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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 graphical algebra with permutations for larger N.(a) We express permutations by the conventional ‘cycle’ notation of the permutation group. Taking into account the global permutation symmetry every operator in our algebra can be summarized by a sum over post-permutations on σ1. (b) Shows an example expressed as a simple three-cycle [(123) … (N)]σ1.
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f6: The graphical algebra with permutations for larger N.(a) We express permutations by the conventional ‘cycle’ notation of the permutation group. Taking into account the global permutation symmetry every operator in our algebra can be summarized by a sum over post-permutations on σ1. (b) Shows an example expressed as a simple three-cycle [(123) … (N)]σ1.

Mentions: Computing the closure polynomial involves (sums over) simple repeated multiplications of terms like σiσj. Each such multiplication will reduce to a generic term like Sijσj. Iterating this leads to a sequence of post-swap operations, or equivalently a single post-permutation operation. This means that a typical term from ρn will consist of a sum over terms shown in Fig. 6. We denote permutations using the conventional cyclic notation, so that (123) ... (N) denotes shifting 1 → 2, 2 → 3 and 3 → 1, etc. To represent this as a unitary operator permuting the ports we surround it by square brackets, e.g., [(123) ... (N)].


Higher-dimensional performance of port-based teleportation
The graphical algebra with permutations for larger N.(a) We express permutations by the conventional ‘cycle’ notation of the permutation group. Taking into account the global permutation symmetry every operator in our algebra can be summarized by a sum over post-permutations on σ1. (b) Shows an example expressed as a simple three-cycle [(123) … (N)]σ1.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f6: The graphical algebra with permutations for larger N.(a) We express permutations by the conventional ‘cycle’ notation of the permutation group. Taking into account the global permutation symmetry every operator in our algebra can be summarized by a sum over post-permutations on σ1. (b) Shows an example expressed as a simple three-cycle [(123) … (N)]σ1.
Mentions: Computing the closure polynomial involves (sums over) simple repeated multiplications of terms like σiσj. Each such multiplication will reduce to a generic term like Sijσj. Iterating this leads to a sequence of post-swap operations, or equivalently a single post-permutation operation. This means that a typical term from ρn will consist of a sum over terms shown in Fig. 6. We denote permutations using the conventional cyclic notation, so that (123) ... (N) denotes shifting 1 → 2, 2 → 3 and 3 → 1, etc. To represent this as a unitary operator permuting the ports we surround it by square brackets, e.g., [(123) ... (N)].

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.