General Bayesian theories and the emergence of the exclusivity principle

TL;DR

This paper develops a Bayesian framework for understanding quantum theory and shows that the exclusivity principle naturally emerges from Bayesian consistency, potentially characterizing quantum correlations.

Contribution

It introduces a general Bayesian approach to reconstruct quantum theory and proves the exclusivity principle arises from Bayesian consistency in this framework.

Findings

Ideal experiments satisfy the exclusivity principle in Bayesian theories.

Quantum correlations can be characterized by Bayesian consistency conditions.

The framework unifies classical and quantum probability theories.

Abstract

We address the problem of reconstructing quantum theory from the perspective of an agent who makes bets about the outcomes of possible experiments. We build a general Bayesian framework that can be used to organize the agent's beliefs and update them when new information becomes available. Our framework includes as special cases classical and quantum probability theory, as well as other forms of probabilistic reasoning that may arise in future physical theories. Building on this framework, we develop a notion of ideal experiment, which in quantum theory coincides with the notion of projective measurement. We then prove that, in every general Bayesian theory, ideal experiments must satisfy the exclusivity principle, a property of projective measurements that plays a central role in the characterization of quantum correlations. Our result suggests that the set of quantum correlations may be completely characterized in terms of Bayesian consistency conditions.

Equations

$$p(F|E\beta )=\frac{p(E\cap F|\beta )}{p(E|\beta )}.$$

$$\sum _{l}p(F_{i,l}|\mathcal{B}\beta )=p(E_i|\mathcal{A}\beta )\forall i,\forall \beta \in \mathsf{B}.$$

$$p(F_{i,l}|\mathcal{B}\beta )=p(F_{i,l}|\mathcal{B}{\mathcal{A}}^{\prime }{E}_i\mathcal{A}\beta )p(E_i|\mathcal{A}\beta )$$

$$\sum _{n}p(E_n|{\mathcal{A}}_n\beta )\le 1$$

$$P_{E_i}:=\sum _{j\in E_i}|j⟩⟨j|,$$

$$p(E_i|\rho )=\mathrm{tr}(P_{E_i}\rho ),$$

$$E_i\rho$$

$${\mathcal{M}}_{E_i}:\rho \mapsto {\mathcal{M}}_{E_i}(\rho ):=E_i\rho$$

$$p(F|E\rho )$$

$$=\frac{\mathrm{tr}(P_F{P}_E\rho P_E)}{\mathrm{tr}(P_E\rho )}$$

$$=\frac{\mathrm{tr}(P_E{P}_F{P}_E\rho )}{\mathrm{tr}(P_E\rho )}$$

$$=\frac{\mathrm{tr}(P_{E\cap F}\rho )}{\mathrm{tr}(P_E\rho )}$$

$$=\frac{p(E\cap F|\rho )}{p(E|\rho )},$$

$$p(E_i|\mathcal{A}\rho )=\mathrm{tr}[{\mathcal{A}}_i(\rho )]=:p({\mathcal{A}}_i|\rho ),$$

$${\rho }_i^{\prime }=\frac{{\mathcal{A}}_i(\rho )}{\mathrm{tr}[{\mathcal{A}}_i(\rho )]}.$$

$$\rho$$

$$=P_E\rho P_E$$

$$=P_E\rho P_E/\mathrm{tr}(P_E\rho ),$$

$$\displaystyle P_{\rho,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E_{i},F_{j}):=p\Big{(}F_{j}\Big{|}({\mathcal{B}}\circ{\mathcal{M}}_{E_{i}}\circ{\mathcal{A}})(\rho)\,\Big{)}\,p\Big{(}E_{i}\Big{|}{\mathcal{A}}(\rho)\,\Big{)}$$

$$\displaystyle P_{\rho,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E_{i},F_{j})=\operatorname{tr}\left[P_{F_{j}}{\mathcal{B}}\Big{(}P_{E_{i}}\,{\mathcal{A}}(\rho)\,P_{E_{i}}\big{)}\right]\,,$$

$$({\mathcal{M}}_F\circ \mathcal{B}\circ {\mathcal{M}}_E\circ \mathcal{A})(\rho )=({\mathcal{M}}_F\circ \mathcal{B}\circ \mathcal{A})(\rho ).$$

$$P_{\rho ,\mathcal{A},\mathbb{E},\mathcal{B},\mathbb{F}}(E_i,F_j)=\mathrm{tr}[P_{F_j}\mathcal{B}(P_{E_i}{\rho }^{\prime }{P}_{E_i})],$$

$$\mathrm{tr}[P_F\mathcal{B}(P_{E_i}{\rho }^{\prime }{P}_{E_i})]=0\forall i\ne i_0.$$

$$\sum _j\mathrm{tr}\left[(P_F{B}_j{P}_{E_i}\sqrt{{\rho }^{\prime }})(\sqrt{{\rho }^{\prime }}{P}_{E_i}{B}_j^{†}{P}_F)\right]=0\forall i\ne i_0,$$

$$P_F{B}_j{P}_{E_i}{\sqrt{\rho }}^{\prime }=0\forall j,\forall i\ne i_0$$

$$P_F\mathcal{B}({\rho }^{\prime })P_F$$

$$=\sum _j{P}_F{B}_j{P}_{E_{i_0}}{\rho }^{\prime }{P}_{E_{i_0}}{B}_j^{†}{P}_F$$

$$=P_F\mathcal{B}(P_E{\rho }^{\prime }{P}_E)P_F,$$

$$({\mathcal{M}}_F\circ \mathcal{B})({\rho }^{\prime })=({\mathcal{M}}_F\circ \mathcal{B}\circ {\mathcal{M}}_E)({\rho }^{\prime }).$$

$$p(F_i|\mathcal{C}\beta )=p(E_i|\beta )\forall i,\forall \beta \in \mathsf{B}$$

$$F_i\mathcal{C}\beta =\mathcal{C}{E}_i\beta$$

$$p(F_i|\mathcal{C}\beta )=\sum _{l}p(F_{i,l}|\mathcal{C}\beta )=p(E_i|\beta ),$$

$$p(F_i|\mathcal{C}{E}_i\beta )=p(E_i|E_i\beta )=\frac{p(E_i\cap E_i|\beta )}{p(E_i|\beta )}=1,$$

$$F_i\mathcal{C}{E}_i\beta =\mathcal{C}{E}_i\beta .$$

Full text

*Introduction.—*Quantum theory portrays a world where the outcomes of individual measurements cannot be predicted with certainty. And yet, the quantum predictions are strikingly accurate and explain an astonishingly broad range of phenomena. The reason for this broad applicability still remains controversial. Does the quantum formalism describe how Nature works at the fundamental level? Or it is just a mathematical tool for guessing the outcomes of our experiments?

Albeit with a variety of nuances, different interpretations of quantum theory tend to favor either one or the other view. For example, Everett’s interpretation Everett57 holds that the quantum framework refers to a multitude of universes interfering with each other. On the other hand of the spectrum, QBism, which originally stood for quantum Bayesianism FS13 , views quantum theory as a set of rules that constrain how agents should make bets about the outcomes of their experiments.

Different interpretations are reflected into different ways to understand quantum correlations. Since Bell bell1964einstein , it has been known that quantum correlations are incompatible with the intuitive worldview known as local realism. But intriguingly, the quantum violations of Bell’s inequalities are not maximal: more general theories compatible with relativistic causality could in principle lead to larger violations khalfin1992quantum ; popescu1994quantum ; rastall1985locality . Following up on this observation, various physical principles have been proposed to explain the quantum bounds on correlations brassard2006limit ; linden2007noquantum ; pawlowski2009information ; navascues2010glance ; fritz2013local . Behind this approach lies the idea that the quantum bounds should be explained in terms of principles constraining how Nature behaves. However, this is not the only option. Instead, one could search for principles constraining how agents assign probabilities to the outcomes of their experiments. This approach has remained mostly unexplored so far, partly due to the lack of a suitable framework, and partly due to a widespread belief that quantum correlations require new physical principles. Even within QBism, the Born rule is regarded as “an empirical addition to the laws of Bayesian probability” FS13 , rather than a consequence of Bayesian probability itself.

In this Rapid Communication we demonstrate that a surprisingly large portion of the set of quantum correlations follows directly from elementary Bayesian conditions. We first build a general Bayesian framework, describing the ways in which an agent can update its beliefs when a new piece of information becomes available. The framework assumes only basic Bayesian laws, such as the validity of the rule of conditional probability, and the consistency of beliefs at different moments of time. Surprisingly, we find that these rather minimalistic assumptions imply the validity of the exclusivity principle, a feature of quantum theory that characterizes a large portion of the set of quantum correlations CSW10 ; Cabello13 ; Yan13 ; ATC14 ; Cabello15 ; Henson15 ; AFLS15 ; Cabello18 .

The exact statement of the exclusivity principle will be given later in the paper. For the moment, the crucial observation is that the exclusivity principle does not hold for arbitrary experiments. In quantum theory, it holds for projective measurements, but it fails to hold for certain nonprojective measurements AFLS15 ; CY16 . This means that, in order to formulate the exclusivity principle in a general physical theory, one has first to extend the notion of projective measurement beyond the quantum framework. Such extension is far from straightforward, because there are multiple inequivalent notions that generalize of the notion of projective measurement in quantum theory CY16 . On top of that, once a choice is made, the exclusivity principle may or may not hold, depending on the theory under consideration. For example, Ref. CSW10 showed that, for a certain generalization of the notion of projective measurement, the exclusivity principle is violated by theories that predict superstrong correlations such as Popescu-Rohrlich boxes popescu1994quantum .

Theories that satisfy the exclusivity principle have a remarkable property: under the natural assumption that two statistically independent experiments can be performed in parallel, the correlations arising in every Bell and Kochen-Specker contextuality scenario must be contained in the quantum set Cabello18 . This means that, under a mild assumption, the quantum set for these contextualty scenarios can be characterized as the largest set of correlations compatible with the exclusivity principle.

In this paper we derive the exclusivity principle from Bayesian consistency conditions. We first introduce a general framework for Bayesian theories. We then formulate a general notion of an ideal experiment, which in the special case of quantum theory coincides with the notion of projective measurement. Our first result is that ideal experiments exist in every general Bayesian theory (Theorem 1). Our second result is that the outcome probabilities arising from ideal experiments satisfy the exclusivity principle (Theorem 2). Complemented with results of Ref. Cabello18 , our results imply that the quantum set of correlations for any Bell or Kochen-Specker contextuality scenario can be derived just from elementary consistency conditions on the agent’s probability assignment. This conclusion surpasses the expectations of QBism: rather than being an empirical addition FS13 , the Born rule appears as a consequence of Bayesian consistency conditions alone.

*Beliefs and probabilities.—*Consider the situation of an agent who makes bets about the outcomes of experiments on a given physical system. We start from a basic class of experiments, which we call principal experiments. The outcomes of the principal experiments form a single sample space ${\mathrm{X}}$, equipped with a $\sigma$-algebra of events $\Sigma$, namely a collection of subsets of ${\mathrm{X}}$ satisfying the properties (i) ${\mathrm{X}}\in\Sigma$, (ii) $E\in\Sigma$ implies $({\mathrm{X}}\setminus E)\in\Sigma$, and (iii) $(E_{i})_{i}\subset\Sigma$ implies $\bigcup_{i}E_{i}\in\Sigma$. In typical cases, ${\mathrm{X}}$ is a finite set and $\Sigma$ is the power set of ${\mathrm{X}}$. A principal experiment corresponds to a partition ${\mathbb{E}}=(E_{i})_{i}$ of the sample space ${\mathrm{X}}$ into disjoint events. For brevity, we will identify the experiment and the corresponding partition.

In making a bet, the agent will rely on its beliefs, including beliefs on the laws of physics, or beliefs on the prior history of the physical system involved in the bet. We denote by ${\mathsf{B}}$ the set of all possible beliefs. For a given belief $\beta$, the agent assigns a probability distribution $p:E\mapsto p(E|\beta)$ satisfying the usual conditions (i) $p(E|\beta)\geq 0$ for all events $E$, (ii) $p(\bigcup_{i}E_{i}|\beta)=\sum_{i}p(E_{i}|\beta)$ whenever all $E_{i}$ are mutually disjoint, and (iii) $p({\mathrm{X}}|\beta)=1$. Here we make the standard assumption that the probability of an event $E$ is independent of the specific experiment ${\mathbb{E}}$ in which $E$ arises.

We stress that the belief $\beta$ determines the probability assignment $p(E|\beta)$, but not vice versa: in general, a belief contains much more than just the outcome probabilities of principal experiments. For example, we will see that in quantum theory a belief is described by a density matrix, while the principal experiments are commuting projective measurements, corresponding to projectors that are diagonal with respect of a fixed standard basis. The probability assignment $p(E|\beta)$ depends only on the diagonal entries of the density matrix $\beta$, and therefore it is not sufficient to determine the whole matrix.

*Bayesian updates.—*Suppose that the agent receives a guarantee that an event $E$ is the case. As a consequence, the agent will update its old belief $\beta$ into a new belief, which we denote as $\beta^{\prime}=E\beta$. Again, we make the standard assumption that the new belief is independent of the specific experiment in which $E$ arises.

The point of updating beliefs is to compute conditional probability distributions. We demand that the probability assignment for the updated belief $E\beta$ is given by the rule of conditional probabilities:

[TABLE]

The rule of conditional probabilities implies several properties of the update map $\beta\mapsto E\beta$. For example, it implies that, once the agent updates its belief based on the event $E$, the agent becomes certain of the event $E$. Indeed, one has $p(E|E\beta)=1$, which follows from letting $E=F$ in Eq. (1).

We stress that the update map does not represent a physical process on the observed system, but rather an operation internal to the agent. In the following, we formulate two conditions that the update should satisfy in order for the agent to be consistent with its beliefs.

*Forward consistency.—*Suppose that the agent is certain of the event $E$, namely $p(E|\beta)=1$. In this case, a guarantee that $E$ is the case does not add any new information, and should not lead to any update:

Forward consistency constrains the agent in how it updates the belief forward in time. A simple consequence of this constraint is that the total event $E={\mathrm{X}}$ does not lead to any update, namely ${\mathrm{X}}\beta=\beta,\,\forall\beta\in{\mathsf{B}}$.

*Actions.—*So far, we considered the situation where an agent bets directly on the occurrence of a certain event. More generally, the conditions under which a bet is made can be altered by some action. We use “action” broadly, including situations in which the agent lets the system evolve under its natural dynamics.

We denote the set of all actions as ${\mathsf{Act}}$, and we assume that it is a monoid, meaning that (i) actions can be composed with one another, (ii) the composition is associative, and (iii) there exists an identity action kraemer2018operational ; chiribella2018agents . When an action ${\mathcal{A}}$ is performed, the agent generally changes its belief to a new belief $\beta^{\prime}={\mathcal{A}}\beta$. We assume that the belief change satisfies the conditions (i) $({\mathcal{A}}{\mathcal{B}})\beta={\mathcal{A}}\,({\mathcal{B}}\beta),\forall\beta\in{\mathsf{B}}\,,\forall{\mathcal{A}},{\mathcal{B}}\in{\mathsf{Act}}$, and (ii) ${\mathcal{I}}\beta=\beta\,,\forall\beta\in{\mathsf{B}}$, where ${\mathcal{I}}$ is the identity action.

*Backward consistency.—*When actions are included in the picture, a new class of sequential experiments arise. A sequential experiment consists of a sequence of actions interspersed by principal experiments. For example, $({\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}})$ represents a sequence consisting of an action ${\mathcal{A}}$, followed by a principal experiment ${\mathbb{E}}$, by another action ${\mathcal{B}}$, and by another principal experiment ${\mathbb{F}}$. Crucially, the belief $\beta$ supplies the agent with a joint probability distribution $P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E,F):=p(F|{\mathcal{B}}E{\mathcal{A}}\beta)\,p(E|{\mathcal{A}}\beta)$ if $p(E|{\mathcal{A}}\beta)\neq 0$, and zero otherwise.

When the marginal probability ${P}_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(F):=\sum_{E\in\mathbb{E}}P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E,F)$ is non-zero, one can define the conditional probability $P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E|F):=P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E,F)/P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(F)$, which quantifies the agent’s confidence in retrodicting that the earlier outcome must have been $E$, given that the later outcome is $F$. We say that the event $F$ is unaffected by the experiment $\mathbb{E}$ if the probability of $F$ in the sequential experiment $({\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}})$ coincides with the probability of $F$ in the sequential experiment $({\mathcal{A}},{\mathcal{B}},{\mathbb{F}})$. In formula, $P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(F)=P_{\beta,{\mathcal{A}},{\mathcal{B}},{\mathbb{F}}}(F):=p(F|{\mathcal{B}}{\mathcal{A}}\beta)$.

If $P_{\beta,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E|F)=1$ and $F$ is unaffected by the experiment $\mathbb{E}$, we say that the event $F$ implies the event $E$. In this case, it is natural to require that the update for event $E$ it is already included in the update for event $F$. This idea motivates the following axiom:

Axioms 1, 2, and 3 define a set of theories, which we call general Bayesian theories (GBTs). In Appendix A we show that quantum theory is an example of GBT, with the principal experiments corresponding to projective measurements on a fixed standard basis and the updates following Lüders’ rule.

*Ideal experiments.—*We now show that every GBT contains a special class of ideal experiments that leave the agent with the option of gathering more refined information in the future.

We say that an experiment $({\mathcal{B}},{\mathbb{F}})$ with partition ${\mathbb{F}}=(F_{i,l})_{i,l}$ is a refinement of another experiment $({\mathcal{A}},{\mathbb{E}})$ with partition ${\mathbb{E}}=(E_{i})_{i}$ if

[TABLE]

The experiment $({\mathcal{A}},{\mathbb{E}})$ is sequentially refinable if there exists an action ${\mathcal{A}}^{\prime}$ such that, for every refinement $({\mathcal{B}},{\mathbb{F}})$ and for every initial belief $\beta$, the probability of the event $F_{i,l}$ in the experiment $({\mathcal{B}},{\mathbb{F}})$ is equal to the joint probability of the events $(E_{i},F_{i,l})$ in the sequential experiment $({\mathcal{A}},{\mathbb{E}},{\mathcal{A}}^{\prime},{\mathcal{B}},{\mathbb{F}})$. In formula,

[TABLE]

for every refinement $({\mathcal{B}},{\mathbb{F}})$, for every event $F_{i,l}\in{\mathbb{F}}$, and for every belief $\beta\in{\mathsf{B}}$ with $p(E_{i}|{\mathcal{A}}\beta)\neq 0$. In other words, the coarse-grained experiment $({\mathcal{A}},{\mathbb{E}})$ does not alter the probability assignment for the fine-grained experiment $({\mathcal{B}},{\mathbb{F}})$, provided that the agent performs the action ${\mathcal{A}}^{\prime}$ between them. Intuitively, if an experiment does not disturb any of its refinements then this experiment alters the belief as little as possible, this being a central property of projective measurements in quantum theory CY14 ; CY16 ; MK14 .

Consider now the family of experiments of the form $({\mathcal{A}},{\mathbb{E}})$ where the action ${\mathcal{A}}$ is fixed and the partition ${\mathbb{E}}$ is variable. If there exists an action ${\mathcal{A}}^{\prime}$ such that condition (31) holds for all partitions ${\mathbb{E}}$, then we call each experiment $({\mathcal{A}},{\mathbb{E}})$ ideal. Such experiments exist in every theory:

The theorem is proved in Appendix B. In Appendix C, we prove that the set of ideal experiments in quantum theory coincides with the set of projective measurements.

The emergence of the exclusivity principle.— The exclusivity principle, as originally introduced in the literature CSW10 ; Cabello13 ; Yan13 ; ATC14 ; Cabello15 ; Henson15 ; AFLS15 , refers to scenarios where multiple alternative experiments share some of their outcome, as in the example of Figure 1. Let $\cal S$ be the set of experiments under consideration, and, for each experiment $\cal E\in\cal S$, let ${O}_{\cal E}$ be the set of its possible outcomes. The outcomes of all experiments are assigned probabilities, with the constraint that the probabilities should sum up to 1 for every experiment, namely $\sum_{x\in{O}_{\cal E}}p(x)=1,\,\forall\cal E\in\cal S$. A set of outcomes $O$, possibly belonging to different experiments, is called pairwise exclusive if, for every pair of outcomes $\{x_{1},x_{2}\}\subseteq O$ there exist an experiment ${\cal E}\in\cal S$ such that both $x_{1}$ and $x_{2}$ belong to ${O}_{\cal E}$. A set of experiments $\cal S$ satisfies the exclusivity principle if the condition $\sum_{x\in O}p(x)\leq 1$ holds for every set $O$ of pairwise exclusive outcomes. This is a non-trivial property, and is not satisfied by all probability assignments. In the example of Figure 1, the assignment $p(a)=1/2$, $p(b)=p(d)=3/8$, $p(c)=p(f)=1/8$, $p(e)=1/4$ satisfies the condition $p(a)+p(d)+p(f)=p(d)+p(b)+p(e)=p(a)+p(b)+p(c)=1$ but violates the exclusivity principle.

We now show how to translate the exclusivity principle in the language of GBTs. The principle applies to outcomes of ideal experiments. By an outcome of an ideal experiment $\cal E=({\mathcal{A}},\mathbb{E})$ we mean a pair $x=({\mathcal{A}},E)$ with $E\in\mathbb{E}$. The set of outcomes of the experiment $\cal E$ is then $O_{\cal E}:=\{({\mathcal{A}},E)~{}|~{}E\in\mathbb{E}\}$.

To say that two experiments share an outcome, as in Figure 1, we need a criterion to identify outcomes of different experiments. For two experiments $\cal E=({\mathcal{A}},\mathbb{E})$ and $\cal E^{\prime}=({\mathcal{A}}^{\prime},\mathbb{E}^{\prime})$, we say that two outcomes $({\mathcal{A}},E)\in O_{\cal E}$ and $({\mathcal{A}}^{\prime},E^{\prime})\in O_{\cal E^{\prime}}$ are equivalent, denoted $({\mathcal{A}},E)\simeq({\mathcal{A}}^{\prime},E^{\prime})$, if they have the same probability for every possible belief, namely $p(E|{\mathcal{A}}\beta)=p(E^{\prime}|{\mathcal{A}}^{\prime}\beta)\,,\forall\beta\in{\mathsf{B}}$. Diagrams like the one in Figure 1 arise when equivalent outcomes are identified.

Two outcomes $({\mathcal{A}},E)\in O_{\cal E}$ and $({\mathcal{A}}^{\prime},E^{\prime})\in O_{\cal E^{\prime}}$, possibly arising from different experiments ${\mathcal{E}}\not={\mathcal{E}}^{\prime}$, are mutually exclusive if $({\mathcal{A}},E)$ and $({\mathcal{A}}^{\prime},E^{\prime})$ are equivalent to two distinct outcomes of a single experiment $\cal F$, namely $({\mathcal{A}},E)\simeq x$ and $({\mathcal{A}}^{\prime},E^{\prime})\simeq x^{\prime}$ for $\{x,x^{\prime}\}\subseteq O_{\cal F}$ and $x\not=x^{\prime}$. A set of outcomes $O$ (of generally different experiments) is pairwise exclusive if, for every pair of outcomes $\{x,x^{\prime}\}\subset O$, $x$ and $x^{\prime}$ are mutually exclusive.

We are now ready to translate the exclusivity principle in the framework of general Bayesian theories. The principle states that for every set $O=\{({\mathcal{A}}_{n},E_{n})\}$ of pairwise exclusive outcomes of ideal experiments, the condition

[TABLE]

must be satisfied for every belief $\beta\in{\mathsf{B}}$.

The central result of our paper is that the bound (4) holds in every GBT:

The proof is based on the refinability property of ideal experiments, which allows us to construct a sequence of binary experiments with the property that a subset of outcomes is equivalent to the original set of pairwise exclusive outcomes. Since all the outcomes in the pairwise exclusive set can be turned into outcomes of a single measurement, the bound (4) then follows from the normalization of the probability distribution for that measurement. The details of the proof are provided in Appendix D.

Combining our derivation of the exclusivity principle with the result of Ref. Cabello18 , we obtain that the set of quantum correlations can be characterized completely in terms of Bayesian consistency conditions:

The proof of Corollary 1 follows from a proof in Cabello18 , by replacing Lemma 1 in Cabello18 with our Theorem 2 and Assumption 2 in Cabello18 with the assumption that any two experiments can be performed in parallel with statistically independent joint distribution.

*Conclusion.—*No interpretation of quantum theory can be considered satisfactory unless one can derive the formalism of the theory from the key elements of the interpretation Fuchs . This problem is especially pressing for QBism, which views quantum theory as a set of rules that constrain how agents make bets about the outcomes of their experiments. Can the quantum formalism be obtained from Bayesian consistency conditions alone?

To address this question, we have introduced a framework for general Bayesian theories. The core of our framework are three consistency conditions on how the agent updates its beliefs: the law of conditional probabilities, forward consistency, and backward consistency. These conditions are sufficient to prove the existence of a set of ideal experiments, which in quantum theory coincide with the projective measurements. We showed that the correlations arising from ideal experiments must satisfy the exclusivity principle, which implies that the set of correlations for every Bell or Kochen-Specker contextuality scenario is equal to the corresponding set in quantum theory, provided that any two statistically independent ideal experiments admit a joint realization.

Our result shows that the set of quantum correlations in Bell and Kochen-Specker contextuality scenarios can be derived from elementary consistency rules on how an agent should bet on the outcomes of future experiments. The problem of whether the whole quantum formalism can be recovered solely from Bayesian consistency conditions is still open. Unlike earlier reconstructions of quantum theory Hardy01 ; CDP11 ; DB11 ; MM11 ; Hardy11 ; Wilce ; HW that adopted an interpretation agnostic approach, a fully Bayesian reconstruction would have the potential to shed new light on the interpretation of quantum theory.

Acknowledgments.— We thank Debbie Leung, Hans J. Briegel, Časlav Brukner, Christopher A. Fuchs, Rüdiger Schack, and Matthew Pusey for discussions during the development of this work. This work is supported by the Foundational Questions Institute through grant FQXi-RFP3-1325, the National Natural Science Foundation of China through grant 11675136, the Hong Kong Research Grant Council through grant 17300317, the Spanish Ministry of Science, Innovation and Universities (MICINN), the Spanish Ministry of Economy, Industry and Competitiveness (MINECO), the European Regional Development Fund FEDER through Grants No. FIS2017-89609-P and No. FIS2015-67161-P, the ERC (Starting Grant No. 258647/GEDENTQOPT and Consolidator Grant 683107/TempoQ), and the Basque Government (Grant No. IT986-16). This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

Appendix A Quantum theory as a general Bayesian theory

Here we specify how quantum theory fits in the framework of general Bayesian theories.

A.1 Principal experiments, beliefs, and updates

For a quantum system with Hilbert space ${\cal H}={{\mathbb{C}}}^{d}$, the sample space is the set ${\mathrm{X}}=\{1,\dots,d\}$, and the possible events are subsets of ${\mathrm{X}}$.

Any given partition of ${\mathrm{X}}$ into disjoint subsets, denoted by ${\mathbb{E}}=(E_{i})_{i}$, is associated to a projective measurement with orthogonal projectors $\{P_{E_{i}}\}$, defined as

[TABLE]

where $\{|j\rangle\}$ is a fixed orthonormal basis for the Hilbert space $\cal H$. In other words, the principal experiments are projective measurements that are diagonal in a fixed basis. We call these measurements the principal measurements.

The set of beliefs is the set of density operators ${\mathsf{B}}:=\{\rho\in L({\cal H})~{}|~{}\rho\geq 0\,,\operatorname{tr}(\rho)=1\}$. For the principal experiment ${\mathbb{E}}=(E_{i})_{i}$, the probability assigned to outcome $i$ is

[TABLE]

where $\rho\in{\mathsf{B}}$ is the initial belief. If the outcome $i$ is guaranteed to be the case, and if $p(E_{i}|\rho)\not=0$, then the belief $\rho$ is updated according to Lüders’ rule

[TABLE]

We call the non-linear map

[TABLE]

the update map. Note that the update map is only defined for $\operatorname{tr}(P_{E_{i}}\,\rho)\not=0$.

Note that Axiom 1 holds: for every pair of events $E\in{\mathrm{X}}$ and $F\in{\mathrm{X}}$ such that $p(E|\rho)\not=0$, one has

[TABLE]

where the first equality follows from Eqs. (6) and (8) with $E_{i}=E$, the second equality follows from Eq. (7) with $E_{i}=E$, the third equality follows from the ciclic property of the trace, the fourth equality follows from the relation $P_{E}P_{F}P_{E}=P_{E\cap F}$, and the last equality follows from Eq. (6), applied with $E_{i}=E$ and with $E_{i}=E\cap F$.

A.2 Actions and composite experiments

The set of actions ${\mathsf{Act}}$ is the set of all possible quantum channels (completely positive trace-preserving maps) from $L({\cal H})$ to itself. The action of a channel ${\mathcal{A}}$ on an input state $\rho$ can be written in the Kraus representation ${\mathcal{A}}(\rho)=\sum_{i}A_{i}\rho A_{i}$, where $(A_{i})_{i}\subset L(\cal H)$ is a set of Kraus operators, satisfying the normalization condition $\sum_{i}A_{i}^{\dagger}A_{i}=I$. The quantum channels form a monoid, with the identity channel ${\mathcal{I}}:\rho\mapsto{\mathcal{I}}(\rho)=\rho$ being the identity element.

A composite experiment $({\mathcal{A}},\mathbb{E})$ is described by a set of completely positive trace non-increasing maps $({\mathcal{A}}_{i})_{i}$, defined by the relation ${\mathcal{A}}_{i}(\rho):=P_{E_{i}}{\mathcal{A}}(\rho)P_{E_{i}}$, $\forall\rho$. The probability assigned to the outcome $i$ is

[TABLE]

and the updated state is

[TABLE]

Note that the sum $\sum_{i}{\mathcal{A}}_{i}$ is a trace-preserving map, as it follows from Equation (9). In general, a set of completely positive maps $({\mathcal{A}}_{i})_{i}$ such that $\sum_{i}{\mathcal{A}}_{i}$ is trace-preserving is called a quantum instrument.

The set of actions includes all unitary channels ${\mathcal{U}}$ of the form ${\mathcal{U}}(\rho)=U\rho U^{\dagger}$, where $U$ is a generic unitary operator. Unitary channels represent the reversible actions in quantum theory. Combined with the principal measurements, the unitary channels generate all projective measurements on the given quantum system.

A.3 Forward consistency

Let $\rho$ be a density matrix, $E\subseteq\{1,\dots,d\}$ be an event, $P_{E}=\sum_{j\in E}|j\rangle\langle j|$ be the corresponding projector, and $\overline{P}_{E}:=I-P_{E}$ be its orthogonal complement. The condition that the event $E$ happens with certainty on $\rho$ is $\operatorname{tr}(P_{E}\rho)=1$. Forward consistency is the statement that, under this condition, the updated state ${\mathcal{M}}_{E}(\rho)=P_{E}\rho P_{E}/\operatorname{tr}(P_{E}\rho)$ is equal to the original state $\rho$.

The proof is simple. The certainty condition $\operatorname{tr}(P_{E}\rho)=1$ is equivalent to $\operatorname{tr}(\overline{P}_{E}\rho\overline{P}_{E})=0$. Since this equation is of the form $\operatorname{tr}(AA^{\dagger})=0$ with $A=\overline{P}_{E}\sqrt{\rho}$, we have $A=0$ and thus $\overline{P}_{E}\sqrt{\rho}=0$ and $\sqrt{\rho}\,\overline{P}_{E}=0$. This leads to

[TABLE]

the last equality following from the certainty condition $\operatorname{tr}(P_{E}\rho)=1$. Since $P_{E}\rho P_{E}/\operatorname{tr}(P_{E}\rho)$ is the updated state through event $E$ [cf. Equation (7)], the above relation can be rewritten as $\rho=E\rho$. Hence, quantum theory satisfies forward consistency.

A.4 Backward consistency

Backward consistency refers to sequential experiments of the form $({\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}})$, where ${\mathcal{A}}$ and ${\mathcal{B}}$ are two actions, and ${\mathbb{E}}$ and ${\mathbb{F}}$ are two partitions of the sample space ${\mathrm{X}}$. In the quantum case, the two actions ${\mathcal{A}}$ and ${\mathcal{B}}$ are quantum channels, and ${\mathrm{X}}=\{1,\dots,d\}$.

For a given density matrix $\rho$, the joint probability distribution of the events $E_{i}\in{\mathbb{E}}$ and $F_{j}\in{\mathbb{F}}$ is

[TABLE]

for p\big{(}E_{i}|{\mathcal{A}}(\rho)\,\big{)}\not=0, and ${\sf Prob}(E_{i},F_{j})=0$ otherwise.

Using Eqs. (6) and (7), the joint probability distribution can be expressed as

[TABLE]

both for p\big{(}E_{i}|{\mathcal{A}}(\rho)\,\big{)}\not=0 and for p\big{(}E_{i}|{\mathcal{A}}(\rho)\,\big{)}=0.

Let $E=E_{i_{0}}$ and $F=F_{j_{0}}$ be two given events in $\mathbb{E}$ and $\mathbb{F}$, respectively. In the following, we will show that $P_{\rho,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E_{i},F_{j})(E|F)=1$ implies the condition

[TABLE]

By proving this condition, we will prove that quantum theory satisfies backward consistency (in fact, we will prove a slightly stronger result, since the condition (14) holds without the need of imposing that the event $F$ is unaffected by the experiment $\mathbb{E}$).

Let us prove condition (14). For brevity, we define $\rho^{\prime}:={\mathcal{A}}(\rho)$, so that Eq. (13) becomes

[TABLE]

The condition $P_{\rho,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E|F)=1$ is equivalent to the condition $P_{\rho,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E_{i}|F)=0$ for every $i\not=i_{0}$. Hence, we also have $P_{\rho,{\mathcal{A}},{\mathbb{E}},{\mathcal{B}},{\mathbb{F}}}(E_{i},F)=0$ for every $i\not=i_{0}$, namely

[TABLE]

Using a Kraus representation ${\mathcal{B}}(\cdot)=\sum_{j}B_{j}\cdot B_{j}^{\dagger}$ and the relation $P_{E}=P_{E}^{2}$, we obtain

[TABLE]

which implies

[TABLE]

Using this relation, we obtain

[TABLE]

or equivalently,

[TABLE]

Recalling the definition $\rho^{\prime}:={\mathcal{A}}(\rho)$, we then obtain the desired Eq. (14).

Appendix B Proof of Theorem 1

In this section, we adopt the following definition:

Definition 1**.**

An action ${\mathcal{A}}$ is ideal if the experiment $({\mathcal{A}},\mathbb{E})$ is ideal for every partition $\mathbb{E}$.

Using this definition, Theorem 1 of the main text can be reformulated as

**Theorem

2’****.**

The identity action is ideal.

To prove this statement, we will use the following lemma:

Lemma 1**.**

Let $({\mathcal{C}},{\mathbb{F}})$ be a refinement of $({\mathcal{I}},{\mathbb{E}})$, with ${\mathbb{F}}=(F_{i,l})_{i,l}$ and ${\mathbb{E}}=(E_{i})_{i}$, and let $F_{i}$ be the event defined by $F_{i}:=\bigcup_{l}F_{i,l}$. Then,

[TABLE]

and the equality

[TABLE]

holds for every belief $\beta\in B$ and for every for every outcome $i$ such that $p(E_{i}|\beta)\neq 0$.

Proof. Since $({\mathcal{C}},{\mathbb{F}})$ is a refinement of $({\mathcal{I}},{\mathbb{E}})$, one has

[TABLE]

where the first equality follows from the additivity of the probabilities of disjoint events. Hence, Eq. (21) holds.

Now, suppose that one has $p(E_{i}|\beta)\neq 0$. Applying Eq. (21) to the initial belief $E_{i}\beta$, we obtain the condition

[TABLE]

the second equality following from Eq. (1) in the main text. Since the agent is certain that the event $F_{i}$ will occur, forward consistency implies that no update should take place on the belief ${\mathcal{C}}E_{i}\beta$. Hence, we obtain the relation

[TABLE]

The next step is to apply backward consistency. Consider the joint probability distribution $P_{\beta,{\mathbb{E}},{\mathcal{C}},{\mathbb{F}}}(E_{j},F_{k}):=p(F_{k}|{\mathcal{C}}E_{j}\beta)\,p(E_{j}|\beta)$. For every pair $(E_{j},F_{k})$ such that $p(E_{j}|\beta)\not=0$, one has

[TABLE]

where the second equality follows from Eq. (23), the third equality follows from the rule of conditional probabilities (Axiom 1 in the main text), the fourth equality follows from the fact that the events $(E_{j})_{j}$ are mutually disjoint.

Eq. (26) implies that the experiment $\mathbb{E}$ does not affect the probability of the event $F_{i}$:

[TABLE]

where the fourth equality follows from Eq. (26), and the fifth equality follows from Eq. (23).

Moreover, Eq. (26) implies the condition $P_{\beta,{\mathbb{E}},{\mathcal{C}},{\mathbb{F}}}(E_{i}|F_{i})=1$. Note that the conditional probability is well defined because the marginal $P_{\beta,{\mathbb{E}},{\mathcal{C}},{\mathbb{F}}}(F_{i})$ is non-zero. Indeed, Eq. (27) implies the condition $P_{\beta,{\mathbb{E}},{\mathcal{C}},{\mathbb{F}}}(F_{i})=p(F_{i}|{\mathcal{C}}\beta)=p(E_{i}|\beta)\not=0$.

Hence, the hypotheses for the applications of backward consistency are satisfied. By applying backward consistency, we obtain the relation

[TABLE]

Combining Eqs. (25) and (28), we obtain Eq. (22). ∎

We are now ready to provide the proof of Theorem 2’.

Proof of Theorem 2’. For an arbitrary partition ${\mathbb{E}}=(E_{i})_{i}$ we consider an arbitrary refinement $({\mathcal{C}},{\mathbb{F}})$ of $({\mathcal{I}},{\mathbb{E}})$ with ${\mathbb{F}}=(F_{i,l})_{i,l}$. Let $F_{i}$ be the event defined by $F_{i}:=\bigcup_{l}F_{i,l}$. Using the rule of conditional probabilities, we obtain for $p(F_{i}|{\mathcal{C}}\beta)\neq 0$,

[TABLE]

the last equality following from Eqs. (22) and (21). If $p(F_{i}|{\mathcal{C}}\beta)=0$, then Eq. (21) implies $p(E_{i}|\beta)=0$. Eq. (29) is exactly the condition for sequential refinability, Eq. (3) from the main text with ${\mathcal{A}}={\mathcal{A}}^{\prime}={\mathcal{I}}$. Since these considerations hold for any partition ${\mathbb{E}}$ and any refinement of $({\mathcal{I}},{\mathbb{E}})$, the action ${\mathcal{I}}$ is ideal. ∎

In addition, we now prove the a strengthening of Theorem 2’, extending its statement to arbitrary reversible actions. Here, we call an action ${\mathcal{A}}$ reversible if there exists another action ${\mathcal{A}}^{-1}$ that acts as its inverse, namely ${\mathcal{A}}^{-1}{\mathcal{A}}\beta={\mathcal{A}}{\mathcal{A}}^{-1}\beta=\beta,\,\forall\beta\in{\mathsf{B}}$.

Corollary 2**.**

If an action ${\mathcal{A}}$ is reversible, then ${\mathcal{A}}$ is ideal.

Proof. We need to show that for every partition ${\mathbb{E}}=(E_{i})_{i}$, for every refinement $({\mathcal{B}},{\mathbb{F}})$ of $({\mathcal{A}},{\mathbb{E}})$, and for every belief $\beta$, the condition of sequential refinability, Eq. (3) from the main text, is satisfied.

The condition that $({\mathcal{B}},{\mathbb{F}})$ is a refinement of $({\mathcal{A}},{\mathbb{E}})$ reads

[TABLE]

The condition of sequential refinability is that there exists some suitable action ${\mathcal{A}}^{\prime}$ such that

[TABLE]

for every refinement $({\mathcal{B}},{\mathbb{F}})$ of $({\mathcal{A}},{\mathbb{E}})$, for every event $F_{i,l}\in{\mathbb{F}}$, and for every belief $\beta\in{\mathsf{B}}$ with $p(E_{i}|{\mathcal{A}}\beta)\neq 0$.

Since ${\mathcal{A}}$ is reversible, all elements of ${\mathsf{B}}$ can be obtained as ${\mathcal{A}}\beta$. Hence, $\beta^{\prime}:={\mathcal{A}}\beta$ is a generic element of ${\mathsf{B}}$. Setting ${\mathcal{C}}:={\mathcal{B}}{\mathcal{A}}^{-1}$, the condition that $({\mathcal{B}},{\mathbb{F}})$ is a refinement of $({\mathcal{A}},{\mathbb{E}})$ becomes

[TABLE]

This condition is equivalent to the fact that $({\mathcal{C}},{\mathbb{F}})$ is a refinement of $({\mathcal{I}},{\mathbb{E}})$. Since ${\mathcal{I}}$ is an ideal action (Theorem 2’), the experiment $({\mathcal{I}},{\mathbb{E}})$ is ideal, and satisfies the sequential refinability condition

[TABLE]

for every refinement $({\mathcal{C}},{\mathbb{F}})$ of $({\mathcal{I}},{\mathbb{E}})$, for every event $F_{i,l}\in{\mathbb{F}}$, and for every $\beta^{\prime}\in{\mathsf{B}}$ with $p(E_{i}|\beta^{\prime})\neq 0$.

Recalling the definitions $\beta^{\prime}:={\mathcal{A}}\beta$ and ${\mathcal{C}}:={\mathcal{B}}{\mathcal{A}}^{-1}$, we can express the above condition as

[TABLE]

for every refinement $({\mathcal{C}},{\mathbb{F}})$ of $({\mathcal{I}},{\mathbb{E}})$, for every event $F_{i,l}\in{\mathbb{F}}$, and for every $\beta\in{\mathsf{B}}$ with $p(E_{i}|{\mathcal{A}}\beta^{\prime})\neq 0$.

Comparing Eq. (34) with Eq. (31), we can see that Eq. (31) holds with ${\mathcal{A}}^{\prime}={\mathcal{A}}^{-1}$.

Hence the condition for sequential refinability is satisfied for $({\mathcal{A}},{\mathbb{E}})$ applied to the experiment $({\mathcal{B}},{\mathbb{F}})$. Since the above argument holds for every partition ${\mathbb{E}}$ and the reversing action is always ${\mathcal{A}}^{\prime}$, the action ${\mathcal{A}}$ is ideal.∎

Appendix C Ideal experiments in quantum theory are projective measurements

Here we show that our notion of an ideal experiment coincides with the notion of a projective measurement in quantum theory.

We consider general quantum measurements, described by quantum instruments, that is, collections $({\mathcal{A}}_{i})_{i}$ of completely positive trace non-increasing maps, with the property that the sum $\sum_{i}{\mathcal{A}}_{i}$ is trace-preserving. For a quantum system in the state $\rho$, the probability of the outcome $i$ is

[TABLE]

where ${\mathcal{A}}_{i}^{\dagger}$ is the adjoint of the map ${\mathcal{A}}_{i}$, uniquely defined by the condition $\operatorname{tr}[O{\mathcal{A}}_{i}(\rho)]=\operatorname{tr}[{\mathcal{A}}_{i}^{\dagger}(O)\,\rho]$, for all operators $\rho$ and $O$. Note that the operators $(A_{i})_{i}$ form a positive operator-valued measure (POVM), namely $A_{i}\geq 0$ and $\sum_{i}A_{i}=I$, the second condition following from the fact that the map $\sum_{i}{\mathcal{A}}_{i}$ is trace-preserving.

A quantum instrument $({\mathcal{B}}_{i,j})_{i,j}$ is a refinement of the quantum instrument $({\mathcal{A}}_{i})_{i}$ if one has

[TABLE]

or equivalently,

[TABLE]

with $B_{i,j}:={\mathcal{B}}_{i,j}^{\dagger}(I)$. Since this equation must hold for every state $\rho$, it can be simplified as

[TABLE]

A quantum instrument is sequentially refinable if there exists a quantum channel ${\mathcal{A}}^{\prime}$ such that, for every refinement $({\mathcal{B}}_{i,j})_{i,j}$, one has

[TABLE]

where $\rho_{i}^{\prime}$ is the conditional state $\rho_{i}^{\prime}:={\mathcal{A}}_{i}(\rho)/\operatorname{tr}[{\mathcal{A}}(\rho)]$.

With these definitions, we have the following lemma:

Lemma 2**.**

*If a quantum instrument $({\mathcal{A}}_{i})_{i}$ is sequentially refinable, then all POVM operators $A_{i}:={\mathcal{A}}_{i}^{\dagger}(I)$ are projectors. *

Proof. Equation (39) can be rewritten as

[TABLE]

with ${\mathcal{A}}_{i}^{\prime}:={\mathcal{A}}^{\prime}{\mathcal{A}}_{i}$. Equivalently, we have

[TABLE]

Now, the above equation holds for every POVM $(B_{i,j})_{i,j}$ that refines the POVM $(A_{i})_{i}$, meaning that Eq. (38) holds. Note that every non-negative operator $B_{i}$ with support contained in the support of $A_{i}$ can be rescaled in such a way that $B_{i}\leq A_{i}$. Hence, one can define a refinement $(B_{i,j})_{i,j}$ with $B_{i,0}=B_{i}$ and $B_{i,1}=A_{i}-B_{i}$ and apply condition (41) to it, thus obtaining

[TABLE]

where ${\mathsf{Supp}}(A_{i})$ and ${\mathsf{Supp}}(B_{i})$ denote the supports of $A_{i}$ and $B_{i}$, respectively.

On the other hand, the map ${\mathcal{A}}_{i}^{\prime{\dagger}}$ annihilates every operator $C$ with support in the orthogonal complement of ${\mathsf{Supp}}(A_{i})$, namely

[TABLE]

The proof of Equation (43) is as follows. By definition, one has

[TABLE]

the second equality following from the fact that the map $\sum_{i}{\mathcal{A}}_{i}^{\prime}$ is trace-preserving, and the last equality following from Equation (42) in the special case $B_{i}=A_{i}$. In order for Equation (C) to hold, we must have

[TABLE]

Now, every operator $C$ with support in the orthogonal complement \big{(}{\mathsf{Supp}}(A_{i})\big{)}^{\perp} is proportional to an operator $C^{\prime}$ such that $C^{\prime}\leq I-A_{i}$. Writing $C$ as $C=\lambda\,C^{\prime}$ for some constant $\lambda>0$, we have

[TABLE]

which proves Equation (43).

To conclude the proof, let us define $\Pi_{i}$ to be the projector on the support of $A_{i}$. Combining Equations (43) and (42), we obtain

[TABLE]

Note that, by definition, one has ${\mathcal{A}}_{i}^{\prime{\dagger}}(I)={\mathcal{A}}_{i}^{\dagger}{\mathcal{A}}^{\dagger}(I)={\mathcal{A}}_{i}^{\dagger}(I)$, because ${\mathcal{A}}$ is trace-preserving and therefore ${\mathcal{A}}^{\dagger}(I)=I$.

Summarizing, we have shown that every operator $A_{i}:={\mathcal{A}}_{i}^{\dagger}(I)$ is a projector. Moreover, the projectors $(A_{i})_{i}$ must be orthogonal. Indeed, for every $i\not=j$, one has

[TABLE]

which implies $\operatorname{tr}(A_{i}A_{j})=0$, or equivalently $\operatorname{tr}[(A_{i}A_{j})(A_{j}A_{i})]=0$. Hence, one must have $A_{i}A_{j}=0$.

In conclusion, the POVM operators $(A_{i})_{i}$ are orthogonal projectors. ∎

Lemma 3**.**

If a quantum instrument $({\mathcal{A}}_{i})_{i}$ is sequentially refinable and ${\mathcal{A}}^{\prime}$ is the quantum channel in the refinability condition (39), then the quantum instrument defined by ${\mathcal{A}}_{i}^{\prime}:={\mathcal{A}}^{\prime}{\mathcal{A}}_{i}$ has the Lüders form ${\mathcal{A}}_{i}^{\prime}(\cdot)=A_{i}\cdot A_{i}$ for every outcome $i$.

Proof. Using Equation (43) with $C=A_{i}^{\perp}:=I-A_{i}$, we obtain the relation

[TABLE]

Let ${\mathcal{A}}_{i}^{\prime}(\rho)=\sum_{j}A_{i,j}^{\prime}\rho A_{i,j}^{\prime{\dagger}}$ be a Kraus representation for ${\mathcal{A}}_{i}^{\prime}$. Inserting it in Equation (48) we obtain

[TABLE]

and therefore

[TABLE]

Since $A_{i}^{\perp}$ is a projector, the above equation can be rewritten as

[TABLE]

which implies $A_{i,j}^{\prime{\dagger}}A_{i}^{\perp}=A_{i}^{\perp}A_{i,j}^{\prime}=0$ for every $i$ and $j$.

For a generic operator $O$, we then have

[TABLE]

having used Equation (42) with $B_{i}=A_{i}OA_{i}$. ∎

In summary, ideal experiments in quantum mechanics are projective measurements, with state update given by Lüders rule. The converse is also true:

Lemma 4**.**

Every projective measurement, with state update given by Lüders’ rule, is sequentially refinable.

Proof. Let $(A_{i})_{i}$ be a projective POVM (i.e. a set of projectors such that $\sum_{i}A_{i}=I$), $({\mathcal{A}}_{i})_{i}$ be the quantum instrument defined by ${\mathcal{A}}_{i}(\rho):=A_{i}\rho A_{i}$, $\forall\rho$, and $({\mathcal{B}}_{ij})_{i,j}$ be a quantum instrument that refines $(A_{i})_{i}$, meaning that the corresponding POVM $(B_{i,j})_{i,j}$, with $B_{i,j}:={\mathcal{B}}^{\dagger}_{i,j}(I)$, satisfies Equation (38). Then, one has

[TABLE]

with $\rho_{i}^{\prime}={\mathcal{A}}_{i}(\rho)/\operatorname{tr}[{\mathcal{A}}_{i}(\rho)]$. Hence, the sequential refinability condition (Equation 4 of the main text) holds with ${\mathcal{A}}^{\prime}={\mathcal{I}}$. ∎

In summary, we have proven the following theorem:

Theorem 3**.**

The set of ideal experiments in quantum theory coincides with the set of projective measurements, with state update given by Lüders rule.

Appendix D Proof of Theorem 2

The proof of Theorem 2 follows the line of argument in Ref. CY14 .

Lemma 5**.**

Let $({\mathcal{A}}_{1},E_{1})$ and $({\mathcal{A}}_{2},E_{2})$ be two mutually exclusive ideal outcomes and let ${\mathcal{A}}_{1}^{\prime}$ be the reversing action in condition (4) of the main text. Then, one has the following implications:

If $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)\neq 0$, then

[TABLE] 2. 2.

If $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)=0$, then $p(E_{2}|{\mathcal{A}}_{2}\beta)=0$.

Proof. Since $({\mathcal{A}}_{1},E_{1})$ and $({\mathcal{A}}_{2},E_{2})$ are mutually exclusive, there exists an experiment $({\mathcal{B}},{\mathbb{F}})$ and two events $F_{1},F_{2}\in\mathbb{F}$ satisfying $p(F_{1}|{\mathcal{B}}\beta)=p(E_{1}|{\mathcal{A}}_{1}\beta)$ and $p(F_{2}|{\mathcal{B}}\beta)=p(E_{2}|{\mathcal{A}}_{2}\beta)$ for all beliefs $\beta$. The experiment $({\mathcal{B}},{\mathbb{F}})$ is a refinement of the experiment $({\mathcal{A}}_{1},{\mathbb{E}}_{1})$ with ${\mathbb{E}}_{1}=(E_{1},\overline{E}_{1})$. This holds since for any belief $\beta$ we have

[TABLE]

and

[TABLE]

Now, there are two possibilities: either $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)\neq 0$ or $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)=0$. For $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)\neq 0$, the sequential refinability of the experiment $({\mathcal{A}}_{1},{\mathbb{E}}_{1})$ implies the equality

[TABLE]

for every belief $\beta$ with $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)\neq 0$. Since $p(F_{2}|{\mathcal{B}}\beta)=p(E_{2}|{\mathcal{A}}_{2}\beta)$ this yields Eq. (54).

If $p(\overline{E}_{1}|{\mathcal{A}}_{1}\beta)=0$, then Eq. (56) yields immediately $p(E_{2}|{\mathcal{A}}_{2}\beta)=p(F_{2}|{\mathcal{B}}\beta)=0$. ∎

We are now ready to prove the following formulation of Theorem 2:

**Theorem

3’****.**

Let $\set{({\mathcal{A}}_{n},E_{n})}{n=1,\dotsc,k}$ be a set of ideal outcomes which are pairwise mutually exclusive. Then the condition

[TABLE]

is satisfied for every belief $\beta$.

Proof. The idea of the proof is to construct a sequential experiment with the property that a subset of outcomes of this experiment have the same probabilities of the ideal outcomes $\set{({\mathcal{A}}_{n},E_{n})}{n=1,\dotsc,k}$. Since the sum of the probabilities over all outcomes of the sequential measurement is 1, this will imply Eq. (58).

We define ${\mathbb{E}}_{i}=(E_{i},\overline{E}_{i})$, and denote by ${\mathcal{A}}^{\prime}_{i}$ denote the reversing action of $A_{i}$, as in Eq. (4) of the main text. Consider the sequential experiment defined by the following procedure:

• (I)

Move to step ($R_{1}$).

• ($R_{i}$)

Perform the action $({\mathcal{A}}_{i},{\mathbb{E}}_{i},{\mathcal{A}}^{\prime}_{i})$. If the event $E_{i}$ takes place, then move to step ($T_{i}$). If $i=k$ and the event $\overline{E}_{i}$ takes place, then move to step ($T_{k+1}$). Otherwise, move to step ($R_{i+1}$).

• ($T_{s}$)

Report outcome $s$ and terminate.

The above procedure defines a sequential experiment with outcome $s\in\{1,2,\dotsc,k+1\}$. We denote by $q_{\beta}(s)$ the probability assigned to the outcome $s$ when the initial belief is $\beta$.

We now prove the equality

[TABLE]

For $s=1$, the equality (59) follows by the definition of the sequential experiment, which has $q_{\beta}(1)=p(E_{1}|{\mathcal{A}}_{1}\beta)$.

To deal with the $s>1$ case, we define the sequence of beliefs $\beta_{0},\beta_{1},\dotsc,\beta_{k}$, via the relations $\beta_{0}:=\beta$ and

[TABLE]

With this definition, the belief $\beta_{i-1}$ is the belief of the agent at the beginning of step ($R_{i}$). For the purpose of this proof, if for some $j$ the probability $p(\overline{E}_{j}|{\mathcal{A}}_{j}\beta_{j-1})$ is zero and the updated belief ${\mathcal{A}}^{\prime}_{j}\,\overline{E}_{j}\,{\mathcal{A}}_{j}\beta_{j-1}$, then $\beta_{j}$ can be chosen to be an arbitrary belief.

The probability of an outcome $1<s\leq k$ is

[TABLE]

To prove Eq. (59) in the case $s\geq 2$, we apply Lemma 5 to the mutually exclusive ideal outcomes $({\mathcal{A}}_{s},E_{s})$ and $({\mathcal{A}}_{j},E_{j})$ for $j\in\{1,\dots,s-1\}$. The key observation is that Lemma 5 implies the equality

[TABLE]

valid for every $j\in\{1,\dots,s-1\}$. For $p(\overline{E}_{j}|{\mathcal{A}}_{j}\beta_{j-1})\not=0$, Eq. (62) follows from the definition $\beta_{j}:={\mathcal{A}}_{j}^{\prime}\overline{E}_{j}{\mathcal{A}}_{j}\beta_{j-1}$, and from the first implication in Lemma 5. For $p(\overline{E}_{j}|{\mathcal{A}}_{j}\beta_{j-1})\not=0$, Eq. (62) follows from the second implication in Lemma 5.

By using Eq. (62), we obtain

[TABLE]

where the third equality follows from Eq. (62) with $j=s-1$, and the subsequent iterations follow from Eq. (62) with $j$ starting from $s-2$ and going down to $j=1$.

Hence, we have proven Eq. (59). Using Eq. (59), it is now easy to prove Eq. (58), and therefore the validity of the exclusivity principle. Since the probabilities $(q_{\beta}(j)\,)_{j=1}^{k+1}$ sum up to 1, one simply has

[TABLE]

∎