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Higher-dimensional performance of port-based teleportation

View Article: PubMed Central - PubMed

ABSTRACT

Port-based teleportation (PBT) is a variation of regular quantum teleportation that operates without a final unitary correction. However, its behavior for higher-dimensional systems has been hard to calculate explicitly beyond dimension d = 2. Indeed, relying on conventional Hilbert-space representations entails an exponential overhead with increasing dimension. Some general upper and lower bounds for various success measures, such as (entanglement) fidelity, are known, but some become trivial in higher dimensions. Here we construct a graph-theoretic algebra (a subset of Temperley-Lieb algebra) which allows us to explicitly compute the higher-dimensional performance of PBT for so-called “pretty-good measurements” with negligible representational overhead. This graphical algebra allows us to explicitly compute the success probability to distinguish the different outcomes and fidelity for arbitrary dimension d and low number of ports N, obtaining in addition a simple upper bound. The results for low N and arbitrary d show that the entanglement fidelity asymptotically approaches N/d2 for large d, confirming the performance of one lower bound from the literature.

No MeSH data available.


The entanglement fidelity and its upper and lower bounds of PBT for small N.Here we give a log-log plot of Fe versus d plot for N = 2, 3, 4 computed using our graphical-algebraic techniques. The analytic form for the upper and lower bounds are given in the text. As d2 ≫ N increases, the lower bound asymptotes towards the straight-line upper bound with gradient −2 in this log-log plot.
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f3: The entanglement fidelity and its upper and lower bounds of PBT for small N.Here we give a log-log plot of Fe versus d plot for N = 2, 3, 4 computed using our graphical-algebraic techniques. The analytic form for the upper and lower bounds are given in the text. As d2 ≫ N increases, the lower bound asymptotes towards the straight-line upper bound with gradient −2 in this log-log plot.

Mentions: We give a trivial upper bond to the entanglement fidelity because the success probability is itself upper-bounded by one. A lower bound has been derived3 to the entanglement fidelity is . Figure 3 demonstrates that the lower bound corresponds to a good approximation for lower N and our trivial upper bound appears to be reached asymptotically for higher dimensions. Note that for a general number of ports satisfying d ≫ N, we have ; and hence the actual entanglement fidelity for PBT is tightly constrained in these circumstances for asymptotically large dimensionalities.


Higher-dimensional performance of port-based teleportation
The entanglement fidelity and its upper and lower bounds of PBT for small N.Here we give a log-log plot of Fe versus d plot for N = 2, 3, 4 computed using our graphical-algebraic techniques. The analytic form for the upper and lower bounds are given in the text. As d2 ≫ N increases, the lower bound asymptotes towards the straight-line upper bound with gradient −2 in this log-log plot.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5015100&req=5

f3: The entanglement fidelity and its upper and lower bounds of PBT for small N.Here we give a log-log plot of Fe versus d plot for N = 2, 3, 4 computed using our graphical-algebraic techniques. The analytic form for the upper and lower bounds are given in the text. As d2 ≫ N increases, the lower bound asymptotes towards the straight-line upper bound with gradient −2 in this log-log plot.
Mentions: We give a trivial upper bond to the entanglement fidelity because the success probability is itself upper-bounded by one. A lower bound has been derived3 to the entanglement fidelity is . Figure 3 demonstrates that the lower bound corresponds to a good approximation for lower N and our trivial upper bound appears to be reached asymptotically for higher dimensions. Note that for a general number of ports satisfying d ≫ N, we have ; and hence the actual entanglement fidelity for PBT is tightly constrained in these circumstances for asymptotically large dimensionalities.

View Article: PubMed Central - PubMed

ABSTRACT

Port-based teleportation (PBT) is a variation of regular quantum teleportation that operates without a final unitary correction. However, its behavior for higher-dimensional systems has been hard to calculate explicitly beyond dimension d = 2. Indeed, relying on conventional Hilbert-space representations entails an exponential overhead with increasing dimension. Some general upper and lower bounds for various success measures, such as (entanglement) fidelity, are known, but some become trivial in higher dimensions. Here we construct a graph-theoretic algebra (a subset of Temperley-Lieb algebra) which allows us to explicitly compute the higher-dimensional performance of PBT for so-called “pretty-good measurements” with negligible representational overhead. This graphical algebra allows us to explicitly compute the success probability to distinguish the different outcomes and fidelity for arbitrary dimension d and low number of ports N, obtaining in addition a simple upper bound. The results for low N and arbitrary d show that the entanglement fidelity asymptotically approaches N/d2 for large d, confirming the performance of one lower bound from the literature.

No MeSH data available.