Dense covariant coding and estimation of SU(2) operations

This paper addresses quantum statistical estimation of operators $U\ensuremath{\in}\mathrm{SU}(2)$ acting on $C{P}^{3}$ as $\ensuremath{\psi}\ensuremath{\mapsto}(U\ensuremath{\bigotimes}I)\ensuremath{\psi}$ where $\ensuremath{\psi}\ensuremath{\in}{C}^{2}\ensuremath{\bigotimes}{C}^{2}.$ This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cram\'er-Rao lower bound takes the minimum, and is achievable, if and only if \ensuremath{\psi} is a maximally entangled state. We next show that an SU(2) orbit on $C{P}^{3}$ equipped with the standard Riemannian structure is isometric to $\mathrm{SU}(2)/{\ifmmode\pm\else\textpm\fi{}I}\ensuremath{\cong}\mathrm{SO}(3)$ if and only if \ensuremath{\psi} is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.This paper addresses quantum statistical estimation of operators $U\ensuremath{\in}\mathrm{SU}(2)$ acting on $C{P}^{3}$ as $\ensuremath{\psi}\ensuremath{\mapsto}(U\ensuremath{\bigotimes}I)\ensuremath{\psi}$ where $\ensuremath{\psi}\ensuremath{\in}{C}^{2}\ensuremath{\bigotimes}{C}^{2}.$ This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cram\'er-Rao lower bound takes the minimum, and is achievable, if and only if \ensuremath{\psi} is a maximally entangled state. We next show that an SU(2) orbit on $C{P}^{3}$ equipped with the standard Riemannian structure is isometric to $\mathrm{SU}(2)/{\ifmmode\pm\else\textpm\fi{}I}\ensuremath{\cong}\mathrm{SO}(3)$ if and only if \ensuremath{\psi} is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.

Physical Review ADec 12, 200170 citations

Parallel treatment of estimation of SU(2) and phase estimation

Masahito Hayashi

Physics Letters AFeb 1, 200651 citations

Application of su(2) algebra

Guozhen Wu

Elsevier eBooksJan 1, 20050 citations

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

E. Bagán, M. Baig, R. Muñoz-Tapia

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.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.

arXiv:quant-ph/0303019Physical Review AMay 14, 200431 citations

Quantum reverse engineering and reference-frame alignment without nonlocal correlations

E. Bagán, M. Baig, R. Muñoz-Tapia

The estimation of unknown qubit elementary gates and the alignment of reference frames are formally the same problem. Using quantum states made out of $N$ qubits, we show that the theoretical precision limit for both problems, which behaves as $1∕{N}^{2}$ , 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.The estimation of unknown qubit elementary gates and the alignment of reference frames are formally the same problem. Using quantum states made out of $N$ qubits, we show that the theoretical precision limit for both problems, which behaves as $1∕{N}^{2}$, 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.

arXiv:quant-ph/0405082Physical Review ASep 7, 200464 citations

Correlation operators in many-particle physics

Jürgen Bosse

Zeitschrift für Physik A Hadrons and NucleiFeb 1, 19720 citations

This paper addresses quantum statistical estimation of operators $U\ensuremath{\in}\mathrm{SU}(2)$ acting on

C{P}^{3}

as $\ensuremath{\psi}\ensuremath{\mapsto}(U\ensuremath{\bigotimes}I)\ensuremath{\psi}$ where $\ensuremath{\psi}\ensuremath{\in}{C}^{2}\ensuremath{\bigotimes}{C}^{2}.$ This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cram\'er-Rao lower bound takes the minimum, and is achievable, if and only if \ensuremath{\psi} is a maximally entangled state. We next show that an SU(2) orbit on

C{P}^{3}

equipped with the standard Riemannian structure is isometric to $\mathrm{SU}(2)/{\ifmmode\pm\else\textpm\fi{}I}\ensuremath{\cong}\mathrm{SO}(3)$ if and only if \ensuremath{\psi} is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally 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

N

qubits, we show that the theoretical precision limit for both problems, which behaves as

1∕{N}^{2}

, 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.