Statistical estimation of the quality of quantum-tomography protocols

We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole families of polarization states of qubits and ququarts including pure, mixed, entangled, and separable.

[1]  A. Holevo.  Statistical structure of quantum theory , 2001 .

[2]  S P Kulik,et al.  Qutrit state engineering with biphotons. , 2004, Physical review letters.

[3]  A Gilchrist,et al.  Measuring entangled qutrits and their use for quantum bit commitment. , 2004, Physical review letters.

[4]  Alessandro Zavatta,et al.  Tomographic reconstruction of the single-photon Fock state by high-frequency homodyne detection , 2004, quant-ph/0406090.

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

[6]  C. Schwemmer,et al.  Permutationally invariant quantum tomography. , 2010, Physical review letters.

[7]  A. Ibort,et al.  An introduction to the tomographic picture of quantum mechanics , 2009, 0904.4439.

[8]  Zhuo-Liang Cao,et al.  Optimal reconstruction of the states in qutrit systems , 2010 .

[9]  I. Walmsley,et al.  Optimal experiment design for quantum state tomography: Fair, precise, and minimal tomography , 2009, 0911.4310.

[10]  E. Knill,et al.  Diluted maximum-likelihood algorithm for quantum tomography , 2006, quant-ph/0611244.

[11]  A. Uhlmann.  Transition Probability (Fidelity) and Its Relatives , 2011, 1106.0979.

[12]  Z. Hradil.  Quantum-state estimation , 1996, quant-ph/9609012.

[13]  Yu. I. Bogdanov,et al.  Unified statistical method for reconstructing quantum states by purification , 2009 .

[14]  B. Sanders,et al.  Quantum process tomography with coherent states , 2010, 1009.3307.

[15]  M. Genovese.  Research on hidden variable theories: A review of recent progresses , 2005, quant-ph/0701071.

[16]  M. Paris,et al.  Properties of entangled photon pairs generated by a CW laser with small coherence time: theory and experiment , 2008, 0802.3131.

[17]  M. Bellini,et al.  Tomographic test of Bell's inequality for a time-delocalized single photon , 2006, quant-ph/0602150.

[18]  Z. Hradil,et al.  Quantum tomography as normalization of incompatible observations , 1998 .

[19]  Takayoshi Kobayashi,et al.  Remote preparation of qutrit states with biphotons , 2007 .

[20]  S A Babichev,et al.  Homodyne tomography characterization and nonlocality of a dual-mode optical qubit. , 2004, Physical review letters.

[21]  Marco Genovese,et al.  Review on qudits production and their application to Quantum Communication and Studies on Local Realism , 2007, 0711.1288.

[22]  MAXIMUM-LIKELIHOOD ESTIMATION OF PHOTON-NUMBER DISTRIBUTION FROM HOMODYNE STATISTICS , 1997, physics/9712043.

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

[24]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[25]  Giacomo Mauro D'Ariano,et al.  Generating qudits with d=3,4 encoded on two-photon states , 2005, quant-ph/0503227.

[26]  C. Kurtsiefer,et al.  Experimental Polarization State Tomography using Optimal Polarimeters , 2006, quant-ph/0603126.

[27]  Matteo G. A. Paris,et al.  A two-step MaxLik-MaxEnt strategy to infer photon distribution from on/off measurement at low quantum efficiency , 2005 .

[28]  A. K. Gavrichenko,et al.  Information aspects of “which-path” interference experiments with microparticles , 2010 .

[29]  Marco Genovese,et al.  Experimental estimation of entanglement at the quantum limit. , 2009, Physical review letters.

[30]  G. Vallone,et al.  Experimental quantum process tomography of non-trace-preserving maps , 2010, 1008.5334.

[31]  G. D’Ariano,et al.  Parameter estimation in quantum optics , 2000, quant-ph/0004066.

[32]  J. Miller.  Numerical Analysis , 1966, Nature.

[33]  A. K. Gavrichenko,et al.  Statistical reconstruction of mixed states of polarization qubits , 2011 .

[34]  S. Kulik,et al.  Statistical reconstruction of the quantum states of three-level optical systems , 2003 .

[35]  S. Olivares,et al.  State reconstruction by on/off measurements , 2009, 0903.0104.

[36]  A. Lvovsky,et al.  Quantum state reconstruction of the single-photon Fock state. , 2001, Physical Review Letters.

[37]  Alexei Gilchrist,et al.  Choice of measurement sets in qubit tomography , 2007, 0706.3756.

[38]  V. I. Man'ko,et al.  Generalized tomographic maps , 2008, 0802.4140.

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

[40]  Sergei P. Kulik,et al.  Polarization state of a biphoton: Quantum ternary logic , 1999 .

[41]  Meng Khoon Tey,et al.  Statistical reconstruction of qutrits , 2004 .

[42]  Marco Genovese,et al.  Experimental reconstruction of photon statistics without photon counting. , 2005, Physical review letters.

[43]  M. Paris,et al.  Optimal estimation of entanglement in optical qubit systems , 2011, 1102.3684.

[44]  Y. S. Teo,et al.  Quantum-state reconstruction by maximizing likelihood and entropy. , 2011, Physical review letters.

[45]  A. Vaziri,et al.  Triggered qutrits for quantum communication protocols. , 2004, Physical review letters.

[46]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[47]  K J Resch,et al.  Manipulating biphotonic qutrits. , 2007, Physical review letters.

[48]  Yu. I. Bogdanov,et al.  Optimization of a quantum tomography protocol for polarization qubits , 2010 .

[49]  Gleb Maslennikov,et al.  Polarization states of four-dimensional systems based on biphotons , 2006 .

[50]  Yu I Bogdanov,et al.  Statistical estimation of the efficiency of quantum state tomography protocols. , 2010, Physical review letters.

[51]  D. Kaszlikowski,et al.  Minimal qubit tomography , 2004, quant-ph/0405084.

[52]  Dmitri Mogilevtsev,et al.  Diagonal element inference by direct detection , 1998 .