Singlets and the related spin concepts of doublets and triplets occur frequently in atomic physics and nuclear physics, where one often needs to determine the total spin of a collection of particles. Since the only observed fundamental particle with zero spin is the extremely inaccessible Higgs boson, singlets in everyday physics are necessarily composed of sets of particles whose individual spins are non-zero, e.g. 1/2 or 1.
While for angular momentum states the singlet-style terminology is seldom used beyond triplets (spin=1), it has proven historically useful for describing much larger particle groups and subgroups that share certain features and are distinguished from each other by quantum numbers beyond spin. An example of this broader use of singlet-style terminology is the nine-member "nonet" of the pseudoscalar mesons.
The simplest possible angular momentum singlet is a set (bound or unbound) of two spin 1/2 (fermion) particles that are oriented so that their spin directions ("up" and "down") oppose each other; that is, they are antiparallel.
The simplest possible bound particle pair capable of exhibiting the singlet state is positronium, which consists of an electron and positron (antielectron) bound by their opposite electric charges. The electron and positron in positronium can also have identical or parallel spin orientations, which results in an experimentally distinct form of positronium with a spin 1 or triplet state.
An unbound singlet consists of a pair of entities small enough to exhibit quantum behavior (e.g. particles, atoms, or small molecules), not necessarily of the same type, for which four conditions hold:
Any spin value can be used for the pair, but the entanglement effect will be strongest both mathematically and experimentally if the spin magnitude is as small as possible, with the maximum possible effect occurring for entities with spin 1/2 (such as electrons and positrons). Early thought experiments for unbound singlets usually assumed the use of two antiparallel spin 1/2 electrons. However, actual experiments have tended to focus instead on using pairs of spin 1 photons. While the entanglement effect is somewhat less pronounced with such spin 1 particles, photons are easier to generate in correlated pairs and (usually) easier to keep in an unperturbed quantum state.
The ability of positronium to form both singlet and triplet states is described mathematically by saying that the product of two doublet representations (meaning the electron and positron, which are both spin 1/2 doublets) can be decomposed into the sum of an adjoint representation (the triplet or spin 1 state) and a trivial representation (the singlet or spin 0 state). While the particle interpretation of the positronium triplet and singlet states is arguably more intuitive, the mathematical description enables precise calculations of quantum states and probabilities.
It is important to realize that particles in singlet states need not be locally bound to each other. For example, when the spin states of two electrons are correlated by their emission from a single quantum event that conserves angular momentum, the resulting electrons remain in a shared singlet state even as their separation in space increases indefinitely over time, provided only that their angular momentum states remain unperturbed. In Dirac notation this distance-indifferent singlet state is usually represented as:
The possibility of spatially extended unbound singlet states has considerable historical and even philosophical importance, since considering such states eventually led to experimental exploration and verification of what is now called quantum entanglement. Quantum entanglement is the ability of quantum systems to maintain relationships that appear to violate the principle of locality, which Albert Einstein considered fundamental and defended throughout his life. Along with Podolsky and Rosen, Einstein proposed the EPR paradox thought experiment to help define his concerns with the non-locality of spatially distributed singlets, using it as a way to assert that quantum mechanics was incomplete.
The difficulty captured by the EPR thought experiment was that by perturbing the angular momentum state of either of the two particles in a spatially distributed singlet state, the quantum state of the remaining particle appears to be "instantaneously" altered, even if the two particles have over time become separated by light years of distance. A critical insight made decades later by John Stewart Bell, who was a strong advocate of Einstein's locality-first perspective, showed that his Bell's theorem could be used to assess the existence or non-existence of singlet entanglement experimentally. The irony was that instead of disproving entanglement, which was Bell's hope, subsequent experiments instead established the reality of entanglement. In fact, there now exist commercial quantum encryption devices whose operation depends fundamentally on the existence and behavior of spatially extended singlets.
A weaker form of Einstein's locality principle remains intact, which is this: Classical, history-setting information cannot be transmitted faster than the speed of light c, not even by using quantum entanglement events. This weaker form of locality is less conceptually elegant than Einstein's absolute locality, but is sufficient to prevent the emergence of causality paradoxes.