
Maximal quantum entanglement is the highest possible entanglement between quantum systems.Maximal quantum entanglement represents the strongest possible quantum correlation between two or more systems, where the state of each subsystem is completely undefined until a measurement is made on another. For two qubits, the maximally entangled states are the Bell states, such as the singlet state where the spins are perfectly anti-correlated. For larger systems, maximal entanglement is more complex — for example, a system of four particles can be maximally entangled in a way that is robust against particle loss, a property known as 'maximally entangled set'. This concept is central to quantum information theory because maximally entangled states are the ideal resources for quantum teleportation, superdense coding, and many quantum cryptographic protocols.
The concept of maximal entanglement dates back to the early days of quantum mechanics, with the Einstein-Podolsky-Rosen paradox (1935) highlighting the extreme correlations. The formal definition for multi-particle systems was developed in the 2000s, particularly through the work on entanglement measures and the maximally entangled set.
Maximal entanglement is defined by the property that the reduced density matrix of any subsystem is maximally mixed — meaning all information about the individual subsystem is erased. For two particles, this means the entropy of each particle is maximal. For multiple particles, the definition becomes more nuanced: a state is maximally entangled if it cannot be transformed into another entangled state using local operations and classical communication (LOCC) with certainty. This leads to the concept of 'maximally entangled sets' — states that are optimal resources for entanglement manipulation. In practice, maximal entanglement is achieved in the laboratory by carefully controlling interactions, such as in spontaneous parametric down-conversion for photons or in trapped-ion systems.
Maximally entangled states are used in quantum key distribution, quantum teleportation experiments, and as benchmarks for quantum computers. They also appear in high-energy physics, where top quark pairs produced at colliders are expected to be maximally entangled.
The spin states of top quark-antiquark pairs produced at the Large Hadron Collider are predicted to be maximally entangled, providing a new way to test quantum mechanics at high energies.
| p_max | variable | maximum momentum — momentum kilogram metre per second |
| e | variable | electric charge — electric charge coulomb |
| B | variable | magnetic field strength — magnetic field tesla |
| L | variable | characteristic length scale — length metre |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| P | variable | Probability distribution over x — probability distribution |
| Q | variable | Probability distribution over x — probability distribution |
| x | variable | Outcome index |
| ∑ | operator | Summation over the discrete sample space — summation |
| |·| | operator | Takes the non-negative magnitude — absolute value |
| G | variable | gravitational constant — gravitational constant cubic metre per kilogram per second squared |
| M | variable | mass of the object — mass kilogram |
| R | variable | radius of the object — radius metre |
| c | variable | speed of light in vacuum — speed of light metre per second |
| L_Edd | variable | Eddington luminosity — luminosity watt |
| M | variable | Mass of the object — mass kilogram |
| κ | variable | Opacity of the stellar material — opacity square metre per kilogram |
| G | constant | Constant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| π | constant | Mathematical constant pi — pi = 3.14159 |














