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arXiv:0811.1209 · Physical Review Letters

Localized Closed Timelike Curves Can Perfectly Distinguish Quantum States

Todd A. Brun, Jim Harrington, Mark M. Wilde

Published May 27, 2009• 62 citations
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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.

DOI
10.1103/physrevlett.102.210402

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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.
The causal self-consistency condition for closed timelike curves can give rise to nonlinear interactions on chronology-respecting qubits. We demonstrate that particular unitary interactions between closed timelike curve qubits and chronology-respecting qubits allow perfect distinguishability of nonorthogonal states, and provide a constructive proof for an arbitrary number of nonorthogonal states. This has a number of highly significant consequences. For example, an adversary with access to closed timelike curves can break the B92, BB84, and SARG04 quantum key distribution protocols, or any prepare-and-measure quantum key distribution scheme. Our result also implies that a party with access to closed timelike curves can violate the Holevo bound by accessing more than log(N) bits of information from an N-dimensional quantum state. In principle, he can transmit an arbitrarily large amount of classical information with a quantum system of fixed size. We discuss the implications of this for quantum cloning.
Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which represents a valid quantification of resources given the ability to construct compact regions of closed timelike curves. The notion of self-consistent evolution for quantum computers whose components follow closed timelike curves, as pointed out by Deutsch [Phys. Rev. D 44, 3197 (1991)], implies that the evolution of the chronology respecting components which interact with the closed timelike curve components is nonlinear. We demonstrate that this nonlinearity can be used to efficiently solve computational problems which are generally thought to be intractable. In particular we demonstrate that a quantum computer which has access to closed timelike curve qubits can solve NP\mathit{NP}NP-complete problems with only a polynomial number of quantum gates.
We study the nature of entanglement in presence of closed timelike curves (CTCs) and we observe that qubits traveling along a CTC allow unambiguous discrimination of Bell states with Local Operations and Classical Communications (LOCC), and can be further generalized to allow unambiguous discrimination of any finite set of bipartite entangled states using just LOCC, that is otherwise known to be impossible. This enables one to create entanglement using just LOCC. Furthermore, we point out how the existence of CTCs would imply entanglement can be polygamous, and thus contradicting the prior known notions of monogamy of entanglement.
Quantum bit commitment and Quantum Key Distribution (QKD) are the building blocks of secure quantum cryptographic protocols. Of the numerous QKD schemes, the BB84 protocol was the first one proposed and it is also the most popular. Bennett and Brassard, in their 1984 paper, proposed both quantum bit commitment and the BB84 scheme of QKD. However, their scheme for bit commitment was later shown to be vulnerable to an EPR attack. Also, in a world with Deutschian Closed Timelike Curves (D-CTCs), the BB84 protocol can be broken as the eavesdropper can intercept whatever key the two parties communicate with each other using a circuit which detects non-orthogonal quantum states. Later, the idea of this circuit was extended using Postselected Closed Timelike Curves (P-CTCs), a second variant of CTCs. In this paper, we first propose a secure quantum bit commitment protocol using the Deutschian model of closed timelike curves. Then, we take up the BB84 protocol and propose a secure version of the protocol, transforming it to an entanglement based scheme using the postselected model of closed timelike curves. In short, we show that in the presence of Closed Timelike Curves, unconditionally secure quantum bit commitment and secure quantum key distribution are both possible.
We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTC-assisted computer is presented with a labeled mixture of states to be distinguished--the most natural formulation--we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combination of its output on the mixture's pure components. Similarly, it is not clear that CTC assistance or nonlinear evolution help solve hard problems if computation is defined as we recommend, as correctly evaluating a function on a labeled mixture of orthogonal inputs.
Recently, the quantum information processing power of closed timelike curves has been discussed. Because the most widely accepted model for quantum closed timelike curve interactions contains ambiguities, different authors have been able to reach radically different conclusions as to the power of such interactions. By tracing the information flow through such systems we are able to derive equivalent circuits with unique solutions, thus allowing an objective decision between the alternatives to be made. We conclude that closed timelike curves, if they exist and are well described by these simple models, would be a powerful resource for quantum information processing.