Perfect state distinguishability and computational speedups with postselected closed timelike curves

TL;DR

This paper explores the computational and measurement capabilities of postselected closed timelike curves (P-CTCs), demonstrating their ability to distinguish quantum states and solve complex problems efficiently, with implications for quantum theory and computation.

Contribution

It shows that a single qubit passing through a P-CTC can perform any postselected measurement and can efficiently solve certain computational problems, highlighting differences from other CTC models.

Findings

P-CTCs can distinguish linearly independent quantum states.

P-CTCs cannot distinguish linearly dependent states.

Explicit circuits using P-CTCs can factor integers and solve NP problems efficiently.

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 (P-CTC) is sufficient to do any postselected quantum measurement, and we discuss an important difference between "Deutschian" CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC 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 P-CTCs 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 P-CTCs is thus weaker than that of D-CTCs 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 P-CTCs. 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.