Bell States
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quantum information science Quantum information science is a field that combines the principles of quantum mechanics with information theory to study the processing, analysis, and transmission of information. It covers both theoretical and experimental aspects of quantum phys ...
, the Bell's states or EPR pairs are specific
quantum states In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
of two
qubits In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—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, ...
that represent the simplest examples of
quantum entanglement Quantum entanglement is the phenomenon where the quantum state of each Subatomic particle, particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic o ...
. The Bell's states are a form of entangled and
normalized Normalization or normalisation refers to a process that makes something more normal or regular. Science * Normalization process theory, a sociological theory of the implementation of new technologies or innovations * Normalization model, used in ...
basis vector In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...
s. This normalization implies that the overall probability of the particles being in one of the mentioned states is 1: \langle \Phi, \Phi \rangle = 1. Entanglement is a basis-independent result of
superposition In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
. 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 "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell's state the two qubits are in initially. Bell's states can be generalized to certain quantum states of multi-qubit systems, such as the
GHZ state The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base uni ...
for three or more subsystems. Understanding of Bell's states is useful in analysis of quantum communication, such as
superdense coding In quantum information theory, superdense coding (also referred to as ''dense coding'') is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the ass ...
and
quantum teleportation Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from on ...
. These mechanisms cannot transmit
information Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...
faster than the speed of light, a result known as the
no-communication theorem In physics, the no-communication theorem (also referred to as the no-signaling principle) is a no-go theorem in quantum information theory. It asserts that during the measurement of an entangled quantum state, it is impossible for one observer ...
.


Bell states

The Bell states are four specific maximally entangled
quantum states In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
of two
qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—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 syste ...
s. They are in a superposition of 0 and 1a linear combination of the two states. Their entanglement means the following: The qubit held by Alice (subscript "A") can be in a superposition of 0 and 1. If Alice measured her qubit in the standard basis, the outcome would be either 0 or 1, each with probability 1/2; if Bob (subscript "B") also measured his qubit, the outcome would be the same as for Alice. Thus, Alice and Bob would each seemingly have random outcome. Through communication they would discover that, although their outcomes separately seemed random, these were perfectly correlated. This perfect correlation at a distance is special: maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement. Hence, following
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
,
Boris Podolsky Boris Yakovlevich Podolsky (; June 29, 1896 – November 28, 1966) was a Russian-American physicist of Jewish descent, noted for his work with Albert Einstein and Nathan Rosen on entangled wave functions and the EPR paradox. Education In 1 ...
, and
Nathan Rosen Nathan Rosen (; March 22, 1909 – December 18, 1995) was an American and Israeli physicist noted for his study on the structure of the hydrogen molecule and his collaboration with Albert Einstein and Boris Podolsky on entangled wave functions and ...
in their famous 1935 " EPR paper", there is something missing in the description of the qubit pair given abovenamely this "agreement", called more formally a hidden variable. In his famous paper of 1964, John S. Bell showed by simple
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
arguments that these correlations (the one for the 0, 1 basis and the one for the +, − basis) cannot ''both'' be made perfect by the use of any "pre-agreement" stored in some hidden variablesbut that quantum mechanics predicts perfect correlations. In a more 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 (a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as 2\sqrt. Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables".


Bell basis

Four specific two-qubit states with the maximal value of 2\sqrt are designated as "Bell states". They are known as the four ''maximally entangled two-qubit Bell states'' and form a maximally entangled basis, known as the Bell basis, of the four-dimensional
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
for two qubits: : , \Phi^+\rangle = \frac \big(, 0\rangle_A \otimes , 0\rangle_B + , 1\rangle_A \otimes , 1\rangle_B\big) \qquad (1) : , \Phi^-\rangle = \frac \big(, 0\rangle_A \otimes , 0\rangle_B - , 1\rangle_A \otimes , 1\rangle_B\big) \qquad (2) : , \Psi^+\rangle = \frac \big(, 0\rangle_A \otimes , 1\rangle_B + , 1\rangle_A \otimes , 0\rangle_B\big) \qquad (3) : , \Psi^-\rangle = \frac \big(, 0\rangle_A \otimes , 1\rangle_B - , 1\rangle_A \otimes , 0\rangle_B\big) \qquad (4)


Creating Bell states via quantum circuits

Although there are many possible ways to create entangled Bell states through quantum circuits, the simplest takes a computational basis as the input, and contains a
Hadamard gate The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
and a
CNOT gate In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-''X'' gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based qu ...
(see picture). As an example, the pictured quantum circuit takes the two qubit input , 00\rangle and transforms it to the first Bell state , \Phi^+\rangle. Explicitly, the Hadamard gate transforms , 00\rangle into a
superposition In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
of (, 0\rangle, 0\rangle + , 1\rangle, 0\rangle) \over \sqrt. This will then act as a control input to the CNOT gate, which only inverts the target (the second qubit) when the control (the first qubit) is 1. Thus, the CNOT gate transforms the second qubit as follows \frac = , \Phi^+\rangle. For the four basic two-qubit inputs, , 00\rangle, , 01\rangle, , 10\rangle, , 11\rangle, the circuit outputs the four Bell states ( listed above). More generally, the circuit transforms the input in accordance with the equation , \beta(x,y)\rangle = \left ( \frac \right ), where \bar is the negation of y.


Properties of Bell states

The result of a measurement of a single qubit in a Bell state is indeterminate, but upon measuring the first qubit in the ''z''-basis, the result of measuring the second qubit is guaranteed to yield the same value (for the \Phi Bell states) or the opposite value (for the \Psi Bell states). This implies that the measurement outcomes are correlated. John Bell was the first to prove that the measurement correlations in the Bell State are stronger than could ever exist between classical systems. This hints that quantum mechanics allows information processing beyond what is possible with classical mechanics. In addition, the Bell states form an orthonormal basis and can therefore be defined with an appropriate measurement. Because Bell states are entangled states, information on the entire system may be known, while withholding information on the individual subsystems. For example, the Bell state is a
pure state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
, but the reduced density operator of the first qubit is a mixed state. The mixed state implies that not all the information on this first qubit is known. Bell States are either symmetric or antisymmetric with respect to the subsystems. Bell states are maximally entangled in the sense that its reduced density operators are maximally mixed, the multipartite generalization of Bell states in this spirit is called the absolutely maximally entangled (AME) state.


Bell state measurement

The Bell measurement is an important concept in
quantum information science Quantum information science is a field that combines the principles of quantum mechanics with information theory to study the processing, analysis, and transmission of information. It covers both theoretical and experimental aspects of quantum phys ...
: It is a joint quantum-mechanical measurement of two
qubit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—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 syste ...
s that determines which of the four Bell states the two qubits are in. A helpful example of
quantum measurement In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability ...
in the Bell basis can be seen in quantum computing. If a
CNOT gate In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-''X'' gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based qu ...
is applied to qubits A and B, followed by a
Hadamard gate The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
on qubit A, a measurement can be made in the computational basis. The CNOT gate performs the act of un-entangling the two previously entangled qubits. This allows the information to be converted from quantum information to a measurement of classical information. Quantum measurement obeys two key principles. The first, the principle of deferred measurement, states that any measurement can be moved to the end of the circuit. The second principle, the principle of implicit measurement, states that at the end of a quantum circuit, measurement can be assumed for any unterminated wires. The following are applications of Bell state measurements: Bell state measurement is the crucial step in
quantum teleportation Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from on ...
. The result of a Bell state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends. Experiments that utilize so-called "linear evolution, local measurement" techniques cannot realize a complete Bell state measurement. Linear evolution means that the detection apparatus acts on each particle independent of the state or evolution of the other, and local measurement means that each particle is localized at a particular detector registering a "click" to indicate that a particle has been detected. Such devices can be constructed from, for example: mirrors, beam splitters, and wave platesand are attractive from an experimental perspective because they are easy to use and have a high measurement
cross-section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture and engineering 3D * Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) ...
. For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques. This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as
teleportation Teleportation is the hypothetical transfer of matter or energy from one point to another without traversing the physical space between them. It is a common subject in science fiction and fantasy literature. Teleportation is often paired with tim ...
. If a Bell state is measured from this ambiguous class, the teleportation event fails. Entangling particles in multiple qubit variables, such as (for photonic systems)
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
and a two-element subset of
orbital angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed sy ...
states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other. Leveraging so-called hyper-entangled systems thus has an advantage for teleportation. It also has advantages for other protocols such as
superdense coding In quantum information theory, superdense coding (also referred to as ''dense coding'') is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the ass ...
, in which hyper-entanglement increases the channel capacity. In general, for hyper-entanglement in n variables, one can distinguish between at most 2^ - 1 classes out of 4^n Bell states using linear optical techniques.


Bell state correlations

Independent measurements made on two qubits that are entangled in Bell states positively correlate perfectly if each qubit is measured in the relevant basis. For the , \Phi^+\rangle state, this means selecting the same basis for both qubits. If an experimenter chose to measure both qubits in a , \Phi^-\rangle Bell state using the same basis, the qubits would appear positively correlated when measuring in the \ basis, anti-correlated in the \ basis, and partially (probabilistically) correlated in other bases. The , \Psi^+\rangle correlations can be understood by measuring both qubits in the same basis and observing perfectly anti-correlated results. More generally, , \Psi^+\rangle can be understood by measuring the first qubit in basis b_1, the second qubit in basis b_2 = X.b_1, and observing perfectly positively correlated results.


Applications


Superdense coding

Superdense coding In quantum information theory, superdense coding (also referred to as ''dense coding'') is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the ass ...
allows two individuals to communicate two bits of classical information by only sending a single qubit. The basis of this phenomenon is the entangled states or Bell states of a two qubit system. In this example, Alice and Bob are very far from each other, and have each been given one qubit of the entangled state. , \psi \rangle = \frac. In this example, Alice is trying to communicate two bits of classical information, one of four two bit strings: '00', '01', '10',or '11'. If Alice chooses to send the two bit message '01', she would perform the X gate to her qubit. Similarly, if Alice wants to send '10', she would apply the phase flip Z; if she wanted to send '11', she would apply the iYgate to her qubit; and finally, if Alice wanted to send the two bit message '00', she would do nothing to her qubit. Alice performs these
quantum 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. Quantum logic gates are the building blocks of quantu ...
transformations locally, transforming the initial entangled state , \psi\rangle into one of the four Bell states. The steps below show the necessary quantum gate transformations, and resulting Bell states, that Alice needs to apply to her qubit for each possible two bit message she desires to send to Bob. 00: I = \begin 1 & 0 \\ 0 & 1 \end \longrightarrow , \psi \rangle = \frac\equiv , \rangle 01: X = \begin 0 & 1 \\ 1 & 0 \end\longrightarrow , \psi \rangle = \frac\equiv , \rangle 10: Z = \begin 1 & 0 \\ 0 & -1 \end\longrightarrow , \psi \rangle = \frac\equiv , \rangle 11: iY = ZX = \begin 0 & 1 \\ -1 & 0 \end\longrightarrow , \psi \rangle = \frac\equiv , \rangle. After Alice applies the desired transformations to her qubit, she sends it to Bob. Bob then performs a measurement on the Bell state, which projects the entangled state onto one of the four two-qubit basis vectors, one of which will coincide with the original two bit message Alice was trying to send.


Quantum teleportation

Quantum teleportation Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from on ...
is the transfer of a quantum state over a distance. It is facilitated by entanglement between A, the giver, and B, the receiver of this quantum state. This process has become a fundamental research topic for quantum communication and computing. More recently, scientists have been testing its applications in information transfer through optical fibers. The process of quantum teleportation is defined as the following: Alice and Bob share an EPR pair and each took one qubit before they became separated. Alice must deliver a qubit of information to Bob, but she does not know the state of this qubit and can only send classical information to Bob. It is performed step by step as the following: # Alice sends her qubits through a
CNOT gate In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-''X'' gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based qu ...
. # Alice then sends the first qubit through a
Hadamard gate The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
. # Alice measures her qubits, obtaining one of four results, and sends this information to Bob. # Given Alice's measurements, Bob performs one of four operations on his half of the EPR pair and recovers the original quantum state. The following quantum circuit describes teleportation:


Quantum cryptography

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 soluti ...
is the use of quantum mechanical properties in order to encode and send information safely. The theory behind this process is the fact that it is impossible to measure a quantum state of a system without disturbing the system. This can be used to detect eavesdropping within a system. The most common form of
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 soluti ...
is
quantum key distribution Quantum key distribution (QKD) is a secure communication method that implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which then can b ...
. It enables two parties to produce a shared random secret key that can be used to encrypt messages. Its private key is created between the two parties through a public channel. Quantum cryptography can be considered a state of entanglement between two multi-dimensional systems, also known as two-
qudit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—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, ...
(quantum digit) entanglement.


See also

*
Bell test experiments A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the exp ...
*
Bell's inequality Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measuremen ...
*
EPR paradox EPR may refer to: Science and technology * EPR (nuclear reactor), European Pressurised-Water Reactor * EPR paradox (Einstein–Podolsky–Rosen paradox), in physics * Earth potential rise, in electrical engineering * East Pacific Rise, a mid-ocea ...
*
GHZ state The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base uni ...
*
Dicke state In quantum optics and quantum information, a Dicke state is a quantum state defined by Robert H. Dicke in connection to spontaneous radiation processes taking place in an ensemble of two-state atoms. A Dicke state is the simultaneous eigenstate of ...
*
Superdense coding In quantum information theory, superdense coding (also referred to as ''dense coding'') is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the ass ...
*
Quantum teleportation Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from on ...
*
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 soluti ...
* Quantum circuits *
Bell diagonal state Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory. Definition The Bell diagonal state is defined as the probabilistic mixture of Bell states: : , \phi^+\ ...


Notes


References

*
pp. 75
* . {{DEFAULTSORT:Bell State Quantum information science Quantum states