Higher-dimensional performance of port-based teleportation
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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. |
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Mentions: Since the graphs represent operators, they may be added and multiplied like operators. Specifically, to multiply two operators denoted by graphs, the graph for the leftmost operator is placed above that of the rightmost operator and the lines of their respective subsystems are joined in the natural manner (see Fig. 5). Any loop reduces to a trace of the identity operator on a d-dimensional subsystem, i.e., the factor d. Other simplifications come from internally rearranging the lines. Finally, the resultant operators can always be placed into a standard form by at most pre- and post-multiplication by permutation operators on the ports (i.e., those subsystems in Bob’s hand labelled 1, …, N). Some of these rules are shown in Fig. 5 for N = 2. |
View Article: PubMed Central - PubMed
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.