Quantum information

For the journal, seenpj Quantum Information.

Optical lattices use lasers to separate rubidium atoms (red) for use as information bits in neutral-atom quantum processors—prototype devices which designers are trying to develop into full-fledged quantum computers.

Quantum informationis the information of thestateof aquantum system.It is the basic entity of study inquantum information theory,and can be manipulated usingquantum information processingtechniques. Quantum information refers to both the technical definition in terms ofVon Neumann entropyand the general computational term.

It is an interdisciplinary field that involvesquantum mechanics,computer science,information theory,philosophyandcryptographyamong other fields.Its study is also relevant to disciplines such ascognitive science,psychologyandneuroscience.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 is required in order to quantify the observation, making this crucial to thescientific method.Inquantum mechanics,due to theuncertainty principle,non-commutingobservablescannot 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 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.

Information is something physical that is encoded in the state of a quantum system.Whilequantum mechanicsdeals with examining properties of matter at the microscopic level,quantum information sciencefocuses on extracting information from those properties,andquantum computationmanipulates and processes information – performs logical operations – usingquantum information processingtechniques.

Quantum information, like classical information, can be processed usingdigital computers,transmittedfrom one location to another, manipulated withalgorithms,and analyzed with computer science andmathematics.Just like the basic unit of classical information is the bit, quantum information deals withqubits.Quantum information can be measured using Von Neumann entropy.

Recently, the field ofquantum computinghas become an active research area because of the possibility to disrupt modern computation, communication, andcryptography.

History and development

Development from fundamental quantum mechanics

The history of quantum information theory began at the turn of the 20th century whenclassical physicswas revolutionized intoquantum physics.The theories of classical physics were predicting absurdities such as theultraviolet 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 was born.

Quantum mechanicswas formulated bySchrödingerusing wave mechanics andHeisenbergusing 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 the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.

See:Dynamical Pictures

Evolution of:	Picture()
	Schrödinger(S)	Heisenberg(H)	Interaction(I)
Ket state	$|\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle$	constant	$|\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle$
Observable	constant	$A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }$	$A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }$
Density matrix	$\rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }$	constant	$\rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }$

Development from communication

In 1960s,Stratonovich,Helstromand Gordonproposed 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 Holevoobtained an upper bound of communication speed in the transmission of a classical message via a quantum channel.

Development from atomic physics and relativity

In the 1970s, techniques for manipulating single-atom quantum states, such as theatom trapand thescanning 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 utilized coarser, simultaneous control over 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 disproveEinstein'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 the speed of light, disproving Einstein's theory. However, theno-cloning theoremshowed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory.

Development from cryptography

See also:Quantum cryptography

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 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, a way of communicating secretly at long distances using theBB84quantum 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.

Development from computer science and mathematics

See also:Quantum supremacyandQuantum algorithm

With the advent ofAlan Turing's revolutionary ideas of a programmable computer, orTuring machine,he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine.This is known as theChurch–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 calledMoore's law.This 'law' is a projective trend that states that the number of transistors in anintegrated circuitdoubles 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 used 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 was developed byDavid DeutschandRichard Jozsa,known as theDeutsch–Jozsa algorithm.This problem however held little to no practical applications.Peter Shorin 1994 came up with a very important and practicalproblem,one of finding the prime factors of an integer. Thediscrete logarithmproblem as it was called, could be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers are more powerful than Turing machines.

Development from information theory

Around the time computer science was making a revolution, so was information theory and communication, throughClaude Shannon.Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and noisy channel coding theorem. He also showed thaterror correcting codescould be used to protect information being sent.

Quantum information theory also followed a similar trajectory, Ben Schumacher in 1995 made an analogue to Shannon's noiseless coding theorem using thequbit.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.

Qubits and information theory

Quantum information differs strongly from classical information, epitomized by thebit,in many striking and unfamiliar ways. While the fundamental unit of classical information is thebit,the most basic unit of quantum information is thequbit.Classical information is measured usingShannon entropy,while the quantum mechanical analogue isVon Neumann entropy.Given astatistical ensembleof quantum mechanical systems with thedensity matrix$\rho$,it is given by$S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).$Many of the same entropy measures in classicalinformation theorycan also be generalized to the quantum case, such as Holevo entropyand theconditional quantum entropy.

Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on theBloch sphere.Despite being continuously valued in this way, a qubit is thesmallestpossible unit of quantum information, and despite the qubit state being continuous-valued, it isimpossibletomeasurethe value precisely. Five famous theorems describe the limits on manipulation of quantum information.

1. no-teleportation theorem,which states that a qubit cannot be (wholly) converted into classical bits; that is, it cannot be fully "read".
2. no-cloning theorem,which prevents an arbitrary qubit from being copied.
3. no-deleting theorem,which prevents an arbitrary qubit from being deleted.
4. no-broadcast theorem,which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (e.g.viaquantum teleportation).
5. no-hiding theorem,which demonstrates the conservation of quantum information.

These theorems are proven fromunitarity,which according toLeonard Susskindis the technical term for the statement that quantum information within the universe is conserved.The five theorems open up possibilities in quantum information processing.

Quantum information processing

The state of a qubit contains all of its information. This state is frequently expressed as a vector on theBloch sphere.This state can be changed by applyinglinear transformationsorquantum gatesto them. Theseunitary transformationsare described as rotations on the Bloch Sphere. While classical gates correspond to the familiar operations ofBoolean logic,quantum gates are physicalunitary operators.

• Due to the volatility of quantum systems and the impossibility of copying states, the storing of quantum information is much more difficult than storing classical information. Nevertheless, with the use ofquantum error correctionquantum information can still be reliably stored in principle. The existence of quantum error correcting codes has also led to the possibility offault-tolerantquantum computation.
• Classical bits can be encoded into and subsequently retrieved from configurations of qubits, through the use of quantum gates. By itself, a single qubit can convey no more than one bit of accessible classical information about its preparation. This isHolevo's theorem.However, insuperdense codinga sender, by acting on one of twoentangledqubits, can convey two bits of accessible information about their joint state to a receiver.
• Quantum information can be moved about, in aquantum channel,analogous to the concept of a classicalcommunications channel.Quantum messages have a finite size, measured in qubits; quantum channels have a finitechannel capacity,measured in qubits per second.
• Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue ofShannonentropy,called thevon Neumann entropy.
• In some cases,quantum algorithmscan be used to perform computations faster than in any known classical algorithm. The most famous example of this isShor's algorithmthat can factor numbers in polynomial time, compared to the best classical algorithms that take sub-exponential time. As factorization is an important part of the safety ofRSA encryption,Shor's algorithm sparked the new field ofpost-quantum cryptographythat tries to find encryption schemes that remain safe even when quantum computers are in play. Other examples of algorithms that demonstratequantum supremacyincludeGrover's search algorithm,where the quantum algorithm gives a quadratic speed-up over the best possible classical algorithm. Thecomplexity classof problems efficiently solvable by aquantum computeris known asBQP.
• Quantum key distribution(QKD) allows unconditionally secure transmission of classical information, unlike classical encryption, which can always be broken in principle, if not in practice. Do note that certain subtle points regarding the safety of QKD are still hotly debated.

The study of all of the above topics and differences comprises quantum information theory.

Relation to quantum mechanics

Quantum mechanicsis 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 aphotonin alinear optical quantum computer,an ion in atrapped ion quantum computer,or it might be a large collection of atoms as in asuperconducting 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 ofdensity matricesover thecomplex numbers.Another important difference with quantum mechanics is that while quantum mechanics often studiesinfinite-dimensionalsystems such as aharmonic oscillator,quantum information theory is concerned with both continuous-variable systemsand finite-dimensional systems.

Entropy and information

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 theory

Classical information is based on the concepts of information laid out byClaude Shannon.Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit.

Shannon entropy

Main article:Entropy (information theory)

See also:Shannon's source coding theorem

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 a measure of information gained after making said measurement.

Shannon entropy, written as a functional of a discrete probability distribution,$P(x_{1}),P(x_{2}),...,P(x_{n})$associated with events$x_{1},...,x_{n}$,can be seen as the average information associated with this set of events, in units of bits:

$$H(X)=H[P(x_{1}),P(x_{2}),...,P(x_{n})]=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})$$

This definition of entropy can be used to quantify 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.

Rényi entropy

Main article:Rényi entropy

TheRényi entropyis a generalization of Shannon entropy defined above. The Rényi entropy of order r, written as a function of a discrete probability distribution,$P(a_{1}),P(a_{2}),...,P(a_{n})$,associated with events$a_{1},...,a_{n}$,is defined as:

$$H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})$$

for$0<r<\infty$and$r\neq 1$.

We arrive at the definition of Shannon entropy from Rényi when$r\rightarrow 1$,ofHartley entropy(or max-entropy) when$r\rightarrow 0$,andmin-entropywhen$r\rightarrow \infty$.

Quantum information theory

Quantum information theory is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made.

Von Neumann entropy

Main article:Von Neumann entropy

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 bydensity operators$\rho$:

$$S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},$$

where$\lambda _{i}$are the eigenvalues of$\rho$.

Von Neumann entropy plays a role in quantum information similar to the role Shannon entropy plays in classical information.

Applications

Quantum communication

Quantum communicationis 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 codingandquantum teleportationare 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 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.

Quantum key distribution

Main article:Quantum key distribution

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 is that it is impossible to copy a quantum key because of theno-cloning theorem.If someone tries to read encoded data, the quantum state being transmitted will change. This could be used to detect eavesdropping.

BB84

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 a third party to another for use in one-time pad encryption.

E91

E91was 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:

1. The entangled states are perfectly correlated which means that if Alice and Bob both measure their particles having either a vertical or horizontal polarization, they always get the same answer with 100% probability. The same is true if they both measure any other pair of complementary (orthogonal) polarizations. This necessitates that the two distant parties have exact directionality synchronization. However, from quantum mechanics theory the quantum state is completely random so that it is impossible for Alice to predict if she will get vertical polarization or horizontal polarization results.
2. Any attempt at eavesdropping by Eve destroys this quantum entanglement such that Alice and Bob can detect.

B92

B92 is a simpler version of BB84.

The main difference between B92 and BB84:

• B92 only needs two states
• BB84 needs 4 polarization states

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 or not he measured it correctly.

Quantum computation

Main article:Quantum computing

The most widely used model in quantum computation is thequantum circuit,which are based on the quantum bit "qubit".Qubit is somewhat analogous to thebitin classical computation. Qubits can be in a 1 or 0quantum state,or they can be in asuperpositionof the 1 and 0 states. However, when qubits are measured the result of the measurement is always either a 0 or a 1; theprobabilitiesof these two outcomes depend on thequantum statethat the qubits were in immediately prior to the measurement.

Any quantum computation algorithm can be represented as a network ofquantum logic gates.

Quantum decoherence

Main article:Quantum decoherence

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 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 error correction

Main article:Quantum error correction

QECis used inquantum computingto protect quantum information from errors due todecoherenceand otherquantum noise.Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.

Peter Shorfirst discovered this method of formulating aquantum error correcting codeby storing the information of one qubit onto a highly entangled state ofancilla qubits.A quantum error correcting code protects quantum information against errors.

Journals

Many journals publish research inquantum information science,although only a few are dedicated to this area. Among these are:

• International Journal of Quantum Information
• npj Quantum Information
• Quantum
• Quantum Information & Computation
• Quantum Information Processing
• Quantum Science and Technology

See also

• Categorical quantum mechanics
• Einstein's thought experiments
• Interpretations of quantum mechanics
• Positive Operator Valued Measure (POVM)
• Quantum clock
• Quantum entanglement
• Quantum foundations
• Quantum information science
• Quantum statistical mechanics
• Qubit
• Qutrit
• Typical subspace