Bayes+Hilbert=QM

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 [87] 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 transactions, 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 , 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 g0 and g(ω) 0 for each ω Ω. We say that g is negative if g0
and g(ω) 0 for each ω ΩK 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 (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 [78].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 = {ω1,n}, with ωi
denoting the elementary event “detection along i”. Mathematically, it means that the quantum
system is measured along its eigenvectors, i.e., the projectors and ωi 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 ; the space of all Hermitian n×n
matrices is denoted by hn×n.
3.
By accepting a gamble G, Alice commits herself to receive γi utiles if the outcome of the
experiment eventually happens to be ωi. The value γi is defined from G and Π* as follows
Πi*GΠi*= γiΠi* 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 hn×n constitutes the set of positive
gambles, whereas the set of negative gambles is similarly given by all gambles G hn×n
h such that G 0. Alice examines the gambles in hn×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 (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 GεI ∈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 hn×n , i.e., GR = Tr(GR) with G,R hn×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 hn×n is structurally and formally equivalent to Kolmogorov’s first and second axioms about
probabilities on n [1, Sec. 2]. In fact, they can be both derived via duality from a coherent set of desirable
gambles on hn×n and, respectively, 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.

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