Perfect State Distinguishability and Computational Speedups with Postselected Closed Timelike Curves
Abstract
Bennett and Schumacher's postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch's model. We show that even a single qubit passing through a postselected CTC (PCTC) is sufficient to do any postselected quantum measurement with certainty, and we discuss an important difference between "Deutschian" CTCs (DCTCs) and PCTCs in which the future existence of a PCTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett and Smith, we explicitly show how a party assisted by PCTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of PCTCs is thus weaker than that of DCTCs because the Holevo bound still applies to circuits using them, regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving PCTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers.
 Publication:

Foundations of Physics
 Pub Date:
 March 2012
 DOI:
 10.1007/s1070101196010
 arXiv:
 arXiv:1008.0433
 Bibcode:
 2012FoPh...42..341B
 Keywords:

 Postselected closed timelike curves;
 State distinguishability;
 Paradoxical computation;
 Quantum Physics;
 General Relativity and Quantum Cosmology
 EPrint:
 15 pages, 4 figures