Score operators of a qubit with applications

The score operators of a quantum system are the symmetric logarithmic derivatives of the system’s parametrically defined quantum state. Score operators are central to the calculation of the quantum Fisher information (QFI) associated with the state of the system, and the QFI determines the maximum precision with which the state parameters can be estimated. We give a simple, explicit expression for score operators of a qubit and apply this expression in a series of settings. We treat in detail the task of identifying a quantum Pauli channel from the state of its qubit output, and we show that a “balanced” probe state is highly robust for this purpose. The QFI for this task is a matrix, and we study its determinant, for which we establish a Cramér-Rao inequality.

[1]  Jaroslav Rehacek,et al.  Maximum-likelihood methods in quantum mechanics , 2004 .

[2]  S. Boixo,et al.  Operational interpretation for global multipartite entanglement. , 2007, Physical review letters.

[3]  D. Harville.  Matrix Algebra From a Statistician's Perspective , 1998 .

[4]  D. Petz.  Quantum Information Theory and Quantum Statistics , 2007 .

[5]  Harry L. Van Trees,et al.  Detection, Estimation, and Modulation Theory: Radar-Sonar Signal Processing and Gaussian Signals in Noise , 1992 .

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

[7]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[8]  Michael R. Frey,et al.  Pauli channels exhibit a transition effect in memory estimation above a parametric threshold , 2010, Defense + Commercial Sensing.

[9]  D. James,et al.  Qubit quantum state tomography , 2004 .

[10]  Akio Fujiwara,et al.  Estimation of a generalized amplitude-damping channel , 2004 .

[11]  W. G. Bickley,et al.  Matrix and other direct methods for the solution of systems of linear difference equations , 1960, Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences.

[12]  H. Yuen,et al.  Review of 'Quantum Detection and Estimation Theory' (Helstrom, C. W.; 1976) , 1977, IEEE Transactions on Information Theory.

[13]  M. Paris.  Quantum estimation for quantum technology , 2008, 0804.2981.

[14]  Hiroshi Imai,et al.  Quantum parameter estimation of a generalized Pauli channel , 2003 .

[15]  Richard A. Johnson,et al.  Applied Multivariate Statistical Analysis , 1983 .

[16]  Holger Mack,et al.  Enhanced estimation of a noisy quantum channel using entanglement , 2001 .

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

[18]  U. Fano.  Description of States in Quantum Mechanics by Density Matrix and Operator Techniques , 1957 .

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

[20]  Giacomo Mauro D'Ariano,et al.  2 Quantum Tomographic Methods , 2004 .

[21]  Michael R. Frey,et al.  Quantum Fisher information and the qudit depolarization channel , 2009, Defense + Commercial Sensing.

[22]  P. Slater.  Applications of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems , 1996 .

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

[24]  Č. Brukner,et al.  CAN ENTANGLEMENT BE EXTRACTED FROM MANY BODY SYSTEMS , 2007 .

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

[26]  David Daems.  Entanglement-enhanced transmission of classical information in Pauli channels with memory: Exact solution , 2007 .