No-free-information principle in general probabilistic theories

Teiko Heinosaari1, Leevi Leppäjärvi1, and Martin Plávala2

1QTF Centre of Excellence, Department of Physics and Astronomy, University of Turku, Turku 20014, Finland
2Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, Slovakia

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In quantum theory, the no-information-without-disturbance and no-free-information theorems express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. As a particular class of state spaces we consider the polygon state spaces, in which we demonstrate our results and show that while the no-information-without-disturbance principle always holds, the validity of the no-free-information principle depends on the parity of the number of vertices of the polygons.

We have introduced a new information-theoretic principle, the no-free-information principle, and compared it to the previously known no-broadcasting and no-information-without-disturbance principles. The no-broadcasting principle states that information cannot be copied, whilst the no-information-without-disturbance principle states that if we want to obtain any information about a system, we must disturb it, thereby changing the state it is in. The newly introduced no-free-information principle states that there is no type of information that can be learned about a system every time we perform any other measurement on it.

By noting that in quantum theory these principles hold, we formalized this problem in a more general class of theories, namely within the framework of general probabilistic theories (GPTs). These theories form a wide class of operational theories where many of the key features of quantum theory can be formulated more generally. By including both quantum and classical theory, as well as countless toy theories, the study of GPTs allows us to compare these theories based on their respective features and quantify their properties.

In the presented manuscript, we have mathematically characterized the principles in question within the GPT framework, tying our work with previously known results. We show that all three principles are strictly different, which means that there are operationally valid theories (even non-classical theories) where one can obtain non-trivial information about a system without disturbing it and where one can always choose to get some fixed non-trivial information when performing any other measurement. We have presented several examples of theories to demonstrate our claims.

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[1] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher. Noncommuting mixed states cannot be broadcast. Phys. Rev. Lett., 76: 2818–2821, 1996. 10.1103/​PhysRevLett.76.2818.

[2] P. Busch. ``No Information Without Disturbance": Quantum Limitations of Measurement. In J. Christian and W. Myrvold, editors, Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. Springer-Verlag, 2009. 10.1007/​978-1-4020-9107-0_13.

[3] T. Heinosaari and M. Ziman. The Mathematical Language of Quantum Theory. From Uncertainty to Entanglement. Cambridge University Press, 2012. 10.1017/​CBO9781139031103.

[4] H. Barnum, J. Barrett, M. Leifer, and A. Wilce. Cloning and Broadcasting in Generic Probabilistic Theories. 2006. URL http:/​/​​abs/​quant-ph/​0611295.

[5] H. Barnum, J. Barrett, M. Leifer, and A. Wilce. Generalized No-broadcasting theorem. Phys. Rev. Lett., 99: 240501, 2007. 10.1103/​PhysRevLett.99.240501.

[6] G. Chiribella, G. M. D'Ariano, and P. Perinotti. Probabilistic theories with purification. Phys. Rev. A, 81: 062348, 2010. 10.1103/​PhysRevA.81.062348.

[7] H. Barnum and A. Wilce. Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci., 270: 3–15, 2011. 10.1016/​j.entcs.2011.01.002.

[8] C. Pfister and S. Wehner. An information-theoretic principle implies that any discrete physical theory is classical. Nat. Commun., 4: 1851, 2013. 10.1038/​ncomms2821.

[9] G. C. Wick, A. S. Wightman, and E. P. Wigner. The intrinsic parity of elementary particles. Phys. Rev., 88: 101–105, 1952. 10.1103/​PhysRev.88.101.

[10] G. Kimura, K. Nuida, and H. Imai. Distinguishability measures and entropies for general probabilistic theories. Rep. Math. Phys., 66: 175 – 206, 2010. 10.1016/​S0034-4877(10)00025-X.

[11] A. J. Short and S. Wehner. Entropy in general physical theories. New J. Phys., 12: 033023, 2010. 10.1088/​1367-2630/​12/​3/​033023.

[12] I. Namioka and R. Phelps. Tensor products of compact convex sets. Pacific J. Math., 31: 469–480, 1969. 10.2140/​pjm.1969.31.469.

[13] S. N. Filippov, T. Heinosaari, and L. Leppäjärvi. Necessary condition for incompatibility of observables in general probabilistic theories. Phys. Rev. A, 95: 032127, 2017. 10.1103/​PhysRevA.95.032127.

[14] T. Heinosaari and T. Miyadera. Incompatibility of quantum channels. J. Phys. A. Math. Gen., 50: 135302, 2017. 10.1088/​1751-8121/​aa5f6b.

[15] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts. Nonlocal correlations as an information-theoretic resource. Phys. Rev. A, 71: 022101, 2005. 10.1103/​PhysRevA.71.022101.

[16] L. Guerini, J. Bavaresco, M. T. Cunha, and A. Acín. Operational framework for quantum measurement simulability. J. Math. Phys., 58: 092102, 2017. 10.1063/​1.4994303.

[17] M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín. Simulating Positive-Operator-Valued Measures with Projective Measurements. Phys. Rev. Lett., 119: 190501, 2017. 10.1103/​PhysRevLett.119.190501.

[18] S. N. Filippov, T. Heinosaari, and L. Leppäjärvi. Simulability of observables in general probabilistic theories. Phys. Rev. A, 97: 062102, 2018. 10.1103/​PhysRevA.97.062102.

[19] M. Oszmaniec, F. B. Maciejewski, and Z. Puchała. All quantum measurements can be simulated using projective measurements and postselection. 2018. URL http:/​/​​abs/​1807.08449.

[20] M. Plávala. All measurements in a probabilistic theory are compatible if and only if the state space is a simplex. Phys. Rev. A, 94: 042108, 2016. 10.1103/​PhysRevA.94.042108.

[21] P. Janotta, C. Gogolin, J. Barrett, and N. Brunner. Limits on nonlocal correlations from the structure of the local state space. New J. Phys., 13: 063024, 2011. 10.1088/​1367-2630/​13/​6/​063024.

[22] R. T. Rockafellar. Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press, 1997.

Cited by

[1] Abraham Westerbaan, Bas Westerbaan, and John van de Wetering, "The three types of normal sequential effect algebras", Quantum 4, 378 (2020).

[2] Thomas D. Galley and Lluis Masanes, "How dynamics constrains probabilities in general probabilistic theories", Quantum 5, 457 (2021).

[3] Giacomo Mauro D'Ariano, Paolo Perinotti, and Alessandro Tosini, "Information and disturbance in operational probabilistic theories", Quantum 4, 363 (2020).

[4] Sergey N. Filippov, Stan Gudder, Teiko Heinosaari, and Leevi Leppäjärvi, "Operational Restrictions in General Probabilistic Theories", Foundations of Physics 50 8, 850 (2020).

[5] Martin Plávala, "General probabilistic theories: An introduction", arXiv:2103.07469.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-20 05:19:21) and SAO/NASA ADS (last updated successfully 2021-10-20 05:19:22). The list may be incomplete as not all publishers provide suitable and complete citation data.

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