Abstract
Recently, many measures have been put forward to quantify the coherence of quantum states relative to a given basis. We extend the relationship between mutually unbiased basis (MUBs) and quantum coherence to a higher dimension. Results include arbitrary complete sets of MUBs from \(\mathbb {C}^2\) to \(\mathbb {C}^4\), and the form of arbitrary \(2 \times 2\) Unitary matrix and any density matrix of qubit states in respect of complete sets of MUBs. We construct a set of three MUBs by tensor product and further think of complete sets of five MUBs in \(\mathbb {C}^4\). Taking the Bell diagonal state as an example, we analyze the coherence of quantum states under MUBs and calculate the corresponding upper and lower bounds. The results show that in addition to selecting the unbiased basis which is often used, we can consider more sets of MUBs, which may be helpful to the analysis of quantum states.
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References
Bandyopadhyay, S.: A new proof for the existence of mutually unbiased bases. Algorithmica 34(4), 512–528 (2002). https://doi.org/10.1007/s00453-002-0980-7
Ivonovic, I.D.: Geometrical description of quantal state determination. J. Phys. A: Gen. Phys. 14(12), 3241–3245 (1981)
Wootters, W.K.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363–381 (1989)
Spengler, C.: Entanglement detection via mutually unbiased bases. Phys. Rev. A 86(2), 022311 (2012)
Melko, R.G.: Restricted Boltzmann machines in quantum physics. Nat. Phys. 15(9), 887–892 (2019)
Ryan-Anderson, C.: Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X 11(4), 041058 (2021)
Durt, T.: On mutually unbiased bases. Int. J. Quantum. Inform. 8(4), 535–640 (2010)
Horodecki, P.: Five open problems in quantum information theory. PRX Quantum 3(1), 010101 (2022)
Luo, S.: Partial coherence with application to the monotonicity problem of coherence involving skew information. Phys. Rev. A 96(2), 022136 (2017)
Baumgratz, T.: Quantifying coherence. Phys. Rev. Lett. 113(14), 140401 (2014)
Napoli, C.: Robustness of coherence: an operational and observable measure of quantum coherence. Phys. Rev. Lett. 116(15), 150502 (2016)
Chen, B.: Notes on modified trace distance measure of coherence. Quantum Inf. Process. 17(5), 1–9 (2018). https://doi.org/10.1007/s11128-018-1879-9
Pires, D.: Geometric lower bound for a quantum coherence measure. Phys. Rev. A 91(4), 042330 (2015)
Hu, M.-L.: Quantum coherence and geometric quantum discord. Phys. Rep. 762, 1–100 (2018)
Mu, H.: Quantum uncertainty relations of two quantum relative entropies of coherence. Phys. Rev. A 102(2), 022217 (2020)
Luo, Y.: Inequivalent multipartite coherence classes and two operational coherence monotones. Phys. Rev. A 99(4), 042306 (2019)
Lian, Y.: Protocol of deterministic coherence distillation and dilution of pure states. Laser Phys. Lett. 17(8), 085201 (2020)
Ren, R.: Tighter sum uncertainty relations based on metric-adjusted skew information. Phys. Rev. A 104(5), 052414 (2021)
Ye, M.: Operational characterization of weight-based resource quantifiers via exclusion tasks in general probabilistic theories. Quantum Inf. Process. 20(9), 1–28 (2021). https://doi.org/10.1007/s11128-021-03251-5
Vinjanampathy, S.: Quantum thermodynamics. Contemp. Phys. 57(4), 545–579 (2016)
Kosloff, R.: Quantum thermodynamics: a dynamical viewpoint. Entropy 15(6), 2100–2128 (2013)
Lambert, N.: Quantum biology. Nat. Phys. 9(1), 10–18 (2013)
Cao, J.: Quantum biology revisited. Sci. Adv. 6(14), eaaz4888 (2020)
McFadden, J.: The origins of quantum biology. Proc. R. Soc. Lond. Ser. A 474(2220), 20180674 (2018)
Solenov, D.: The potential of quantum computing and machine learning to advance clinical research and change the practice of medicine. Mo. Med. 115(5), 463–467 (2018)
Hassanzadeh, P.: Towards the quantum-enabled technologies for development of drugs or delivery systems. J. Controlled Release 324, 260–279 (2020)
Streltsov, A.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115(2), 020403 (2015)
Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edn. Cambridge University Press, Cambridge (2017)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. Cambridge University Press, New York (2011)
Winter, A.: Operational resource theory of coherence. Phys. Rev. Lett. 116(12), 120404 (2016)
Acknowledgements
This paper was supported by National Science Foundation of China (Grant Nos: 12071271, 11671244, 62001274), the Higher School Doctoral Subject Foundation of Ministry of Education of China (Grant No: 20130202110001) and the Research Funds for the Central Universities (GK202003070). Special thanks to Professor Yuanhong Tao for her enlightening academic report. Useful suggestions given by Dr. Ruonan Ren, Dr. Ping Li, Dr. Mingfei Ye and Yongxu Liu are also acknowledged.
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Ma, X., Li, Y. (2022). Coherence of Quantum States Based on Mutually Unbiased Bases in \(\mathbb {C}^4\). In: Cai, Z., Chen, Y., Zhang, J. (eds) Theoretical Computer Science. NCTCS 2022. Communications in Computer and Information Science, vol 1693. Springer, Singapore. https://doi.org/10.1007/978-981-19-8152-4_3
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DOI: https://doi.org/10.1007/978-981-19-8152-4_3
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