Misplaced Pages

#877122

129-445: It is an interdisciplinary field that involves quantum mechanics , computer science , information theory , philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science , psychology and neuroscience . Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement

258-475: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that

387-404: A {\displaystyle a} of a C*-algebra A {\displaystyle {\mathcal {A}}} is called positive if a = x ∗ x {\displaystyle a=x^{*}x} for some x {\displaystyle x} . Positivity of a map is defined accordingly. This characterization is not universally accepted; the quantum instrument

516-520: A 1 ) , P ( a 2 ) , . . . , P ( a n ) {\displaystyle P(a_{1}),P(a_{2}),...,P(a_{n})} , associated with events a 1 , . . . , a n {\displaystyle a_{1},...,a_{n}} , is defined as: H r ( A ) = 1 1 − r log 2 ⁡ ∑ i = 1 n P r (

645-692: A i ) {\displaystyle H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})} for 0 < r < ∞ {\displaystyle 0<r<\infty } and r ≠ 1 {\displaystyle r\neq 1} . We arrive at the definition of Shannon entropy from Rényi when r → 1 {\displaystyle r\rightarrow 1} , of Hartley entropy (or max-entropy) when r → 0 {\displaystyle r\rightarrow 0} , and min-entropy when r → ∞ {\displaystyle r\rightarrow \infty } . Quantum information theory

774-454: A harmonic oscillator , quantum information theory is concerned with both continuous-variable systems and finite-dimensional systems. Entropy measures the uncertainty in the state of a physical system. Entropy can be studied from the point of view of both the classical and quantum information theories. Classical information is based on the concepts of information laid out by Claude Shannon . Classical information, in principle, can be stored in

903-407: A quantum operation , whose properties we now summarize. Let H A {\displaystyle H_{A}} and H B {\displaystyle H_{B}} be the state spaces (finite-dimensional Hilbert spaces ) of the sending and receiving ends, respectively, of a channel. L ( H A ) {\displaystyle L(H_{A})} will denote

1032-486: A basis to encode quantum information for purposes such as quantum cryptography . The channel is capable of transmitting not only basis states (e.g. | 0 ⟩ {\displaystyle |0\rangle } , | 1 ⟩ {\displaystyle |1\rangle } ) but also superpositions of them (e.g. | 0 ⟩ + | 1 ⟩ {\displaystyle |0\rangle +|1\rangle } ). The coherence of

1161-415: A bit of binary strings. Any system having two states is a capable bit. Shannon entropy is the quantification of the information gained by measuring the value of a random variable. Another way of thinking about it is by looking at the uncertainty of a system prior to measurement. As a result, entropy, as pictured by Shannon, can be seen either as a measure of the uncertainty prior to making a measurement or as

1290-407: A channel can be viewed as a linear operator, it is tempting to use the natural operator norm . In other words, the closeness of Φ {\displaystyle \Phi } to the ideal channel Λ {\displaystyle \Lambda } can be defined by However, the operator norm may increase when we tensor Φ {\displaystyle \Phi } with

1419-563: A channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators: that is unital and completely positive ( CP ). The operator spaces can be viewed as finite-dimensional C*-algebras . Therefore, we can say a channel is a unital CP map between C*-algebras: Classical information can then be included in this formulation. The observables of

SECTION 10

#1733104635878

1548-506: A classical computer hence showing that quantum computers should be more powerful than Turing machines. Around the time computer science was making a revolution, so was information theory and communication, through Claude Shannon . Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and noisy channel coding theorem . He also showed that error correcting codes could be used to protect information being sent. Quantum information theory also followed

1677-545: A classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions C ( X ) {\displaystyle C(X)} on some set X {\displaystyle X} . We assume X {\displaystyle X} is finite so C ( X ) {\displaystyle C(X)} can be identified with the n -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with entry-wise multiplication. Therefore, in

1806-460: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,

1935-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}

2064-450: A large collection of atoms as in a superconducting quantum computer . Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers . Another important difference with quantum mechanics is that while quantum mechanics often studies infinite-dimensional systems such as

2193-455: A large number of quantum systems. The development of viable single-state manipulation techniques led to increased interest in the field of quantum information and computation. In the 1980s, interest arose in whether it might be possible to use quantum effects to disprove Einstein's theory of relativity . If it were possible to clone an unknown quantum state, it would be possible to use entangled quantum states to transmit information faster than

2322-441: A linear map Φ {\displaystyle \Phi } between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend Φ {\displaystyle \Phi } uniquely to the full space of operators. This leads to the adjoint map Φ ∗ {\displaystyle \Phi ^{*}} , which describes

2451-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement

2580-427: A map are sometimes abbreviated CPTP . In the literature, sometimes the fourth property is weakened so that Φ {\displaystyle \Phi } is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP. Density matrices acting on H A only constitute a proper subset of the operators on H A and same can be said for system B . However, once

2709-401: A map between Hilbert spaces, we obtain its adjoint Φ {\displaystyle \Phi } given by While Φ {\displaystyle \Phi } takes states on A to those on B , Φ ∗ {\displaystyle \Phi ^{*}} maps observables on system B to observables on A . This relationship is same as that between

SECTION 20

#1733104635878

2838-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector

2967-499: A measure of information gained after making said measurement. Shannon entropy, written as a functional of a discrete probability distribution, P ( x 1 ) , P ( x 2 ) , . . . , P ( x n ) {\displaystyle P(x_{1}),P(x_{2}),...,P(x_{n})} associated with events x 1 , . . . , x n {\displaystyle x_{1},...,x_{n}} , can be seen as

3096-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and

3225-422: A particle to a possibly distant receiver. Consequently, the teleportation process is a quantum channel. The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver. Teleportation occurs by a joint measurement of the sent particle and the remaining entangled particle. This measurement results in classical information which must be sent to

3354-467: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to

3483-485: A programmable computer, or Turing machine , he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine. This is known as the Church–Turing thesis . Soon enough, the first computers were made, and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called Moore's law . This 'law'

3612-408: A quantum mechanical effect F i {\displaystyle F_{i}} . F i {\displaystyle F_{i}} 's are assumed to be positive operators acting on appropriate state space and ∑ i F i = I {\textstyle \sum _{i}F_{i}=I} . (Such a collection is called a POVM .) In the Heisenberg picture,

3741-470: A qubit contains all of its information. This state is frequently expressed as a vector on the Bloch sphere. This state can be changed by applying linear transformations or quantum gates to them. These unitary transformations are described as rotations on the Bloch sphere. While classical gates correspond to the familiar operations of Boolean logic , quantum gates are physical unitary operators . The study of

3870-444: A similar trajectory, Ben Schumacher in 1995 made an analogue to Shannon's noiseless coding theorem using the qubit . A theory of error-correction also developed, which allows quantum computers to make efficient computations regardless of noise and make reliable communication over noisy quantum channels. Quantum information differs strongly from classical information, epitomized by the bit , in many striking and unfamiliar ways. While

3999-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of

Quantum information - Misplaced Pages Continue

4128-545: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of

4257-433: A third party to another for use in one-time pad encryption. E91 was made by Artur Ekert in 1991. His scheme uses entangled pairs of photons. These two photons can be created by Alice, Bob, or by a third party including eavesdropper Eve. One of the photons is distributed to Alice and the other to Bob so that each one ends up with one photon from the pair. This scheme relies on two properties of quantum entanglement: B92

4386-491: A way of communicating secretly at long distances using the BB84 quantum cryptographic protocol. The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of an eavesdropper in a secure communication line will immediately let the two parties trying to communicate know of the presence of the eavesdropper. With the advent of Alan Turing 's revolutionary ideas of

4515-468: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of

4644-408: Is unital , that is, Φ ∗ ( I ) = I {\displaystyle \Phi ^{*}(I)=I} . Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel. So far we have only defined quantum channel that transmits only quantum information. As stated in the introduction, the input and output of

4773-415: Is CP and unital. Notice that Ψ ( f ⊗ I ) {\displaystyle \Psi (f\otimes I)} gives precisely the observable map. The map describes the overall state change. Suppose two parties A and B wish to communicate in the following manner: A performs the measurement of an observable and communicates the measurement outcome to B classically. According to

4902-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of

5031-554: Is a projective trend that states that the number of transistors in an integrated circuit doubles every two years. As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference. This led to the advent of quantum computing, which uses quantum mechanics to design algorithms. At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem

5160-582: Is a simpler version of BB84. The main difference between B92 and BB84: Like the BB84, Alice transmits to Bob a string of photons encoded with randomly chosen bits but this time the bits Alice chooses the bases she must use. Bob still randomly chooses a basis by which to measure but if he chooses the wrong basis, he will not measure anything which is guaranteed by quantum mechanics theories. Bob can simply tell Alice after each bit she sends whether he measured it correctly. The most widely used model in quantum computation

5289-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes

Quantum information - Misplaced Pages Continue

5418-689: Is called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics. Quantum mechanics Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but

5547-447: Is called the Kraus rank of Ψ {\displaystyle \Psi } . A channel with Kraus rank 1 is called pure . The time evolution is one example of a pure channel. This terminology again comes from the channel-state duality. A channel is pure if and only if its dual state is a pure state. In quantum teleportation , a sender wishes to transmit an arbitrary quantum state of

5676-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in

5805-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate

5934-399: Is defined by A measure-and-prepare channel can not be the identity map. This is precisely the statement of the no teleportation theorem , which says classical teleportation (not to be confused with entanglement-assisted teleportation ) is impossible. In other words, a quantum state can not be measured reliably. In the channel-state duality , a channel is measure-and-prepare if and only if

6063-541: Is finite-dimensional, Ψ {\displaystyle \Psi } is a unital CP map between spaces of matrices By Choi's theorem on completely positive maps , Ψ {\displaystyle \Psi } must take the form where N ≤ nm . The matrices K i are called Kraus operators of Ψ {\displaystyle \Psi } (after the German physicist Karl Kraus , who introduced them). The minimum number of Kraus operators

6192-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes

6321-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In

6450-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,

6579-1018: Is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made. One interpretation of Shannon entropy was the uncertainty associated with a probability distribution. When we want to describe the information or the uncertainty of a quantum state, the probability distributions are simply replaced by density operators ρ {\displaystyle \rho } : S ( ρ ) ≡ − t r ( ρ   log 2 ⁡   ρ ) = − ∑ i λ i   log 2 ⁡   λ i , {\displaystyle S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},} where λ i {\displaystyle \lambda _{i}} are

SECTION 50

#1733104635878

6708-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making

6837-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately

6966-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then

7095-535: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent

7224-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,

7353-409: Is required in order to quantify the observation, making this crucial to the scientific method . In quantum mechanics , due to the uncertainty principle , non-commuting observables cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of

7482-471: Is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in a Frobenius algebra or Frobenius category . For a purely quantum system, the time evolution, at certain time t , is given by where U = e − i H t / ℏ {\displaystyle U=e^{-iHt/\hbar }} and H

7611-413: Is that it is impossible to copy a quantum key because of the no-cloning theorem . If someone tries to read encoded data, the quantum state being transmitted will change. This could be used to detect eavesdropping. The first quantum key distribution scheme, BB84 , was developed by Charles Bennett and Gilles Brassard in 1984. It is usually explained as a method of securely communicating a private key from

7740-472: Is the Hamiltonian and t is the time. Clearly this gives a CPTP map in the Schrödinger picture and is therefore a channel. The dual map in the Heisenberg picture is Consider a composite quantum system with state space H A ⊗ H B . {\displaystyle H_{A}\otimes H_{B}.} For a state the reduced state of ρ on system A , ρ , is obtained by taking

7869-411: Is the quantum circuit , which are based on the quantum bit " qubit ". Qubit is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state , or they can be in a superposition of the 1 and 0 states. However, when qubits are measured, the result of the measurement is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that

SECTION 60

#1733104635878

7998-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation

8127-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because

8256-501: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}

8385-535: The Internet . Terminologically, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation viewed not merely as the reduced dynamics of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps ) We will assume for

8514-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators

8643-411: The partial trace of ρ with respect to the B system: The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. In the Heisenberg picture, the dual map of this channel is where A is an observable of system A . An observable associates a numerical value f i ∈ C {\displaystyle f_{i}\in \mathbb {C} } to

8772-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called

8901-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to

9030-399: The 1980s. However, around the same time another avenue started dabbling into quantum information and computation: Cryptography . In a general sense, cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another. Bennett and Brassard developed a communication channel on which it is impossible to eavesdrop without being detected,

9159-540: The Heisenberg picture is where Ψ i {\displaystyle \Psi _{i}} is defined in the following way: Factor F i = M i 2 {\displaystyle F_{i}=M_{i}^{2}} (this can always be done since elements of a POVM are positive) then Ψ i ( A ) = M i A M i {\displaystyle \;\Psi _{i}(A)=M_{i}AM_{i}} . We see that Ψ {\displaystyle \Psi }

9288-417: The Heisenberg picture, if the classical information is part of, say, the input, we would define B {\displaystyle {\mathcal {B}}} to include the relevant classical observables. An example of this would be a channel Notice L ( H B ) ⊗ C ( X ) {\displaystyle L(H_{B})\otimes C(X)} is still a C*-algebra. An element

9417-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on

9546-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system

9675-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

9804-400: The Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa. It can be directly checked that if Φ {\displaystyle \Phi } is assumed to be trace preserving, Φ ∗ {\displaystyle \Phi ^{*}}

9933-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses

10062-413: The Schrödinger equation for the particle in a box are or, from Euler's formula , 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 general dynamics of a qubit . An example of classical information is a text document transmitted over

10191-468: The above classical state to the density matrix The total operation is the composition Channels of this form are called measure-and-prepare or in Holevo form. In the Heisenberg picture, the dual map Φ ∗ = Φ 1 ∗ ∘ Φ 2 ∗ {\displaystyle \Phi ^{*}=\Phi _{1}^{*}\circ \Phi _{2}^{*}}

10320-447: The above topics and differences comprises quantum information theory. Quantum mechanics is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a photon in a linear optical quantum computer , an ion in a trapped ion quantum computer , or it might be

10449-525: The action of Φ {\displaystyle \Phi } in the Heisenberg picture : The spaces of operators L ( H A ) and L ( H B ) are Hilbert spaces with the Hilbert–Schmidt inner product. Therefore, viewing Φ : L ( H A ) → L ( H B ) {\displaystyle \Phi :L(H_{A})\rightarrow L(H_{B})} as

10578-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of

10707-439: The assumption that Alice and Bob have a pre-shared Bell state , dense coding transfers two classical bits from Alice to Bob by using one qubit, again under the same assumption, that Alice and Bob have a pre-shared Bell state. One of the best known applications of quantum cryptography is quantum key distribution which provide a theoretical solution to the security issue of a classical key. The advantage of quantum key distribution

10836-583: The average information associated with this set of events, in units of bits: H ( X ) = H [ P ( x 1 ) , P ( x 2 ) , . . . , P ( x n ) ] = − ∑ i = 1 n P ( x i ) log 2 ⁡ P ( x i ) {\displaystyle H(X)=H[P(x_{1}),P(x_{2}),...,P(x_{n})]=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})} This definition of entropy can be used to quantify

10965-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy

11094-434: The basic unit of classical information is the bit, quantum information deals with qubits . Quantum information can be measured using Von Neumann entropy. Recently, the field of quantum computing has become an active research area because of the possibility to disrupt modern computation, communication, and cryptography . The history of quantum information theory began at the turn of the 20th century when classical physics

11223-418: The capacity of a channel Φ {\displaystyle \Phi } , we need to compare it with an "ideal channel" Λ {\displaystyle \Lambda } . For instance, when the input and output algebras are identical, we can choose Λ {\displaystyle \Lambda } to be the identity map. Such a comparison requires a metric between channels. Since

11352-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for

11481-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in

11610-577: The corresponding observable map Ψ {\displaystyle \Psi } maps a classical observable to the quantum mechanical one In other words, one integrates f against the POVM to obtain the quantum mechanical observable. It can be easily checked that Ψ {\displaystyle \Psi } is CP and unital. The corresponding Schrödinger map Ψ ∗ {\displaystyle \Psi ^{*}} takes density matrices to classical states: where

11739-416: The corresponding state is separable . Actually, all the states that result from the partial action of a measure-and-prepare channel are separable, and for this reason measure-and-prepare channels are also known as entanglement-breaking channels. Consider the case of a purely quantum channel Ψ {\displaystyle \Psi } in the Heisenberg picture. With the assumption that everything

11868-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}

11997-481: The effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map Φ {\displaystyle \Phi } with pure quantum input ρ ∈ L ( H ) {\displaystyle \rho \in L(H)} and with output space C ( X ) ⊗ L ( H ) {\displaystyle C(X)\otimes L(H)} : Let The dual map in

12126-678: The eigenvalues of ρ {\displaystyle \rho } . Von Neumann entropy plays a role in quantum information similar to the role Shannon entropy plays in classical information. Quantum communication is one of the applications of quantum physics and quantum information. There are some famous theorems such as the no-cloning theorem that illustrate some important properties in quantum communication. Dense coding and quantum teleportation are also applications of quantum communication. They are two opposite ways to communicate using qubits. While teleportation transfers one qubit from Alice and Bob by communicating two classical bits under

12255-545: The family of operators on H A . {\displaystyle H_{A}.} In the Schrödinger picture , a purely quantum channel is a map Φ {\displaystyle \Phi } between density matrices acting on H A {\displaystyle H_{A}} and H B {\displaystyle H_{B}} with the following properties: The adjectives completely positive and trace preserving used to describe

12384-621: The fundamental unit of classical information is the bit , the most basic unit of quantum information is the qubit . Classical information is measured using Shannon entropy , while the quantum mechanical analogue is Von Neumann entropy . Given a statistical ensemble of quantum mechanical systems with the density matrix ρ {\displaystyle \rho } , it is given by S ( ρ ) = − Tr ⁡ ( ρ ln ⁡ ρ ) . {\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).} Many of

12513-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,

12642-406: The identity map on some ancilla. To make the operator norm even a more undesirable candidate, the quantity may increase without bound as n → ∞ . {\displaystyle n\rightarrow \infty .} The solution is to introduce, for any linear map Φ {\displaystyle \Phi } between C*-algebras, the cb-norm The mathematical model of

12771-652: The inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalized functionals , and invoking the Riesz representation theorem , we can put The observable map, in the Schrödinger picture, has a purely classical output algebra and therefore only describes measurement statistics. To take the state change into account as well, we define what is called a quantum instrument . Let { F 1 , … , F n } {\displaystyle \{F_{1},\dots ,F_{n}\}} be

12900-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior

13029-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;

13158-473: The message he receives, B prepares his (quantum) system in a specific state. In the Schrödinger picture, the first part of the channel Φ {\displaystyle \Phi } 1 simply consists of A making a measurement, i.e. it is the observable map: If, in the event of the i -th measurement outcome, B prepares his system in state R i , the second part of the channel Φ {\displaystyle \Phi } 2 takes

13287-404: The moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional. The memoryless in the section title carries the same meaning as in classical information theory : the output of a channel at a given time depends only upon the corresponding input and not any previous ones. Consider quantum channels that transmit only quantum information. This is precisely

13416-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It

13545-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude

13674-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of

13803-396: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }

13932-518: The philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements. See: Dynamical Pictures In the 1960s, Ruslan Stratonovich , Carl Helstrom and Gordon proposed a formulation of optical communications using quantum mechanics. This was the first historical appearance of quantum information theory. They mainly studied error probabilities and channel capacities for communication. Later, Alexander Holevo obtained an upper bound of communication speed in

14061-402: The physical resources required to store the output of an information source. The ways of interpreting Shannon entropy discussed above are usually only meaningful when the number of samples of an experiment is large. The Rényi entropy is a generalization of Shannon entropy defined above. The Rényi entropy of order r, written as a function of a discrete probability distribution, P (

14190-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For

14319-445: The qubit state being continuous-valued, it is impossible to measure the value precisely. Five famous theorems describe the limits on manipulation of quantum information. These theorems are proven from unitarity , which according to Leonard Susskind is the technical term for the statement that quantum information within the universe is conserved. The five theorems open possibilities in quantum information processing. The state of

14448-445: The qubits were in immediately prior to the measurement. Any quantum computation algorithm can be represented as a network of quantum logic gates . If a quantum system were perfectly isolated, it would maintain coherence perfectly, but it would be impossible to test the entire system. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; this process

14577-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that

14706-476: The receiver to complete the teleportation. Importantly, the classical information can be sent after the quantum channel has ceased to exist. Experimentally, a simple implementation of a quantum channel is fiber optic (or free-space for that matter) transmission of single photons . Single photons can be transmitted up to 100 km in standard fiber optics before losses dominate. The photon's time-of-arrival ( time-bin entanglement ) or polarization are used as

14835-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and

14964-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present

15093-472: The same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and the conditional quantum entropy . Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the Bloch sphere . Despite being continuously valued in this way, a qubit is the smallest possible unit of quantum information, and despite

15222-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while

15351-402: The speed of light, disproving Einstein's theory. However, the no-cloning theorem showed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory. Despite all the excitement and interest over studying isolated quantum systems and trying to find a way to circumvent the theory of relativity, research in quantum information theory became stagnant in

15480-401: The state is maintained during transmission through the channel. Contrast this with the transmission of electrical pulses through wires (a classical channel), where only classical information (e.g. 0s and 1s) can be sent. Before giving the definition of channel capacity, the preliminary notion of the norm of complete boundedness , or cb-norm of a channel needs to be discussed. When considering

15609-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then

15738-873: The system is an eigenstate of the observable. Since any two non-commuting observables are not simultaneously well-defined, a quantum state can never contain definitive information about both non-commuting observables. Data can be encoded into the quantum state of a quantum system as quantum information . While quantum mechanics deals with examining properties of matter at the microscopic level, quantum information science focuses on extracting information from those properties, and quantum computation manipulates and processes information – performs logical operations – using quantum information processing techniques. Quantum information, like classical information, can be processed using digital computers , transmitted from one location to another, manipulated with algorithms , and analyzed with computer science and mathematics . Just like

15867-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space

15996-442: The transmission of a classical message via a quantum channel . In the 1970s, techniques for manipulating single-atom quantum states, such as the atom trap and the scanning tunneling microscope , began to be developed, making it possible to isolate single atoms and arrange them in arrays. Prior to these developments, precise control over single quantum systems was not possible, and experiments used coarser, simultaneous control over

16125-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of

16254-519: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained

16383-480: Was born. Quantum mechanics was formulated by Erwin Schrödinger using wave mechanics and Werner Heisenberg using matrix mechanics . The equivalence of these methods was proven later. Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von Neumann formulated quantum theory using operator algebra in a way that it described measurement as well as dynamics. These studies emphasized

16512-461: Was developed by David Deutsch and Richard Jozsa , known as the Deutsch–Jozsa algorithm . This problem however held little to no practical applications. Peter Shor in 1994 came up with a very important and practical problem , one of finding the prime factors of an integer. The discrete logarithm problem as it was called, could theoretically be solved efficiently on a quantum computer but not on

16641-417: Was revolutionized into quantum physics . The theories of classical physics were predicting absurdities such as the ultraviolet catastrophe , or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypotheses to classical physics. Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics

#877122