Research ArticleNo Access

Quantum detection problem for fusion frames

A. Rahimi

https://orcid.org/0000-0003-2095-6811

Department of Mathematics, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran

E-mail Address: rahimi@maragheh.ac.ir

Search for more papers by this author

A. Ghasemzadeh

Department of Mathematics, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran

E-mail Address: abdollah-gz@yahoo.com

Search for more papers by this author

B. Daraby

https://orcid.org/0000-0001-6872-8661

Department of Mathematics, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran

E-mail Address: bdaraby@maragheh.ac.ir

Search for more papers by this author

, and

Firdous A. Shah

https://orcid.org/0000-0001-8461-869X

Department of Mathematics, University of Kashmir, South Campus, Anantnag 192 101, Jammu and Kashmir, India

E-mail Address: fashah@uok.edu.in

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219887824500671Cited by:1 (Source: Crossref)

Abstract

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

Keywords:

AMSC: 53D22, 42B10, 42C40, 65R10, 44A35

References

1. F. Akrami, P. G. Casazza, M. A. Hasankhani Fard and A. Rahimi, A note on norm retrievable real Hilbert space frames, J. Math. Anal. Appl. 517 (2023) 126620. Crossref, Web of Science, Google Scholar
2. R. Balan, Equivalence relations and distances between Hilbert frames, Proc. Am. Math. Soc. 127(8) (1999) 2353–2366. Crossref, Web of Science, Google Scholar
3. P. Balazs, D. Bayer and A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Phys. A: Math. Theor. 45(24) (2012) 244023. Crossref, Web of Science, Google Scholar
4. C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phys. Rev. Lett. 68 (1992) 3121–3124. Crossref, Web of Science, Google Scholar
5. S. Botelho-Andrade, P. G. Casazza, D. Cheng, J. Haas and T. T. Tran, The quantum detection problem: a survey, in Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics, Vol. 2, QTS-X/LT-XII, June 2017, Varna, Bulgaria, pp. 337–352 (Springer, Singapore, 2018). Google Scholar
6. S. Botelho-Andrade, P. G. Casazza, D. Cheng and T. T. Tran, The solution to the frame quantum detection problem, J. Fourier Anal. Appl. 25 (2019) 2268–2323. Crossref, Web of Science, Google Scholar
7. S. Botelho-Andrade, P. G. Casazza, D. Cheng, J. Haas and T. T. Tran, The quantum detection problem: A survey, Springer Proc. Math. Stat. 255 (2017) 337–352. Google Scholar
8. P. Busch, P. Lahti, J. P. Pellonpa and K. Ylinen, Quantum Measurement (Springer, Berlin, 2016). Crossref, Google Scholar
9. P. G. Casazza, The art of frame theory, Taiwanese J. Math. 4(2) (2000) 1–127. Crossref, Web of Science, Google Scholar
10. P. G. Casazza, Modern tools for Weyl-Heisenberg (Gabor) frame theory, Adv. Imag. Elect. Phys. 115 (2000) 1–127 Google Scholar
11. P. G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math. Amer. Math. Soc. 345 (2004) 87–113. Crossref, Google Scholar
12. P. G. Casazza and G. Kutyniok, Finite Frames (Birkhauser, 2013). Crossref, Google Scholar
13. O. Christensen, An Introduction to Frames and Riesz Bases (Birkhauser, Boston, 2016). Google Scholar
14. O. Christensen and A. Rahimi, Frame properties of wave packet systems in $L^{2} (R^{d})$ $L^{2} (R^{d})$ , Adv. Comp. Math. 29(2) (2008) 101–111. Crossref, Web of Science, Google Scholar
15. A. Conca, D. Edidin, M. Hering and C. Vinzant, An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal. 38 (2015) 346–356 Crossref, Web of Science, Google Scholar
16. I. Daubechies, A. Grossman and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27(5) (1986) 1271–1283. Crossref, Web of Science, Google Scholar
17. R. J. Duffin and A. C. Schaeffer, A class of non-harmonic Fourier series, Trans. Am. Math. Soc. 72 (1952) 341–366. Crossref, Web of Science, Google Scholar
18. A. Ghaani-Farashahi, Generalized wavelet transforms over finite fields, Linear Multilinear Algeb. 68(8) (2020) 1585–1604. Crossref, Web of Science, Google Scholar
19. A. Ghaani Farashahi, Theoretical frame properties of wave-packet matrices over prime fields, Linear Multilinear Algeb. 65(12) (2017) 2508–2529. Crossref, Web of Science, Google Scholar
20. A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algeb. Appl. 489 (2016) 75–92. Crossref, Web of Science, Google Scholar
21. A. Ghaani Farashahi, Wave packet transform over finite fields, Elect. J. Linear Algeb. 30 (2015) 507–529. Crossref, Web of Science, Google Scholar
22. D. Han, Q. Hu, L. Rui and H. Wang, Quantum injectivity of multi-window Gabor frames in finite dimensions, Ann. Funct. Anal. 13 (2022) 59, https://doi.org/10.1007/s43034-022-00208-2 Crossref, Web of Science, Google Scholar
23. D. Han, Q. Hu and R. Liu, Injective continuous frames and quantum detections, Banach J. Math. Anal. 15 (2021) 1–24. Crossref, Web of Science, Google Scholar
24. D. Han, K. Kornelson and D. Larson, Frames for Undergraduates, Vol. 40 (American Mathematical Society, 2007). Crossref, Google Scholar
25. D. Han and D. R. Larson, Frames, bases and group representations, Memoirs AMS 697 (2000). Google Scholar
26. G. Hong and P. Li, On the continuous frame quantum detection problem, Results Math. 78(2) (2023) 64. Crossref, Web of Science, Google Scholar
27. P. D. Lax and R. Phillips, Scattering Theory (Academic Press, New York, 1967). Google Scholar
28. P. A. Lynn, An Introduction to the Analysis and Processing of Signals, 2nd edn. (Macmillan, London, 1982). Crossref, Google Scholar
29. D. Mclaren, S. Plosker and C. Ramsey, On operator-valued measures, arXiv:1801.00331 (2017). Google Scholar
30. G. J. Murphy, C∗-Algebras and Operator Theory (Academic Press, Cambridge, 1990). Google Scholar
31. M. Paris and J. Rehacek, Quantum State Estimation, Vol. 649 (Springer, Berlin, 2004). Crossref, Google Scholar
32. R. Schatten, Norm Ideals of Completely Continuous Operators, Vol. 27 (Springer, Berlin, 2013). Google Scholar
33. W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322(1) (2006) 437–452. Crossref, Web of Science, Google Scholar
34. A. Smith, C. A. Riofro, B. E. Anderson, H. Sosa-Martinez, I. H. Deutsch and P. S. Jessen, Quantum state tomography by continuous measurement and compressed sensing, Phys. Rev. A. 87 (2013) 030102. Crossref, Web of Science, Google Scholar
35. V. S. Sunder, Operators on Hilbert Space, Vol. 71 (Springer, Berlin/Heidelberg, Germany, 2016). Crossref, Google Scholar
36. K. Zhu, Operator Theory in Function Spaces, (No. 138) (American Mathematical Society, 2007). Crossref, Google Scholar