Editing Quantum information (section)

==Qubits and information theory==
Quantum information differs strongly from classical information, epitomized by the [[bit]], in many striking and unfamiliar ways. While the fundamental unit of classical information is the [[bit]], the most basic unit of quantum information is the [[qubit]]. Classical information is measured using [[Entropy (information theory)|Shannon entropy]], while the quantum mechanical analogue is [[Von Neumann entropy]]. Given a [[statistical ensemble]] of quantum mechanical systems with the [[density matrix]] <math>\rho</math>, it is given by <math> S(\rho) = -\operatorname{Tr}(\rho \ln \rho).</math><ref name="Nielsen2010" /> Many of the same entropy measures in classical [[information theory]] can also be generalized to the quantum case, such as Holevo entropy<ref>{{cite web|url=http://www.mi.ras.ru/~holevo/eindex.html|title=Alexandr S. Holevo|website=Mi.ras.ru|access-date=4 December 2018}}</ref> and the [[conditional quantum entropy]].

Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the [[Bloch sphere]]. Despite being continuously valued in this way, a qubit is the ''smallest'' possible unit of quantum information, and despite the qubit state being continuous-valued, it is [[EPR paradox|impossible]] to [[quantum measurement|measure]] the value precisely. Five famous theorems describe the limits on manipulation of quantum information.<ref name="Nielsen2010" /> 
# [[no-teleportation theorem]], which states that a qubit cannot be (wholly) converted into classical bits; that is, it cannot be fully "read". 
# [[no-cloning theorem]], which prevents an arbitrary qubit from being copied.
# [[no-deleting theorem]], which prevents an arbitrary qubit from being deleted. 
# [[no-broadcast theorem]], which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (''e.g.'' via [[quantum teleportation]]).
# [[no-hiding theorem]], which demonstrates the conservation of quantum information.

These theorems are proven from [[Unitarity (physics)|unitarity]], which according to [[Leonard Susskind]] is the technical term for the statement that quantum information within the universe is conserved.{{r|Susskind2014|p=94|quote=The minus first law says that information is never lost. If two identical isolated systems start out in different states, they stay in different states. Moreover, in the past they were also in different states. On the other hand, if two identical systems are in the same state at some point in time, then their histories and their future evolutions must also be identical. Distinctions are conserved. The quantum version of the minus first law has a name — unitarity.}} The five theorems open possibilities in quantum information processing.