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arXiv:quant-ph/0002088 · Physical Review A

Optimal quantum teleportation with an arbitrary pure state

Konrad Banaszek

Published Jul 13, 2000• 85 citations
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Abstract

We present an elementary derivation of the maximum fidelity attainable in teleportation using a single copy of two d-level systems in an arbitrary pure state. This derivation provides a complete set of necessary and sufficient conditions for optimal teleportation protocols. We also discuss the information on the teleported particle that is revealed in course of the protocol using a nonmaximally entangled state.

DOI
10.1103/physreva.62.024301

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We present an elementary derivation of the maximum fidelity attainable in teleportation using a single copy of two d-level systems in an arbitrary pure state. This derivation provides a complete set of necessary and sufficient conditions for optimal teleportation protocols. We also discuss the information on the teleported particle that is revealed in course of the protocol using a nonmaximally entangled state.
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Abstract Quantum teleportation with an arbitrary two-qubit state can be appropriately characterized in terms of maximal fidelity and fidelity deviation. The former quantifies optimality of the process and is defined as the maximal average fidelity achievable within the standard protocol and local unitary strategies, whereas the latter, defined as the standard deviation of fidelity over all input states, is a measure of fidelity fluctuations. The maximal fidelity for a two-qubit state is known and is given by a simple formula that can be exactly computed, but no such formula is known for the fidelity deviation. In this paper, we derive an exact computable formula for the fidelity deviation in optimal quantum teleportation with an arbitrary state of two qubits. From this formula, we obtain the dispersion-free condition, also known as the universality condition: the condition that all input states are teleported equally well and provide a necessary and sufficient condition for a state to be both useful (maximal fidelity larger than the classical bound) and universal (zero fidelity deviation). We also show that for any given maximal fidelity, larger than the classical bound, there always exist dispersion-free or universal states and argue that such states are the most desirable ones within the set of useful states. We illustrate these results with well-known families of two-qubit states: pure entangled states, Bell-diagonal states, and subsets of X states.
A conclusive teleportation protocol of a d -dimensional two-particle unknown quantum state using three d -dimensional particles in an arbitrary pure state is proposed. A sender teleports the unknown state conclusively to a receiver by using the positive operator valued measure(POVM) and introducing an ancillary qudit to perform the generalized Bell basis measurement. We calculate the optimal teleportation fidelity. We also discuss and analyse the reason why the information on the teleported state is lost in the course of the protocol.
We investigate two-copy scenario of quantum teleportation based on Bell measurements. The detailed protocol is presented and the general expression of the corresponding optimal teleportation fidelity is derived, which is given by the two-copy fully entangled fraction that is invariant under local unitary transformations. We prove that under a specific case of the protocol, which is significant for improving the optimal fidelity, the set of states with their two-copy fully entangled fractions bounded by a threshold value that required for useful two-copy teleportation is convex and compact. Hence the witness operators exist to separate states that are useful for two-copy teleportation from the rest ones. Moreover, we show that the optimal fidelity of two-copy teleportation surpasses that of the original one copy teleportation.
Quantum teleportation of qudits is revisited. In particular, we analyze the case where the quantum channel corresponds to a nonmaximally entangled state and show that the success of the protocol is directly related to the problem of distinguishing nonorthogonal quantum states. The teleportation channel can be seen as a coherent superposition of two channels, one of them being a maximally entangled state, thus leading to perfect teleportation, and the other, corresponding to a nonmaximally entangled state living in a subspace of the d-dimensional Hilbert space. The second channel leads to a teleported state with reduced fidelity. We calculate the average fidelity of the process and show its optimality.