## 2022 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality Technical Report 2022. Abstract | Links | BibTeX | Tags: Quantum mechanics @techreport{Benavoli2022, title = {Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality}, author = {Benavoli , Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {https://arxiv.org/abs/2203.04124}, year = {2022}, date = {2022-09-05}, abstract = {We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs). This formulation provides a direct interpretation of density matrices as quasi-moment matrices. Using QEOs, we will provide a series of representation theorems, a' la de Finetti, relating a classical probability mass function (satisfying certain symmetries) to a quasi-expectation operator. We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. Although particles indistinguishability is considered a truly "weird" quantum phenomenon, it is not special. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT. Using this connection, we will rederive the first and second quantisation in QT for bosons through the classical statistical concept of exchangeable random variables. Using this approach, we will show how classical reality emerges in QT as the number of identical bosons increases (similar to what happens for finitely exchangeable sequences of rolls of a classical dice).}, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {techreport} } We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs). This formulation provides a direct interpretation of density matrices as quasi-moment matrices. Using QEOs, we will provide a series of representation theorems, a' la de Finetti, relating a classical probability mass function (satisfying certain symmetries) to a quasi-expectation operator. We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. Although particles indistinguishability is considered a truly "weird" quantum phenomenon, it is not special. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT. Using this connection, we will rederive the first and second quantisation in QT for bosons through the classical statistical concept of exchangeable random variables. Using this approach, we will show how classical reality emerges in QT as the number of identical bosons increases (similar to what happens for finitely exchangeable sequences of rolls of a classical dice). |

## 2021 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco The Weirdness Theorem and the Origin of Quantum Paradoxes Journal Article Foundations of Physics, 51 (95), 2021. Abstract | Links | BibTeX | Tags: Quantum mechanics @article{Benavoli2021f, title = {The Weirdness Theorem and the Origin of Quantum Paradoxes}, author = {Benavoli , Alessio and Facchini, Alessandro and Zaffalon, Marco}, url = {https://link.springer.com/article/10.1007/s10701-021-00499-w}, doi = {10.1007/s10701-021-00499-w}, year = {2021}, date = {2021-09-28}, journal = {Foundations of Physics}, volume = {51}, number = {95}, abstract = {We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics.}, keywords = {Quantum mechanics}, pubstate = {published}, tppubtype = {article} } We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics. |

Benavoli, Alessio; Facchini, Alessandro; Zaffalon, Marco Quantum indistinguishability through exchangeable desirable gambles Inproceedings ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR, 2021. Abstract | Links | BibTeX | Tags: desirability, Quantum mechanics @inproceedings{benavoli2021quantum, title = {Quantum indistinguishability through exchangeable desirable gambles}, author = {Alessio Benavoli and Alessandro Facchini and Marco Zaffalon}, url = {https://arxiv.org/abs/2105.04336}, year = {2021}, date = {2021-06-01}, booktitle = {ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR}, abstract = {Two particles are identical if all their intrinsic properties, such as spin and charge, are the same, meaning that no quantum experiment can distinguish them. In addition to the well known principles of quantum mechanics, understanding systems of identical particles requires a new postulate, the so called symmetrization postulate. In this work, we show that the postulate corresponds to exchangeability assessments for sets of observables (gambles) in a quantum experiment, when quantum mechanics is seen as a normative and algorithmic theory guiding an agent to assess her subjective beliefs represented as (coherent) sets of gambles. Finally, we show how sets of exchangeable observables (gambles) may be updated after a measurement and discuss the issue of defining entanglement for indistinguishable particle systems. }, keywords = {desirability, Quantum mechanics}, pubstate = {published}, tppubtype = {inproceedings} } Two particles are identical if all their intrinsic properties, such as spin and charge, are the same, meaning that no quantum experiment can distinguish them. In addition to the well known principles of quantum mechanics, understanding systems of identical particles requires a new postulate, the so called symmetrization postulate. In this work, we show that the postulate corresponds to exchangeability assessments for sets of observables (gambles) in a quantum experiment, when quantum mechanics is seen as a normative and algorithmic theory guiding an agent to assess her subjective beliefs represented as (coherent) sets of gambles. Finally, we show how sets of exchangeable observables (gambles) may be updated after a measurement and discuss the issue of defining entanglement for indistinguishable particle systems. |

## 2019 |

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. |

## 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

## 2022 |

Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality Technical Report 2022. |

## 2021 |

The Weirdness Theorem and the Origin of Quantum Paradoxes Journal Article Foundations of Physics, 51 (95), 2021. |

Quantum indistinguishability through exchangeable desirable gambles Inproceedings ISIPTA'21 Int. Symposium on Imprecise Probability: Theories and Applications, PJMLR, 2021. |

## 2019 |

Computational Complexity and the Nature of Quantum Mechanics 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. |