## 2019 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Bernstein's socks and polynomial-time provable coherence Technical Report 2019. Abstract | Links | BibTeX | Tags: bounded rationality, desirability, Quantum mechanics @techreport{Benavoli2019d, title = {Bernstein's socks and polynomial-time provable coherence}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {https://arxiv.org/abs/1903.04406}, year = {2019}, date = {2019-03-12}, abstract = {We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which non-negativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins. }, keywords = {bounded rationality, desirability, Quantum mechanics}, pubstate = {published}, tppubtype = {techreport} } We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this paper we continue to investigate properties of this class of models. In particular we verify that the space of Bernstein polynomials in which non-negativity is specified by the Krivine-Vasilescu certificate is yet another instance of this theory. As a consequence, we show how it is possible to construct in it a thought experiment uncovering entanglement with classical (hence non quantum) coins. |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Computational Complexity and the Nature of Quantum Mechanics Technical Report 2019. Abstract | Links | BibTeX | Tags: Quantum mechanics @techreport{Benavoli2019bb, title = {Computational Complexity and the Nature of Quantum Mechanics}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco }, url = {https://arxiv.org/abs/1902.04569}, year = {2019}, date = {2019-02-14}, abstract = {Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation. }, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {techreport} } Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation. |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Computational Complexity and the Nature of Quantum Mechanics (Extended version) Technical Report 2019. Abstract | Links | BibTeX | Tags: Quantum mechanics, Sum-of-squares polynomials @techreport{Benavoli2019c, title = {Computational Complexity and the Nature of Quantum Mechanics (Extended version)}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco }, url = {https://arxiv.org/abs/1902.03513}, year = {2019}, date = {2019-02-11}, abstract = {Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation. }, keywords = {Quantum mechanics, Sum-of-squares polynomials}, pubstate = {published}, tppubtype = {techreport} } Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation. |

## 2017 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Bayes + Hilbert = Quantum Mechanics Conference Proceedings of the 14th International Conference on Quantum Physics and Logic (QPL 2017), Nijmegen, The Netherlands, 3-7 July, 2017. Links | BibTeX | Tags: Quantum mechanics @conference{Benavoli2017m, title = {Bayes + Hilbert = Quantum Mechanics}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco }, url = {http://qpl.science.ru.nl/papers/QPL_2017_paper_4.pdf}, year = {2017}, date = {2017-07-05}, booktitle = {Proceedings of the 14th International Conference on Quantum Physics and Logic (QPL 2017), Nijmegen, The Netherlands, 3-7 July}, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {conference} } |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics Journal Article Foundations of Physics, 47 (7), pp. 991–1002, 2017, ISSN: 1572-9516. Abstract | Links | BibTeX | Tags: Quantum mechanics @article{benavoli2016f, title = {A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {http://arxiv.org/abs/1606.03615}, doi = {10.1007/s10701-017-0097-0}, issn = {1572-9516}, year = {2017}, date = {2017-01-01}, journal = {Foundations of Physics}, volume = {47}, number = {7}, pages = {991--1002}, abstract = {Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for n = 2 . The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration.}, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {article} } Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for n = 2 . The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should be excluded from consideration. |

## 2016 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Quantum mechanics: The Bayesian theory generalized to the space of Hermitian matrices Journal Article Physics Review A, 94 , pp. 042106, 2016. Abstract | Links | BibTeX | Tags: Quantum mechanics @article{benavoli2016d, title = {Quantum mechanics: The Bayesian theory generalized to the space of Hermitian matrices}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {http://arxiv.org/abs/1605.08177 }, doi = {10.1103/PhysRevA.94.042106}, year = {2016}, date = {2016-10-01}, journal = {Physics Review A}, volume = {94}, pages = {042106}, publisher = {American Physical Society}, abstract = {We consider the problem of gambling on a quantum experiment and enforce rational behavior by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalized to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalized Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalization (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.}, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {article} } We consider the problem of gambling on a quantum experiment and enforce rational behavior by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalized to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalized Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalization (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers. |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Quantum rational preferences and desirability Inproceedings Proceedings of The 1st International Workshop on “Imperfect Decision Makers: Admitting Real-World Rationality”, NIPS 2016, 2016. Abstract | Links | BibTeX | Tags: Quantum mechanics @inproceedings{benavoli2016h, title = {Quantum rational preferences and desirability}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {http://arxiv.org/abs/1610.06764}, year = {2016}, date = {2016-01-01}, booktitle = {Proceedings of The 1st International Workshop on “Imperfect Decision Makers: Admitting Real-World Rationality”, NIPS 2016}, journal = {ArXiv e-prints 1610.06764}, abstract = {We develop a theory of quantum rational decision making in the tradition of Anscombe and Aumann's axiomatisation of preferences on horse lotteries. It is essentially the Bayesian decision theory generalised to the space of Hermitian matrices. Among other things, this leads us to give a representation theorem showing that quantum complete rational preferences are obtained by means of expected utility considerations. }, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {inproceedings} } We develop a theory of quantum rational decision making in the tradition of Anscombe and Aumann's axiomatisation of preferences on horse lotteries. It is essentially the Bayesian decision theory generalised to the space of Hermitian matrices. Among other things, this leads us to give a representation theorem showing that quantum complete rational preferences are obtained by means of expected utility considerations. |

# Publications

## 2019 |

Bernstein's socks and polynomial-time provable coherence Technical Report 2019. |

Computational Complexity and the Nature of Quantum Mechanics Technical Report 2019. |

Computational Complexity and the Nature of Quantum Mechanics (Extended version) Technical Report 2019. |

## 2017 |

Bayes + Hilbert = Quantum Mechanics Conference Proceedings of the 14th International Conference on Quantum Physics and Logic (QPL 2017), Nijmegen, The Netherlands, 3-7 July, 2017. |

A Gleason-Type Theorem for Any Dimension Based on a Gambling Formulation of Quantum Mechanics Journal Article Foundations of Physics, 47 (7), pp. 991–1002, 2017, ISSN: 1572-9516. |

## 2016 |

Quantum mechanics: The Bayesian theory generalized to the space of Hermitian matrices Journal Article Physics Review A, 94 , pp. 042106, 2016. |

Quantum rational preferences and desirability Inproceedings Proceedings of The 1st International Workshop on “Imperfect Decision Makers: Admitting Real-World Rationality”, NIPS 2016, 2016. |