Editing T-symmetry (section)

===Time reversal in quantum mechanics===

[[Image:parity 1drep.png|frame|Two-dimensional representations of [[parity (physics)|parity]] are given by a pair of quantum states that go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all [[irreducible representation]]s of parity are one-dimensional. '''Kramers' theorem''' states that time reversal need not have this property because it is represented by an anti-unitary operator.]]

This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly,
# that it must be represented as an anti-unitary operator,
# that it protects non-degenerate quantum states from having an [[electric dipole moment]],
# that it has two-dimensional representations with the property {{nowrap|''T''<sup>2</sup> {{=}} −1}} (for [[fermion]]s).

The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of [[quantum states]] into each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate the theorem that all [[abelian group]]s be represented by one-dimensional irreducible representations. The reason it does this is that it is represented by an anti-unitary operator. It thus opens the way to [[spinor]]s in quantum mechanics.

On the other hand, the notion of quantum-mechanical time reversal turns out to be a useful tool for the development of physically motivated [[quantum computing]] and [[Quantum simulator|simulation]] settings, providing, at the same time, relatively simple tools to assess their [[Computational complexity|complexity]]. For instance, quantum-mechanical time reversal was used to develop novel [[boson sampling]] schemes<ref name="Chakhmakhchyan2017">{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title=Boson sampling with Gaussian measurements|journal=Physical Review A|date=2017|volume=96|issue=3|page=032326 |doi=10.1103/PhysRevA.96.032326|arxiv=1705.05299|bibcode=2017PhRvA..96c2326C|s2cid=119431211}}</ref> and to prove the duality between two fundamental optical operations, [[beam splitter#Quantum mechanical description|beam splitter]] and [[Squeezed coherent state#Operator representation|squeezing]] transformations.<ref>{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title= Simulating arbitrary Gaussian circuits with linear optics|journal=Physical Review A|date=2018|volume=98|issue=6|page=062314|doi=10.1103/PhysRevA.98.062314|arxiv=1803.11534|bibcode=2018PhRvA..98f2314C|s2cid=119227039}}</ref>