Bayesian estimation of one-parameter qubit gates

We address estimation of one-parameter unitary gates for qubit systems and seek optimal probes and measurements. Single- and two-qubit probes are analysed in detail focusing on precision and stability of the estimation procedure. Bayesian inference is employed and compared with the ultimate quantum limits to precision, taking into account the biased nature of the Bayes estimator in the non-asymptotic regime. In addition, through the evaluation of the asymptotic a posteriori distribution for the gate parameter and a comparison with the results of Monte Carlo simulated experiments, we show that asymptotic optimality of the Bayes estimator is actually achieved after a limited number of runs. The robustness of the estimation procedure against fluctuations of the measurement settings is investigated and the use of entanglement to improve the overall stability of the estimation scheme is also analysed in some detail.

[1]  G. D’Ariano,et al.  Maximum-likelihood estimation of the density matrix , 1999, quant-ph/9909052.

[2]  T. Ralph,et al.  Quantum process tomography of a controlled-NOT gate. , 2004, Physical review letters.

[3]  Rauch,et al.  Quantum phase in interferometry. , 1996, Physical review letters.

[4]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[5]  H. V. Trees,et al.  Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking , 2007 .

[6]  Quantum state measurement by realistic heterodyne detection. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[7]  Thomas L. Marzetta,et al.  Detection, Estimation, and Modulation Theory , 1976 .

[8]  Giacomo Mauro D'Ariano,et al.  Imprinting complete information about a quantum channel on its output state. , 2003, Physical review letters.

[9]  H. Vincent Poor,et al.  An Introduction to Signal Detection and Estimation , 1994, Springer Texts in Electrical Engineering.

[10]  S. Braunstein,et al.  Geometry of Quantum States , 1995 .

[11]  Osamu Hirota,et al.  "Quantum Communication, Computing, and Measurement" , 2012 .

[12]  P. Zoller,et al.  Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate , 1996, quant-ph/9611013.

[13]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[14]  Hradil.  Estimation of counted quantum phase. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[15]  Debbie W. Leung,et al.  Realization of quantum process tomography in NMR , 2000, quant-ph/0012032.

[16]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[17]  E. Knill,et al.  Optimal quantum measurements of expectation values of observables , 2006, quant-ph/0607019.

[18]  Asymptotic Estimation of a Shift Parameter of a Quantum State , 2003, quant-ph/0307225.

[19]  L. Peliti,et al.  Error threshold in simple landscapes , 1996, cond-mat/9610028.

[20]  A Smerzi,et al.  Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry. , 2007, Physical review letters.

[21]  H. V. Trees.  Detection, Estimation, And Modulation Theory , 2001 .

[22]  Matteo G. A. Paris,et al.  Quorum of observables for universal quantum estimation , 2001 .

[23]  S. Lloyd,et al.  Quantum metrology. , 2005, Physical review letters.

[24]  G M D'Ariano,et al.  Using entanglement improves the precision of quantum measurements. , 2001, Physical review letters.

[25]  D Malakoff,et al.  Bayes Offers a 'New' Way to Make Sense of Numbers , 1999, Science.

[26]  Aephraim M. Steinberg,et al.  Diagnosis, prescription, and prognosis of a bell-state filter by quantum process tomography. , 2003, Physical review letters.

[27]  Aephraim M. Steinberg,et al.  Quantum process tomography on vibrational states of atoms in an optical lattice , 2003, quant-ph/0312210.

[28]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[29]  Andrew G. White,et al.  Measurement of qubits , 2001, quant-ph/0103121.

[30]  C. R. Rao,et al.  Minimum variance and the estimation of several parameters , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[31]  R. Gill,et al.  State estimation for large ensembles , 1999, quant-ph/9902063.

[32]  M. Hotta,et al.  Quantum Estimation by Local Observables , 2004 .

[33]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[34]  Isaac L. Chuang,et al.  Prescription for experimental determination of the dynamics of a quantum black box , 1997 .

[35]  Akio Fujiwara,et al.  Quantum channel identification problem , 2001 .

[36]  Hiroshi Nagaoka,et al.  Quantum Fisher metric and estimation for pure state models , 1995 .

[37]  G. Milburn,et al.  Generalized uncertainty relations: Theory, examples, and Lorentz invariance , 1995, quant-ph/9507004.

[38]  S. Braunstein,et al.  Geometry of Quantum States a , 1995 .

[39]  Ruediger Schack,et al.  Unknown Quantum States and Operations, a Bayesian View , 2004, quant-ph/0404156.

[40]  Sergio Boixo,et al.  Generalized limits for single-parameter quantum estimation. , 2006, Physical review letters.

[41]  Generation of entangled photon pairs using small-coherence-time continuous wave pump lasers. , 2008, Applied optics.

[42]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[43]  Daniel M. Greenberger,et al.  Fundamental Problems in Quantum Theory , 1995 .

[44]  Andrew G. Glen,et al.  APPL , 2001 .

[45]  O. Barndorff-Nielsen,et al.  Fisher information in quantum statistics , 1998, quant-ph/9808009.

[46]  J. G. Gander,et al.  An introduction to signal detection and estimation , 1990 .

[47]  Alberto Porzio,et al.  Quantum tomography as a tool for the characterization of optical devices , 2001, quant-ph/0110110.

[48]  Improving quantum interferometry by using entanglement , 2001, quant-ph/0110105.

[49]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .