QM is based on four main axioms, which were derived after a long process of trial and

error. The motivations for the axioms are not always clear and even to experts the basic axioms of QM often

appear counter-intuitive. In a recent paper [1], we have shown that:

- It is possible to derive quantum mechanics from a single principle of self-consistency or, in other

words, that QM laws of Nature are logically consistent; - QM is just the Bayesian theory generalised to the complex Hilbert space.

To obtain these results we have generalised the theory of desirable gambles (TDG) to complex numbers.

TDG was originally introduced by Williams, and later reconsidered by Walley, to justify in a subjective way a

very general form of probability theory.

[**Theory of desirable gambles**]

In classical subjective, or Bayesian, probability, there is a well-established way to check whether the

probability assignments of a certain subject, whom we call Alice, about the result of an uncertain experiment is

valid, in the sense that they are self-consistent. The idea is to use these probability assignments to define

odds—the inverses of probabilities—about the results of the experiment (e.g., Head or Tail in the case of a coin

toss); and then show that there is no way to make Alice a sure loser in the related betting system, that is, to make

her lose money no matter the outcome of the experiment. Historically this is also referred to as the

impossibility to make a Dutch book or that the assessments are coherent; and Alice in these conditions is

regarded as a rational subject. De Finetti [3] showed that Kolmogorov’s probability axioms can

be derived by imposing the principle of coherence alone on a subject’s odds about an uncertain

experiment.

Williams and Walley [8, 7] have later shown that it is possible to justify probability in a simpler and more

elegant way. Their approach is also more general than de Finetti’s, because coherence is defined

purely as logical consistency without any explicit reference to probability (which is also what allows

coherence to be generalised to other domains, such as quantum mechanics); the idea is to work in the

dual space of gambles. To understand this framework, we consider an experiment whose outcome

ω belongs to a certain space of possibilities Ω (e.g., Head or Tail). We can model Alice’s beliefs

about ω by asking her whether *she accepts engaging in certain risky transaction*s, called **gambles**,

whose outcome depends on the actual outcome of the experiment. Mathematically, a gamble is a

bounded real-valued function on Ω, g : Ω →ℝ, which is interpreted as an uncertain reward in a

linear utility scale. If Alice accepts a gamble g, this means that she commits herself to receive g(ω) utiles (euros)

if the outcome of the experiment eventually happens to be the event ω ∈Ω. Since g(ω) can be negative, Alice can

also lose utiles. Therefore Alice’s acceptability of a gamble depends on her knowledge about the

experiment.

The set of gambles that Alice accepts—let us denote it by K —is called her set of desirable gambles. We say

that a gamble g is positive if g≠0 and g(ω) ≥0 for each ω ∈Ω. We say that g is negative if g≠0

and g(ω) ≤0 for each ω ∈Ω. K is said to be coherent when it satisfies the following minimal

requirements:^{2}

- D1
- Any positive gamble g must be desirable for Alice (g∈K), given that it may increase Alice’s capital

without ever decreasing it (accepting partial gain). - D2
- Any negative gamble g must not be desirable for Alice (g∈K), given that it may only decrease

Alice’s capital without ever increasing it (avoiding partial loss). - D3
- If Alice finds g and h to be desirable (g,h ∈K ), then also λg+ νh must be desirable for her

(λg+νh ∈K), for any 0 < λ,ν ∈ℝ (linearity of the utility scale).

In spite of their simple character, these axioms alone define a very general theory of probability. *De Finetti’s*

* (Bayesian) theory is the particular case obtained by additionally imposing some regularity (continuity)*

* requirement and especially completeness, that is, the idea that a subject should always be capable of*

comparing options [7, 8].In this case, probability is derived from K via (mathematical) **duality**.

[**QM**]

In [1] we have extended desirability to QM. To introduce this extension, we first have to define what is a

gamble in a quantum experiment and how the payoff for the gamble is computed. To this end, we consider an

experiment relative to an n-dimensional quantum system and two subjects: the gambler (Alice) and the

bookmaker. The** n-dimensional quantum system** is prepared by the bookmaker in some quantum state. We

assume that Alice has her personal knowledge about the experiment (possibly no knowledge at

all).

- 1.
- The bookmaker announces that he will measure the quantum system along its n orthogonal

directions and so the outcome of the measurement is an element of Ω = {ω_{1},…,ω_{n}}, with ω_{i}

denoting the elementary event “detection along i”. Mathematically, it means that the quantum

system is measured along its eigenvectors, i.e., the projectors and ω_{i}is the event “indicated” by

the i-th projector. - 2.
- Before the experiment, Alice declares the set of gambles she is willing to accept. Mathematically,

a gamble G on this experiment is a n×n Hermitian matrix in ℂ; the space of all Hermitian n×n

matrices is denoted by ℂ_{h}^{n×n}. - 3.
- By accepting a gamble G, Alice commits herself to receive γ
_{i}∈ℝ utiles if the outcome of the

experiment eventually happens to be ω_{i}. The value γ_{i}is defined from G and Π^{*}as follows

Π_{i}^{*}GΠ_{i}^{*}= γ_{i}Π_{i}^{*}for i = 1,…,n. It is a real number since G is Hermitian.

The subset of all positive semi-definite and non-zero (PSDNZ) matrices in ℂ_{h}^{n×n} constitutes the set of positive

gambles, whereas the set of negative gambles is similarly given by all gambles G ∈ℂ_{h}^{n×n}

h such that G ≩ 0. Alice examines the gambles in **ℂ _{h}^{n×n}** and comes up with the subset K of the gambles that she finds desirable. Alice’s

rationality is then characterised by simply applying the

**rational axioms of the theory of desirable gambles**to the

space of hermitian matrices:

- S1
- Any PSDNZ (positive gamble) G must be desirable for Alice (G ∈K), given that it may increase

Alice’s utiles without ever decreasing them (accepting partial gain). - S2
- Any G ≩ 0 (negative gamble) must not be desirable for Alice (G in K ), given that it may only

decrease Alice’s utiles without ever increasing them (avoiding partial loss). - S3
- If Alice finds G and H desirable (G,H ∈K), then also λG + νH must be desirable for her

(λG+νH ∈K), for any 0 < λ,ν ∈ℝ (linearity of the utility scale).

From a geometric point of view, a coherent set of desirable gambles K is a convex cone without its apex and that

contains all PSDNZ matrices (and thus it is disjoint from the set of all matrices such that G ≩ 0). We may also

assume that K satisfies the following additional property:

- S4
- if G ∈K then either G ≰ 0 or G–εI ∈K for some strictly positive real number ε (openness).

This property is not necessary for rationality, but it is technically convenient as it precisely isolates the kind of

models we use in QM (as well as in classical probability) [1]. The openness condition (S4) has a gambling

interpretation too: it means that we will only consider gambles that are strictly desirable for Alice; these are the

gambles for which Alice expects gaining something—even an epsilon of utiles. For this reason, K is called set

of strictly desirable gambles (SDG) in this case.

An SDG is said to be maximal if there is no larger SDG containing it. In [1, Theorem IV.4], we have shown

that maximal SDGs and density matrices are one-to-one. The mapping between them is obtained through the

standard inner product in** ℂ _{h}^{n×n}** , i.e., G⋅R = Tr(G

^{†}R) with G,R ∈ℂ

_{h}

^{n×n}

h via a representation theorem [1, TheoremIV.4].

This result has several consequences. First, it provides a gambling interpretation of the first axiom of QM on

density operators. Second, it shows that density operators are coherent, since the dual of ρ is a valid

SDG. This also implies that QM is self-consistent—a gambler that uses QM to place bets on a

quantum experiment cannot be made a partial (and, thus, sure) loser. Third, the first axiom of QM

on ℂ_{h}^{n×n} is structurally and formally equivalent to Kolmogorov’s first and second axioms about

probabilities on ℝ^{n} [1, Sec. 2]. In fact, they can be both derived via duality from a coherent set of desirable

gambles on ℂ_{h}^{n×n} and, respectively, ℝ^{n}. In [1] we have also derived Born’s rule and the other three

axioms of QM as **a consequence of rational gambling on a quantum experiment ****and show that that**

** measurement, partial tracing and tensor product are equivalent to the probabilistic notions of**

** Bayes’ rule, marginalisation and independence. **Finally, as an additional consequence of the

aforementioned representation result, in [2] we have shown that a subject who uses dispersion-free

probabilities to accept gambles on a quantum experiment can always be made a sure loser: she

loses utiles no matter the outcome of the experiment. We say that dispersion-free probabilities are

incoherent, which means that they are logically inconsistent with the axioms of QM. Moreover, we have

proved that it is possible to derive a stronger version of Gleason’s theorem that holds in any finite dimension (hence even for n = 2),

through much a simpler proof, which states that all coherent

probability assignments in QM must be obtained as the trace of the product of a projector and a density

operator.

A list of relevant bibliographic references, as well as a comparison between our approach and similar

approaches like QBism [4] and Pitowsky’s quantum gambles [6], can be found in [1].

### References

[1] Alessio Benavoli, Alessandro Facchini & Marco Zaffalon (2016): Quantum mechanics: The Bayesian

theory generalized to the space of Hermitian matrices. Phys. Rev. A 94, p. 042106. Available at

https://arxiv.org/pdf/1605.08177.pdf.

[2] Alessio

Benavoli, Alessandro Facchini & Marco Zaffalon (2017): A Gleason-Type Theorem for Any Dimension Based

on a Gambling Formulation of Quantum Mechanics. Foundations of Physics, pp. 1–12, doi:10.H.B1007/H.B

s10701-_017-_0097-_0H.B. Available at https://arxiv.org/pdf/1606.03615.pdf.

[3] B. de Finetti (1937): La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri

Poincaré 7, pp. 1–68. English translation in [5].

[4] Christopher A Fuchs & Ruediger Schack (2013): Quantum-Bayesian coherence. Reviews of Modern

Physics 85(4), p. 1693.

[5] H. E. Kyburg Jr. & H. E. Smokler, editors (1964): Studies in Subjective Probability. Wiley, New York.

Second edition (with new material) 1980.

[6] Itamar Pitowsky (2003): Betting on the outcomes of measurements: a Bayesian theory of quantum

probability. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern

Physics 34(3), pp. 395–414.

[7] P. Walley (1991): Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, New York.

[8] P. M. Williams (1975): Notes on conditional previsions. Technical Report, School of Mathematical and

Physical Science, University of Sussex, UK.