Abstract
Operator-valued frames are natural generalization of frames that have been used in many applied areas such as quantum computing, packets encoding and sensor networks. We focus on developing the theory about operator-valued frame generators for projective unitary representations of finite or countable groups which can be viewed as the theory of quantum channels with group structures. We present new results for operator-valued frames concerning (general and structured) dilation property, orthogonal frames, frame representation and dual frames. Our results are complementary to some of the recent work of Kaftal, Larson and Zhang, and in some cases our treatment is more elementary and transparent.
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Benedetto J J, Colella D. Wavelet analysis of spectrogram seizure chips. In: Proc SPIE Wavelet Appl in Signal and Image Proc. III, vol 2569. San Diego, CA: SPIE, 1994
Bodmann B. Optimal linear transmission by loss-insensitive packet encoding. Appl Comput Harmon Anal, 2007, 22: 274–285
Casazza P, Han D, Larson D. Frames in Banach spaces. Contemp Math Amer Math Soc, 1999, 247: 149–182
Casazza P, Kovavvević J. Equal-norm tight frames with erasures. Adv Comput Math, 2003, 18: 387–430
Casazza P, Kutyniok G, Li S. Fusion frames and distributed processing. Appl Comput Harmon Anal, 2008, 25: 114–132
Choi M D. Completely positive linear maps on complex matrices. Linear Algebr Appl, 1975, 10: 285–290
Deaubechies I. Ten Lectures on Wavelets. Philadephia: SIAM, 1992
Eldar Y, Blcskei H. Geometrically uniform frames. IEEE Trans Inform Theory, 2003, 49: 993–1006
Frank M, Larson D. Frames in Hilbert C*-modules and C*-algebras. J Operator Theory, 2002, 48: 273–314
Gabardo J P, Han D. Frame representations for group-like unitary operator systems. J Operator Theory, 2003, 49: 223–214
Han D. Wandering vector multipliers for unitary gruops. Trans Amer Math Soc, 2001, 353: 3347–3370
Han D. Approximations for Gabor and wavelet frames. Trans Amer Math Soc, 2003, 355: 3329–3342
Han D. Frame representations and Parseval duals with applications to Gabor frames. Trans Amer Math Soc, 2008, 360: 3307–3326
Han D. Dilations and completions for Gabor systems. J Fourier Anal Appl, 2009, 15: 201–217
Han D. The existence of tight Gabor duals for Gabor frames and subspace Gabor frames. J Funct Anal, 2009, 256: 129–148
Han D, Larson D. Frames, Bases and Group Representations. Mem Amer Math Soc, 147. Providence, RI: Amer Math Soc, 2000
Kaftal V, Larson D, Zhang S. Operator-valued frames. Trans Amer Math Soc, 2009, 361: 6349–6385
Kadison R, Ringrose J. Fundamentals of the Theory of Operator Algebras, Vol. I and II. New York: Academic Press, Inc, 1983 and 1985
Kalra D. Complex equiangular cyclic frames and erasures. Linear Algebra Appl, 2006, 419: 373–399
Kribs D. A brief introduction to operator quantum error correction. In: Operator Theory, Operator Algebras, and Applications, Contemp Math, 414. Providence, RI: Amer Math Soc, 2006, 27–34
Lopez J, Han D. Optimal dual frames for erasures. Linear Algebra Appl, 2010, 432: 471–482
Pasieka A, Kribs D, Laflamme R, et al. On the geometric interpretation of single qubit quantum operations on the Bloch sphere. Acta Appl Math, 2009, 108: 697–707
Sarovar M, Milburn G. Optimal estimation of one-parameter quantum channels. J Phys A, 2006, 39: 8487–8505
Sun W. G-frames and g-Riesz bases. J Math Anal Appl, 332: 437–452
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Dedicated to Professor Richard V. Kadison’s 85th Birthday
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Han, D., Li, P., Meng, B. et al. Operator valued frames and structured quantum channels. Sci. China Math. 54, 2361–2372 (2011). https://doi.org/10.1007/s11425-011-4292-8
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DOI: https://doi.org/10.1007/s11425-011-4292-8