In this paper, we show that a result precisely analogous to the traditional quantum no-cloning theorem holds in classical mechanics. This classical no-cloning theorem does not prohibit classical cloning, we argue, because it is based on a too-restrictive definition of cloning. Using a less popular, more inclusive definition of cloning, we give examples of classical cloning processes. We also prove that a cloning machine must be at least as complicated as the object it is supposed to clone.
REFERENCES
1.
Abramsky
, S.
, “No-Cloning in categorical quantum mechanics
,” in Semantic Techniques in Quantum Computation
edited by Gay
, S.
and Mackie
, I.
(Cambridge University Press
, Cambridge
, 2010
), pp. 1
–28
.2.
Baez
, J.
, “Quantum quandaries: a category-theoretic perspective
,” in The Structural Foundations of Quantum Gravity
, edited by Rickles
, D.
, French
, S.
, and Saatsi
, J. T.
(Oxford University Press
, Oxford
, 2006
), pp. 240
–266
.3.
Baez
, J.
and Hoffnung
, A.
, Classical mechanics: The Hamiltonian approach, (2008
), see http://math.ucr.edu/home/baez/classical/#hamiltonian.4.
Coecke
, B.
, “Introducing categories to the practicing physicist
,” in What is Category Theory?
edited by Sica
, G.
(Polimetrica
, Monza, Italy
, 2006
), 45
–74
.5.
da Silva
, A. C.
, Lectures on Symplectic Geometry
(Springer-Verlag
, Berlin, Heidelberg
, 2008
).6.
Daffertshofer
, A.
, Plastino
, A. R.
, and Plastino
, A.
, Phys. Rev. Lett.
88
(21
), 210601
(2002
).7.
Pati
, A. K.
and Braunstein
, S. L.
, “Can arbitrary quantum systems undergo self-replication
?” in Quantum Aspects of Life
, edited by Abbott
, D.
, Davies
, P. C. W.
, and Pati
, A. K.
(Imperial College Press
, Covent Garden, London
, 2008
), pp. 223
–231
.8.
Scarani
, V.
, Iblisdir
, S.
, Gisin
, N.
, and Acin
, A.
, Rev. Mod. Phys.
77
(4
), 1225
–1256
(2005
).9.
Walker
, T. A.
, “Relationships between Quantum and Classical Information,” Ph.D. dissertation, University of York
, 2008
.10.
This proposition remains true even if U is only required to be a linear isometry (see Definition 5).
11.
The analogy will be made precise in Sec. IV A.
12.
This proposition remains true even if ϕ is only required to be a symplectic map (see Definition 6).
13.
There appear to be two definitions here, but they are both special cases of a single, more general definition, as explained in Sec. IV A.
14.
The propositions that depend on this definition remain true even if U is only required to be a linear isometry (see Definition 5).
15.
The propositions that depend on this definition remain true even if ϕ is only required to be a symplectic map (see Definition 6).
16.
The propositions that depend on this definition remain true even if U is only required to be a linear isometry (see Definition 5).
17.
The propositions that depend on this definition remain true even if ϕ is only required to be a symplectic map (see Definition 6).
18.
When open systems are taken into consideration, some transformations have to be represented by non-unitary maps, leading to a more complicated categorical setting. One possible choice is the category whose objects and arrows are complex Hilbert spaces and bounded linear maps, discussed in Ref. 2.
© 2012 American Institute of Physics.
2012
American Institute of Physics
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