Development from communication

In 1960s, Stratonovich, Helstrom and Gordon^[20] 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.^[20][21][22] Later, Alexander Holevo obtained an upper bound of communication speed in the transmission of a classical message via a quantum channel.^[23][24]

Development from atomic physics and relativity

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 utilized coarser, simultaneous control over a large number of quantum systems.^[2] 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 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.^[2]

Development from 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.^[2]

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 the BB84 quantum cryptographic protocol.^[25] 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

With the advent of Alan Turing's revolutionary ideas of a programmable computer, or Turing machine, he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine.^[26][27] 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' is a projective trend that states that the number of transistors in an integrated circuit doubles every two years.^[28] 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 by David Deutsch and Richard Jozsa, known as the Deutsch–Jozsa algorithm. This problem however held little to no practical applications.^[2] 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 be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers are more powerful than Turing machines.

Shannon entropy

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.^[2]

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

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

This definition of entropy can be used to quantify 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.^[35]

Rényi entropy

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, ${\displaystyle P(a_{1}),P(a_{2}),...,P(a_{n})}$, associated with events ${\displaystyle a_{1},...,a_{n}}$, is defined as:^[37]

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

for ${\displaystyle 0<r<\infty }$ and ${\displaystyle r\neq 1}$.

We arrive at the definition of Shannon entropy from Rényi when ${\displaystyle r\rightarrow 1}$, of Hartley entropy (or max-entropy) when ${\displaystyle r\rightarrow 0}$, and min-entropy when ${\displaystyle r\rightarrow \infty }$.

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 by density operators ${\displaystyle \rho }$:

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

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

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

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.^[2]

E91

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:

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.^[38]

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.^[39]