Optimal two-qubit tomography based on local and global measurements: Maximal robustness against errors as described by condition numbers
Open accessAbstract
We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel--Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all the 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized Pauli operators, where twelve of them can be locally measured, and the other four require nonlocal Bell measurements. Our method corresponds to selectively measuring, one by one, all the real and imaginary elements of an unknown two-qubit density matrix. We describe two experimentally feasible setups of this protocol for the optimal reconstruction of two photons in an unknown polarization state using conventional detectors and linear-optical elements. Moreover, we define the operators for the optimal reconstruction of the states of multiqubit or multilevel (qudit) systems.
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Tomography procedures involving some global quantum measurements have increased error robustness relative to using only local qubit measurements, and thus can be completed in less time.
“tomography procedures involving some global quantum measurements have increased error robustness relative to using only local qubit measurements, and thus can be completed in less time.20”
From Scalable on-chip quantum state tomography (2018) · cites this paper as [20]
Section VI (p. 8) says the approach combining local and global measurements is more robust against errors than a local-only approach. Section IV (p. 4) describes its condition-number-based robustness, but the paper does not establish that the procedure can therefore be completed in less time; searches for shorter measurement time and faster tomography did not find that link. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThe experimentally realized device has a condition number of approximately 5, which is better than the condition number of approximately 9 for typical tomography.
“The experimentally realized device has a condition number of ≃5, which is better than the condition number of ≃9 for typical tomography.20”
From Scalable on-chip quantum state tomography (2018) · cites this paper as [20]
The Introduction (p. 1) says that a smaller condition number makes tomography more robust, supporting the direction of the comparison. But the cited paper does not establish a condition number of approximately 5 for the experimentally realized device or approximately 9 for typical tomography; its Experimental Setups section (p. 8) says an experimental realization of its proposed protocol will be presented elsewhere. (LLM, conf=0.78)
Experimental — accuracy not yet measured
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Cites
James, D. F. V., Kwiat, P. G., Munro, W. J., & White, A. G. (2001) · Physical Review A · doi:10.1103/physreva.64.052312 · Cited by 2,176 · cited 2× · see citing works
“[15] based on 16 projectors given explicitly in Appendix A.”
MethodIterative algorithm for reconstruction of entangled states
Řeháček, J., Hradil, Z., & Ježek, M. (2001) · Physical Review A · doi:10.1103/physreva.63.040303 · Cited by 130 · see citing works
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Research works printed but not linked: 7
- [1]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2001).
- [2]M. G. A. Paris and J. ˇ Reh´ aˇ cek (eds.), Quantum State Estimation , Lecture Notes in Physics, Vol. 649 (Springer, Berlin, 2004).
- [43]Yu. I. Bogdanov, S. P. Kulik, E. V. Moreva, I. V. Tikhonov, and A. K. Gavrichenk, 'Optimization of a Quantum Tomography Protocol for Polarization Qubits,' JETP Letters 91 , 686 (2010).
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- [47]G. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1989).
- [50]S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, 'A new proof for the existence of mutually unbiased bases,' Algorithmica 34 , 512 (2002), arXiv:quantph/0103162v3.
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Scalable on-chip quantum state tomography
Titchener, J. G., Gräfe, M., Heilmann, R., Solntsev, A. S., Szameit, A., & Sukhorukov, A. A. (2018) · npj Quantum Information · doi:10.1038/s41534-018-0063-5 · Cited by 61 · see citing works
“The experimentally realized device has a condition number of ' 5, which is better than the condition number of ' 9 for typical to tomography [19].”
Background
- Authors
- Miranowicz, A., Bartkiewicz, K., Peřina, J., Koashi, M., Imoto, N., & Nori, F.
- Affiliations
- RIKEN (JP)
- Adam Mickiewicz University in Poznań (PL)
- Palacký University Olomouc (CZ)
- Photon Science Center of the University of Tokyo (JP)
- The University of Osaka (JP)
- University of Michigan (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2014-12-18
- Volume
- 90
- Issue
- 6
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1990062941
- Cited by
- 57
- References
- 2
- Citations by year
- 2025: 4 · 2024: 8 · 2023: 3 · 2022: 3 · 2021: 5 · 2020: 5 · 2019: 7 · 2018: 4 · 2017: 6 · 2016: 9 · 2015: 3
- FWCI
- 5.98
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Narodowe Centrum Nauki (DEC-2011/03/B/ST2/01903, DEC-2011/02/A/ST2/00305, DEC-2013/11/D/ST2/02638)
- Foundation For Polish Science
- Ministry of Education, Youth and Sports (LO1305)
- Japan Society for the Promotion of Science London (25247068)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- quantum state tomography, condition number, linear inversion, local measurements, Bell measurement, mutually unbiased bases, error resilience, linear optics, qudits, multi-qubit states
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