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arXiv:quant-ph/9605025 · Presented at

How much does it cost to teleport?

H. F. Chau, Hoi‐Kwong Lo

Published May 16, 1996• 4 citations
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Abstract

We show that the entropy of entanglement of a state characterizes its ability to teleport. In particular, in order to teleport faithfully an unknown quantum $N$-state, the two users must share an entangled state with at least $\log_2 N$ bits entropy of entanglement. We also note that the maximum capacity for a mixed state ${\cal M}$ to teleport equals the maximum amount of entanglement entropy that can be distilled out from ${\cal M}$. Our result, therefore, provides an alternative interpretation for entanglement purification.

DOI
10.48550/arxiv.quant-ph/9605025

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We show that the entropy of entanglement of a state characterizes its ability to teleport. In particular, in order to teleport faithfully an unknown quantum NNN-state, the two users must share an entangled state with at least log⁡2N\log_2 Nlog2​N bits entropy of entanglement. We also note that the maximum capacity for a mixed state M{\cal M}M to teleport equals the maximum amount of entanglement entropy that can be distilled out from M{\cal M}M. Our result, therefore, provides an alternative interpretation for entanglement purification.
The amount of entanglement necessary to teleport quantum states drawn from general ensemble {pi,ρi}\{p_i,ρ_i\}{pi​,ρi​} is derived. The case of perfect transmission of individual states and that of asymptotically faithful transmission are discussed. Using the latter result, we also derive the optimum compression rate when the ensemble is compressed into qubits and bits.
A new interpretation of entanglement entropy is proposed: entanglement entropy of a pure state with respect to a division of Hilbert space into two subspaces 1 and 2 is an amount of information which can be transmitted through 1 and 2 from a system interacting with 1 to another system interacting with 2. The transmission medium is the quantum entanglement between 1 and 2. In order to support the interpretation, suggestive arguments are given: variational principles in entanglement thermodynamics and quantum teleportation. It is shown that a quantum state having maximal entanglement entropy plays an important role in quantum teleportation. Hence the entanglement entropy is, in some sense, an index of the efficiency of quantum teleportation. Finally, the implications for the information loss problem and Hawking radiation are discussed.
Two separated observers, by applying local operations to a supply of not-too-impure entangled states (e.g., singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity (e.g., near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield D(M)D\left(M\right)D(M) of pure singlets $(|{\ensuremath{\Psi}}^{\ensuremath{-}}〉)$ distillable from mixed states MMM, showing D(M)>0D\left(M\right)>0D(M)>0 if $〈{\ensuremath{\Psi}}^{\ensuremath{-}}|M|{\ensuremath{\Psi}}^{\ensuremath{-}}〉>\frac{1}{2}$.
Quantum teleportation with additional a priori information about the input state achieves higher fidelity than teleportation of a completely unknown state. However, perfect teleporation of two nonorthogonal input states requires the same amount of entanglement as perfect teleportation of an unknown state, namely one ebit. We analyze how well two-state teleportation can be achieved using every degree of pure-state entanglement. We find the highest fidelity of ``teleportation'' that can be achieved with only classical communication but no shared entanglement. A two-state telecloning scheme is constructed.
The ability to teleport entanglement through maximally entangled mixed states as defined by concurrence and linear entropy is studied. We show how the teleported entanglement depends on the quality of the quantum channel used, as defined through its entanglement and mixedness, as well as the form of the target state to be teleported. We present new results based on the fidelity of the teleported state as well as an experimental setup that is immediately implementable with currently available technology.