Quantum information

The history of quantum information began at the turn of the 20th century when classical physics 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 hypothesis 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.[2]

Quantum mechanics was formulated by Schrodinger using wave mechanics and Heisenberg using matrix mechanics. The equivalence of these methods was proven later.[10] 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.[11] These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.

See: Dynamical Pictures

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

In 1960s, Stratonovich, Helstrom and Gordon[12] 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.[12][13] Later, Holevo obtained an upper bound of communication speed in the transmission of a classical message via a quantum channel.[14][15]

In the 1970s techniques of manipulating single quantum states such as the atom trap and the scanning tunneling microscope were starting to get developed. Isolating single atoms and moving them around to fashion an array of atoms at will was starting to become a reality. Prior to these developments, complete control over single quantum systems was not possible, and techniques involved a somewhat coarse level of control over a large number of quantum systems, none of which individually were directly accessible.[2] Extracting and manipulating information stored in individual atoms naturally started to become an interesting avenue, and quantum information and computation was starting to get developed.

In the 1980s, interest arose in whether it might be possible to use quantum effects to signal faster than light, an attempt of disproving Einstein's theory of relativity. If cloning an unknown quantum state were possible then Einstein's theory could be disproved. However, it turns out that quantum states could not, in general, be cloned. The no-cloning theorem is one of the earliest results of quantum information.[2]

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 eavesdrop without being detected, a way of communicating secretly at long distances using the BB84 quantum cryptographic protocol.[16] The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of a eavesdropper in a secure communication line will immediately let the two parties trying to communicate would know of the presence of the eavesdropper.

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.[17][18] 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.[19] 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.

See also: Quantum Supremacy, Quantum algorithms

Around the time computer science was making a revolution, so was information theory and communication, through Claude Shannon.[20][21][22] 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 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.[2]

Classical information is based on the concepts of information laid out by Claude Shannon. Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit.[28]

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 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.[29]

See also: Shannon's noiseless channel coding theorem

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:[28]

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

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

We arrive at the definition of Shannon entropy from Rényi when ${\displaystyle r\rightarrow 1}$

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.[28]

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 swapped out by density operators ${\displaystyle \rho }$.

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

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

Von Neumann plays a similar role in quantum information that Shannon entropy does in classical information