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| Until recently, most studies on [[time travel]] are based upon [[classical physics|classical]] [[general relativity]]. Coming up with a quantum version of time travel requires us to figure out the time evolution equations for [[density state]]s in the presence of [[closed timelike curve]]s (CTC).
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| [[Igor Dmitriyevich Novikov|Novikov]]<ref>{{cite journal
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| | last1 = Friedman
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| | first1 = John
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| | authorlink1 = John Fiedman
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| | last2 = Morris
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| | first2 = Michael
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| | authorlink2 = Mike Morris (physicist)
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| | last3 = Novikov
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| | first3 = Igor
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| | authorlink3 = Igor Dmitriyevich Novikov
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| | last4 = Echeverria
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| | first4 = Fernando
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| | authorlink4 = Fernando Echeverria
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| | last5 = Klinkhammer
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| | first5 = Gunnar
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| | authorlink5 = Gunnar Klinkhammer
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| | last6 = Thorne
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| | first6 = Kip
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| | authorlink6 = Kip Thorne
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| | last7 = Yurtsever
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| | first7 = Ulvi
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| | authorlink7 = Ulvi Yurtsever
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| | date = 15 September 1990
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| | title = Cauchy problem in spacetimes with closed timelike curves
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| | journal = Physical Review
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| | volume = 42
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| | issue = 6
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| | series = D
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| | pages = 1915–1930
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| | doi = 10.1103/PhysRevD.42.1915
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| | url = http://link.aps.org/doi/10.1103/PhysRevD.42.1915
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| |bibcode = 1990PhRvD..42.1915F }}
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| </ref> had conjectured that once quantum mechanics is taken into account, [[Novikov self-consistency principle|self-consistent]] solutions always exist for all time machine configurations, and initial conditions. However, it has been noted such solutions are not unique in general, in violation of [[determinism]], [[unitarity]] and [[linearity]].
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| The application of self-consistency to quantum mechanical time machines has taken two main routes. [[Novikov self-consistency principle|Novikov's rule]] applied to the density matrix itself gives the Deutsch prescription. Applied instead to the state vector, the same rule gives nonunitary physics with a dual description in terms of post-selection.
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| == Deutsch's prescription ==
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| In 1991, [[David Deutsch]]<ref>{{cite journal
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| | last1 = Deutsch
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| | first1 = David
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| | authorlink1 = David Deutsch
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| | date = 15 Nov 1991
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| | title = Quantum mechanics near closed timelike lines
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| | journal = Physical Review
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| | volume = 44
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| | issue = 10
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| | series = D
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| | pages = 3197–3217
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| | doi = 10.1103/PhysRevD.44.3197
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| | url = http://link.aps.org/doi/10.1103/PhysRevD.44.3197
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| |bibcode = 1991PhRvD..44.3197D }}
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| </ref> came up with a proposal for the time evolution equations, with special note as to how it resolves the [[grandfather paradox]] and nondeterminism. However, his resolution to the grandfather paradox is considered unsatisfactory to some people, because it states the time traveller reenters another [[Many-worlds interpretation|parallel universe]], and that the actual quantum state is a [[quantum superposition]] of states where the time traveller does and does not exist. | |
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| He made the simplifying assumption that we can split the quantum system into a subsystem A external to the closed timelike curve, and a CTC part. Also, he assumed that we can combine all the time evolution between the exterior and the CTC into a single [[unitary operator]] ''U''. This presupposes the [[Schrödinger picture]]. We have a [[tensor product]] for the combined state of both systems. He makes the further assumption there is no correlation between the initial density state of A and the density state of the CTC. This assumption is not time-symmetric, which he tried to justify by appealing to measurement theory and the second law of thermodynamics. He proposed that the density state restricted to the CTC is a fixed-point of
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| :<math>\rho_{\text{CTC}} = \text{Tr}_A \left[ U \left( \rho_A \otimes \rho_{\text{CTC}} \right) U^\dagger\right]</math>.
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| He showed that such [[Renormalization group|fixed points]] always exist. He justified this choice by noting the [[expectation value]] of any CTC observable will match after a loop. However, this could lead to "multivalued" histories if memory is preserved around the loop. In particular, his prescription is incompatible with [[path integral formulation|path integrals]] unless we allow for multivalued fields. Another point to note is in general, we have more than one fixed point, and this leads to [[Indeterminism|nondeterminism]] in the time evolution. He suggested the solution to use is the one with the [[maximum entropy principle|maximum entropy]]. The final external state is given by <math>\text{Tr}_{\text{CTC}} \left[ U \left( \rho_A \otimes \rho_{\text{CTC}} \right) U^\dagger\right]</math>. Pure states can evolve into mixed states.
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| This leads to seemingly paradoxical resolutions to the grandfather paradox. Assume the external subsystem is irrelevant, and only a qubit travels in the CTC. Also assume during the course around the time machine, the value of the qubit is flipped according to the unitary operator
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| :<math>U = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}</math>. | |
| The most general fixed-point solution is given by
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| :<math>\rho_{\text{CTC}} = \begin{pmatrix}\frac{1}{2} & a\\a & \frac{1}{2}\end{pmatrix}</math> | |
| where ''a'' is a real number between <math>-1/2</math> and <math>1/2</math>. This is an example of the nonuniqueness of solutions. The solution maximizing the [[von Neumann entropy]] is given by <math>a=0</math>. We can think of this as a mixture (not superposition) between the states <math>\left( \left| 0 \right\rangle + \left| 1 \right\rangle \right)/\sqrt{2}</math> and <math>\left( \left| 0 \right\rangle - \left| 1 \right\rangle \right)/\sqrt{2}</math>. This leads to an interesting interpretation that if the qubit starts off with a value of 0, it will end up with a value of 1, and vice versa, but this should not be problematic according to Deutsch because the qubit ends up in a different parallel universe in the [[many worlds interpretation]].
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| Later researchers have noted that if his prescription turned out to be right, computers in the vicinity of a time machine can solve [[PSPACE-complete]] problems.<ref>{{cite journal
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| | last1 = Aaronson
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| | first1 = Scott
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| | authorlink1 = Scott Aaronson
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| | last2 = Watrous
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| | first2 = John
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| | authorlink2 = John Watrous (computer scientist)
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| |date=Feb 2009
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| | title = Closed Timelike Curves Make Quantum and Classical Computing Equivalent
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| | journal = Proceedings of the Royal Society
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| | volume = 465
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| | issue = 2102
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| | series = A
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| | pages = 631–647
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| | doi = 10.1098/rspa.2008.0350
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| | url = http://rspa.royalsocietypublishing.org/content/465/2102/631
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| | archiveurl = http://arxiv.org/abs/0808.2669
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| | archivedate = 19 Aug 2008
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| |bibcode = 2009RSPSA.465..631A |arxiv = 0808.2669 }}
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| </ref>
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| == Lloyd's prescription ==
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| An alternative proposal was later presented by [[Seth Lloyd]]<ref>{{cite journal
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| | last1 = Lloyd
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| | first1 = Seth
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| | authorlink1 = Seth Lloyd
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| | last2 = Maccone
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| | first2 = Lorenzo
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| | authorlink2 = Lorenzo Maccone
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| | last3 = Garcia-Patron
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| | first3 = Raul
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| | authorlink3 = Raul Garcia-Patron
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| | last4 = Giovannetti
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| | first4 = Vittorio
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| | authorlink4 = Giovannetti Vittorio
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| | last5 = Shikano
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| | first5 = Yutaka
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| | authorlink5 = Yutaka Shikano
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| | last6 = Pirandola
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| | first6 = Stefano
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| | authorlink6 = Stefano Pirandola
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| | last7 = Rozema
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| | first7 = Lee A.
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| | authorlink7 = Lee A. Rozema
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| | last8 = Darabi
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| | first8 = Ardavan
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| | authorlink8 = Ardavan Darabi
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| | last9 = Soudagar
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| | first9 = Yasaman
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| | authorlink9 = Yasaman Soudagar
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| | last10 = Shalm
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| | first10 = Lynden K.
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| | authorlink10 = Lynden K. Shalm
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| | last11 = Steinberg
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| | first11 = Aephraim M.
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| | authorlink11 = Aephraim M. Steinberg
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| | date = 27 January 2011
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| | title = Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency
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| | journal = Physical Review Letters
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| | volume = 106
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| | issue = 4
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| | pages = 040403
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| | doi = 10.1103/PhysRevLett.106.040403
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| | url = http://prl.aps.org/abstract/PRL/v106/i4/e040403
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| | archiveurl = http://arxiv.org/abs/1005.2219
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| | archivedate = 12 May 2010
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| |bibcode = 2011PhRvL.106d0403L |arxiv = 1005.2219 }}</ref><ref>{{cite journal
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| | last1 = Lloyd
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| | first1 = Seth
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| | authorlink1 = Seth Lloyd
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| | last2 = Maccone
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| | first2 = Lorenzo
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| | authorlink2 = Lorenzo Maccone
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| | last3 = Garcia-Patron
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| | first3 = Raul
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| | authorlink3 = Raul Garcia-Patron
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| | last4 = Giovannetti
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| | first4 = Vittorio
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| | authorlink4 = Giovannetti Vittorio
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| | last5 = Shikano
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| | first5 = Yutaka
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| | authorlink5 = Yutaka Shikano
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| | date = 19 July 2010
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| | title = The quantum mechanics of time travel through post-selected teleportation
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| | arxiv = 1007.2615
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| |bibcode = 2010arXiv1007.2615L }}
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| </ref> based upon [[post-selection]] and path integrals. In particular, the path integral is over single-valued fields, leading to self-consistent histories. He assumed it is ill-defined to speak of the actual density state of the CTC itself, and we should only focus upon the density state outside the CTC. His proposal for the time evolution of the external density state is
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| :<math>\rho_f = \frac{C\rho_i C^\dagger}{\text{Tr}\left[ C\rho_i C^\dagger\right]}</math>, where <math>C = \text{Tr}_{\text{CTC}}\left[ U \right]</math>.
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| If <math>\text{Tr}\left[ C\rho_i C^\dagger\right]=0</math>, no solution exists due to [[destructive interference]] in the path integral. For instance, the grandfather paradox has no solution, and leads to an inconsistent state. If a solution exists, it is clearly unique. Now, [[quantum computer]]s using time machines can only solve [[PP-complete]] problems.
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| == Entropy and computation == | |
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| The same description of CTC physics was derived independently in 2001 by Michael Devin, and applied to thermodynamics.<ref>{{cite thesis |first=Michael |last=Devin |title=Thermodynamics of Time Machines(unpublished) |publisher=University of Arkansas |year=2001}}
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| </ref><ref>{{cite journal
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| | last1 = Devin
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| | first1 = Michael
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| | title = Thermodynamics of Time Machines
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| |arxiv = 1302.3298 }}</ref> The same model with the introduction of a noise term allowing for inexact periodicity, allows the grandfather paradox to be resolved, and clarifies the computational power a time machine assisted computer. Each time traveling qubit has an associated negentropy, given approximately by the logarithm of the noise of the communication channel. Each use of the time machine can be used to extract as much work from a thermal bath. In a brute force search for a randomly generated password, the entropy of the unknown string can be effectively reduced by a similar amount. Because the negentropy and computational power diverge as the noise term goes to zero, complexity class may not be the best way to describe the capabilities of time machines.
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| == Nonlinear time evolution ==
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| [[Charles H. Bennett (computer scientist)|Bennett]]<ref>{{cite journal
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| | last1 = Charles
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| | first1 = Bennett
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| | authorlink1 = Charles H. Bennett (computer scientist)
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| | last2 = Leung
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| | first2 = Debbie
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| | authorlink2 = Debbie Leung
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| | last3 = Smith
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| | first3 = Graeme
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| | authorlink3 = Graeme Smith
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| | last4 = Smolin
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| | first4 = John
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| | authorlink4 = John A. Smolin
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| | date = 21 Oct 2009
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| | title = Can Closed Timelike Curves or Nonlinear Quantum Mechanics Improve Quantum State Discrimination or Help Solve Hard Problems?
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| | journal = Physical Review Letters
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| | volume = 103
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| | issue = 17
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| | pages = 2009
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| | doi = 10.1103/PhysRevLett.103.170502
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| | url = http://link.aps.org/doi/10.1103/PhysRevLett.103.170502
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| | archiveurl = http://arxiv.org/abs/0908.3023
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| | archivedate = 15 Oct 2009
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| | bibcode=2009PhRvL.103q0502B
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| |arxiv = 0908.3023 }}
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| </ref> et al. have pointed some of the problems which arise when unitarity breaks down and quantum evolution becomes nonlinear.
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| == See also ==
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| * [[Novikov self-consistency principle]]
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| * [[grandfather paradox]]
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| * [[ontological paradox]]
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| == References ==
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| {{reflist}}
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| {{Time travel}}
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| {{DEFAULTSORT:Quantum Mechanics Of Time Travel}}
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| [[Category:Time travel]]
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum gravity]]
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