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arXiv:1206.4836 · Physical Review A

Deduction of an upper bound on the success probability of port-based teleportation from the no-cloning theorem and the no-signaling principle

Damián Pitalúa-García

Published Apr 12, 2013• 7 citations
arXivHTMLPDF

Abstract

In port-based teleportation, Alice teleports an unknown quantum state $|\ensuremath{\psi}\ensuremath{\rangle}$ to one of $N$ ports at Bob's site. Alice applies a measurement and sends Bob the outcome $k$. Bob only needs to select the $k$th port in order to obtain $|\ensuremath{\psi}\ensuremath{\rangle}$. We present a theorem in the spirit of the no-cloning theorem, which says that it is impossible to extract any information from an unknown quantum state if only a single copy of it is provided and if the state remains unchanged. We use this theorem and the no-signaling principle to prove an upper bound on the success probability of port-based teleportation.

DOI
10.1103/physreva.87.040303

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In port-based teleportation, Alice teleports an unknown quantum state ∣\ensuremathψ\ensuremath⟩|\ensuremath{\psi}\ensuremath{\rangle}∣\ensuremathψ\ensuremath⟩ to one of NNN ports at Bob's site. Alice applies a measurement and sends Bob the outcome kkk. Bob only needs to select the kkkth port in order to obtain ∣\ensuremathψ\ensuremath⟩|\ensuremath{\psi}\ensuremath{\rangle}∣\ensuremathψ\ensuremath⟩. We present a theorem in the spirit of the no-cloning theorem, which says that it is impossible to extract any information from an unknown quantum state if only a single copy of it is provided and if the state remains unchanged. We use this theorem and the no-signaling principle to prove an upper bound on the success probability of port-based teleportation.
The scheme of quantum teleportation, where Bob has multiple (N)(N)(N) output ports and obtains the teleported state by simply selecting one of the NNN ports, is thoroughly studied. We consider both the deterministic version and probabilistic version of the teleportation scheme aiming to teleport an unknown state of a qubit. Moreover, we consider two cases for each version: (i) the state employed for the teleportation is fixed to a maximally entangled state and (ii) the state is also optimized as well as Alice's measurement. We analytically determine the optimal protocols for all the four cases and show the corresponding optimal fidelity or optimal success probability. All these protocols can achieve the perfect teleportation in the asymptotic limit of $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The entanglement properties of the teleportation scheme are also discussed.
This PhD thesis contains a general introduction and three main chapters. Chapter 2 investigates Bell inequalities that generalize the CHSH and Braunstein-Caves inequalities. Chapter 3 shows a derivation of an upper bound on the success probability of a class of quantum teleportation protocols, denoted as port-based teleportation, from the no-cloning theorem and the no-signalling principle. Chapter 4 introduces the principle of quantum information causality. Chapter 2 considers the predictions of quantum theory and local hidden variable theories (LHVT) for the correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by a given angle θ. The predictions of LHVT correspond to binary colourings of the Bloch sphere with antipodal points oppositely coloured. We show a Bell inequality for all θ, which generalizes the CHSH and the Braunstein-Caves inequalities in the sense that the measurement choices are not restricted to be in a finite set, but are constrained only by the angle θ. We motivate and explore the hypothesis that for a continuous range of θ > 0, the maximum correlation (anticorrelation) is obtained by assigning to one qubit the colouring with one hemisphere black and the other white, and assigning the same (reverse) colouring to the other qubit. We describe numerical tests that are consistent with this hypothesis and bound the range of θ. Chapter 3 shows a derivation of an upper bound on the success probability of port-based teleportation from the no-cloning theorem and the no-signalling principle. Chapter 4 introduces the principle of quantum information causality, a quantum version of the information causality principle. The quantum information causality principle states the maximum amount of quantum information that a transmitted quantum system can communicate as a function of its dimension, independently of any quantum physical resources previously shared by the communicating parties. These principles reduce to the no-signalling principle if no systems are transmitted. We present a new quantum information task, the quantum information causality game, whose success probability is upper bounded by the new principle, and show that an optimal strategy to perform it combines the quantum teleportation and superdense coding protocols with a task that has classical inputs.
Telecloning is a protocol introduced by Murao et al. [Phys. Rev. A 59, 156 (1999)] to distribute copies of an unknown quantum state to many receivers in a way that beats the trivial ``clone-and-teleport'' protocol. In the last decade, a new type of teleportation called port-based teleportation, in which the receiver can recover the state simply by looking at the correct port without having to actively perform correction operations, has been widely studied. In this paper, we consider the analog of telecloning, in which conventional teleportation is replaced by the port-based variant. To achieve this, we generalize the optimal measurement used in port-based teleportation and develop a new one that achieves port-based telecloning. Numerical results show that, in certain cases, the proposed protocol is strictly better than the trivial clone-and-teleport approach.
We suggest a method for teleporting an unknown quantum state. In this method the sender Alice first uses a Controlled-Not operation on the particle in the unknown quantum state and an ancillary particle which she wants to send to the receiver Bob. Then she sends ancillary particle to Bob. When Alice is informed by Bob that the ancillary particle is received, she performs a local measurement on the particle and sends Bob the outcome of the local measurement via a classical channel. Depending on the outcome Bob can restore the unknown quantum state, which Alice destroyed, on the ancillary particle successfully. As an application of this method we propose a quantum secure direct communication protocol.
Quantum teleportation, a cornerstone of quantum information science, enables the transfer of an unknown quantum state from a sender (Alice) to a receiver (Bob) using entanglement and classical communication. A fundamental principle of quantum mechanics, the no-cloning theorem, dictates that an arbitrary unknown quantum state cannot be perfectly duplicated. This paper introduces the concept of Bell-Certified Unclonable Teleportation, a novel protocol designed to intrinsically integrate the verification of entanglement quality and the preservation of unclonability within the teleportation process itself. By leveraging the unique properties of Bell states and their measurements, this protocol aims to provide a certification mechanism that assures both the successful transfer of the quantum state and the impossibility of its unauthorized replication during or after transmission. We explore how Bell state analysis, coupled with specific classical communication strategies, can serve as a robust signature for the integrity and security of the teleported state, thereby enhancing the trustworthiness of quantum communication channels. The proposed scheme offers a theoretical framework for building more secure and verifiable quantum communication primitives, addressing critical concerns in quantum cryptography and the future development of quantum networks.