reach an object given a certain time. Nothing can travel faster than the speed of light, so only objects in the “past” part of an object’s light-cone can possibly transfer information to that object.
This model holds in the first part of the experiment, but once the researchers had tuned the interaction ranges of the ions, they found that the speeds of the quasiparticles were such that they could no longer be described in terms of light-cones.
The experiment is significant not only for its findings, which agree closely with prediction (and are the first time entanglement due to quasiparticles has been observed), but also because it lays the groundwork for many future avenues of study.
Experiments like this one, involving many-body systems, are crucial to our understanding of a wide range of quantum phenomena.


Alice and Bob
Alice and Bob are two commonly used placeholder names. They are used for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party B encrypted by Party B's public key." Following the alphabet, the specific names have evolved into common parlance within these fields—helping technical topics to be explained in a more understandable fashion.
These placeholder names are used for convenience and easier understanding. For example, if a writer wants to explain encrypted emails, the explanation might be:
- 1. Alice gets Bob's public key from the company directory.
- 2. Alice sends a message to Bob encrypted with Bob's public key.
- 3. Bob can use his secret key to unscramble it.
Every reader can intuitively figure out that they themselves could do the same thing as Bob or Alice.
Following the alphabet, the specific names have evolved into common parlance within these fields—helping technical topics to be explained in a more understandable fashion.
- 4. Then Dave decrypts the email he got, and gives a copy to Gena.
- 5. Then Erin decrypts the email she got, and gives a copy to Heather.
In cryptography and computer security, there are a number of widely used names for the participants in discussions and presentations about various protocols. The names are conventional, somewhat self-suggestive, sometimes humorous, and effectively act as metasyntactic variables.
In typical implementations of these protocols, it is understood that the actions attributed to characters such as Alice or Bob need not always be carried out by human parties directly, but also by a trusted automated agent (such as a computer program) on their behalf.
Cast of characters
This list is drawn mostly from the book Applied Cryptography by Bruce Schneier. Alice and Bob are archetypes in cryptography; Eve is also common. Names further down the alphabet are less common.
- Alice and Bob. Generally, Alice wants to send a message to Bob. These names were used by Ron Rivest in the 1978 Communications of the ACM article presenting the RSA cryptosystem, and in A Method for Obtaining Digital Signatures and Public-Key Cryptosystems published April 4, 1977, revised September 1, 1977, as technical Memo LCS/TM82. Rivest denies that these names have any relation to the 1969 movie Bob & Carol & Ted & Alice, as occasionally suggested by others.
- Carol, Carlos or Charlie, as a third participant in communications.
- Chuck, as a third participant usually of malicious intent.
- Craig, the password cracker (usually encountered in situations with stored hashed/salted passwords).
- Dan or Dave, a fourth participant.
- Erin, a fifth participant. (It's rare to see Erin; E is usually reserved for Eve.)
- Eve, an eavesdropper, is usually a passive attacker. While she can listen in on messages between Alice and Bob, she cannot modify them. In quantum cryptography, Eve may also represent the environment.
- Frank, a sixth participant (and so on alphabetically).
- Mallet or Mallory, a malicious attacker (less commonly called Trudy, an intruder.); unlike the passive Eve, this one is the active man-in-the-middle attacker who can modify messages, substitute his/her own messages, replay old messages, and so on. The difficulty of securing a system against Mallet/Mallory is much greater than against Eve.
- Oscar, an opponent, similar to Mallet/Mallory but not necessarily malicious. Could be white-hat but still wants to crack, modify, substitute, or replay messages.
- Peggy, a prover, and Victor, a verifier, often must interact in some way to show that the intended transaction has actually taken place. They are often found in zero-knowledge proofs. Alternate names for the prover and the verifier are Pat and Vanna after Pat Sajak and Vanna White, the hosts of Wheel of Fortune.
- Sybil, an attacker who marshals a large number of pseudonymous identities, e.g. to subvert a reputation system. See Sybil attack.
- Trent, a trusted arbitrator, is some kind of neutral third party, whose exact role varies with the protocol under discussion.
- Walter, a warden, may be needed to guard Alice and Bob in some respect, depending on the protocol being discussed.
- Wendy, a whistleblower, is an insider threat with privileged information.
Although an interactive proof system is not quite a cryptographic protocol, it is sufficiently related to mention the cast of characters its literature features:
- Arthur and Merlin: In interactive proof systems, the prover has unbounded computational ability and is hence associated with Merlin, the powerful wizard. He claims the truth of a statement, and Arthur, the wise king, questions him to verify the claim. These two characters also give the name for two complexity classes, namely MA and AM.
- A similar pair of characters is Paul and Carole. The characters were introduced in the solution of the Twenty Questions problem, where "Paul", who asked questions, stood for Paul Erdős and "Carole", who answered them, was an anagram of "oracle". They were further used in certain combinatorial games in the roles of Pusher and Chooser respectively, and have since been used in various roles.
- Newton, David E. (1997). Encyclopedia of Cryptography. Santa Barbara California: Instructional Horizons, Inc. p. 10.
- RFC 4949
- "Security's inseparable couple". Network World. February 7, 2005.
- Tanenbaum, Andrew S. (2007), Distributed Systems: Principles and Paradigms, Pearson Prentice Hall, p. 171;399–402, ISBN 978-0-13-239227-3
- Bruce Schneier (1994), Applied Cryptography: Protocols, Algorithms, and Source Code in C, Wiley, ISBN 9780471597568, p. 44: "Mallet can intercept Alice's database inquiry, and substitute his own public key for Alice's. He can do the same to Bob."
- Charles L. Perkins et al. (2000), Firewalls: 24seven, Network Press, ISBN 9780782125290, p. 130: "Mallet maintains the illusion that Alice and Bob are talking to each other rather than to him by intercepting the messages and retransmitting them."
- Brian LaMacchia (2002), .NET Framework Security, Addison-Wesley, ISBN 9780672321849, p. 616: "Mallet represents an active adversary that not only listens to all communications between Alice and Bob but can also modify the contents of any communication he sees while it is in transit."
- Shlomi Dolev, ed. (2009), Algorithmic Aspects of Wireless Sensor Networks, Springer, ISBN 9783642054334, p. 67: "We model key choices of Alice, Bob and adversary Mallet as independent random variables A, B and M [...]"
- Bruce Schneier (1996), Applied Cryptography: Protocols, Algorithms, and Source Code in C, Second Edition, Wiley, ISBN 9780471117094, p. 23: Table 2.1: Dramatis Personae
- Carsten Lund et al. (1992). "Algebraic Methods for Interactive Proof Systems". J. ACM (ACM) 39 (4): 859–868. doi:10.1145/146585.146605.
- Spencer, Joel; Winkler, Peter (1992), Three Thresholds for a Liar, Combinatorics, Probability and Computing 1 (01): 81–93, doi:10.1017/S0963548300000080
- Muthukrishnan, S. (2005), Data Streams: Algorithms and Applications, Now Publishers, p. 3, ISBN 978-1-933019-14-7
- C.H. Lindsey, Regulation of Investigatory Powers Bill: Some Scenarios, 2000
- A Method for Obtaining Digital Signatures and Public-Key Cryptosystems
- The Alice and Bob After-Dinner Speech, given at the Zurich Seminar, April 1984, by John Gordon
- Geek Song: "Alice and Bob"
- Alice and Bob jokes (mainly Quantum Computing-related)
- Alice and Bob: IT's inseparable couple
- A short history of Bobs (story and slideshow) in the computing industry, from Alice & Bob to Microsoft Bob and Father of Ethernet Bob Metcalfe
- Alice and Bob en Français


First Electronic Quantum Processor Created
2009 07 01
A team led by Yale University researchers has created the first rudimentary solid-state quantum processor, taking another step toward the ultimate dream of building a quantum computer.
The two-qubit processor is the first solid-state quantum processor that resembles a conventional computer chip and is able to run simple algorithms. (Credit: Blake Johnson/Yale University)
They also used the two-qubit superconducting chip to successfully run elementary algorithms, such as a simple search, demonstrating quantum information processing with a solid-state device for the first time. Their findings appeared in Nature's advanced online publication June 28.
"Our processor can perform only a few very simple quantum tasks, which have been demonstrated before with single nuclei, atoms and photons," said Robert Schoelkopf, the William A. Norton Professor of Applied Physics & Physics at Yale. "But this is the first time they've been possible in an all-electronic device that looks and feels much more like a regular microprocessor."
Working with a group of theoretical physicists led by Steven Girvin, the Eugene Higgins Professor of Physics & Applied Physics, the team manufactured two artificial atoms, or qubits ("quantum bits"). While each qubit is actually made up of a billion aluminum atoms, it acts like a single atom that can occupy two different energy states. These states are akin to the "1" and "0" or "on" and "off" states of regular bits employed by conventional computers. Because of the counterintuitive laws of quantum mechanics, however, scientists can effectively place qubits in a "superposition" of multiple states at the same time, allowing for greater information storage and processing power.
For example, imagine having four phone numbers, including one for a friend, but not knowing which number belonged to that friend. You would typically have to try two to three numbers before you dialed the right one. A quantum processor, on the other hand, can find the right number in only one try.
"Instead of having to place a phone call to one number, then another number, you use quantum mechanics to speed up the process," Schoelkopf said. "It's like being able to place one phone call that simultaneously tests all four numbers, but only goes through to the right one."
These sorts of computations, though simple, have not been possible using solid-state qubits until now in part because scientists could not get the qubits to last long enough. While the first qubits of a decade ago were able to maintain specific quantum states for about a nanosecond, Schoelkopf and his team are now able to maintain theirs for a microsecond—a thousand times longer, which is enough to run the simple algorithms.
To perform their operations, the qubits communicate with one another using a "quantum bus"—photons that transmit information through wires connecting the qubits—previously developed by the Yale group.
The key that made the two-qubit processor possible was getting the qubits to switch "on" and "off" abruptly, so that they exchanged information quickly and only when the researchers wanted them to, said Leonardo DiCarlo, a postdoctoral associate in applied physics at Yale's School of Engineering & Applied Science and lead author of the paper.
Next, the team will work to increase the amount of time the qubits maintain their quantum states so they can run more complex algorithms. They will also work to connect more qubits to the quantum bus. The processing power increases exponentially with each qubit added, Schoelkopf said, so the potential for more advanced quantum computing is enormous. But he cautions it will still be some time before quantum computers are being used to solve complex problems.
"We're still far away from building a practical quantum computer, but this is a major step forward."
Authors of the paper include Leonardo DiCarlo, Jerry M. Chow, Lev S. Bishop, Blake Johnson, David Schuster, Luigi Frunzio, Steven Girvin and Robert Schoelkopf (all of Yale University), Jay M. Gambetta (University of Waterloo), Johannes Majer (Atominstitut der Österreichischen Universitäten) and Alexandre Blais (Université de Sherbrooke).
Article source: ScienceDaily.com
Jim Elvidge - Programmed Reality, The Power of 10, Science & The Soul
Nick Begich - Mind Control & Emerging Technologies
A short Introduction to Quantum Computation
Is Quantum Mechanics Controlling Your Thoughts?
How Time-Traveling Could Affect Quantum Computing
Nano-Diamonds Might Lead to Quantum Computing
'Light trap' is a Step Towards Quantum Memory
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More precisely, quantum teleportation is a quantum protocol by which a qubit a (the basic unit of quantum information) can be transmitted exactly (in principle) from one location to another. The prerequisites are a conventional communication channel capable of transmitting two classical bits (i.e. one of four states), and an entangled pair (b,c) of qubits, with b at the origin and c at the destination. (So whereas b and c are intimately related, a is entirely independent of them other than being initially colocated with b.) The protocol has three steps: measure a and b jointly to yield two classical bits; transmit the two bits to the other end of the channel (the only potentially time-consuming step, due to speed-of-light considerations); and use the two bits to select one of four ways of recovering c. The upshot of this protocol is to permute the original arrangement ((a,b),c) to ((b′,c′),a), that is, a moves to where c was and the previously separated qubits of the Bell pair turn into a new Bell pair (b′,c′) at the origin.
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:
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.
Similarly, it has also been shown formally that classical teleportation, aka. option 3, is impossible; this is called the no teleportation theorem. This is another way to say that quantum information cannot be measured reliably.
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.
Alice applies a unitary operation on the qubits AC and measures the result to obtain two classical bits. In this process, the two qubits are destroyed. Bob's qubit, B, now contains information about C; however, the information is somewhat randomized. More specifically, Bob's qubit B is in one of four states uniformly chosen at random and Bob cannot obtain any information about C from his qubit.
Alice provides her two measured qubits, which indicate which of the four states Bob possesses. Bob applies a unitary transformation which depends on the qubits he obtains from Alice, transforming his qubit into an identical copy of the qubit C.
Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be written generally as:
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 (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
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):
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):
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.
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 :
to recover the state.
to his qubit.
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.
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
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.
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.
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 dimensional state space. To teleport, Alice makes a partial measurement on the two particles in her possession in some entangled basis on the dimensional subsystem. This measurement has equally probable outcomes, which are then communicated to Bob classically. Bob recovers the desired state by sending his particle through an appropriate unitary gate.
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
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 Schrödinger picture.
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 .
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 Ψi 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
holds. The left hand side of the equation is:
where Ψi* is the adjoint of Ψi in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes
The success criterion for teleportation has the expression


On December 17, 1903, at Kitty Hawk, North Carolina, the 1903 Wright Flyer became the first powered, heavier-than-air machine to achieve controlled, sustained flight with a pilot aboard.
On this date in 1969, Neil Armstrong, aboard the Apollo 11 Lunar Lander, along with Buzz Aldrin, touched down on the surface of the moon. Michael Collins waited aboard the Command Module, orbiting the moon.
You run and you run to catch up with the sun but it's sinking
Racing around to come up behind you again.
The sun is the same in a relative way but you're older,
Shorter of breath and one day closer to death.
"Time" by Mason, Waters, Wright, Gilmour
Through the insights of the brightest minds, we begin to see evidence of Tachyons, particles that travel backwards in time, and through experimentation with the LHC, we begin to see elementary particles.
Mind you, the theory was a proper theory in the sense that it was mathematically consistent, and also because it predicted certain observable consequences-namely, that if tachyons existed they would emit a certain type of radiation (Cerenkov radiation) in a vacuum. This radiation was searched for, and none was found. So, after a flurry of excitement, physicists lost interest in tachyons and went on to more massive hypotheses, such as black holes. As far as physicists are concerned, tachyons do not exist. (Committee for Skeptical Inquiry)
There is a Canadian company called D-Wave which has the first commercially available quantum computer, and is set to release a 512 qbit version by the end of this year. Add Artifical Intelligence to this computing capability, and the possibilities are mind-boggling.
Traditional computer process information as bits that can be a 0 or a 1. Quantum computers utilize the potential of quantum mechanics by making its bits a 0, a 1, or a 0 and a 1 simultaneously. This “superposition” lets it do many calculations at once, where a traditional computer can only perform one.
It would appear the only restrictions on this technology are the limitations imposed by the speed of light, but future experiments may produce instantaneous transfers.
The experiment was carried out by scientists at Hefei National Laboratory for Physical Sciences in Anhui, China. During its course, the scientists took two quantum entangled particles. One was sent to a distant quantum memory node while the other was present at the lab. The scientists then altered the state of the photon in the lab and it directly affected the state of the distant photon. This is a very exciting development for the world of quantum computing as well as those researching on faster modes of communication transmission. If indeed this progress can be translated into greater, more sophisticated system, it would mean that we can create the fastest data-transmission machines in the near future.
Will it be possible in the future to be placed inside one of these machines, have your whole body mapped into digital form, and transmitted by Quantum Tunneling to another location in space and time?
An MRI system can create axial images as well as sagitall (slicing the bread side-to-side lengthwise) and coronal (think of the layers in a layer cake) images, or any degree in between, without the patient ever moving.
While it is true that a Tachyon is a putative particle, at one point in time the Higgs Boson was also just theoretical particle, until recent discovery by the LHC at Cern. The detection of a Tachyon would by it's very nature be very difficult to prove, a particle that resides just above the speed of light.
Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate of unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year. (Isaac Newton, cited in Philosophy of Physics: Space and Time by Tim Maudlin, pg 13.)
Space and time are the framework within which the mind is constrained to construct its experience of reality. (Immanuel Kant)
he seems to be describing a "Star Trek" transporter, not a time machine
However, no such radiation was discovered by any test, so the uniform conclusion of physicists is that tachyons do not exist
we have invented measurement of space in order to be able to quantify distances, we have invented measurement of time in order to be able to quantify durations. By this perspective, time is not really anything -- it is merely the intellectual imposition of order.
This theory has a wide range of consequences which have been experimentally verified, including counter-intuitive ones such as length contraction, time dilation and relativity of simultaneity.
1. Once transmitted digitally, how is the body re-assembled into it's original biological form?
2. How does the body return, if the transmitting unit is at the origin point?
In truth, a teleporter, which I'm sure my opponent has no problem with, would still transmit across time, but in such in a tighter field, and only moments into the future.
To measure a particle traveling faster than the speed of light requires equipment that hasn't been devised yet.
So then perhaps Einstein was wrong with Special Relativity, and if time is a mere sequential measurement, then odd effects such as time dilation cannot exist.
'First round both debaters opened very strongly, both using very good scientific logic and the steady progess of mankind as strong bases for an opener, while both maintaining a somewhat tongue-in-cheek approach to Hollywood’s portrayal of time travel and how easy it seems. For the first round, due to a better explanation of the cons of the realities of time travel, the round goes to Adjensen.
Second round, Druid42 attempted to explain how we as humans could go about achieving the possibility of time travel, but as it would seem didn’t have the room to extrapolate the theory properly, instead delivering a more ‘teleportation’ based theory than one of time travel. Adjensen however in the second round recovered somewhat strongly, expanding on his previous post regarding the impossibilities of time manipulation and dimension given any forms of technology. Even though the information in the second round was only marginally more than in the first, the second round goes to Adjensen for a once again more concise reply.
Last round Druid42 began to address the issues his opponent had raised, and gave some very strong retorts to Adjensen’s ideas in the second round. He recovered strongly by bringing the facts to the table and opening the possibility of how time travel could actually work.
However, Adjensen had one last card to play, and this statement;
The inevitable plot hole of any time travel story is that with such a device, anything in any time can be done. Screwed something up? No worries, just go back five minutes earlier and inform yourself of the error. That didn't work, either? Go back five minutes before that. Repeat until you get it right, because you always have five more minutes.
Is very convincing to the con stance of time travel.
A very difficult debate to judge given the complex nature of the topic, but adjensen is the winner on this one.'
Although both contestants brought up valid points, I feel that Druid42 has prevailed. While Druid42 has shown that the future is full of possibilities and such a machine/concept is a future possibility, adjensen maintained a present sense of technology not looking toward the future of possibilities. Their arguments are based on what it is we know and understand today without giving leeway to what we may understand tomorrow.
As we do not understand how such a machine would impact our past/future we can only argue against by the standards we now have the ability to grasp and I don't feel that they made a good enough argument to nullify the possibility as a future occurrence.
wont profess to understand exactly what these two great debaters were talking about, but overall the case adjensen built seemed to be more coherent. I actually also learned something about the possibilities of time travel from adjensen who was debating against time travel. This part especially sold me to adjensens side:
"Why is the timeframe of the receipt of a time machine of no relevance? Because once it exists, ever and anywhere, it exists always and everywhere. It doesn't matter if such a device isn't invented for a million billion years, because it takes away any limitations on where and when it can be".
He did well in showing the logical inconsistences inherent in Druids side.
What I liked best about Druids debate is the idea of first proving that instant movement of matter through space looks to become a possibility in the future and then extrapolating from that the idea of movement through time. He buffered this point with "Rose's Law", showing the exponential progress of mankind. This was a brilliant move that swayed me to his side of the debate for a short while.
I would prefer Druids side to be true (wouldn't we all?) but at the end of the Debate I feel that adjensen made the slightly stronger case.


Technologies that exploit the unique weirdness of quantum mechanics could debut in the very near future, thanks to the groundbreaking work of a huge European research consortium.
Unbreakable cryptography, unimaginable simulations of profoundly complex problems and super-fast networks are just some of the promise held out by quantum computing. And now European scientists are poised to deliver on that promise, thanks to the work of the Qubit Applications (QAP) project.
The integrated project has cherry-picked major obstacles in the path of quantum computing, problems that could have immediate applications and could command a ready market.
Chief among them is quantum cryptography. “Quantum computing, when it arrives, could make all current cryptographic technology obsolete,” notes QAP co-coordinator Professor Ian Walmsley.
Thankfully, researchers have developed quantum cryptography to deal with that issue.
“Quantum cryptography over short distances was demonstrated in a previous project,” explains Walmsley. “The problem is, it only works over a short distance.”
Weaving entangled webs
That is because quantum cryptography relies on entanglement. Entanglement is a concept that explains how two or more particles exhibit correlation – a relationship if you like – that would be impossible to explain unless you supposed that they belonged to the same entity, even though they might be separated by vast distance.
Imagine you were playing a game of quantum coin flipping with a colleague: you are heads and the colleague tails. You are two distinct individuals, but if the coin comes up heads your colleague loses, and you win. There is a correlation between the coin tossing. Now, with a quantum coin, it is heads the colleague wins and tails you win at the same time.
This is the extra bit that quantum mechanics gives us, and which we use in secure communications, suggests Walmsley.
That explains, with a little inaccuracy, the concept of entanglement, and it is at the core of quantum key distribution, or QKD. It is far too complex to break quantum encryption by brute force, and it is immune to eavesdropping because, at the quantum level, the act of observing an object changes the object observed. It means that encryption is guaranteed by the laws of physics.
The technique was demonstrated in Vienna 2008, but it works only over short distances. EU-funded QAP hopes to develop a quantum repeater that can maintain entanglement over large distances. It has already had considerable success up to the 200km range, and growing.
Ideal information carrier
Maintaining entanglement over long distances – so essential to QKD, but also communications and networks – is the most immediate and compelling application in the QAP programme, but it is far from the only one. Many other areas of work show signs of progress, too. Storage and memory are essential for quantum computing.
It is not too difficult to encode a piece of information on a photon, which is an ideal information carrier because of its high speed and weak interaction with the environment.
It is difficult to store that information for any length of time, so QAP is developing ways of transferring quantum information from photons to and from atoms and molecules for storage, and the project is making steady progress.
Similarly, QAP’s work to develop quantum networks is progressing well. One team within the overall research effort has managed to develop a reliable way to calibrate and test detectors, a prime element in the network system.
“This is important because it will be essential to develop reliable methods to test results if work on quantum networks is to progress,” notes Walmsley. The research group has submitted a patent application for this work.
Quantum simulation, too, offers some tantalising opportunities. The primary goal of QAP’s Quantum Simulations and Control subproject is to develop and advance experimental systems capable of simulating quantum systems whose properties are not approachable on classical computers.
Imagine, for example, trying to model superconducting theory. It is hugely complex, and classic computers are quickly overwhelmed by the size of the problem.
But quantum methods are inherently capable of dealing with far greater complexity, because of the nature of the qubit, or quantum bit. Classical, digital bits operate on the basis of on or off, yes or no. But quantum bits can be yes, no, or both. It takes classical computing from 2D, into the 3D information world.
One could say that, while classical computers attack problems linearly, quantum computers attack problems exponentially. As a result, with just a few qubits, it is possible to do incredibly large computations, and that means that quantum simulation of complex problems could be a medium-term application.
“We are not saying we will solve all the problems in the area of simulation, but we will make a good start,” warns Walmsley.
That defines QAP nicely: a kick-start for quantum applications in Europe.
The QAP project received funding from the ICT strand of the EU’s Sixth Framework Programme for research.
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New evidence that plants get their energy using quantum entanglement
2014 01 20
By George Dvorsky | io9
Biophysicists theorize that plants tap into the eerie world of quantum entanglement during photosynthesis. But the evidence to date has been purely circumstantial. Now, scientists have discovered a feature of plants that cannot be explained by classical physics alone — but which quantum mechanics answers quite nicely.
The fact that biological systems can exploit quantum effects is quite astounding. In a way, they’re like mini-quantum computers capable of scanning all possible options in order to choose the most efficient paths or solutions. For plants, this means the ability to make the most of the energy they receive and then deliver that energy from leaves with near perfect efficiency.
But for this to work, plants require the capacity to work in harmony with the wild, wacky, and extremely small world of quantum phenomena. The going theory is that plants have light-gathering macromolecules in their cells that can transfer energy via molecular vibrations — vibrations that have no equivalents in classical physics. Most of these light-gathering macromolecules are comprised of chromophores attached to proteins. These macromolecules carry out the first step of photosynthesis by capturing sunlight and efficiently transferring the energy.
Previous inquiries suggested that this energy is transferred in a wave-like manner, but it was a process that could still be explained by classical physics.
In Perfect Quantum Harmony
In the new study, however, UCL researchers identified a specific feature in biological systems that can only be predicted by quantum physics. The team learned that the energy transfer in the light-harvesting macromolecules is facilitated by specific vibrational motions of the chromophores.
"We found that the properties of some of the chromophore vibrations that assist energy transfer during photosynthesis can never be described with classical laws, and moreover, this non-classical behaviour enhances the efficiency of the energy transfer," noted supervisor and co-author Alexandra Olaya-Castro in a statement.
The vibrations in question are periodic motions of the atoms within a molecule. It’s similar to how an object moves when it’s attached to a spring. Sometimes, the energy of two vibrating chromophores match the energy difference between the electronic transitions of chromophores. The result is a coherent exchange of a single quantum of energy.
Read the full article at: io9.com
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A possible application is the development of a super-fast computer and highly precise clocks that could be the future basis for a new standard of time
Serge Haroche and David Wineland have opened the door to a new era of experimentation with quantum physics by demonstrating the direct observation of individual quantum systems without destroying them. Through their ingenious laboratory methods they have managed to measure and control very fragile quantum states, enabling their field of research to take the very first steps towards building a new type of super fast computer, based on quantum physics. These methods have also led to the construction of extremely precise clocks that could become the future basis for a new standard of time, with more than hundred-fold greater precision than present-day caesium clocks.
For single particles of light or matter, the laws of classical physics cease to apply and quantum physics takes over. But single particles are not easily isolated from their surrounding environment and they lose their mysterious quantum properties as soon as they interact with the outside world.
Both Laureates work in the field of quantum optics studying the fundamental interaction between light and matter.
In David Wineland’s laboratory in Boulder, Colorado, electrically charged atoms or ions are kept inside a trap by surrounding them with electric fields.
One of the secrets behind Wineland’s breakthrough is the mastery of the art of using laser beams and creating laser pulses. A laser is used to put the ion in its lowest energy state and thus enabling the study of quantum phenomena with the trapped ion. A carefully tuned laser pulse can be used to put the ion in a superposition state, which is a simultaneous existence of two distinctly different states.
For instance, the quantum superposition of the ion’s energy states can be studied by using the laser pulse to nudge the ion halfway between the high- and low-energy levels.
Controlling single photons
Serge Haroche and his research group employ a different method to reveal the mysteries of the quantum world. In their laboratory in Paris microwave photons bounce back and forth inside a small cavity between two mirrors, about three centimetres apart. The mirrors are made of superconducting material and are cooled to a temperature just above absolute zero. These superconducting mirrors are so reflective that a single photon can bounce back and forth inside the cavity for almost a tenth of a second before it is lost or absorbed.
During its long life time, many quantum manipulations can be performed with the trapped photon. Haroche uses specially prepared atoms, so-called Rydberg atoms to both control and measure the microwave photon in the cavity. A Rydberg atom has a radius of about 125 nanometres which is roughly 1,000 times larger than typical atoms. The Rydberg atoms are sent into the cavity one by one at a carefully chosen speed, so that the interaction with the microwave photon occurs in a well-controlled manner.
The Rydberg atom traverses and exits the cavity, leaving the microwave photon behind. But the interaction between the photon and the atom creates a change in the phase of quantum state of the atom: if you think of the atom’s quantum state as a wave, the peaks and the dips of the wave become shifted. This phase shift can be measured when the atom exits the cavity, thereby revealing the presence or absence of a photon inside the cavity. With no photon there is no phase shift. Haroche can thus measure a single photon without destroying it.
Physics in the quantum world has some inherent uncertainty or randomness to it. One example of this contrary behaviour is superposition, where a quantum particle can be in several different states simultaneously.
Why do we never become aware of these strange facets of our world? Why can we not observe a superposition of quantum marble in our every-day life? The Austrian physicist and Nobel Laureate (Physics 1933) Erwin Schrödinger battled with this question. Like many other pioneers of quantum theory, he struggled to understand and interpret its implications. As late as 1952, he wrote: “We never experiment with just one electron or atom or (small) molecule. In thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences...”
In order to illustrate the absurd consequences of moving between the micro-world of quantum physics and our every-day macro-world, Erwin Schrödinger described a thought experiment with a cat: Schrödinger’s cat is completely isolated from the outside world inside a box. The cat must be in a superposition state of being both dead and alive.
The box also contains a bottle of deadly cyanide which is released only after the decay of some radioactive atom, also inside the box.
The radioactive decay is governed by the laws of quantum mechanics, according to which the radioactive material is in a superposition state of both having decayed and not yet decayed. Therefore the cat must also be in a superposition state of being both dead and alive. Now, if you peek inside the box, you risk killing the cat because the quantum superposition is so sensitive to interaction with the environment that the slightest attempt to observe the cat would immediately ‘collapse’ the ‘cat-state’ to one of the two possible outcomes — dead or alive. Instead of Schrödinger’s cat, Haroche and Wineland trap quantum particles and put them in cat-like superposition states. These quantum objects are not really macroscopic as a cat, but they are still quite large by quantum standards.
Inside Haroche’s cavity microwave photons are put in cat-like states with opposite phases at the same time, like a stopwatch with a needle that spins both clockwise and counterclockwise simultaneously. The microwave field inside the cavity is then probed with Rydberg atoms. The result is another unintelligible quantum effect called entanglement.
Entanglement has also been described by Erwin Schrödinger and can occur between two or more quantum particles that have no direct contact but still can read and affect the properties of each other. Entanglement of the microwave field and Rydberg atoms allowed Haroche to map the life and death of the cat-like state inside his cavity, following it step by step, atom by atom, as it underwent a transition from the quantum superposition of states to a well defined state of classical physics.
A possible application of ion traps that many scientists dream of is the quantum computer. In present-day classical computers the smallest unit of information is a bit that takes the value of either 1 or 0. In a quantum computer, however, the basic unit of information — a quantum bit or qubit — can be 1 and 0 at the same time.
Two quantum bits can simultaneously take on four values — 00, 01, 10 and 11 — and each additional qubit doubles the amount of possible states. For n quantum bits there are 2 possible states, and a quantum computer of only 300 qubits could hold 2 values simultaneously.
Wineland’s group was the first in the world to demonstrate a quantum operation with two quantum bits. Since control operations have already been achieved with a few qubits, there is no reason to believe that it should not be possible to achieve such operations with many more qubits.
However, to build such a quantum computer one has to satisfy two opposing requirements: the qubits need to be adequately isolated from their environment in order not to destroy their quantum properties, yet they must also be able to communicate with the outside world in order to pass on the results of their calculations. David Wineland and his team of researchers have also used ions in a trap to build a clock that is a hundred times more precise than the caesium-based atomic clocks which are currently the standard for our measurement of time. Time is kept by setting, or synchronizing all clocks against one standard. Caesium clocks operate in the microwave range whereas Wineland’s ion clocks use visible light — hence their name: optical clocks.
An optical clock can consist of just one ion or two ions in a trap. With two ions, one is used as the clock and the other is used to read the clock without destroying its state, or causing it to miss a tick. The precision of an optical clock is better than one part in 10 — if one had started to measure time at the beginning of the universe in the Big Bang about 14 billion years ago, the optical clock would only have been off by about five seconds today.
With such precision, some extremely subtle and beautiful phenomena of nature have been observed, such as changes in the flow of time, or minute variations of gravity, the fabric of space-time. According to Einstein’s theory of relativity, time is affected by motion and gravity.
The higher the speed and the stronger the gravity, the slower the passage of time. We may not be aware of these effects, but they have in fact become part of our everyday life. When we navigate with the GPS we rely on time signals from satellites with clocks that are routinely calibrated, because gravity is somewhat weaker several hundred kilometres altitude.
With an optical clock it is possible to measure a difference in the passage of time when the clock’s speed is changed by less than 10 metres per second, or when gravity is altered as a consequence of a difference in height of only 30 centimetres.
[Edited excerpts from “Popular Information” available at the Nobel Prize website]