Until recently, most studies on time travel are based upon classical general relativity. Coming up with a quantum version of time travel requires us to figure out the time evolution equations for density states in the presence of closed timelike curves (CTC).
Novikov[1] had conjectured that once quantum mechanics is taken into account, 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.
The application of self-consistency to quantum mechanical time machines has taken two main routes. 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.
Deutsch's prescription
In 1991, David Deutsch[2] 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 parallel universe, and that the actual quantum state is a quantum superposition of states where the time traveller does and does not exist.
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
\( \rho_{\text{CTC}} = \text{Tr}_A \left[ U \left( \rho_A \otimes \rho_{\text{CTC}} \right) U^\dagger\right]. \)
He showed that such 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 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 nondeterminism in the time evolution. He suggested the solution to use is the one with the maximum entropy. The final external state is given by \( \text{Tr}_{\text{CTC}} \left[ U \left( \rho_A \otimes \rho_{\text{CTC}} \right) U^\dagger\right] \). Pure states can evolve into mixed states.
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
\( U = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}. \)
The most general fixed-point solution is given by
\( \rho_{\text{CTC}} = \begin{pmatrix}\frac{1}{2} & a\\a & \frac{1}{2}\end{pmatrix} \)
where a is a real number between -1/2 and 1/2. This is an example of the nonuniqueness of solutions. The solution maximizing the von Neumann entropy is given by a=0. We can think of this as a mixture (not superposition) between the states \( \left( \left| 0 \right\rangle + \left| 1 \right\rangle \right)/\sqrt{2} \) and \) \left( \left| 0 \right\rangle - \left| 1 \right\rangle \right)/\sqrt{2} \). 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.
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.[3]
However, it was shown in an article by Tolksdorf and Verch that Deutsch's CTC fixed point condition can be fulfilled to arbitrary precision in any quantum system described according to relativistic quantum field theory on spacetimes where CTCs are excluded, casting doubts on whether Deutsch's condition is really characteristic of quantum processes mimicking CTCs in the sense of general relativity.[4]
Lloyd's prescription
An alternative proposal was later presented by Seth Lloyd[5][6] 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
\( \rho_f = \frac{C\rho_i C^\dagger}{\text{Tr}\left[ C\rho_i C^\dagger\right]} \), where \( C = \text{Tr}_{\text{CTC}}\left[ U \right]. \)
If \( \text{Tr}\left[ C\rho_i C^\dagger\right]=0 \), 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 computers using time machines can only solve PP-complete problems.
Entropy and computation
A related description of CTC physics was given in 2001 by Michael Devin, and applied to thermodynamics.[7][8] 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 of 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.
See also
Novikov self-consistency principle
Grandfather paradox
Ontological paradox
References
Friedman, John; Morris, Michael; Novikov, Igor; Echeverria, Fernando; Klinkhammer, Gunnar; Thorne, Kip; Yurtsever, Ulvi (15 September 1990). "Cauchy problem in spacetimes with closed timelike curves" (PDF). Physical Review. D. 42 (6): 1915–1930. Bibcode:1990PhRvD..42.1915F. doi:10.1103/PhysRevD.42.1915. PMID 10013039.
Deutsch, David (15 Nov 1991). "Quantum mechanics near closed timelike lines". Physical Review. D. 44 (10): 3197–3217. Bibcode:1991PhRvD..44.3197D. doi:10.1103/PhysRevD.44.3197. PMID 10013776.
Aaronson, Scott; Watrous, John (Feb 2009). "Closed Timelike Curves Make Quantum and Classical Computing Equivalent". Proceedings of the Royal Society. A. 465 (2102): 631–647.arXiv:0808.2669. Bibcode:2009RSPSA.465..631A. doi:10.1098/rspa.2008.0350.
Tolksdorf, Juergen; Verch, Rainer (2018). "Quantum physics, fields and closed timelike curves: The D-CTC condition in quantum field theory". Communications in Mathematical Physics,. 357 (1): 319–351.arXiv:1609.01496. Bibcode:2018CMaPh.357..319T. doi:10.1007/s00220-017-2943-5.
Lloyd, Seth; Maccone, Lorenzo; Garcia-Patron, Raul; Giovannetti, Vittorio; Shikano, Yutaka; Pirandola, Stefano; Rozema, Lee A.; Darabi, Ardavan; Soudagar, Yasaman; Shalm, Lynden K.; Steinberg, Aephraim M. (27 January 2011). "Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency". Physical Review Letters. 106 (4): 040403.arXiv:1005.2219. Bibcode:2011PhRvL.106d0403L. doi:10.1103/PhysRevLett.106.040403. PMID 21405310.
Lloyd, Seth; Maccone, Lorenzo; Garcia-Patron, Raul; Giovannetti, Vittorio; Shikano, Yutaka (2011). "The quantum mechanics of time travel through post-selected teleportation". Physical Review D. 84 (2): 025007.arXiv:1007.2615. Bibcode:2011PhRvD..84b5007L. doi:10.1103/PhysRevD.84.025007.
Devin, Michael (2001). Thermodynamics of Time Machines(unpublished) (Thesis). University of Arkansas.
Devin, Michael (2013). "Thermodynamics of Time Machines".arXiv:1302.3298 [gr-qc].
vte
Quantum mechanics
Background
Introduction History
timeline Glossary Classical mechanics Old quantum theory
Fundamentals
Bra–ket notation Casimir effect Coherence Coherent control Complementarity Density matrix Energy level
degenerate levels excited state ground state QED vacuum QCD vacuum Vacuum state Zero-point energy Hamiltonian Heisenberg uncertainty principle Pauli exclusion principle Measurement Observable Operator Probability distribution Quantum Qubit Qutrit Scattering theory Spin Spontaneous parametric down-conversion Symmetry Symmetry breaking
Spontaneous symmetry breaking No-go theorem No-cloning theorem Von Neumann entropy Wave interference Wave function
collapse Universal wavefunction Wave–particle duality
Matter wave Wave propagation Virtual particle
Quantum
quantum coherence annealing decoherence entanglement fluctuation foam levitation noise nonlocality number realm state superposition system tunnelling Quantum vacuum state
Mathematics
Equations
Dirac Klein–Gordon Pauli Rydberg Schrödinger
Formulations
Heisenberg Interaction Matrix mechanics Path integral formulation Phase space Schrödinger
Other
Quantum
algebra calculus
differential stochastic geometry group Q-analog
List
Interpretations
Bayesian Consistent histories Cosmological Copenhagen de Broglie–Bohm Ensemble Hidden variables Many worlds Objective collapse Quantum logic Relational Stochastic Transactional
Experiments
Afshar Bell's inequality Cold Atom Laboratory Davisson–Germer Delayed-choice quantum eraser Double-slit Elitzur–Vaidman Franck–Hertz experiment Leggett–Garg inequality Mach-Zehnder inter. Popper Quantum eraser Quantum suicide and immortality Schrödinger's cat Stern–Gerlach Wheeler's delayed choice
Science
Measurement problem QBism
Quantum
biology chemistry chaos cognition complexity theory computing
Timeline cosmology dynamics economics finance foundations game theory information nanoscience metrology mind optics probability social science spacetime
Technologies
Quantum technology
links Matrix isolation Phase qubit Quantum dot
cellular automaton display laser single-photon source solar cell Quantum well
laser
Extensions
Dirac sea Fractional quantum mechanics Quantum electrodynamics
links Quantum geometry Quantum field theory
links Quantum gravity
links Quantum information science
links Quantum statistical mechanics Relativistic quantum mechanics De Broglie–Bohm theory Stochastic electrodynamics
Related
Quantum mechanics of time travel Textbooks
Category Category Portal Physics Portal WikiProject Physics WikiProject Commons page Commons
vte
Time travel
General terms and concepts
Chronology protection conjecture Closed timelike curve Novikov self-consistency principle Self-fulfilling prophecy Quantum mechanics of time travel
Time travel in fiction
Timelines in fiction
in science fiction in games
Temporal paradoxes
Grandfather paradox Causal loop
Parallel timelines
Alternative future Alternate history Many-worlds interpretation Multiverse Parallel universes in fiction
Philosophy of space and time
Butterfly effect Determinism Eternalism Fatalism Free will Predestination
Spacetimes in general relativity that
can contain closed timelike curves
Alcubierre metric BTZ black hole Gödel metric Kerr metric Krasnikov tube Misner space Tipler cylinder van Stockum dust Traversable wormholes
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
