Editing Singlet state (section)

== Singlets and entangled states ==
Particles in singlet states do not need to 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 [[bra–ket notation|Dirac notation]] this distance-indifferent singlet state is usually represented as:

:<math>\frac{1}{\sqrt{2}}\left( \left|\uparrow \downarrow \right\rangle -  \left|\downarrow \uparrow \right\rangle\right).</math>

The possibility of spatially extended unbound singlet states has considerable historical and even philosophical importance, since considering such states contributed importantly to the theoretical and experimental exploration and verification of what is now called [[quantum entanglement]].  Along with [[Boris Podolsky|Podolsky]] and [[Nathan Rosen|Rosen]], [[Albert Einstein|Einstein]] proposed the [[EPR paradox]] thought experiment to help define his concerns with what he viewed as the non-locality of spatially separated entangled particles, using it in an argument that quantum mechanics was incomplete.  In 1951 David Bohm formulated a version of the "paradox" using spin singlet states.<ref>Bohm, D. (1951). Quantum Theory, Prentice-Hall, Englewood Cliffs, page 29, and Chapter 5 section 3, and Chapter 22 Section 19.</ref>

The difficulty captured by the EPR-Bohm thought experiment was that by measuring a spatial component of the angular momentum of either of two particles that have been prepared in a spatially distributed singlet state, the quantum state of the remaining particle, conditioned on the measurement result obtained, appears to be "instantaneously" altered, even if the two particles have over time become separated by light years of distance.  Decades later, [[John Stewart Bell]], who was a strong advocate of Einstein's locality-first perspective, proved [[Bell's theorem]] and showed that it 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{{citation needed|date=February 2022}}, subsequent experiments instead established the reality of entanglement. In fact, there now exist commercial [[Quantum cryptography|quantum encryption]] devices whose operation depends fundamentally on the existence and behavior of spatially extended singlets.{{citation needed|date=September 2018}}

A weaker form of Einstein's locality principle remains intact, which is this: Classical information cannot be transmitted faster than the speed of light ''c'', not even by using quantum entanglement events. This form of locality is weaker than the notion of "Einstein locality" or "local realism" used in the EPR and Bell's Theorem papers, but is sufficient to prevent the emergence of [[Causality (physics)|causality]] paradoxes.