In this paper we review a number of recent developments in the study of quantum tomography which is one of the useful methods for quantum state estimation and quantum information acquisition, having sparked explosion of interest in recent years. The quantum process tomography is also analyzed. At the same time, some success experiments and applications of quantum tomography are introduced. Finally, a number of open problems and future directions in this field are proposed.
KeywordsQuantum TomographyQuantum Information AcquisitionQuantum State EstimationReview and Expectation
G. M. D’Ariano, “Quantum Tomography: General Theory and New Experiments,” Fortschr Phys, Vol. 48, No. 5-7, 2000, pp. 579-588.
G. M. D’Ariano, M. D. Laurentis, et al., “Quantum Tomography as a Tool for the Characterization of Optical Devices,” Journal of Optics B: Quantum and Semiclassical Optics, Vol. 4, No. 3, 2002, pp. S127-S132.
G. M. D’Ariano, M. G. A. Paris and M. F. Sacchi, “Quantum Tomography,” Advances in Imaging and Electron Physics, Academic Press Inc., Vol. 128, 2003, pp. 205- 308.
L. M. Artiles, R. D. Gill and M. I. Guta, “An Invitation to Quantum Tomography,” Journal of the Royal Statistical Society Series B, Vol. 67, No. 1, 2005, pp. 109-134.
Z. Hradil, J. Rehacek, et al., “Qubit Quantum State Tomography,” Lecture Notes in Physics, Vol. 649, 2004, pp. 113–145.
M. G. A. Paris and J. Rehácek, “Quantum State Estimation,” Springer, Berlin, 2004.
W. K. Wootters and W. H. Zurek, “A Single Quantum cannot be Cloned,” Nature, Vol. 299, No. 5886, 1982, pp. 802-803.
W. Heisenberg, “Uber den Anschaulichen Inhalt der Quantentheoretischen Kinematik und Mechanik, ” Zeits- chrift für Physik, Vol. 43, 1927, pp. 172-198.
G. M. D’Ariano and H. P. Yuen, “Impossibility of Measuring the Wave Function of a Single Quantum System,” Physical Review Letters, Vol. 76, No. 16, 1996, pp. 2832- 2835.
U. Fano, “Description of States in Quantum Mechanics by Density Matrix and Operator Techniques,” Reviews of Modern Physics, Vol. 29, No. 1, 1957, pp. 74-93.
Q. A. Turchette, C. J. Hood, et al., “Measurement of Conditional Phase Shifts for Quantum Logic,” Physical Review Letters, Vol. 75, No. 25, 1995, pp. 4710-4713.
J. F. Poyatos, J. I. Cirac and P. Zoller, “Complete Characterization of a Quantum Process: The Two-Bit Quantum Gate,” Physical Review Letters, Vol. 78, No. 2, 1997, pp. 390-393.
G. M. D’Ariano and P. L. Presti, “Quantum Tomography for Measuring Experimentally the Matrix Elements of an Arbitrary Quantum Operation,” Physical Review Letters, Vol. 86, No. 19, 2001, pp. 4195-4198.
D. T. Smithey, M. Beck, et al., “Measurement of the Wigner Distribution and The Density Matrix of a Light Mode Using Optical Homodyne Tomography: Application to Squeezed States and The Vacuum,” Physical Review Letters, Vol. 70, No. 9, 1993, pp. 1244-1247.
M. G. Raymer, M. Beck and D. McAlister, “Complex Wave-Field Reconstruction Using Phase-Space Tomography,” Physical Review Letters, Vol. 72, No. 8, 1994, pp. 1137-1140.
T. J. Dunn, I. A. Walmsley and S. Mukamel, “Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography,” Physical Review Letters, Vol. 74, No. 6, 1995, pp. 884-887.
V. Buzek, R. Derka, et al., “Reconstruction of Quantum States of Spin Systems: From Quantum Bayesian Inference to Quantum Tomography,” Annals of Physics, Vol. 266, No. 2, 1998, pp. 454-496.
T. Coudreau, L. Vernac, et al., “Quantum Tomography of a Laser Beam Interacting with Cold Atoms,” Europhysics Letters, Vol. 46, No. 6, 1999, pp. 722-727.
O. V. Man’ko, “Optical Tomography and Measuring Quantum States of an Ion in a Paul Trap and in a Penning Trap,” Proceedings of the SPIE-The International Society for Optical Engineering, Orlando, Vol. 3736, 1999, pp. 68-75.
V. A. Andreev and V. I. Man’Ko, “Quantum Tomography of Spin States and the Einstein-Podolsky-Rosen Para- dox,” Journal of Optics B: Quantum and Semiclassical Optics, Vol. 2, No. 2, 2000, pp. 122-125.
M. Beck, “Quantum State Tomography with Array Detectors,” Physical Review Letters, Vol. 84, No. 25, 2000, pp. 5748-5751.
A. Luis, “Quantum Tomography of Input-Output Processes,” Physical Review A, Vol. 62, No. 5, 2000, pp. (054 302)1-4.
M. G. Raymer and A. C. Funk, “Quantum-State Tomography of Two-Mode Light Using Generalized Rotations in Phase Space,” Physical Review A, Vol. 61, No. 1, 2000, pp. (015801)1-3.
A. M. Childs, I. L. Chuang and D. W. Leung, “Realization of Quantum Process Tomography in NMR,” Physical Review A, Vol. 64, No. 1, 2001, pp. (012314)1-7.
J. S. Lee, “The Quantum State Tomography on an NMR System,” Physics Letters A, Vol. 305, No. 6, 2002, pp. 349-353.
G. M. D’riano, C. Macchiavello and M. G. A. Paris, “Detection of the Density Matrix through Optical Homodyne Tomography without Filtered Back Projection,” Physical Review A, Vol. 50, No. 5, 1994, pp. 4298-4302.
G. M. D’riano, U. Leonhardt and H. Paul, “Homodyne Detection of the Density Matrix of the Radiation Field,” Physical Review A, Vol. 52, No. 3, 1995, pp. (R)1801- 1804.
M. Munroe, D. Boggavarapu, et al., “Photon-Number Statistics from the Phase-Averaged Quadrature-Field Distribution: Theory and Ultrafast Measurement,” Physical Review A, Vol. 52, No. 2, 1995, pp. (R)924-927.
S. Schiller, G. Breitenbach, et al., “Quantum Statistics of the Squeezed Vacuum by Measurement of the Density Matrix in the Number State Representation,” Physical Review Letters, Vol. 77, No. 14, 1996, pp. 2933-2936.
S. Wallentowitz and W. Vogel, “Reconstruction of the Quantum Mechanical State of a Trapped Ion,” Physical Review Letters, Vol. 75, No. 16, 1995, pp. 2932-2935.
T. J. Dunn, I. A. Walmsley and S. Mukamel, “Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography,” Physical Review Letters, Vol. 74, No. 6, 1995, pp. 884-887.
C. Kurtsiefer, T. Pfau and J. Mlynek, “Measurement of the Wigner Function of an Ensemble of Helium Atoms,” Nature, Vol. 386, No. 6621, 1997, pp. 150-153.
D. Leibfried, D. M. Meekhof, et al., “Experimental Determination of the Motional Quantum State of a Trapped Atom,” Physical Review Letters, Vol. 77, No. 21, 1996, pp. 4281-4285.
M. A. Nielsen, E. Knill and R. Laflamme, “Complete Quantum Teleportation Using Nuclear Magnetic Resonance,” Nature, Vol. 396, No. 6706, 1998, pp. 52-55.
J. B. Altepeter, D. Branning, et al., “Ancilla-Assisted Quantum Process Tomography,” Physical Review Letters, Vol. 90, No. 19, 2003, pp. (193601)1-4.
F. D. Martini, A. Mazzei, et al., “Exploiting Quantum Parallelism of Entanglement for a Complete Experimental Quantum Characterization of a Single-Qubit Device,” Physical Review A, Vol. 67, No. 6, 2003, pp. (062307)1-5.
L. E. Ballentine, “Quantum Mechanics: a Modern Development,” World Scientific Publishing Company, Singapore, 1998.
K. E. Cahill and R. J. Glauber, “Ordered Expansions in Boson Amplitude Operators,” Physical Review, Vol. 177, No. 5, 1969, pp. 1857-1881.
C. T. Lee, “Theorem on Nonclassical States,” Physical Review A, Vol. 52, No. 4, 1995, pp. 3374-3376.
A. I. Lvovsky, “Iterative Maximum-Likelihood Reconstruction in Quantum Homodyne Tomography,” Journal of Optics B: Quantum and Semiclassical Optics, Vol. 6, No. 6, 2004, pp. (S)556-559.
G. M. D’Ariano and M. G. A. Paris, “Adaptive Quantum Homodyne Tomography,” Physical Review A, Vol. 60, No. 1, 1999, pp. 518-528.
H. P. Yuen and V. W. S. Chan, “Noise in Homodyne and Heterodyne-Detection,” Optics Letters, Vol. 8, No. 3, 1983, pp. 177-179.
G. L. Abbas, V. W. S. Chan and S. T. Yee, “Local-Oscill- Ator Excess-Noise Suppression for Homodyne and Heterodyne Detection,” Optics Letters, Vol. 8, No. 8, 1983, pp. 419-421.
M. Beck, D. T. Smithey and M. G. Raymer, “Experimental Determination of Quantum-Phase Distributions Using Optical Homodyne Tomography,” Physical Review A, Vol. 48, No. 2, 1993, pp. (R)890-893.
J. J. Longdell and M. J. Sellars, “Experimental Demonstration of Quantum-State Tomography and Qubit-Qubit Interactions for Rare-Earth-Metal-Ion-Based Solid-State Qubits,” Physical Review A, Vol. 69, No. 3, 2004, pp. (32307)1-5.
Y. X. Liu, L. F. Wei and F. Nori, “Quantum Tomography for Solid-State Qubits,” Europhysics Letters, Vol. 67, No. 6, 2004, pp. 874-880.
Y. Nambu and K. Nakamura, “Experimental Investigation of a Nonideal Two-Qubit Quantum-State Filter by Quantum Process Tomography,” Physical Review Letters, Vol. 94, No. 1, 2005, pp. (010404)1-4.
M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press, Cambridge, 2000.
Y. Nambu, K. Usami, et al., “Generation of Polarization-Entangled Photon Pairs in a Cascade of Two Type-I Crystals Pumped by Femtosecond Pulses,” Physical Review A, Vol. 66, No. 3, 2002, pp. (033816)1-10.
W. Dür and J. I. Cirac, “Nonlocal Operations: Purification, Storage, Compression, Tomography, and Probabilistic Implementation,” Physical Review A, Vol. 64, No. 1, 2001, pp. (012317)1-14.
Z. S. Sazonova and R. Singh, “Detection and Correction of Errors with Quantum Tomography,” Proceedings of the SPIE-The International Society for Optical Engineering, Seattle, Vol. 4750, 2002, pp. 47-53.
D. F. V. James, P. G. Kwiat, et al., “Measurement of Qubits,” Physical Review A, Vol. 64, No. 5, 2001, pp. (052312)1-15.
J. Rehacek and Z. Hradil, “Maxent Assisted Maxlik Qu- antum Tomography,” AIP Conference Proceedings, New York, Vol. 707, 2004, pp. 480-489.
V. Buzek, “Quantum Tomography from Incomplete Data via Maxent Principle,” Quantum State Estimation, Lecture Notes in Physics, Springer-Verlag. Vol. 649. 2004, pp. 189-234.
V. Buzek and G. Drobny, “Quantum Tomography via the MaxEnt Principle,” Journal of Modern Optics, Vol. 47, No. 14-15, 2000, pp. 2823-2839.
K. Banaszek, “Maximum-Likelihood Algorithm for Qu- antum Tomography,” Acta Physica Slovaca, Vol. 49, No. 4, 1999, pp. 633-638.
G. M. D’Ariano, M. Rubin, et al., “Quantum Tomography of the GHZ State,” Fortschr Phys, Vol. 48, No. 5-7, 2000, pp. 599-603.
G. L. Long, H. Y. Yan, et al., “Experimental NMR Realization of a Generalized Quantum Search Algorithm,” Physics Letters A, Vol. 286, No. 2-3, 2000, pp. 121-126.
G. L. Long, H. Y. Yan and Y. Sun, “Analysis of Density Matrix Reconstruction in NMR Quantum Computing,” Journal of Optics B, Vol. 3, No. 6, 2001, pp. 376-381.
I. L. Chuang, N. Gershenfeld, et al., “Bulk Quantum Computation with Nuclear Magnetic Resonance: Theory and Experiment,” Proceedings of the Royal Society A, London, Vol. 454, No. 1969, 1998, pp. 447-467.
X. Ji and B. H. Wildenthal, “Effective Interaction for N = 50 Isotones,” Physical Review C, Vol. 37, No. 3, 1988, pp. 1256-1266.
A. Peres, “Quantum Theory: Concepts and Methods,” Kluwer, Dordrecht, 1995.