Localized Closed Timelike Curves Can Perfectly Distinguish Quantum States

Abstract

We show that qubits traveling along closed timelike curves are a resource that a party can exploit to distinguish perfectly any set of quantum states. As a result, an adversary with access to closed timelike curves can break any prepare-and-measure quantum key distribution protocol. Our result also implies that a party with access to closed timelike curves can violate the Holevo bound.PACS numbers: 03.65. Wj, 03.67.Dd, 03.67.Hk, 04.20.Gz Introduction-The theory of general relativity points to the possible existence of closed timelike curves (CTCs) [1,2]. The grandfather paradox is one criticism raised to their existence, but Deutsch resolved this paradox by presenting a method for finding self-consistent solutions of CTC interactions [3].Recently, several quantum information researchers have assumed that CTCs exist and have examined the consequences of this assumption for computation [4,5,6]. Brun showed that a classical treatment (assuming a lack of contradictions) allows NP-hard problems to be computed with a polynomial number of gates [4]. Bacon followed with a purely quantum treatment that demonstrates the same reduction of NP-hard problems to P, along with a sketch of how to perform this reduction in a fault-tolerant manner [5]. Aaronson and Watrous have recently established that either classical or quantum computers interacting with closed timelike curves can compute any function in PSPACE in polynomial time [6].In this Letter, we show how a party with access to CTCs, or a "CTC-assisted" party, can perfectly distinguish among a set of non-orthogonal quantum states. The result has implications for fundamental protocols in quantum communication because a simple corollary is that a CTC-assisted party can break any prepareand-measure quantum key distribution protocol [7,8,9]. (The security of such a scheme relies on the informationdisturbance tradeoff for identifying quantum states.) Furthermore, the capacity for quantum systems to carry classical information becomes unbounded.Our work here raises fundamental questions concerning the nature of a physical world in which closed timelike curves exist because it challenges the postulate of quantum mechanics that non-orthogonal states cannot be perfectly distinguished. A full theory of quantum gravity would have to resolve this apparent contradiction between the implication of CTCs and the laws of quantum mechanics. Note that any alternative source of nonlinearity would raise similar questions.We structure this Letter as follows. First, we give some background on Deutsch's formalism regarding CTCs in

Replicating the benefits of Deutschian closed timelike curves without breaking causality

“…8 More generally, our results imply that any non-linear quantum map that makes use of CTCs can also be synthesized using only open timelike curves. This follows from the reasoning of Brun et al, 6 which argued that if any two quantum states can be distinguished, then one can effectively implement any map between two quantum states. Related to this is the question of whether OTC enhanced computation has equal computational power to their CTC counterparts.…”

“…In harnessing these open timelike curves, we developed methods to efficiently solve NP-complete problems, and clone unknown quantum states to arbitrary fidelity. Other radical effects commonly attributed to CTCs come as natural consequences, such as distinguishing non-orthogonal states, 6 the capacity to communicate more than a bit of information in a single qubit and the breaking of non-entanglement based cryptographic schemes. 8 More generally, our results imply that any non-linear quantum map that makes use of CTCs can also be synthesized using only open timelike curves.…”

“…6,7,[9][10][11] Note that since the effects of CTCs are non-linear, many conventional assumptions of linear quantum mechanics break. One consequence is that different unravellings of a density operator give different predictions, and therefore must be treated very carefully.…”

“…In cases where the input state is deterministically prepared in a pure state, shot-by-shot, but the particular state may change shotby-shot, there remains some ambiguity. The conventional view is taken by Bacon, 10 Brun et al 6 and Pienaar et al 13 -that subensembles of identical input states should be treated as pure. Others argue that if this sequence is truly random, then the randomness must always be treated as if it arose from the purification of an entangled quantum state.…”

“…Non-orthogonal quantum states can be perfectly distinguished, the uncertainty principle can be violated, and arbitrary unknown quantum states can be cloned to any fixed fidelity. [6][7][8] In harnessing these effects, many problems thought to be intractable for standard quantum computers now field efficient solutions. [9][10][11][12] Though radical, these effects seem somewhat rationalized in the context of requiring broken causality-the sentiment being that they are curiosities that will vanish once causality is imposed.…”