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Simulations of Closed Timelike Curves

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Proposed models of closed timelike curves (CTCs) have been shown to enable powerful information-processing protocols. We examine the simulation of models of CTCs both by other models of CTCs and by physical systems without access to CTCs. We prove that the recently proposed transition probability CTCs (T-CTCs) are physically equivalent to postselection CTCs (P-CTCs), in the sense that one model can simulate the other with reasonable overhead. As a consequence, their information-processing capabilities are equivalent. We also describe a method for quantum computers to simulate Deutschian CTCs (but with a reasonable overhead only in some cases). In cases for which the overhead is reasonable, it might be possible to perform the simulation in a table-top experiment. This approach has the benefit of resolving some ambiguities associated with the equivalent circuit model of Ralph et al. Furthermore, we provide an explicit form for the state of the CTC system such that it is a maximum-entropy state, as prescribed by Deutsch.

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  1. We note that (3.15) demonstrates that the vector \(\sqrt{p_{0}}\left| 0\right\rangle _{A} \otimes \vert \psi _{0}\rangle _{RSCC^{\prime }}\) and the operator \(\text {Tr}_{C_{1}\ldots C_{n}}\{ V_{AC_{1}\ldots C_{n}}\}\) saturate the bound given in [11, Eq. (10)]. That is, to saturate [11, Eq. (10)], we can therein set \(P = \text {Tr}_{C_{1}\ldots C_{n}}\{ V_{AC_{1}\ldots C_{n}}\}\), \(\vert \psi \rangle = \sqrt{p_{0}}\left| 0\right\rangle _{A} \otimes \vert \psi _{0}\rangle _{RSCC^{\prime }}\), the chronology-respecting systems to be \(ARSCC^{\prime }\), and the chronology-violating systems to be \(C_{1}\ldots C_{n}\). We thank John-Mark Allen for pointing this out to us.


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We are especially grateful to Tom Cooney for many enlightening discussions on fixed points of CPTP linear maps. We thank John-Mark Allen for helpful feedback that improved the manuscript. We also acknowledge Jonathan Dowling for his help in obtaining FQXI funds to support this research. Finally, we acknowledge support from the Department of Physics and Astronomy at Louisiana State University and the Foundational Questions Institute (FQXI) for supporting the grant “Closed timelike curves and quantum information processing.”

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Correspondence to Mark M. Wilde.

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Brun, T.A., Wilde, M.M. Simulations of Closed Timelike Curves. Found Phys 47, 375–391 (2017).

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