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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.


The multiplication rule when N = 2.(a) Here we illustrate the multiplication σ1σ1. Loops reduce to a numerical factor d; (b) Here we illustrate the multiplication σ1σ2. Lines may be internally rearranged so long as the ordering (or labelling) of all line ends is unchanged. Note that standard forms can be produced by at most the pre- or post-application of permutations among the ports (i.e., excluding subsystem zero). The result of the multiplication is found to be labelled as σ4 in Fig. 4. This in turn may be written in terms of σ2 as S12σ2 (here S12 is a simple permutation operator which swaps ports 1 and 2).
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f5: The multiplication rule when N = 2.(a) Here we illustrate the multiplication σ1σ1. Loops reduce to a numerical factor d; (b) Here we illustrate the multiplication σ1σ2. Lines may be internally rearranged so long as the ordering (or labelling) of all line ends is unchanged. Note that standard forms can be produced by at most the pre- or post-application of permutations among the ports (i.e., excluding subsystem zero). The result of the multiplication is found to be labelled as σ4 in Fig. 4. This in turn may be written in terms of σ2 as S12σ2 (here S12 is a simple permutation operator which swaps ports 1 and 2).

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.


Higher-dimensional performance of port-based teleportation
The multiplication rule when N = 2.(a) Here we illustrate the multiplication σ1σ1. Loops reduce to a numerical factor d; (b) Here we illustrate the multiplication σ1σ2. Lines may be internally rearranged so long as the ordering (or labelling) of all line ends is unchanged. Note that standard forms can be produced by at most the pre- or post-application of permutations among the ports (i.e., excluding subsystem zero). The result of the multiplication is found to be labelled as σ4 in Fig. 4. This in turn may be written in terms of σ2 as S12σ2 (here S12 is a simple permutation operator which swaps ports 1 and 2).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f5: The multiplication rule when N = 2.(a) Here we illustrate the multiplication σ1σ1. Loops reduce to a numerical factor d; (b) Here we illustrate the multiplication σ1σ2. Lines may be internally rearranged so long as the ordering (or labelling) of all line ends is unchanged. Note that standard forms can be produced by at most the pre- or post-application of permutations among the ports (i.e., excluding subsystem zero). The result of the multiplication is found to be labelled as σ4 in Fig. 4. This in turn may be written in terms of σ2 as S12σ2 (here S12 is a simple permutation operator which swaps ports 1 and 2).
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

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.