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arXiv:quant-ph/0303019 · Physical Review A

Entanglement-assisted alignment of reference frames using a dense covariant coding

E. Bagán, M. Baig, R. Muñoz-Tapia

Published May 14, 2004• 31 citations
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Abstract

We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share a maximally entangled state. We deal with the concrete problem of aligning reference frames or trihedra by means of a quantum system. We find the optimal covariant measurement and compute the corresponding average error, which has a remarkably simple close form. The connection of this procedure with that of estimating unitary transformations on qubits is briefly discussed.

DOI
10.1103/physreva.69.050303

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We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share a maximally entangled state. We deal with the concrete problem of aligning reference frames or trihedra by means of a quantum system. We find the optimal covariant measurement and compute the corresponding average error, which has a remarkably simple close form. The connection of this procedure with that of estimating unitary transformations on qubits is briefly discussed.
I investigate dense coding with a general mixed state on the Hilbert space C d ⊗ C d shared between a sender and receiver. The following result is proved. When the sender prepares the signal states by mutually orthogonal unitary transformations with equal a priori probabilities, the capacity of dense coding is maximized. It is also proved that the optimal capacity of dense coding χ* satisfies E R (ρ) ≤ χ* ≤ E R (ρ) + log 2 d , where E R (ρ) is the relative entropy of entanglement of the shared entangled state.
The estimation of unknown qubit elementary gates and the alignment of reference frames are formally the same problem. Using quantum states made out of NNN qubits, we show that the theoretical precision limit for both problems, which behaves as 1∕N21∕{N}^{2}1∕N2, can be asymptotically attained with a covariant protocol that exploits the quantum correlation of internal degrees of freedom instead of the more fragile entanglement between distant parties. This cuts by half the number of qubits needed to achieve the precision of the dense covariant coding protocol.