Abstract
We describe a particular class of pairs of quantum observables which are extremal in the convex set of all pairs of compatible quantum observables. The pairs in this class are constructed as uniformly noisy versions of two mutually unbiased bases (MUB) with possibly different noise intensities affecting each basis. We show that not all pairs of MUB can be used in this construction, and we provide a criterion for determining those MUB that actually do yield extremal compatible observables. We apply our criterion to all pairs of Fourier conjugate MUB, and we prove that in this case extremality is achieved if and only if the quantum system Hilbert space is odd-dimensional. Remarkably, this fact is no longer true for general non-Fourier conjugate MUB, as we show in an example. Therefore, the presence or the absence of extremality is a concrete geometric manifestation of MUB inequivalence, that already materializes by comparing sets of no more than two bases at a time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Heinosaari, T., Miyadera, T., Ziman, M.: An invitation to quantum incompatibility. J. Phys. A 49, 123001 (2016)
Holevo, A.S.: Statistical definition of observable and the structure of statistical models. Rep. Math. Phys. 22, 385–407 (1985)
Ali, S.T., Carmeli, C., Heinosaari, T., Toigo, A.: Commutative POVMs and fuzzy observables. Found. Phys. 39, 593–612 (2009)
Busch, P., Lahti, P., Pellonpää, J.-P., Ylinen, K.: Quantum Measurement. Springer, Switzerland (2016)
Davies, E.B.: Quantum Theory of Open System. Academic Press, London (1976)
Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)
Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Co., Amsterdam (1982)
Heinosaari, T., Schultz, J., Toigo, A., Ziman, M.: Maximally incompatible quantum observables. Phys. Lett. A 378, 1695–1699 (2014)
Heinosaari, T., Kiukas, J., Reitzner, D.: Noise robustness of the incompatibility of quantum measurements. Phys. Rev. A 92, 022115 (2015)
Busch, P., Heinosaari, T., Schultz, J., Stevens, N.: Comparing the degrees of incompatibility inherent in probabilistic physical theories. EPL 103, 10002 (2013)
Banik, M., Gazi, M.R., Ghosh, S., Kar, G.: Degree of complementarity determines the nonlocality in quantum mechanics. Phys. Rev. A 87, 052125 (2013)
Jenčová, A., Plávala, M.: Conditions on the existence of maximally incompatible two-outcome measurements in general probabilistic theory. Phys. Rev. A 96, 022113 (2017)
Gudder, S.: Compatibility for probabilistic theories. Math. Slovaca 66, 449–458 (2016)
Vidal, G., Tarrach, R.: Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999)
Uola, R., Budroni, C., Gühne, O., Pellonpää, J.-P.: One-to-one mapping between steering and joint measurability problems. Phys. Rev. Lett. 115, 230402 (2015)
Cavalcanti, D., Skrzypczyk, P.: Quantitative relations between measurement incompatibility, quantum steering, and nonlocality. Phys. Rev. A 93, 052112 (2016)
Designolle, S., Skrzypczyk, P., Fröwis, F., Brunner, N.: Quantifying measurement incompatibility of mutually unbiased bases. Phys. Rev. Lett. 122, 050402 (2019)
Haapasalo, E.: Robustness of incompatibility for quantum devices. J. Phys. A 48, 255303 (2015)
Haagerup, U.: Orthogonal maximal abelian \(*\)-subalgebras of the \(n\times n\) matrices and cyclic \(n\)-roots. In: Operator algebras and quantum field theory (Rome, 1996), pp. 296–322. International Press, Cambridge (1997)
Carmeli, C., Heinosaari, T., Toigo, A.: Informationally complete joint measurements on finite quantum systems. Phys. Rev. A 85, 012109 (2012)
Uola, R., Luoma, K., Moroder, T., Heinosaari, T.: Adaptive strategy for joint measurements. Phys. Rev. A 94, 022109 (2016)
Carmeli, C., Heinosaari, T., Toigo, A.: Quantum incompatibility witnesses. Phys. Rev. Lett. 122, 130402 (2019)
Guerini, L., Cunha, M.T.: Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements. J. Math. Phys. 59, 042106 (2018)
Busch, P.: Unsharp reality and joint measurements for spin observables. Phys. Rev. D 33, 2253–2261 (1986)
Tadej, W., Życzkowski, K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn. 13, 133–177 (2006)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Durt, T., Englert, B.-G., Bengtsson, I., Życzkowski, K.: On mutually unbiased bases. Int. J. Quantum Inf. 8, 535–640 (2010)
Lang, S.: Algebra, Volume 211 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (2002)
Ivanović, I.D.: Geometrical description of quantal state determination. J. Phys. A 14, 3241–3245 (1981)
Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)
Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)
Carmeli, C., Schultz, J., Toigo, A.: Covariant mutually unbiased bases. Rev. Math. Phys. 28, 1650009 (2016)
Acknowledgements
We thank Leonardo Guerini and Marcelo Terra Cunha for having pointed out their paper [23] to our attention, and for useful discussions during the development of the present manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Paul Busch.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Carmeli, C., Cassinelli, G. & Toigo, A. Constructing Extremal Compatible Quantum Observables by Means of Two Mutually Unbiased Bases. Found Phys 49, 532–548 (2019). https://doi.org/10.1007/s10701-019-00274-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-019-00274-y