Quantum mechanics of time travel

In 1991, David Deutsch proposed a method to explain how quantum systems interact with closed timelike curves (CTCs) using time evolution equations. This method aims to address paradoxes like the grandfather paradox,{{cite journal

| last1 = Deutsch

| first1 = David

| authorlink1 = David Deutsch

| date = 15 Nov 1991

| title = Quantum mechanics near closed timelike lines

| journal = Physical Review

| volume = 44

| issue = 10

| pages = 3197–3217

| doi = 10.1103/PhysRevD.44.3197

| pmid = 10013776

|bibcode = 1991PhRvD..44.3197D }}

{{Cite journal |last=Lindley |first=David |date=2011-02-04 |title=Time Travel without Regrets |url=https://physics.aps.org/story/v27/st5 |journal=Physics |language=en |volume=27 |issue=4 |pages=5 |doi=10.1103/PhysRevLett.106.040403 |pmid=21405310 |bibcode=2011PhRvL.106d0403L |arxiv=1005.2219 |access-date=2024-07-04 |archive-date=2024-07-16 |archive-url=https://web.archive.org/web/20240716091017/https://physics.aps.org/story/v27/st5 |url-status=live }} which suggests that a time traveler who stops their own birth would create a contradiction. One interpretation of Deutsch's approach is that it allows for self-consistency without necessarily implying the existence of parallel universes.

To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. To describe the combined evolution of both parts over time, he used a unitary operator (U). This approach relies on a specific mathematical framework to describe quantum systems. The overall state is represented by combining the density matrices (ρ) for both parts using a tensor product (⊗).{{Cite web |last=Michael A. Nielsen, Isaac L. Chuang |title=Quantum Computation and Quantum Information |url=https://www.michaelnielsen.org/qcqi/QINFO-book-nielsen-and-chuang-toc-and-chapter1-nov00.pdf |access-date=2024-07-04 |archive-date=2024-04-20 |archive-url=https://web.archive.org/web/20240420221156/https://michaelnielsen.org/qcqi/QINFO-book-nielsen-and-chuang-toc-and-chapter1-nov00.pdf |url-status=live }} While Deutsch's approach does not assume initial correlation between these two parts, this does not inherently break time symmetry.

Deutsch's proposal uses the following key equation to describe the fixed-point density matrix (ρCTC) for the CTC:

$\backslash rho\_\{\backslash text\{CTC\}\}\; =\; \backslash text\{Tr\}\_A\; \backslash left[\; U\; \backslash left(\; \backslash rho\_A\; \backslash otimes\; \backslash rho\_\{\backslash text\{CTC\}\}\; \backslash right)\; U^\backslash dagger\backslash right]$.

The unitary evolution involving both the CTC and the external subsystem determines the density matrix of the CTC as a fixed point, focusing on its state.

Deutsch's proposal ensures that the CTC returns to a self-consistent state after each loop. However, if a system retains memories after traveling through a CTC, it could create scenarios where it appears to have experienced different possible pasts.{{Cite journal |last=Lucas |first=Dunlap |title=The Metaphysics of D-CTCs: On the Underlying Assumptions of Deutsch's Quantum Solution to the Paradoxes of Time Travel |journal= Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|date=2015 |volume=56 |page=39 |doi=10.1016/j.shpsb.2016.09.001 |arxiv=1510.02742 |bibcode=2016SHPMP..56...39D }}

Furthermore, Deutsch's method may not align with common probability calculations in quantum mechanics unless we consider multiple paths leading to the same outcome. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing randomness (nondeterminism). Deutsch suggested using solutions that maximize entropy, aligning with systems' natural tendency to evolve toward higher entropy states.

To calculate the final state outside the CTC, trace operations consider only the external system's state after combining both systems' evolution.

Deutsch's approach has intriguing implications for paradoxes like the grandfather paradox. For instance, if everything except a single qubit travels through a time machine and flips its value according to a specific operator:

:.

Deutsch argues that maximizing von Neumann entropy is relevant in this context. In this scenario, outcomes may mix starting at 0 and ending at 1 or vice versa. While this interpretation can align with many-worlds views of quantum mechanics, it does not necessarily imply branching timelines after interacting with a CTC.{{Cite journal |last=Wallace |first=David |date=2003-09-01 |title=Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation |url=https://www.sciencedirect.com/science/article/pii/S1355219803000364 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |series=Quantum Information and Computation |volume=34 |issue=3 |pages=415–439 |doi=10.1016/S1355-2198(03)00036-4 |arxiv=quant-ph/0303050 |bibcode=2003SHPMP..34..415W |issn=1355-2198 |access-date=2024-07-04 |archive-date=2024-07-16 |archive-url=https://web.archive.org/web/20240716091653/https://www.sciencedirect.com/science/article/abs/pii/S1355219803000364 |url-status=live }}

Researchers have explored Deutsch's ideas further. If feasible, his model might allow computers near a time machine to solve problems beyond classical capabilities; however, debates about CTCs' feasibility continue.{{cite journal

| last1 = Aaronson

| first1 = Scott

| authorlink1 = Scott Aaronson

| last2 = Watrous

| first2 = John

| authorlink2 = John Watrous (computer scientist)

|date=Feb 2009

| title = Closed Timelike Curves Make Quantum and Classical Computing Equivalent

| journal = Proceedings of the Royal Society

| volume = 465

| issue = 2102

| pages = 631–647

| arxiv = 0808.2669

| bibcode = 2009RSPSA.465..631A

| doi = 10.1098/rspa.2008.0350

| s2cid = 745646

}}{{Cite web |last=Billings |first=Lee |title=Time Travel Simulation Resolves "Grandfather Paradox" |url=https://www.scientificamerican.com/article/time-travel-simulation-resolves-grandfather-paradox/ |access-date=2024-07-16 |website=Scientific American |language=en |archive-date=2024-06-23 |archive-url=https://web.archive.org/web/20240623220143/https://www.scientificamerican.com/article/time-travel-simulation-resolves-grandfather-paradox/ |url-status=live }}

Despite its theoretical nature, Deutsch's proposal has faced significant criticism.{{Cite web |date=2015-09-02 |title=A problem with David Deutsch's model of time travel |url=https://conjecturesandrefutations.com/2015/09/02/a-problem-with-david-deutschs-model-of-time-travel/ |access-date=2024-07-16 |website=Conjectures and Refutations |language=en |archive-date=2023-06-04 |archive-url=https://web.archive.org/web/20230604211310/https://conjecturesandrefutations.com/2015/09/02/a-problem-with-david-deutschs-model-of-time-travel/ |url-status=live }} For example, Tolksdorf and Verch demonstrated that quantum systems in spacetimes without CTCs can achieve results similar to Deutsch's criterion with any prescribed accuracy.{{cite journal

| last1 = Tolksdorf

| first1 = Juergen

| authorlink1 =

| last2 = Verch

| first2 = Rainer

| authorlink2 =

|date=2018

| title = Quantum physics, fields and closed timelike curves: The D-CTC condition in quantum field theory

| journal = Communications in Mathematical Physics

| volume = 357

| issue = 1

| series =

| pages = 319–351

| arxiv = 1609.01496

| bibcode =2018CMaPh.357..319T

| doi = 10.1007/s00220-017-2943-5

| s2cid = 55346710

}}{{Cite journal |title=Replicating the benefits of Deutschian closed timelike curves without breaking causality |date=2015 |doi=10.1038/npjqi.2015.7 |url=https://www.nature.com/articles/npjqi20157.pdf |last1=Yuan |first1=Xiao |last2=Assad |first2=Syed M. |last3=Thompson |first3=Jayne |last4=Haw |first4=Jing Yan |last5=Vedral |first5=Vlatko |last6=Ralph |first6=Timothy C. |last7=Lam |first7=Ping Koy |last8=Weedbrook |first8=Christian |last9=Gu |first9=Mile |journal=npj Quantum Information |volume=1 |issue=1 |page=15007 |arxiv=1412.5596 |bibcode=2015npjQI...115007Y |access-date=2024-07-04 |archive-date=2024-07-16 |archive-url=https://web.archive.org/web/20240716090909/https://www.nature.com/articles/npjqi20157.pdf |url-status=live }} This finding challenges claims that quantum simulations of CTCs are related to closed timelike curves as understood in general relativity. Their research also shows that classical systems governed by statistical mechanics could also meet these criteria{{cite journal

| last1 = Tolksdorf

| first1 = Juergen

| authorlink1 =

| last2 = Verch

| first2 = Rainer

| authorlink2 =

|date=2021

| title = The D-CTC condition is generically fulfilled in classical (non-quantum) statistical systems

| journal = Foundations of Physics

| volume = 51

| issue = 93

| series =

| page = 93

| arxiv = 1912.02301

| bibcode =

2021FoPh...51...93T| doi = 10.1007/s10701-021-00496-z

| s2cid = 208637445

}} without invoking peculiarities attributed solely to quantum mechanics. Consequently, they argue that their findings raise doubts about Deutsch's explanation of his time travel scenario using many-worlds interpretations of quantum physics.

Seth Lloyd proposed an alternative approach to time travel with closed timelike curves (CTCs), based on "post-selection" and path integrals.{{cite journal

| last1 = Lloyd

| first1 = Seth

| authorlink1 = Seth Lloyd

| last2 = Maccone

| first2 = Lorenzo

| authorlink2 = Lorenzo Maccone

| last3 = Garcia-Patron

| first3 = Raul

| authorlink3 = Raul Garcia-Patron

| last4 = Giovannetti

| first4 = Vittorio

| authorlink4 = Giovannetti Vittorio

| last5 = Shikano

| first5 = Yutaka

| authorlink5 = Yutaka Shikano

| last6 = Pirandola

| first6 = Stefano

| authorlink6 = Stefano Pirandola

| last7 = Rozema

| first7 = Lee A.

| authorlink7 = Lee A. Rozema

| last8 = Darabi

| first8 = Ardavan

| authorlink8 = Ardavan Darabi

| last9 = Soudagar

| first9 = Yasaman

| authorlink9 = Yasaman Soudagar

| last10 = Shalm

| first10 = Lynden K.

| authorlink10 = Lynden K. Shalm

| last11 = Steinberg

| first11 = Aephraim M.

| authorlink11 = Aephraim M. Steinberg

| date = 27 January 2011

| title = Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency

| journal = Physical Review Letters

| volume = 106

| issue = 4

| article-number = 040403

| doi = 10.1103/PhysRevLett.106.040403

|hdl = 1721.1/63096

| pmid=21405310

| bibcode = 2011PhRvL.106d0403L

| s2cid = 18442086

| url= https://dspace.mit.edu/bitstream/handle/1721.1/63096/Lloyd-2011-Closed%20Timelike%20Curv.pdf

}} Path integrals are a powerful tool in quantum mechanics that involve summing probabilities over all possible ways a system could evolve, including paths that do not strictly follow a single timeline.{{cite journal

| last1 = Lloyd

| first1 = Seth

| authorlink1 = Seth Lloyd

| last2 = Maccone

| first2 = Lorenzo

| authorlink2 = Lorenzo Maccone

| last3 = Garcia-Patron

| first3 = Raul

| authorlink3 = Raul Garcia-Patron

| last4 = Giovannetti

| first4 = Vittorio

| authorlink4 = Giovannetti Vittorio

| last5 = Shikano

| first5 = Yutaka

| authorlink5 = Yutaka Shikano

| year = 2011

| title = The quantum mechanics of time travel through post-selected teleportation

| doi=10.1103/PhysRevD.84.025007

| volume=84

| issue=2

| article-number = 025007

| journal=Physical Review D

| hdl= 1721.1/66971

| s2cid = 15972766

| url= https://dspace.mit.edu/bitstream/handle/1721.1/66971/Lloyd-2011-Quantum%20mechanics%20of%20time%20travel%20through%20post-.pdf

}} Unlike classical approaches, path integrals can accommodate histories involving CTCs, although their application requires careful consideration of quantum mechanics' principles.

He proposes an equation that describes the transformation of the density matrix, which represents the system's state outside the CTC after a time loop:

:$\backslash rho\_f\; =\; \backslash frac\{C\backslash rho\_i\; C^\backslash dagger\}\{\backslash text\{Tr\}\backslash left[\; C\backslash rho\_i\; C^\backslash dagger\backslash right]\}$, where $C\; =\; \backslash text\{Tr\}\_\{\backslash text\{CTC\}\}\backslash left[\; U\; \backslash right]$.

In this equation:

• $\backslash rho\_f$ is the density matrix of the system after interacting with the CTC.
• $\backslash rho\_i$ is the initial density matrix of the system before the time loop.
• $C$ is a transformation operator derived from the trace operation over the CTC, applied to the unitary evolution operator $U$.

The transformation relies on the trace operation, which summarizes aspects of the matrix. If this trace term is zero ($\backslash text\{Tr\}\backslash left[\; C\backslash rho\_i\; C^\backslash dagger\backslash right]=0$), it indicates that the transformation is invalid in that context, but does not directly imply a paradox like the grandfather paradox. Conversely, a non-zero trace suggests a valid transformation leading to a unique solution for the external system's state.

Thus, Lloyd's approach aims to filter out histories that lead to inconsistencies by allowing only those consistent with both initial and final states. This aligns with post-selection, where specific outcomes are considered based on predetermined criteria; however, it does not guarantee that all paradoxical scenarios are eliminated.

Michael Devin (2001) proposed a model that incorporates closed timelike curves (CTCs) into thermodynamics,{{Citation |last=Devin |first=Michael |title=Thermodynamics of Time Machines |date=2013-02-08 |arxiv=1302.3298 }} suggesting it as a potential way to address the grandfather paradox.{{cite thesis |first=Michael |last=Devin |title=Thermodynamics of Time Machines(unpublished) |publisher=University of Arkansas |year=2001}}

{{cite arXiv

| last1 = Devin

| first1 = Michael

| title = Thermodynamics of Time Machines

|eprint = 1302.3298

| class = gr-qc

| year = 2013

}} This model introduces a "noise" factor to account for imperfections in time travel, proposing a framework that could help mitigate paradoxes. In contrast, Carlo Rovelli has argued that thermodynamics inhibits time travel to the past.{{Citation |last=Rovelli |first=Carlo |title=Can we travel to the past? Irreversible physics along closed timelike curves |date=2019-12-11 |arxiv=1912.04702 }}

• Novikov self-consistency principle
• Grandfather paradox
• Causal loop
• Chronology protection conjecture
• Retrocausality

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