Minimal orthonormal bases for pure quantum state estimation

Leonardo Zambrano1, Luciano Pereira2, and Aldo Delgado3

1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
2Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
3Instituto Milenio de Investigación en Óptica y Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile

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We present an analytical method to estimate pure quantum states using a minimum of three measurement bases in any finite-dimensional Hilbert space. This is optimal as two bases are insufficient to construct an informationally complete positive operator-valued measurement (IC-POVM) for pure states. We demonstrate our method using a binary tree structure, providing an algorithmic path for implementation. The performance of the method is evaluated through numerical simulations, showcasing its effectiveness for quantum state estimation.

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[1] M. Paris and J. Řeháček, eds., Quantum State Estimation (Springer Berlin Heidelberg, 2004).

[2] D. F. V. James, P. G. Kwiat, W. J. Munro and A. G. White, Measurement of qubits, Phys. Rev. A 64, 052312 (2001).

[3] R. T. Thew, K. Nemoto, A. G. White and W. J. Munro, Qudit quantum-state tomography, Phys. Rev. A 66, 012303 (2002).

[4] I. D. Ivanovic, Geometrical description of quantal state determination, J. Phys. A Math. Theor. 14, 3241 (1981).

[5] W. K. Wootters and B. D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Phys. 191, 363 (1989).

[6] S. N. Filippov and V. I. Man, Mutually unbiased bases: tomography of spin states and the star-product scheme, Phys. Scr. T143, 014010 (2011).

[7] R. B. A. Adamson and A. M. Steinberg, Improving Quantum State Estimation with Mutually Unbiased Bases, Phys. Rev. Lett. 105, 030406 (2010).

[8] G. Lima et al., Experimental quantum tomography of photonic qudits via mutually unbiased basis, Opt. Express 19, 3542 (2011).

[9] J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171 (2004).

[10] S. T. Flammia, A. Silberfarb and C. M. Caves, Minimal informationally complete measurements for pure states, Found. Phys. 35, 1985 (2005).

[11] T. Durt, C. Kurtsiefer, A. Lamas-Linares and A. Ling, Wigner tomography of two-qubit states and quantum cryptography, Phys. Rev. A 78, 042338 (2008).

[12] Z. E. D. Medendorp et al., Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements, Phys. Rev. A 83, 051801 (2011).

[13] N. Bent et al., Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures, Phys. Rev. X 5, 041006 (2015).

[14] J. Eisert et al., Quantum certification and benchmarking, Nat. Rev. Phys. 2, 382 (2020).

[15] J. Chen et al., Uniqueness of quantum states compatible with given measurement results, Phys. Rev. A 88, 012109 (2013).

[16] Q. P. Stefano, L. Rebón, S. Ledesma and C. Iemmi, Set of 4d–3 observables to determine any pure qudit state, Opt. Lett. 44, 2558 (2019).

[17] D. Ha and Y. Kwon, A minimal set of measurements for qudit-state tomography based on unambiguous discrimination, Quantum Inf. Process. 17, 232 (2018).

[18] Y. Wang, Determination of finite dimensional pure quantum state by the discrete analogues of position and momentum (2021), arXiv:2108.05752.

[19] C. Carmeli, T. Heinosaari, J. Schultz and A. Toigo, How many orthonormal bases are needed to distinguish all pure quantum states?, Eur. Phys. J. D 69, 179 (2015).

[20] L.-L. Sun, S. Yu and Z.-B. Chen, Minimal determination of a pure qutrit state and four-measurement protocol for pure qudit state, J. Phys. A Math. Theor. 53, 075305 (2020).

[21] J.-P. Amiet and S. Weigert, Reconstructing a pure state of a spin s through three Stern-Gerlach measurements, Journal of Physics A: Mathematical and General 32, 2777 (1999).

[22] J. Shang, Z. Zhang and H. K. Ng, Superfast maximum-likelihood reconstruction for quantum tomography, Phys. Rev. A 95, 062336 (2017).

[23] D. Goyeneche et al., Five Measurement Bases Determine Pure Quantum States on Any Dimension, Phys. Rev. Lett. 115, 090401 (2015).

[24] C. Carmeli, T. Heinosaari, M. Kech, J. Schultz and A. Toigo, Stable pure state quantum tomography from five orthonormal bases, EPL 115, 30001 (2016).

[25] L. Zambrano, L. Pereira and A. Delgado, Improved estimation accuracy of the 5-bases-based tomographic method, Phys. Rev. A 100, 022340 (2019).

[26] L. Zambrano et al., Estimation of Pure States Using Three Measurement Bases, Phys. Rev. Applied 14, 064004 (2020).

[27] L. Pereira, L. Zambrano and A. Delgado, Scalable estimation of pure multi-qubit states, npj Quantum Inf. 8, 57 (2022).

[28] D. Ahn et al., Adaptive Compressive Tomography with No a priori Information, Phys. Rev. Lett. 122, 100404 (2019a).

[29] D. Ahn et al., Adaptive compressive tomography: A numerical study, Phys. Rev. A 100, 012346 (2019b).

[30] J. Cariñe et al., Multi-core fiber integrated multi-port beam splitters for quantum information processing, Optica 7, 542 (2020).

[31] D. Martínez et al., Certification of a non-projective qudit measurement using multiport beamsplitters, Nat. Phys. 19, 190 (2023).

[32] A. E. Willner, K. Pang, H. Song, K. Zou and H. Zhou, Orbital angular momentum of light for communications, Appl. Phys. Rev. 8, 041312 (2021).

[33] S. Rojas-Rojas et al., Evaluating the coupling efficiency of OAM beams into ring-core optical fibers, Opt. Express 29, 23381 (2021).

[34] D. O. Akat'ev, A. V. Vasiliev, N. M. Shafeev, F. M. Ablayev and A. A. Kalachev, Multiqudit quantum hashing and its implementation based on orbital angular momentum encoding, Laser Phys. Lett. 19, 125205 (2022).

[35] H.-H. Lu et al., Quantum Phase Estimation with Time-Frequency Qudits in a Single Photon, Adv. Quantum Technol. 3, 1900074 (2020).

[36] Y. Chi et al., A programmable qudit-based quantum processor, Nat. Commun. 13, 1166 (2022).

[37] M. Ringbauer et al., A universal qudit quantum processor with trapped ions, Nat. Phys. 18, 1053 (2022).

[38] J. Řeháček et al., Full Tomography from Compatible Measurements, Phys. Rev. Lett. 103, 250402 (2009).

[39] J. Finkelstein, Pure-state informationally complete and ``really'' complete measurements, Phys. Rev. A 70, 052107 (2004).

[40] Y. Wang and Y. Shang,Pure state `really' informationally complete with rank-1 POVM, Quantum Inf. Process. 17, 51 (2018).

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