Quantum entanglement in the context of "Topological order"

Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction with, for instance, atoms and molecules. Photons have been used to test many of the counter-intuitive predictions of quantum mechanics, such as entanglement and teleportation, and are a useful resource for quantum information processing.

Erwin Rudolf Josef Alexander Schrödinger (/ˈʃroʊdɪŋər/ SHROH-ding-er, German: [ˈʃʁøːdɪŋɐ] ; 12 August 1887 – 4 January 1961), sometimes written as Schroedinger or Schrodinger, was an Austrian–Irish theoretical physicist who developed fundamental results in quantum theory. In particular, he is recognized for devising the Schrödinger equation, an equation that provides a way to calculate the wave function of a system and how it changes dynamically in time. He coined the term "quantum entanglement" in 1935.

In addition, Schrödinger wrote many works on various aspects of physics: statistical mechanics and thermodynamics, physics of dielectrics, color theory, electrodynamics, general relativity, and cosmology, and he made several attempts to construct a unified field theory. In his book, What Is Life?, Schrödinger addressed the problems of genetics, looking at the phenomenon of life from the point of view of physics. He also paid great attention to the philosophical aspects of science, ancient, and oriental philosophical concepts, ethics, and religion. He also wrote on philosophy and theoretical biology. In popular culture, he is best known for his "Schrödinger's cat" thought experiment.

Quantum information science is an interdisciplinary field that combines the principles of quantum mechanics, information theory, and computer science to explore how quantum phenomena can be harnessed for the processing, analysis, and transmission of information. Quantum information science covers both theoretical and experimental aspects of quantum physics, including the limits of what can be achieved with quantum information. The term quantum information theory is sometimes used, but it refers to the theoretical aspects of information processing and does not include experimental research.

At its core, quantum information science explores how information behaves when stored and manipulated using quantum systems. Unlike classical information, which is encoded in bits that can only be 0 or 1, quantum information uses quantum bits or qubits that can exist simultaneously in multiple states because of superposition. Additionally, entanglement—a uniquely quantum linkage between particles—enables correlations that have no classical counterpart.

In condensed matter physics, a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement, fractionalized excitations, and absence of ordinary magnetic order.

The quantum spin liquid state was first proposed by physicist Phil Anderson in 1973 as the ground state for a system of spins on a triangular lattice that interact antiferromagnetically with their nearest neighbors, i.e. neighboring spins seek to be aligned in opposite directions. Quantum spin liquids generated further interest when in 1987 Anderson proposed a theory that described high-temperature superconductivity in terms of a disordered spin-liquid state.

Quantum materials is an umbrella term in condensed matter physics that encompasses all materials whose essential properties cannot be described in terms of semiclassical particles and low-level quantum mechanics. These are materials that present strong electronic correlations or some type of electronic order, such as superconducting or magnetic orders, or materials whose electronic properties are linked to non-generic quantum effects – topological insulators, Dirac electron systems such as graphene, as well as systems whose collective properties are governed by genuinely quantum behavior, such as ultra-cold atoms, cold excitons, polaritons, and so forth. On the microscopic level, four fundamental degrees of freedom – that of charge, spin, orbit and lattice – become intertwined, resulting in complex electronic states; the concept of emergence is a common thread in the study of quantum materials.

Quantum materials exhibit puzzling properties with no counterpart in the macroscopic world: quantum entanglement, quantum fluctuations, robust boundary states dependent on the topology of the materials' bulk wave functions, etc. Quantum anomalies such as the chiral magnetic effect link some quantum materials with processes in high-energy physics of quark-gluon plasmas.

A quantum computer is a (real or theoretical) computer that exploits superposed and entangled states. Quantum computers can be viewed as sampling from quantum systems that evolve in ways that may be described as operating on an enormous number of possibilities simultaneously, though still subject to strict computational constraints. By contrast, ordinary ("classical") computers operate according to deterministic rules. (A classical computer can, in principle, be replicated by a classical mechanical device, with only a simple multiple of time cost. On the other hand (it is believed), a quantum computer would require exponentially more time and energy to be simulated classically.) It is widely believed that a quantum computer could perform some calculations exponentially faster than any classical computer. For example, a large-scale quantum computer could break some widely used public-key cryptographic schemes and aid physicists in performing physical simulations. However, current hardware implementations of quantum computation are largely experimental and only suitable for specialized tasks.

The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in ordinary or "classical" computing. However, unlike a classical bit, which can be in one of two states (a binary), a qubit can exist in a linear combination of two states known as a quantum superposition. The result of measuring a qubit is one of the two states given by a probabilistic rule. If a quantum computer manipulates the qubit in a particular way, wave interference effects amplify the probability of the desired measurement result. The design of quantum algorithms involves creating procedures that allow a quantum computer to perform this amplification.

In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles of states. These arise in quantum mechanics in two different situations:

1. when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of the ensemble of possible preparations; and
2. when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.

Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states (not to be confused with superposed states), such as quantum statistical mechanics, open quantum systems and quantum information.