Non-classical correlations between quantum systems enabling coordination, imaging, or energy extraction.Quantum correlation refers to statistical dependencies between measurements on separate quantum systems that cannot be explained by classical physics. Unlike ordinary correlations, which arise from shared causes or direct influences, quantum correlations can be stronger than any local hidden-variable model permits, a phenomenon famously quantified by Bell inequalities. These correlations are the resource behind quantum entanglement, where the state of one particle is intrinsically linked to another regardless of distance, and they underpin protocols like quantum teleportation and superdense coding.
The concept was formalized in the 1930s through the Einstein-Podolsky-Rosen paradox, and John Bell's 1964 theorem turned it into a testable prediction. Alain Aspect's experiments in the 1980s confirmed the violation of Bell inequalities, establishing quantum correlations as a real physical phenomenon.
Quantum correlations emerge from the superposition principle: a composite system can exist in a state that is not a simple product of individual states, meaning the particles do not have definite properties until measured. When measured, the outcomes are random individually but perfectly or strongly correlated jointly, violating classical bounds. This arises because the wavefunction describes the whole system, not just its parts, and measurement collapses it globally, instantaneously linking the results.
Quantum correlations enable secure quantum key distribution, quantum teleportation, and enhanced imaging techniques like ghost imaging, where an image is formed from photons that never interacted with the object.
Quantum correlations can be so strong that they allow two parties to generate shared random numbers that are provably private, even if an eavesdropper has unlimited computational power.
| q | variable | entropic index characterizing non-extensivity — entropic index |
| p_i | variable | probability of microstate i — probability |
| W | variable | number of accessible microstates — number of states |
| S_q | variable | Tsallis entropy — entropy joule per kelvin |
| k | constant | Boltzmann constant (or a generalized constant depending on context) — boltzmann constant = 1.380649e-23 joule per kelvin |
| ∑ | operator | sum over all microstates — summation |
| [\text{X/Y}] | variable | Abundance ratio of element X to Y relative to solar — abundance ratio |
| [\text{Z/Y}] | variable | Abundance ratio of element Z to Y relative to solar — abundance ratio |
| a | variable | Slope of the correlation — dimensionless |
| b | variable | Intercept of the correlation — dimensionless |
| 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 |
| C(t) | variable | Normalized autocorrelation function of observable A — correlation function |
| A(t) | variable | Value of observable A at time t — observable varies |
| A(0) | variable | Value of observable A at initial time — observable varies |
| ⟨...⟩ | operator | Averages the product over a statistical ensemble of initial conditions — ensemble average |
| S | variable | Action of the theory — action joule second |
| g | variable | Determinant of the spacetime metric tensor — metric determinant |
| φ | variable | Scalar field non-minimally coupled to gravity — scalar field |
| R | variable | Ricci scalar curvature — ricci scalar per metre squared |
| ∂φ | variable | Gradient of the scalar field — field gradient per metre |
| ∫ d⁴x | operator | Integration over four-dimensional spacetime — four dimensional integral |
| √(-g) | operator | Volume element factor for curved spacetime — sqrt negative metric determinant |
| (∂φ)² | operator | Kinetic term of the scalar field, contraction of its gradient with itself — kinetic term |


