'''Continuous-variable quantum information''' is the area of [[quantum information science]] that makes use of [[Observable|physical observables]], like the strength of an [[electromagnetic field]], whose numerical values belong to [[List of continuity-related mathematical topics|continuous]] [[Interval (mathematics)|intervals]].<ref name=":4">{{Cite journal|last=Lloyd|first=Seth|author-link=Seth Lloyd|last2=Braunstein|first2=Samuel L.|author-link2=Samuel L. Braunstein|date=1999-01-01|title=Quantum Computation over Continuous Variables|url=https://link.aps.org/doi/10.1103/PhysRevLett.82.1784|journal=[[Physical Review Letters]]|volume=82|issue=8|pages=1784–1787|arxiv=quant-ph/9810082|doi=10.1103/PhysRevLett.82.1784|bibcode=1999PhRvL..82.1784L}}</ref> One primary application is [[quantum computing]]. In a sense, continuous-variable quantum computation is "analogueanalog", while quantum computation using [[qubit]]s is "digital." In more technical terms, the former makes use of [[Hilbert space]]s that are [[Dimension|infinite-dimensional]], while the Hilbert spaces for systems comprising collections of qubits are finite-dimensional.<ref>{{Cite book|url=|title=Quantum Information with Continuous Variables|last=Braunstein|first=S. L.|last2=Pati|first2=A. K.|date=2012-12-06|publisher=Springer Science & Business Media|year=|isbn=9789401512589|___location=|pages=|language=en|doi=10.1007/978-94-015-1258-9}}</ref><ref>{{Cite journal|last=Braunstein|first=Samuel L.|last2=van Loock|first2=Peter|date=2005-06-29|title=Quantum information with continuous variables|url=https://link.aps.org/doi/10.1103/RevModPhys.77.513|journal=[[Reviews of Modern Physics]]|volume=77|issue=2|pages=513–577|arxiv=quant-ph/0410100|doi=10.1103/RevModPhys.77.513|bibcode=2005RvMP...77..513B}}</ref> One motivation for studying continuous-variable quantum computation is to understand what resources are necessary to make quantum computers more powerful than classical ones.<ref>{{Cite journal|last=Adesso|first=Gerardo|last2=Ragy|first2=Sammy|last3=Lee|first3=Antony R.|date=2014-03-12|title=Continuous Variable Quantum Information: Gaussian States and Beyond|url=http://www.worldscientific.com/doi/abs/10.1142/S1230161214400010|journal=[[Open Systems & Information Dynamics]]|volume=21|issue=1n02|pages=1440001|arxiv=1401.4679|doi=10.1142/S1230161214400010|issn=1230-1612}}</ref>
== Implementation ==
Line 13:
== Classical emulation ==
In all approaches to quantum computing, it is important to know whether a task under consideration can be carried out efficiently by a classical computer. An [[algorithm]] might be described in the language of quantum mechanics, but upon closer analysis, revealed to be implementable using only classical resources. Such an algorithm would not be taking full advantage of the extra possibilities made available by quantum physics. In the theory of quantum computation using finite-dimensional Hilbert spaces, the [[Gottesman–Knill theorem]] demonstrates that there exists a set of quantum processes that can be emulated efficiently on a classical computer. Generalizing this theorem to the continuous-variable case, it can be shown that, likewise, a class of continuous-variable quantum computations can be simulated using only classical analogueanalog computations. This class includes, in fact, some computational tasks that use [[quantum entanglement]].<ref>{{Cite journal|last=Bartlett|first=Stephen D.|last2=Sanders|first2=Barry C.|last3=Braunstein|first3=Samuel L.|last4=Nemoto|first4=Kae|date=2002-02-14|title=Efficient Classical Simulation of Continuous Variable Quantum Information Processes|url=https://link.aps.org/doi/10.1103/PhysRevLett.88.097904|journal=[[Physical Review Letters]]|volume=88|issue=9|pages=097904|arxiv=quant-ph/0109047|doi=10.1103/PhysRevLett.88.097904|bibcode=2002PhRvL..88i7904B}}</ref> When the [[Wigner quasiprobability distribution|Wigner quasiprobability representations]] of all the quantities—states, time evolutions ''and'' measurements—involved in a computation are nonnegative, then they can be interpreted as ordinary probability distributions, indicating that the computation can be modeled as an essentially classical one.<ref name=":3" /> This type of construction can be thought of as a continuum generalization of the [[Spekkens Toy Model]].<ref>{{Cite journal|last=Bartlett|first=Stephen D.|last2=Rudolph|first2=Terry|last3=Spekkens|first3=Robert W.|date=2012-07-10|title=Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction|url=https://link.aps.org/doi/10.1103/PhysRevA.86.012103|journal=[[Physical Review A]]|volume=86|issue=1|pages=012103|arxiv=1111.5057|doi=10.1103/PhysRevA.86.012103|bibcode=2012PhRvA..86a2103B}}</ref>
== Computing continuous functions with discrete quantum systems ==
