In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Indicated by the Greek letter sigma, they are denoted by tau when used in connection with isospin symmetries, they are σ 1 = σ x = σ 2 = σ y = σ 3 = σ z =. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field; each Pauli matrix is Hermitian, together with the identity matrix I, the Pauli matrices form a basis for the vector space of 2 × 2 Hermitian matrices. Hermitian operators represent observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space ℝ3; the Pauli matrices generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for s u, which exponentiates to the special unitary group SU.
The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of ℝ3. All three of the Pauli matrices can be compacted into a single expression: σ a = where i = √−1 is the imaginary unit, δab is the Kronecker delta, which equals +1 if a = b and 0 otherwise; this expression is useful for "selecting" any one of the matrices numerically by substituting values of a = 1, 2, 3, in turn useful when any of the matrices is to be used in algebraic manipulations. The matrices are involutory: σ 1 2 = σ 2 2 = σ 3 2 = − i σ 1 σ 2 σ 3 = = I where I is the identity matrix; the determinants and traces of the Pauli matrices are: det σ i = − 1, tr σ i = 0. From which, we can deduce that the eigenvalues of each σi are ±1. With the inclusion of the identity matrix, I, the Pauli matrices form an orthogonal basis of the real Hilbert space of 2 × 2 complex Hermitian matrices, H 2, the complex Hilbert space of all 2 × 2 matrices, M 2, 2; each of the Pauli matrices has two eigenvalues, +1 and −1.
The corresponding normalized eigenvectors are: ψ x + = 1 2, ψ x − = 1 2 ( 1 − 1
Michael Aaron Nielsen is a quantum physicist, science writer, computer programming researcher living in San Francisco. In 2004 Nielsen was characterized as Australia's "youngest academic" and secured a Federation Fellowship at the University of Queensland, he worked at the Los Alamos National Laboratory, as the Richard Chace Tolman Prize Fellow at Caltech, a Senior Faculty Member at the Perimeter Institute for Theoretical Physics. Nielsen obtained his PhD in physics in 1998 at the University of New Mexico. With Isaac Chuang he is the co-author of a popular textbook on quantum computing. In 2007, Nielsen announced a marked shift in his field of research: from quantum information and computation to “the development of new tools for scientific collaboration and publication”; this work includes "massively collaborative mathematics" projects like the Polymath project with Timothy Gowers. Besides writing books and essays, he has given talks about Open Science, he was a member of the Working Group on Open Data in Science at the Open Knowledge Foundation.
In 2015 Nielsen published the online textbook Neural Networks and Deep Learning. The same year he joined the Recurse Center as a Research Fellow. Since 2017 Nielsen works as a Research Fellow at Y Combinator Research.. Nielsen, Michael A. Reinventing Discovery: The New Era of Networked Science. Princeton, N. J: Princeton University Press. ISBN 0-691-14890-2; this book is based on themes that are covered in his essay on the Future of Science. Nielsen, Michael A.. Neural Networks and Deep Learning. Determination Press
Quantum teleportation is a process by which quantum information can be transmitted from one location to another, with the help of classical communication and 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 quanta, this has not yet been achieved between anything larger than molecules. Although the name is inspired by the teleportation 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 first expounding the idea of quantum teleportation was published by C. H. Bennett, G. Brassard, C.
Crépeau, R. Jozsa, A. Peres, W. K. Wootters in 1993. Quantum teleportation was first realized in single photons being demonstrated in various material systems such as atoms, ions and superconducting circuits; the latest reported record distance for quantum teleportation is 1,400 km by the group of Jian-Wei Pan using the Micius satellite for space-based quantum teleportation. 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 represented using one or zero; the quantum analog of a bit is qubit. Qubits encode a type of information, called quantum information, which differs from "classical" information. For example, quantum information can be neither destroyed. 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 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.
As of 2015, the quantum states of single photons, photon modes, single atoms, atomic ensembles, defect centers in solids, single electrons, superconducting circuits have been employed as information bearers. 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, its two parts shared between two locations. In essence, a certain kind of quantum channel between two sites must be established first, before a qubit can be moved. Teleportation 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, this must be done over ordinary classical communication channels.
The need for such classical channels may, at first, seem disappointing. What's more, Bell states are most shared using photons from lasers, so teleportation could be done, in principle, through open space, i.e. without the need to send the light through cables or optical fibers. The quantum states of single atoms have been teleported. An atom consists of several parts: the qubits in the electronic state or electron shells surrounding the atomic nucleus, the qubits in the nucleus itself, the electrons and neutrons making up the atom. Physicists have teleported the qubits encoded in the electronic state of atoms, it is therefore inaccurate to say "an atom has been teleported". The quantum state of an atom has. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them; the importance of teleporting nuclear state is unclear: nuclear state does affect the atom, e.g. in hyperfine splitting, but whether such state would need to be teleported in some futuristic "practical" application is debatable.
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems. These correlations hold 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 though light hasn't yet had time to travel the distance; 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 algebra, Hilbert spaces and projection matrixes. A qub
The Bell states, a concept in quantum information science, are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The Bell states are a form of normalized basis vectors; this normalization implies that the overall probability of the particle being in one of the mentioned states is 1: ⟨ Φ | Φ ⟩ = 1. Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will collapse it into one of its basis states with a given probability; because of the entanglement, measurement of one qubit will assign one of two possible values to the other qubit where the value assigned depends on which Bell state the two qubits are in. Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 subsystems. Understanding of the Bell states is essential in analysis of quantum communication and quantum teleportation; the no-communication theorem prevents this behavior from transmitting information faster than the speed of light, because there is a need for A to communicate information to B.
The Bell states are four specific maximally entangled quantum states of two qubits. They are in a superposition of 1 -- that is, a linear combination of the two states, their entanglement means the following: The qubit held by Alice can be 0 as well as 1. If Alice measured her qubit in the standard basis, the outcome would be random, either possibility 0 or 1 having probability 1/2, but if Bob measured his qubit, the outcome would be the same as the one Alice got. So, if Bob measured, he would get a random outcome on first sight, but if Alice and Bob communicated, they would find out that, although their outcomes seemed random, they are correlated; this perfect correlation at a distance is special: maybe the two particles "agreed" in advance, when the pair was created, which outcome they would show in case of a measurement. Hence, following Einstein and Rosen in 1935 in their famous "EPR paper", there is something missing in the description of the qubit pair given above—namely this "agreement", called more formally a hidden variable.
In his famous paper of 1964, John S. Bell showed by simple probability theory arguments that these correlations cannot both be made perfect by the use of any "pre-agreement" stored in some hidden variables—but that quantum mechanics predicts perfect correlations. In a more formal and refined formulation known as the Bell-CHSH inequality, it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of local "hidden variable" theory, but certain systems permitted in quantum mechanics can attain values as high as 2 2. Thus, quantum theory violates the idea of local "hidden variables. Four specific two-qubit states with the maximal value of 2 2 are designated as "Bell states", they are known as the four maximally entangled two-qubit Bell states, they form a maximally entangled basis, known as the Bell basis, of the four-dimensional Hilbert space for two qubits: | Φ + ⟩ = 1 2 | Φ − ⟩ = 1 2 | Ψ + ⟩ = 1 2 | Ψ − ⟩ = 1 2. Although there are many possible ways to create entangled Bell states through quantum circuits, the simplest takes a computational basis as the input, contains a Hadamard gate and a CNOT gate.
As an example, the quantum circuit pictures takes the two qubit input | 00 ⟩ and transformed it to the firs
Quantum computing is the use of quantum-mechanical phenomena such as superposition and entanglement to perform computation. A quantum computer is used to perform such computation, which can be implemented theoretically or physically; the field of quantum computing is a sub-field of quantum information science, which includes quantum cryptography and quantum communication. Quantum Computing was started in the early 1980s when Richard Feynman and Yuri Manin expressed the idea that a quantum computer had the potential to simulate things that a classical computer could not. In 1994, Peter Shor shocked the world with an algorithm that had the potential to decrypt all secured communications. There are two main approaches to physically implementing a quantum computer analog and digital. Analog approaches are further divided into quantum simulation, quantum annealing, adiabatic quantum computation. Digital quantum computers use quantum logic gates to do computation. Both approaches use quantum qubits.
Qubits are fundamental to quantum computing and are somewhat analogous to bits in a classical computer. Qubits can be in a 0 quantum state, but they can be in a superposition of the 1 and 0 states. However, when qubits are measured they always give a 0 or a 1 based on the quantum state they were in. Today's physical quantum computers are noisy and quantum error correction is a burgeoning field of research. Quantum supremacy is the next milestone that quantum computing will achieve soon. While there is much hope and research in the field of quantum computing, as of March 2019 there have been no commercially useful algorithms published for today's noisy quantum computers. A classical computer has a memory made up of bits, where each bit is represented by either a one or a zero. A quantum computer, on the other hand, maintains a sequence of qubits, which can represent a one, a zero, or any quantum superposition of those two qubit states. In general, a quantum computer with n qubits can be in any superposition of up to 2 n different states..
A quantum computer operates on its qubits using measurement. An algorithm is composed of a fixed sequence of quantum logic gates and a problem is encoded by setting the initial values of the qubits, similar to how a classical computer works; the calculation ends with a measurement, collapsing the system of qubits into one of the 2 n eigenstates, where each qubit is zero or one, decomposing into a classical state. The outcome can, therefore, be at most n classical bits of information. If the algorithm did not end with a measurement, the result is an unobserved quantum state. Quantum algorithms are probabilistic, in that they provide the correct solution only with a certain known probability. Note that the term non-deterministic computing must not be used in that case to mean probabilistic because the term non-deterministic has a different meaning in computer science. An example of an implementation of qubits of a quantum computer could start with the use of particles with two spin states: "down" and "up".
This is true. A quantum computer with a given number of qubits is fundamentally different from a classical computer composed of the same number of classical bits. For example, representing the state of an n-qubit system on a classical computer requires the storage of 2n complex coefficients, while to characterize the state of a classical n-bit system it is sufficient to provide the values of the n bits, that is, only n numbers. Although this fact may seem to indicate that qubits can hold exponentially more information than their classical counterparts, care must be taken not to overlook the fact that the qubits are only in a probabilistic superposition of all of their states; this means that when the final state of the qubits is measured, they will only be found in one of the possible configurations they were in before the measurement. It is incorrect to think of a system of qubits as being in one particular state before the measurement; the qubits are in a superposition of states before any measurement is made, which directly affects the possible outcomes of the computation.
To better understand this point, consider a classical computer that operates on a three-bit register. If the exact state of the register at a given time is not known, it can be described as a probability distribution over the 2 3 = 8 different three-bit strings 000, 001, 010, 011, 100, 101, 110, 111. If there is no uncertainty over its state it is in one of these states with probability 1. However, if it is a probabilistic computer there is a possibility of it being in any one of a number of different states; the state of a three-qubit quantum computer is described by an eight-dimensional vector (
YouTube is an American video-sharing website headquartered in San Bruno, California. Three former PayPal employees—Chad Hurley, Steve Chen, Jawed Karim—created the service in February 2005. Google bought the site in November 2006 for US$1.65 billion. YouTube allows users to upload, rate, add to playlists, comment on videos, subscribe to other users, it offers a wide variety of corporate media videos. Available content includes video clips, TV show clips, music videos and documentary films, audio recordings, movie trailers, live streams, other content such as video blogging, short original videos, educational videos. Most of the content on YouTube is uploaded by individuals, but media corporations including CBS, the BBC, Hulu offer some of their material via YouTube as part of the YouTube partnership program. Unregistered users can only watch videos on the site, while registered users are permitted to upload an unlimited number of videos and add comments to videos. Videos deemed inappropriate are available only to registered users affirming themselves to be at least 18 years old.
YouTube and its creators earn advertising revenue from Google AdSense, a program which targets ads according to site content and audience. The vast majority of its videos are free to view, but there are exceptions, including subscription-based premium channels, film rentals, as well as YouTube Music and YouTube Premium, subscription services offering premium and ad-free music streaming, ad-free access to all content, including exclusive content commissioned from notable personalities; as of February 2017, there were more than 400 hours of content uploaded to YouTube each minute, one billion hours of content being watched on YouTube every day. As of August 2018, the website is ranked as the second-most popular site in the world, according to Alexa Internet. YouTube has faced criticism over aspects of its operations, including its handling of copyrighted content contained within uploaded videos, its recommendation algorithms perpetuating videos that promote conspiracy theories and falsehoods, hosting videos ostensibly targeting children but containing violent and/or sexually suggestive content involving popular characters, videos of minors attracting pedophilic activities in their comment sections, fluctuating policies on the types of content, eligible to be monetized with advertising.
YouTube was founded by Chad Hurley, Steve Chen, Jawed Karim, who were all early employees of PayPal. Hurley had studied design at Indiana University of Pennsylvania, Chen and Karim studied computer science together at the University of Illinois at Urbana–Champaign. According to a story, repeated in the media and Chen developed the idea for YouTube during the early months of 2005, after they had experienced difficulty sharing videos, shot at a dinner party at Chen's apartment in San Francisco. Karim did not attend the party and denied that it had occurred, but Chen commented that the idea that YouTube was founded after a dinner party "was very strengthened by marketing ideas around creating a story, digestible". Karim said the inspiration for YouTube first came from Janet Jackson's role in the 2004 Super Bowl incident, when her breast was exposed during her performance, from the 2004 Indian Ocean tsunami. Karim could not find video clips of either event online, which led to the idea of a video sharing site.
Hurley and Chen said that the original idea for YouTube was a video version of an online dating service, had been influenced by the website Hot or Not. Difficulty in finding enough dating videos led to a change of plans, with the site's founders deciding to accept uploads of any type of video. YouTube began as a venture capital-funded technology startup from an $11.5 million investment by Sequoia Capital and an $8 million investment from Artis Capital Management between November 2005 and April 2006. YouTube's early headquarters were situated above a pizzeria and Japanese restaurant in San Mateo, California; the domain name www.youtube.com was activated on February 14, 2005, the website was developed over the subsequent months. The first YouTube video, titled Me at the zoo, shows co-founder Jawed Karim at the San Diego Zoo; the video was uploaded on April 23, 2005, can still be viewed on the site. YouTube offered the public a beta test of the site in May 2005; the first video to reach one million views was a Nike advertisement featuring Ronaldinho in November 2005.
Following a $3.5 million investment from Sequoia Capital in November, the site launched on December 15, 2005, by which time the site was receiving 8 million views a day. The site grew and, in July 2006, the company announced that more than 65,000 new videos were being uploaded every day, that the site was receiving 100 million video views per day. According to data published by market research company comScore, YouTube is the dominant provider of online video in the United States, with a market share of around 43% and more than 14 billion views of videos in May 2010. In May 2011, 48 hours of new videos were uploaded to the site every minute, which increased to 60 hours every minute in January 2012, 100 hours every minute in May 2013, 300 hours every minute in November 2014, 400 hours every minute in February 2017; as of January 2012, the site had 800 million unique users a month. It is estimated that in 2007 YouTube consumed as much bandwidth as the entire Internet in 2000. According to third-party web analytics providers and SimilarWeb, YouTube is the second-most visited website in the world, as of December 2016.
Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics and the world as described by classical mechanics concerning quantum entanglement where two or more particles in a quantum state continue to be mutually dependent at large physical separations. This theorem is named after John Stewart Bell. A series of experiments has verified the theorem and showed that quantum entanglement occurs over large distances. Quantum entanglement has profound implications for the outcomes of measurements of quantum systems. In its simplest form, Bell's theorem states: No physical theory of local hidden variables can reproduce all of the predictions of quantum mechanics. Cornell solid-state physicist David Mermin has described the appraisals of the importance of Bell's theorem in the physics community as ranging from "indifference" to "wild extravagance". Lawrence Berkeley particle physicist Henry Stapp declared: "Bell's theorem is the most profound discovery of science."Bell's theorem rules out local hidden variables as a viable explanation of quantum mechanics.
Bell concluded: In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant. Bell summarized one of the possible ways to address the theorem, superdeterminism, in a 1985 BBC Radio interview: There is a way to escape the inference of superluminal speeds and spooky action at a distance, but it involves absolute determinism in the complete absence of free will. Suppose the world is super-deterministic, with not just inanimate nature running on behind-the-scenes clockwork, but with our behavior, including our belief that we are free to choose to do one experiment rather than another predetermined, including the'decision' by the experimenter to carry out one set of measurements rather than another, the difficulty disappears.
There is no need for a faster-than-light signal to tell particle A what measurement has been carried out on particle B, because the universe, including particle A, already'knows' what that measurement, its outcome, will be. In the early 1930s, the philosophical implications of the current interpretations of quantum theory troubled many prominent physicists of the day, including Albert Einstein. In a well-known 1935 paper, Boris Podolsky and co-authors Einstein and Nathan Rosen sought to demonstrate by the EPR paradox that quantum mechanics was incomplete; this provided hope. But that conclusion rested on the reasonable assumptions of locality and realism. In the vernacular of Einstein: locality meant no instantaneous action at a distance; these assumptions were hotly debated in the physics community, notably between Einstein and Niels Bohr. In his groundbreaking 1964 paper, "On the Einstein Podolsky Rosen paradox", physicist John Stewart Bell presented an analogy to EPR's hypothetical paradox.
Using their reasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there. After providing a mathematical formulation of locality and realism based on this, he showed specific cases where this would be inconsistent with the predictions of quantum mechanics theory. In experimental tests following Bell's example, now using quantum entanglement of photons instead of electrons, John Clauser and Stuart Freedman and Alain Aspect et al. demonstrated that the predictions of quantum mechanics are correct in this regard, although relying on additional unverifiable assumptions that open loopholes for local realism. In October 2015, Hensen and co-workers reported that they performed a loophole-free Bell test which might force one to reject at least one of the principles of locality, realism, or freedom-of-choice. Two of these logical possibilities, non-locality and non-realism, correspond to well-developed interpretations of quantum mechanics, have many supporters.
Conclusive experimental evidence of the violation of Bell's inequality would drastically reduce the class of acceptable deterministic theories but would not falsify absolute determinism, described by Bell himself as "not just inanimate nature running on behind-the-scenes clockwork, but with our behaviour, including our belief that we are free to choose to do one experiment rather than another predetermined". However, Bell himself considered absolute determinism an implausible solution. Bell's theorem states that any physical theory that incorporates local realism cannot reproduce all the predictions of quantum mechanical theory; because numerous experiments agree with the predictions of quantum mechanical theory, show differences between correlations that could not be explained by local hidden variables, the experimental results have been taken by many as refuting the concept of local realism as an explanation of the physical phenomena under test. For a hidden variable theory, if Bell's conditions are correct, the results that agree with quantu