
Quantum information is the encoding and processing of data using quantum states like superposition and entanglement.Quantum information studies methods of encoding, transmitting, and processing data using quantum systems. Unlike classical information, quantum information cannot be arbitrarily copied (no-cloning theorem) and can be compressed in new ways. The main resources are superposition and entanglement. They allow the implementation of algorithms that have no classical analogs in efficiency, for example, integer factorization (Shor's algorithm) and database search (Grover's algorithm). Quantum entanglement also provides absolutely secure key distribution (BB84 protocol).
The origin of the field is associated with the analysis of the EPR paradox and Bell's inequalities (1964), which pointed to the nonclassical nature of quantum correlations. In 1981, Feynman proposed using quantum systems to simulate physics. David Deutsch in 1985 described a universal quantum computer and introduced the concept of quantum parallelism. The explosion of interest was sparked by Shor's algorithm (1994) for factorization and the discovery of quantum error-correcting codes (Shor, Steane, 1995–96). Since the 2000s, experiments on ion traps and superconducting circuits have begun.
A quantum computer operates on the state of a register of N qubits, which is a superposition of 2^N basis states. The computation is specified by unitary evolution (quantum gates) that uses interference to amplify the amplitudes of correct answers and suppress incorrect ones. The states are measured at the end, yielding a result with a probability given by the Born rule. Entanglement provides nonlocal correlations, allowing, for example, in teleportation to transfer a quantum state using a previously distributed entangled pair and two classical bits. The superposition principle and interference are the basis for the speedup. The difficulty is that interaction with the environment causes decoherence, destroying quantum properties.
Quantum computers crack modern ciphers in minutes, and in the future will be able to design drugs by simulating molecules.
The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state — nature sets a limit on copying.
| I(A;B) | variable | Mutual information between subsystems A and B — mutual information bit |
| S(A) | variable | Entropy of subsystem A — entropy bit |
| S(B) | variable | Entropy of subsystem B — entropy bit |
| S(A ∪ B) | variable | Entropy of the combined system A and B — entropy bit |
| ∪ | operator | Combines the two subsystems into one joint system — set union |
| Ψ | variable | quantum amplitude (transition amplitude or wavefunction) — probability amplitude varies |
| φ | variable | field configuration (path variable) — field varies |
| S[φ] | variable | action functional of the field configuration — action joule second |
| ℏ | constant | Sets the scale of quantum fluctuations — reduced planck constant = 1.054571817e-34 joule second |
| ∫ 𝒟φ | operator | sums over all possible histories of the field — functional integral |



















