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quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
, a qubit () or quantum bit is a basic unit of
quantum information Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both ...
—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
in which the two levels can be taken as spin up and spin down; or the polarization of a single
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
.


Etymology

The coining of the term ''qubit'' is attributed to
Benjamin Schumacher Benjamin "Ben" Schumacher is an American theoretical physicist, working mostly in the field of quantum information theory. He discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in ...
. In the acknowledgments of his 1995 paper, Schumacher states that the term ''qubit'' was created in jest during a conversation with William Wootters.


Bit versus qubit

A binary digit, characterized as 0 or 1, is used to represent information in classical computers. When averaged over both of its states (0,1), a binary digit can represent up to one bit of Shannon information, where a bit is the basic unit of
information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
. However, in this article, the word bit is synonymous with a binary digit. In classical computer technologies, a ''processed'' bit is implemented by one of two levels of low DC
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
, and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between two
logic level In digital circuits, a logic level is one of a finite number of states that a digital signal can inhabit. Logic levels are usually represented by the voltage difference between the signal and ground, although other standards exist. The range ...
s must be passed as fast as possible, as electrical voltage cannot change from one level to another instantaneously. There are two possible outcomes for the measurement of a qubit—usually taken to have the value "0" and "1", like a bit or binary digit. However, whereas the state of a bit can only be either 0 or 1, the general state of a qubit according to quantum mechanics can be a coherent superposition of both. Moreover, whereas a measurement of a classical bit would not disturb its state, a measurement of a qubit would destroy its coherence and irrevocably disturb the superposition state. It is possible to fully encode one bit in one qubit. However, a qubit can hold more information, e.g., up to two bits using superdense coding. For a system of ''n'' components, a complete description of its state in classical physics requires only ''n'' bits, whereas in quantum physics it requires 2''n'' complex numbers (or a single point in a 2''n''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
).


Standard representation

In quantum mechanics, the general
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of a qubit can be represented by a linear superposition of its two
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
basis states (or basis
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s). These vectors are usually denoted as , 0 \rangle = \bigl begin 1\\ 0 \end\bigr/math> and , 1 \rangle = \bigl begin 0\\ 1 \end\bigr/math>. They are written in the conventional Dirac—or "bra–ket"—notation; the , 0 \rangle and , 1 \rangle are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states, \, together called the computational basis, are said to span the two-dimensional linear vector (Hilbert) space of the qubit. Qubit basis states can also be combined to form product basis states. A set of qubits taken together is called a quantum register. For example, two qubits could be represented in a four-dimensional linear vector space spanned by the following product basis states: , 00 \rangle = \biggl begin 1\\ 0\\ 0\\ 0 \end\biggr/math>, , 01 \rangle = \biggl begin 0\\ 1\\ 0\\ 0 \end\biggr/math>, , 10 \rangle = \biggl begin 0\\ 0\\ 1\\ 0 \end\biggr/math>, and , 11 \rangle = \biggl begin 0\\ 0\\ 0\\ 1 \end\biggr/math>. In general, ''n'' qubits are represented by a superposition state vector in 2''n'' dimensional Hilbert space.


Qubit states

A pure qubit state is a
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deriv ...
superposition of the basis states. This means that a single qubit can be described by a linear combination of , 0 \rangle and , 1 \rangle : : , \psi \rangle = \alpha , 0 \rangle + \beta , 1 \rangle where α and β are the
probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the qu ...
s, that are both
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. When we measure this qubit in the standard basis, according to the Born rule, the probability of outcome , 0 \rangle with value "0" is , \alpha , ^2 and the probability of outcome , 1 \rangle with value "1" is , \beta , ^2. Because the absolute squares of the amplitudes equate to probabilities, it follows that \alpha and \beta must be constrained according to the second axiom of probability theory by the equation : , \alpha , ^2 + , \beta , ^2 = 1. The probability amplitudes, \alpha and \beta, encode more than just the probabilities of the outcomes of a measurement; the ''relative phase'' between \alpha and \beta is for example responsible for quantum interference, as seen in the
two-slit experiment In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanica ...
.


Bloch sphere representation

It might, at first sight, seem that there should be four
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
in , \psi \rangle = \alpha , 0 \rangle + \beta , 1 \rangle\,, as \alpha and \beta are
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint . This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of Hopf coordinates: :\begin \alpha &= e^ \cos\frac, \\ \beta &= e^ \sin\frac. \end Additionally, for a single qubit the ''global phase'' of the state e^ has no physically observable consequences, so we can arbitrarily choose to be real (or in the case that is zero), leaving just two degrees of freedom: :\begin \alpha &= \cos\frac, \\ \beta &= e^ \sin\frac, \end where e^ is the physically significant ''relative phase''. The possible quantum states for a single qubit can be visualised using a Bloch sphere (see picture). Represented on such a 2-sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where , 0 \rangle and , 1 \rangle are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state (, 0 \rangle + , 1 \rangle)/ would lie on the equator of the sphere at the positive X-axis. In the classical limit, a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles. The surface of the Bloch sphere is a
two-dimensional space In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
, which represents the observable state space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles \varphi and \theta.


Mixed state

A pure state is fully specified by a single ket, , \psi\rangle = \alpha , 0\rangle + \beta , 1\rangle,\, a coherent superposition, represented by a point on the surface of the Bloch sphere as described above. Coherence is essential for a qubit to be in a superposition state. With interactions, quantum noise and
decoherence Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
, it is possible to put the qubit in a mixed state, a statistical combination or “incoherent mixture” of different pure states. Mixed states can be represented by points ''inside'' the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles \varphi and \theta , as well as the length r of the vector that represents the mixed state.
Quantum error correction Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing th ...
can be used to maintain the purity of qubits.


Operations on qubits

There are various kinds of physical operations that can be performed on qubits. *
Quantum logic gate In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, ...
s, building blocks for a quantum circuit in a
quantum computer Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
, operate on a set of qubits (a
register Register or registration may refer to: Arts entertainment, and media Music * Register (music), the relative "height" or range of a note, melody, part, instrument, etc. * ''Register'', a 2017 album by Travis Miller * Registration (organ), th ...
); mathematically, the qubits undergo a ( reversible)
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
described by multiplying the quantum gates unitary matrix with the
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
vector. The result from this multiplication is a new quantum state. * Quantum measurement is an irreversible operation in which information is gained about the state of a single qubit, and
coherence Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deriv ...
is lost. The result of the measurement of a single qubit with the state \psi = \alpha , 0\rangle + \beta , 1\rangle will be either , 0\rangle with probability , \alpha, ^2 or , 1\rangle with probability , \beta, ^2. Measurement of the state of the qubit alters the magnitudes of α and β. For instance, if the result of the measurement is , 1\rangle, α is changed to 0 and β is changed to the phase factor e^ no longer experimentally accessible. If measurement is performed on a qubit that is entangled, the measurement may
collapse Collapse or its variants may refer to: Concepts * Collapse (structural) * Collapse (topology), a mathematical concept * Collapsing manifold * Collapse, the action of collapsing or telescoping objects * Collapsing user interface elements ** ...
the state of the other entangled qubits. * Initialization or re-initialization to a known value, often , 0\rangle. This operation collapses the quantum state (exactly like with measurement). Initialization to , 0\rangle may be implemented logically or physically: Logically as a measurement, followed by the application of the Pauli-X gate if the result from the measurement was , 1\rangle. Physically, for example if it is a
superconducting Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
phase qubit In quantum computing, and more specifically in superconducting quantum computing, the phase qubit is a superconducting device based on the superconductor–insulator–superconductor (SIS) Josephson junction, designed to operate as a quantum bit, ...
, by lowering the energy of the quantum system to its
ground state The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
. * Sending the qubit through a
quantum channel In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information ...
to a remote system or machine (an I/O operation), potentially as part of a quantum network.


Quantum entanglement

An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems. The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the , \Phi^+\rangle
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form ...
: :\frac (, 00\rangle + , 11\rangle). In this state, called an ''equal superposition'', there are equal probabilities of measuring either product state , 00\rangle or , 11\rangle, as , 1/\sqrt, ^2 = 1/2. In other words, there is no way to tell if the first qubit has value “0” or “1” and likewise for the second qubit. Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either , 0\rangle or , 1\rangle, i.e., she can now tell if her qubit has value “0” or “1”. Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a , 0\rangle, Bob must measure the same, as , 00\rangle is the only state where Alice's qubit is a , 0\rangle. In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value “0” or “1” — a most surprising circumstance that can not be explained by classical physics.


Controlled gate to construct the Bell state

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the controlled NOT gate (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is , 1\rangle, and otherwise leaves it unchanged. With respect to the unentangled product basis \, it maps the basis states as follows: : , 0 0 \rangle \mapsto , 0 0 \rangle : , 0 1 \rangle \mapsto , 0 1 \rangle : , 1 0 \rangle \mapsto , 1 1 \rangle : , 1 1 \rangle \mapsto , 1 0 \rangle . A common application of the CNOT gate is to maximally entangle two qubits into the , \Phi^+\rangle
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form ...
. To construct , \Phi^+\rangle, the inputs A (control) and B (target) to the CNOT gate are: \frac(, 0\rangle + , 1\rangle)_A and , 0\rangle_B After applying CNOT, the output is the , \Phi^+\rangle Bell State: \frac(, 00\rangle + , 11\rangle).


Applications

The , \Phi^+\rangle
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form ...
forms part of the setup of the superdense coding, quantum teleportation, and entangled
quantum cryptography Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
algorithms. Quantum entanglement also allows multiple states (such as the
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form ...
mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical computer. Many of the successes of quantum computation and communication, such as quantum teleportation and superdense coding, make use of entanglement, suggesting that entanglement is a
resource Resource refers to all the materials available in our environment which are technologically accessible, economically feasible and culturally sustainable and help us to satisfy our needs and wants. Resources can broadly be classified upon thei ...
that is unique to quantum computation. A major hurdle facing quantum computing, as of 2018, in its quest to surpass classical digital computing, is noise in quantum gates that limits the size of quantum circuits that can be executed reliably.


Quantum register

A number of qubits taken together is a qubit register.
Quantum computer Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
s perform calculations by manipulating qubits within a register.


Qudits and qutrits

The term qudit denotes the unit of quantum information that can be realized in suitable ''d''-level quantum systems. A qubit register that can be measured to ''N'' states is identical to an ''N''-level qudit. A rarely used
synonym A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are al ...
for qudit is quNit, since both ''d'' and ''N'' are frequently used to denote the dimension of a quantum system. Qudits are similar to the integer types in classical computing, and may be mapped to (or realized by) arrays of qubits. Qudits where the ''d''-level system is not an exponent of 2 can not be mapped to arrays of qubits. It is for example possible to have 5-level qudits. In 2017, scientists at the National Institute of Scientific Research constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits. In 2022, researchers at the
University of Innsbruck The University of Innsbruck (german: Leopold-Franzens-Universität Innsbruck; la, Universitas Leopoldino Franciscea) is a public research university in Innsbruck, the capital of the Austrian federal state of Tyrol, founded on October 15, 1669. ...
succeeded in developing a universal qudit quantum processor with trapped ions. In the same year, researchers at Tsinghua University's Center for Quantum Information implemented the dual-type qubit scheme in trapped ion quantum computers using the same ion species. Similar to the qubit, the
qutrit A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states. The qutrit is analogous to the classical radix-3 trit, just as ...
is the unit of quantum information that can be realized in suitable 3-level quantum systems. This is analogous to the unit of classical information trit of
ternary computer A ternary computer, also called trinary computer, is one that uses ternary logic (i.e., base 3) instead of the more common binary system (i.e., base 2) in its calculations. This means it uses trits (instead of bits, as most computers do). Types ...
s.


Physical implementations

Any two-level quantum-mechanical system can be used as a qubit. Multilevel systems can be used as well, if they possess two states that can be effectively decoupled from the rest (e.g., ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate two-level systems to various degrees were successfully realized. Similarly to a classical bit where the state of a transistor in a processor, the magnetization of a surface in a
hard disk A hard disk drive (HDD), hard disk, hard drive, or fixed disk is an electro-mechanical data storage device that stores and retrieves digital data using magnetic storage with one or more rigid rapidly rotating platters coated with magn ...
and the presence of current in a cable can all be used to represent bits in the same computer, an eventual quantum computer is likely to use various combinations of qubits in its design. The following is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only.


Qubit storage

In 2008 a team of scientists from the U.K. and U.S. reported the first relatively long (1.75 seconds) and coherent transfer of a superposition state in an electron spin "processing" qubit to a
nuclear spin In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe ...
"memory" qubit. This event can be considered the first relatively consistent quantum data storage, a vital step towards the development of
quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
. In 2013, a modification of similar systems (using charged rather than neutral donors) has dramatically extended this time, to 3 hours at very low temperatures and 39 minutes at room temperature. Room temperature preparation of a qubit based on electron spins instead of nuclear spin was also demonstrated by a team of scientists from Switzerland and Australia. An increased coherence of qubits is being explored by researchers who are testing the limitations of a Ge hole spin-orbit qubit structure.


See also

*
Ancilla bit Ancilla bits are some extra bits being used to achieve some specific goals in computation (e.g. reversible computation). In classical computation, any memory bit can be turned on or off at will, requiring no prior knowledge or extra complexity. ...
*
Bell state The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form ...
,
W state The W state is an entangled quantum state of three qubits which in the bra-ket notation has the following shape : , \mathrm\rangle = \frac(, 001\rangle + , 010\rangle + , 100\rangle) and which is remarkable for representing a specific type of ...
and
GHZ state The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
* Bloch sphere * Physical and logical qubits * Quantum register * Two-state quantum system * The elements of the group U(2) are all possible single-qubit quantum gates * The circle group
U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
define the phase about the qubits basis states


Notes


References


Further reading

* * * * A treatment of two-level quantum systems, decades before the term “qubit” was coined, is found in the third volume of ''
The Feynman Lectures on Physics ''The Feynman Lectures on Physics'' is a physics textbook based on some lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the Cali ...
'
(2013 ebook edition)
in chapters 9-11. * A non-traditional motivation of the qubit aimed at non-physicists is found in '' Quantum Computing Since Democritus'', by
Scott Aaronson Scott Joel Aaronson (born May 21, 1981) is an American theoretical computer scientist and David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. His primary areas of research are quantum computing ...
, Cambridge University Press (2013). * An introduction to qubits for non-specialists, by the person who coined the word, is found in Lecture 21 of ''The science of information: from language to black holes'', by Professor
Benjamin Schumacher Benjamin "Ben" Schumacher is an American theoretical physicist, working mostly in the field of quantum information theory. He discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in ...
, The Great Courses, The Teaching Company (4DVDs, 2015). * A
picture book A picture book combines visual and verbal narratives in a book format, most often aimed at young children. With the narrative told primarily through text, they are distinct from comics, which do so primarily through sequential images. The images ...
introduction to entanglement, showcasing a Bell state and the measurement of it, is found in ''Quantum entanglement for babies'', by Chris Ferrie (2017). . {{Authority control Quantum computing Quantum states Teleportation Units of information Australian inventions