Generalizing teleportation

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Generalized teleportation and entanglement recycling
Sergii Strelchuk,1, ∗ Michał Horodecki,2 and Jonathan Oppenheim3
of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, U.K. 2 Institute for Theoretical Physics and Astrophysics, University of Gdansk, 80-952 Gdansk, Poland ´ ´ 3 Department of Applied Mathematics and Theoretical Physics, University ofCambridge, Cambridge CB3 0WA, U.K. University College of London, Department of Physics & Astronomy, London, WC1E 6BT and London Interdisciplinary Network for Quantum Science
1 Department

arXiv:1209.2683v2 [quant-ph] 13 Dec 2012

We introduce new teleportation protocols which are generalizations of the original teleportation protocols that use the Pauli group [1] and the port-based teleportationprotocols, introduced by Hiroshima and Ishizaka [2], that use the symmetric permutation group. We derive sufficient condition for a set of operations, which in general need not form a group, to give rise to a teleportation protocol and provide examples of such schemes. This generalization leads to protocols with novel properties and is needed to push forward new schemes of computation based onthem. Port-based teleportation protocols and our generalizations use a large resource state consisting of N singlets to teleport only a single qubit state reliably. We provide two distinct protocols which recycle the resource state to teleport multiple states with error linearly increasing with their number. The first protocol consists of sequentially teleporting qubit states, and the secondteleports them in a bulk.

Teleportation lies at the very heart of quantum information theory, being the pivotal primitive in a variety of tasks. Teleportation protocols are a way of sending an unknown quantum state from one party to another using a resource in the form of an entangled state shared between two parties, Alice and Bob, in advance. First, Alice performs a measurement on the state shewants to teleport and her part of the resource state, then she communicates the classical information to Bob. He applies the unitary operation conditioned on that information to obtain the teleported state. A notable use of teleportation is in relation to computing, where it plays a key role enabling universal quantum computation and establishing a strong link between a particular teleportationprotocol and a kind of computation possible to be implemented using it [3]. Recently, Hiroshima and Ishizaka introduced portbased teleportation [2] which has the distinct property that Bob does not need to apply a correction after Alice’s measurement. It is an important primitive for programmable quantum processors [2, 5–7], which rely on an efficient way of storing a unitary transformation and actingit on an arbitrary quantum state. This protocol evades the fundamental limitations of the no-go theorem proved in [7], which states that universal deterministic programmable quantum processors cannot exist. Even though the protocol makes it possible to execute arbitrary instructions deterministically, the result will be inherently noisy. Port-based teleportation has already found its use ininstantaneous non-local quantum computation [8]. In the latter task, using it as the underlying teleportation routine dramatically reduced the amount of entanglement required to perform it. Such computations proved to be instrumental in attack schemes on position-based

quantum cryptography [9–12]. Currently, it is known that the minimum amount of entanglement an adversary needs to perform asuccessful attack on the scheme must be at least linear in the number of communicated qubits [11]. Also, an adversary having access to at most an exponential amount of entanglement can successfully break any position-based cryptography scheme [8]. However, we do not know how much entanglement is necessary to break all schemes of this kind. Any improvement of the underlying teleportation protocol will...
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