¹

²

1 Introduction↩︎

One of the most debated fundamental problems in modern physics is whether the quantum formalism’s description of nature is complete, or if it could be supplemented with additional, or “hidden”, variables [1]–[7]. While hidden-variable models have been constructed which (partially or fully) reproduce the predictions of quantum mechanics [5], [7]–[9], various impossibility theorems, notably Bell’s [10], [11] and Kochen and Specker’s [4], have shown that such attempts inevitably violate certain consistency conditions which we would arguably want from a successful description of reality [4], [10], [12], [13]. These theorems differ in the conditions they impose, but a common feature is their indirect dependence on incompatible observables [4], [11], [14], [15].

In this paper, we explore the quantum state update rule (Lüders’ rule [16]) to establish an equivalence between the pairwise compatibility of all observables in a scenario, and the existence of a specific type of deterministic hidden-variable model for that scenario. The models we consider (called deterministic “underlying-state models” hereafter) are those in which underlying states determine the values of quantum observables, while quantum states act as statistical mixtures of (i.e., probability measures over) these “better defined states” [3]. These are the models typically considered in the contextuality and nonlocality literature [3], [4], [17]–[19]. In Ref. [20], [21], we argue that the result remains valid when the assumption of determinism is dropped, meaning it applies to all ontological models [18], as well as all other hidden-variable models satisfying factorisability (i.e., the union of statistical independence and no superluminal interaction [22]), such as are typically used to test Bell nonlocality [11], [19]. Since we consider Lüders’ rule, our analysis does not apply — at least not straightforwardly — to hidden-variable reformulations of quantum mechanics that (directly [23]) exclude the collapse postulate, such as de Broglie-Bohm pilot wave theory.

To begin with, we argue that, contrary to the typical description of updated states as post-measurement states, updated states are theoretically constructed as pre-measurement states, with the qualifier post-measurement state regarded instead as a merely circumstantial attribute. That is, instead of being updated by propositions in the simple past (“a measurement of \(\hat{A}\) was made, and the value \(\alpha\) was obtained”), we argue that quantum states are theoretically updated to ensure the validity of propositions in the first conditional (“if a measurement is made of \(\hat{A}\), the result will be found to be \(\alpha\)”) [24]. This approach aligns with multiple perspectives on quantum mechanics [25]–[29]. By this account, we argue Lüders’ rule should translate into underlying-state models as conditional probability, because considerations about interactions with measuring apparatuses are automatically removed. Using Bayes’ rule and Kolmogorov’s extension theorem [30], we then show that a set of quantum observables admits a state-updating underlying-state model (where states update via conditional probability) if and only if that set consists of pairwise compatible observables. A single pair of incompatible observables is sufficient to rule out a scenario being representable by a state-updating hidden-variable model. Building on this result, we analyse ontological and metaphysical views typically associated with the absence of hidden variables in quantum mechanics.

2 Quantum state update↩︎

We first argue that updated states in quantum mechanics are constructed as pre-measurement states, with the qualifier post-measurement state, typically assumed to be their defining feature, instead being merely a circumstantial attribute. We show that this corresponds to an objective description of Lüders’ rule. Note that this is an account of the theoretical quantum update mechanism taken as an axiom in standard quantum mechanics. It should not be misunderstood as a particular interpretation of any physical process which occurs when quantum systems are measured in actual experiments.

Let \(\hat{A}\) be an observable in a finite-dimensional quantum system \(\mathfrak{S}\), and let \(\alpha\) be one of its eigenvalues. Consider the proposition “If a measurement is made of \(\hat{A}\), the result will be found to be \(\alpha\)” [24], denoted \([\hat{A} =\alpha]\) hereafter. Let \(\hat{\Pi}(\hat{A}=\alpha) \equiv \chi_{\{\alpha\}}(\hat{A})\) be the corresponding projection (in which \(\chi_{\{\alpha\}}\) is the indicator function of \(\{\alpha\}\) and \(\chi_{\{\alpha\}}(\hat{A})\) is given by the functional calculus [31]), and let \(\hat{\Pi}(\hat{A}=\alpha)(\mathcal{H})\) be the subspace of \(\mathcal{H}\) onto which \(\hat{\Pi}(\hat{A}=\alpha)\) projects, i.e., \[\hat{\Pi}(\hat{A}=\alpha)(\mathcal{H}) = \{\hat{\Pi}(\hat{A}=\alpha)\psi: \psi \in \mathcal{H}\},\] in which \(\mathcal{H}\) denotes the finite-dimensional Hilbert space associated with \(\mathfrak{S}\).

Pure states that lie within the subspace \(\hat{\Pi}(\hat{A}=\alpha)(\mathcal{H})\) ensure the proposition is true, in that they assign probability \(1\) to \([\hat{A}=\alpha]\). This probability is strictly smaller than \(1\) for any other pure state. More broadly, a state \(\hat{\rho}\) (i.e., a density operator) assigns probability \(1\) to \([\hat{A}=\alpha]\) if and only if it is a convex combination of pure states lying in the corresponding subspace.

The orthocomplement of \(\hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\), namely the set \[\hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})^{\perp} \doteq\label{eq:orthocomplement} \{\phi \in \mathcal{H}: \forall_{\psi \in \hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})}\langle \phi\vert \psi \rangle = 0\},\tag{1}\] is the subspace corresponding the proposition \([\hat{A} \neq \alpha]\), which asserts that “If a measurement is made of \(\hat{A}\), the result will be found to be different to \(\alpha\)”. This is because a pure state \(\psi\) satisfies \(P_{\psi}[\hat{A} \neq \alpha]=1\) if and only if \(\psi \in \hat{\Pi}(\hat{A} = \alpha)(\mathcal{S})^{\perp}\), in which, for any state \(\hat{\rho}\) \[P_{\hat{\rho}}[A=\alpha] \equiv \text{tr}\left(\hat{\rho}\hat{\Pi}(\hat{A}=\alpha)\right)\] and \(P_{\hat{\rho}}[A\neq\alpha] \doteq 1-P_{\hat{\rho}}[A=\alpha]\).

These two subspaces have trivial intersection (i.e., their intersection is the zero vector), and their direct sum is the entire Hilbert space [31]. This means that, for any vector \(\psi\in \mathcal{H}\), there exists a unique pair \(\psi_{\alpha} \in \hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\), \(\psi_{\neg\alpha} \in \hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})^{\perp}\), such that \[\psi = \psi_{\alpha} + \psi_{\neg\alpha}.\] Furthermore, \(\psi_{\alpha}\) is the unique element of \(\hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\) that minimises the distance from \(\psi\), in that \[\Vert \psi - \psi_{\alpha}\Vert =\label{eq:minimizeDistance} \min \{\Vert \phi - \psi\Vert: \phi \in \hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\},\tag{2}\] and analogously for \(\psi_{\neg\alpha}\). The vector \(\psi_{\alpha}\) is said to be the orthogonal projection of \(\psi\) on \(\hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\). It follows by construction that \[\psi_{\alpha} = \hat{\Pi}(\hat{A} = \alpha) \psi.\]

Projecting a pure state \(\psi\) onto \(\hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\) is the optimal way of reconstructing \(\psi\) to ensure the validity of the proposition \([\hat{A} =\alpha]\), i.e., to ensure — at the theoretical level — that, if a measurement is made of \(\hat{A}\), the result will be found to be \(\alpha\). This is because, among all pure states (which we can consider as unidimensional subspaces) that assign probability 1 to \([\hat{A} = \alpha]\), the subspace spanned by \(\hat{\Pi}(\hat{A} = \alpha)\psi\) is the closest to \(\psi\), as the very notion of orthogonal projection implies. Hence, \(\hat{\Pi}(\hat{A} = \alpha)\psi\) is the state in \(\hat{\Pi}(\hat{A} = \alpha)(\mathcal{H})\) that best approximates \(\psi\) in a mathematically precise sense. As shown in the Appendix, this reasoning naturally extends to the update of density operators.

We denote by \(T_{[\hat{A}=\alpha]}: \mathcal{S}_{0} \rightarrow\mathcal{S}_{0}\) the mapping given by \[T_{[\hat{A}=\alpha]}(\psi) \doteq \frac{\hat{\Pi}(\hat{A}=\alpha)\psi}{\Vert\hat{\Pi}(\hat{A}=\alpha)\psi\Vert }\] for each \(\psi \in \mathcal{S}_{0}\), where \(\mathcal{S}_{0}\) consists of all pure states of the system.

3 Order-dependent predictions and incompatibility↩︎

We will denote the distribution of an observable \(\hat{B}\) in the state \(T_{[\hat{A} =\alpha]}(\psi)\) by \(P_{\psi}[\hat{B} = \cdot \;|\hat{A} =\alpha]\). That is, for each \(\beta \in\sigma(\hat{B})\) we have \[P_{\psi}[\hat{B} =\beta|\hat{A} =\alpha] \equiv P_{T_{[\hat{A} =\alpha]}(\psi)}[\hat{B} =\beta].\]

Let \(\hat{A}\) and \(\hat{B}\) be (not necessarily compatible) observables, and let \(\psi\) be a pure state. For each \(\alpha\in \sigma(\hat{A})\) and \(\beta \in\sigma(\hat{B})\), let us define \[\begin{align} P_{\psi}[&\hat{A}=\alpha,\hat{B}=\beta] \doteq\tag{3}P_{\psi}[\hat{A}=\alpha]P_{\psi}[\hat{B}=\beta | \hat{A} = \alpha] \\ &=\tag{4}\langle \psi |\hat{\Pi}(\hat{A}=\alpha)\hat{\Pi}(\hat{B}=\beta)\hat{\Pi}(\hat{A}=\alpha) | \psi \rangle \end{align}\] (note that \(\{\hat{\Pi}(\hat{A}=\alpha)\hat{\Pi}(\hat{B} = \beta)\hat{\Pi}(\hat{A}=\alpha): (\alpha,\beta) \in \sigma(\hat{A})\times\sigma(\hat{B})\}\) is a POVM). \(P_{\psi}[\hat{A}=\alpha,\hat{B}=\beta]\) can be thought of as the (theoretically constructed) probability of obtaining values \(\alpha\) and \(\beta\) by measuring \(\hat{A}\) and \(\hat{B}\) in sequence. Compatibility is equivalent to order-independent predictions of sequential measurements:

The proof can be found in the Appendix. Note that, due to Eq. (3 ), Eq. (?? ) is formally equivalent to Bayes’ rule.

4 State-updating Deterministic underlying-state models↩︎

Deterministic underlying-state models, as often used in the contextuality and nonlocality literature [17], [18], [32], are defined as follows:

Associated with a finite-dimensional system \(\mathfrak{S}\) is a measurable space \(\boldsymbol{\Lambda} \equiv (\Lambda,\mathcal{A})\) [30]; the elements of the underlying set \(\Lambda\) are called “hidden variables”, “hidden states”, or “underlying states”. Each underlying state determines the value of all physical quantities of the system, thus observables must be represented by real-valued (measurable) functions on \(\boldsymbol{\Lambda}\). The function \(f_{\hat{A}}\) representing an observable \(\hat{A}\) assigns, to each underlying state \(\lambda\), the value \(f_{\hat{A}}(\lambda)\) that \(\hat{A}\) assumes when the system is in the underlying state \(\lambda\) [4]. To use Bell’s terminology, underlying states are the “better defined states” over which states of \(\mathfrak{S}\) are averages [3]. Each state \(\psi\) of \(\mathfrak{S}\) therefore defines a probability measure \(\mu_{\psi}\) on \(\boldsymbol{\Lambda}\). Given any measurable set \(\Omega \subset \Lambda\), \(\mu_{\psi}(\Omega)\) is the probability that the system is in an underlying state lying in \(\Omega\) [4].

Underlying states assign truth-values to propositions. The proposition \([\hat{A} = \alpha]\) (“If a measurement is made of \(\hat{A}\), the result will be found to be \(\alpha\)”) is rendered true by \(\lambda\) if \(\alpha=f_{\hat{A}}(\lambda)\) (i.e., if \(\alpha\) is the value possessed by \(\hat{A}\) in the underlying state \(\lambda\)), and false otherwise [15], [24], [33]. This is because, whenever a physical quantity \(\hat{A}\) possesses a definite value, a measurement of \(\hat{A}\) must reveal, up to operational constraints such as limited accuracy of instruments, this value to the observer: if a quantity holds a value \(\alpha\), the apparatus’s reported value must be as close to \(\alpha\) as the accuracy of the instrument permits. This has two important consequences.

First, the (theoretically constructed) probability \(P_{\psi}[\hat{A} =\alpha]\) that a measurement of a quantity \(\hat{A}\) returns the value \(\alpha\) is the probability that, at that moment, the system lies in an underlying state that assigns this value to \(\hat{A}\). This means \[\begin{align} P_{\psi}[\hat{A} = \alpha] &=\label{eq:probabilityModel} \mu_{\psi}(\Omega(\hat{A}=\alpha)), \end{align}\tag{5}\] where \(\Omega(\hat{A}=\alpha)\) consists of all underlying states in which \([\hat{A}=\alpha]\) holds, i.e., \[\begin{align} \Omega(\hat{A} =\alpha) &\doteq\label{eq:setProposition} f_{\hat{A}}^{-1}(\{\alpha\})\\ &=\{\lambda \in \Lambda: f_{\hat{A}}(\lambda) =\alpha\}. \end{align}\tag{6}\]

Second, Bayesian inference (i.e., conditional probability) is the optimal way of reconstructing a state to ensure that a proposition \([\hat{A}=\alpha]\) holds. If the system is described by a state \(\mu_{\psi}\), the proposition \([\hat{A} = \alpha]\) — which, under the assumption that underlying states exist, is equivalent to “the physical quantity \(\hat{A}\) has a value, and that value is \(\alpha\)” — conditions \(\mu_{\psi}\) on the set of underlying states for which this claim is true, namely the set \(\Omega(\hat{A}=\alpha)\). Hence, the updated probability measure \(\tau_{[\hat{A} = \alpha]}(\mu_{\psi})\) that reconstructs \(\psi\) to ensure that \([\hat{A}=\alpha]\) holds is given by \[\begin{align} \tau_{[\hat{A} = \alpha]}(\mu_{\psi})( \;\cdot \;) &\doteq\label{eq:updateModel} \frac{\mu_{\psi}( \;\cdot \;\cap \Omega(\hat{A}=\alpha))}{\mu_{\psi}(\Omega(\hat{A}=\alpha))}\nonumber \\ &\equiv \mu_{\psi}( \cdot \;|\Omega(\hat{A}=\alpha)). \end{align}\tag{7}\]

For this to be consistent with the state update mechanism of \(\mathfrak{S}\), the probability measure representing \(T_{[\hat{A} = \alpha]}(\psi)\) must be the measure \(\mu_{\psi}\) conditioned on \(f_{\hat{A}}^{-1}(\{\alpha\})\), i.e., \[\mu_{T_{[\hat{A} = \alpha]}(\psi)} =\label{eq:preLudersBayes} \tau_{[\hat{A} = \alpha]}(\mu_{\psi}).\tag{8}\]

To summarise, let \(\mathfrak{S}\) be a finite-dimensional quantum system, and let \(\mathcal{O}_{S}\) be a non-empty subset of \(\mathcal{O}\). A state-updating deterministic underlying-state model for \(\mathcal{O}_{S}\) consists of a measurable space \(\boldsymbol{\Lambda} \equiv (\Lambda,\mathcal{A})\), a mapping \(\Psi\) assigning pure states of \(\mathfrak{S}\) to probability measures on \(\boldsymbol{\Lambda}\), and a mapping \(\Phi\) assigning observables in \(\mathcal{O}_{S}\) to measurable functions on \(\boldsymbol{\Lambda}\), such that, for each observable \(\hat{A} \in \mathcal{O}_{S}\), value \(\alpha\in \sigma(\hat{A})\), and pure state \(\psi\), Eqs. 5 and 8 are satisfied. This means that the distribution of \(\hat{A}\) w.r.t. \(\psi\) is that of the random variable \(f_{\hat{A}}\equiv \Phi(\hat{A})\) in the probability space defined by \(\mu_{\psi} \equiv \Psi(\psi)\) (Eq. 5 ), and that Lüders’ rule translates to \(\boldsymbol{\Lambda}\) as conditional probability, as encapsulated by the following commutative diagram

(as before, \(\tau_{[\hat{A} = \alpha]}\) denotes the mapping that conditions each probability measure \(\mu:\mathcal{A} \rightarrow[0,1]\)) on \(\Omega(\hat{A}=\alpha)\). When \(\mathcal{O}_{S}\) contains all observables of \(\mathfrak{S}\), we say that \(\mathfrak{M} \equiv (\boldsymbol{\Lambda},\Phi,\Psi)\) is a model for the system \(\mathfrak{S}\).

We define an underlying-state model for a set of observables \(\mathcal{O}_{S}\), and not only for the entire system \(\mathfrak{S}\), for practical reasons. The subscript \(S\) stands for “scenario” and is motivated by the concept of a “measurement scenario”, which appears in operational approaches to quantum foundations [18], [19], [34].

5 Incompatibility obstructs state-updating underlying-state models↩︎

We now show that incompatibility obstructs the existence of state-updating underlying-state models. If a deterministic state-updating underlying-state model exists for a set of observables \(\mathcal{O}_{S}\), then for any pair of observables \(A,B \in \mathcal{O}_{S}\), any pure state \(\psi\) and any values \(\alpha\in\sigma(A)\), \(\beta \in \sigma(B)\), \[\begin{align} P_{\psi}&[\hat{A} = \alpha,\hat{B} =\beta] \doteq P_{\psi}[\hat{A} = \alpha]P_{\psi}[\hat{B} = \beta| \hat{A} = \alpha] \\ &= P_{\psi}[\hat{A} = \alpha]\frac{\mu_{\psi}(\Omega(\hat{A}=\alpha) \cap \Omega(\hat{B}=\beta))}{P_{\psi}[\hat{A}=\alpha]}\\ &=\mu_{\psi}(\Omega(\hat{A}=\alpha) \cap \Omega(\hat{B}=\beta))\\ &=P_{\psi}[\hat{B} =\beta]P_{\psi}[\hat{A} = \alpha|\hat{B} = \beta]\\ &=P_{\psi}[\hat{B} = \beta,\hat{A} = \alpha]. \end{align}\] When this is true, according to Lemma 1, \(A\) and \(B\) are compatible. This proves that a state-updating underlying-state model exists for \(\mathcal{O}_{S}\) only if \(\mathcal{O}_{S}\) is a set of pairwise compatible observables. On the other hand, the most general version of the Kolmogorov extension theorem (see Theorem 2.4.3 of Ref. [30]) ensures that state-updating underlying-state models can be constructed for any set of pairwise compatible observables (see Appendix for details). This leads to the following result.

As shown in Ref. [20], a variant of Proposition 1 also applies to stochastic underlying-state models, whose underlying states assign probabilities — rather than values — to observables. It means that, arguably, incompatibility obstructs stochastic underlying-state models as well (or at least those for which state update corresponds to conditional probability á la Bayes’ rule). We explore the consequences of this result in Ref. [21]. Finally, recall that, regarding contextuality and nonlocality, there is no loss of generality in working with deterministic models [17], [19], [35].

6 Discussion↩︎

If our considerations regarding the quantum state update are correct, Proposition 1 provides a straightforward method for ruling out (deterministic) underlying-state models of quantum mechanics. The proof is independent of specific states and observables, as well as space-time considerations; it establishes a direct connection between compatibility and underlying states, demonstrating that incompatibility is sufficient to obstruct such models.

State-updating underlying-state models are inherently Kochen-Specker noncontextual [17], [36] and Bell-local [10], [19]. This means that, if \(\vec{A} \equiv (\hat{A}_{1},\dots,\hat{A}_{m})\) are pairwise compatible, then, for any \(\vec{\alpha} \equiv (\alpha_{1},\dots,\alpha_{m}) \in \prod_{i=1}^{m}\sigma(\hat{A}_{i})\) and any state \(\psi\), \[P_{\psi}[\vec{A} = \vec{\alpha}] =\label{eq:jointDistributionModelM} \mu_{\psi}(\cap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i})),\tag{9}\] in which, \[\begin{align} P_{\psi}[\vec{A} = \vec{\alpha}] &\doteq\label{eq:jointDistributionM} \prod_{i=1}^{m} P_{\psi}[\hat{A}_{i}=\alpha_{i}|\hat{A}_{1}=\alpha_{1},\dots,\hat{A}_{i-1}=\alpha_{i-1}]\\ &=\left\langle \psi \left| \prod_{i=1}^{m}\hat{\Pi}(\hat{A}_{i}=\alpha_{i}) \psi\right.\right\rangle \end{align}\tag{10}\] (clearly, \(P_{\psi}[\hat{A}_{i}= \cdot \;|\hat{A}_{1}=\alpha_{1},\dots,\hat{A}_{i-1}=\alpha_{i-1}]\) denotes the distribution of \(\hat{A}_{i}\) in the state \((T_{[\hat{A}_{i-1}=\alpha_{i-1}]} \circ \dots \circ T_{[\hat{A}_{1}=\alpha_{1}]})(\psi)\)). Eq. (9 ) follow from straightforward calculations, as shown e.g., in Ref. [20]. In particular, it follows from our analysis of the state update mechanism that contextual deterministic underlying-state models are inconsistent with the canonical postulates of quantum mechanics (which include the collapse postulate).

The view that Lüders’ rule assigns states to post-measurement states is particularly popular in the contemporary quantum foundations community, largely due to the influence of quantum information theory [37] and the prevalence of operational approaches in the field. This view permeates longstanding debates about the assumptions of “ideal measurements”, attempts to reconstruct quantum mechanics, and discussions related to hidden variables [18], [32], [37]–[39]. As shown in Ref. [20], the alternative proposed here —– which agrees with multiple perspectives on quantum mechanics [25]–[28], [40] — can shed light on important topics in quantum foundations, extending beyond the problem of hidden variables.

An important consequence of the correspondence between incompatibility, order-dependent predictions, and constraints on hidden-variable models we establish is that it makes it easier to identify classical parallels to our inability to form underlying-state models. Our work therefore contributes to the debate about whether — and, if so, why — phenomena associated with such models, such as contextuality, can be considered a signature of non-classicality [32], [41]–[44].

In fact, consider classical wave theory, the theory of classical physics in which parallels with quantum mechanics are traditionally drawn [6], [45], [46]. Consider a wave on a string of length \(L\), and let \(V\) and \(N\) denote the physical quantities “phase velocity” and the “number of nodes”, respectively [47]. Following the terminology of quantum mechanics, we can say that travelling waves are eigenstates of \(V\), as they correspond to states with well-defined phase velocities. Analogously, standing waves can be seen as eigenstates of \(N\). The phase velocity of a travelling wave is the eigenvalue of \(V\) it is associated with, while the eigenvalue of \(N\) associated with a standing wave is its number of nodes. We denote by \(\psi_{\pm }\) an eigenstate of \(V\) corresponding to the eigenvalue \(\pm v\), where \(v\) is the phase velocity determined by the medium [47]. Similarly, \(\psi_{n}\) is an eigenstate of \(N\) with \(n \in \mathbb{N}\) nodes. If \(\psi\) is a travelling wave, it is a superposition (i.e., linear combination) of eigenstates of \(N\) corresponding to different eigenvalues, as Fourier’s theorem shows [48]. Conversely, if \(\psi\) is a standing wave, it can be written as a superposition of waves travelling in different directions (i.e., corresponding to different eigenvalues of \(V\)). Recall that every wave is either travelling or standing, and that no standing wave is a travelling wave, and vice versa. It means that the quantities \(V\) and \(N\) have no eigenstates in common, as with incompatible observables in quantum theory.

Now, consider a state update mechanism analogous to Lüders’ rule, which projects waves onto eigenstates. That is, if \(\psi = a_{+} \psi_{+} + a_{-} \psi_{-}\), then \(T_{[V = \pm v]}(\psi) = \psi_{\pm}\), while \(T_{[N=n]}(\psi) = \psi_{n}\) with \(\psi = \sum_{n=1}^{\infty} a_{n} \psi_{n}\) (for simplicity, ignore considerations about the uniqueness of decompositions, collinear eigenvectors, and normalisation). Consider the theoretical predictions of a (hypothetical) sequential measurement of either \(V\) and then \(N\), or \(N\) and then \(V\), on the state \(\psi_{+}\). If we measure \(V\) first, then \(N\), the theoretically constructed prediction of obtaining values \(+v\) and \(n\) is \[\begin{align} P_{\psi_{+}}[V=+v, N = n] &\doteq P_{\psi_{+}}[V=+v]P_{\psi_{+}}[N=n] \\ &=P_{\psi_{+}}[N=n] \end{align}\] If instead we measure \(N\) first, then \(V\), \[\begin{align} P_{\psi_{+}}[N=n,V=+v] &= P_{\psi_{+}}[N=n]P_{\psi_{+}}[V=+v|N=n] \\ &=P_{\psi_{+}}[N=n]P_{\psi_{n}}[V=+v], \end{align}\] in which \(P_{\psi_{+}}[V=+v|N=n] \equiv P_{T_{[N=n]}(\psi_{+})}[V=+v]\). Unless \(P_{\psi_{n}}[V=+v] =1\), \[P_{\psi_{+}}[V=+v,N=n] \neq P_{\psi_{+}}[N=n,V=+v].\] By construction, this update mechanism fits precisely the same description we provided for the quantum state update. Thus, as argued above, it must translate to underlying-state models as conditional probability. On the other hand, the calculations above show that order-dependent theoretical predictions of sequential measurements are to be expected in classical wave systems, thereby obstructing (via Proposition 1) the existence of underlying-state models for them.

To conclude, we would like to emphasise that Proposition 1 corroborates the traditional interpretation of the Kochen-Specker theorem [4], [15], [24], [33], [49]–[51]: quantum mechanics seems inconsistent with mechanistic realism, or naïve realism (as Isham and Butterfield [33], and Döring [24] put it), where all physical quantities of a system are simultaneously possessed by that system.

7 Appendix↩︎

On the Update of Density Operators. In the main text, we showed that the update of pure states under objective propositions (those that specify the measurement outcome) is, by construction, a theoretical mechanism for reconstructing states to ensure the validity of certain propositions about future (i.e., subsequent) measurements of physical quantities. Here, we extend this observation to density operators, which represent states where the experimentalist has limited knowledge of the system’s pure state.

As we know, the state (i.e., density operator) corresponding to the projected vector \(\hat{\Pi}(\hat{A} = \alpha)\psi\) is the rank-1 projection \[\begin{align} T_{[\hat{A} = \alpha]}(\vert \psi \rangle \langle \psi \vert) &\doteq \left|\frac{\hat{\Pi}(\hat{A} = \alpha)\psi}{\Vert \hat{\Pi}(\hat{A} = \alpha)\Vert}\right\rangle \left\langle\frac{\hat{\Pi}(\hat{A} = \alpha)\psi}{\Vert \hat{\Pi}(\hat{A} = \alpha)\Vert}\right| \\ &=\frac{\hat{\Pi}(\hat{A} = \alpha)\vert \psi \rangle \langle \psi \vert\hat{\Pi}(\hat{A} = \alpha)}{P_{\psi}[\hat{A} = \alpha]}. \end{align}\] With a slight abuse of notation, let’s denote also by \(\mathcal{S}_{0}\) the set of rank-1 projections on \(\mathcal{H}\).

The straightforward way of extending the mapping \(\mathcal{S}_{0} \ni \vert \psi \rangle \langle \psi \vert \mapsto T_{[\hat{A} = \alpha]}(\vert \psi \rangle \langle \psi \vert) \in \mathcal{S}_{0}\) to the set \(\mathcal{S}\) of all states (i.e., density operators) is by defining \[T_{[\hat{A} = \alpha]}(\hat{\rho}) \doteq\label{eq:firstLuders} \frac{\hat{\Pi}(\hat{A} = \alpha)\hat{\rho}\hat{\Pi}(\hat{A} = \alpha)}{P_{\hat{\rho}}[\hat{A}=\alpha]}\tag{11}\] for each state \(\hat{\rho}\). As well know, this is Lüders’ rule [16], an improvement of von Neumann’s collapse postulate [45]. Let’s show that our analysis of the state update rule remains valid under this extension.

As one can easily check, \(T_{[\hat{A} = \alpha]}(\hat{\rho})\) ensures that \([\hat{A}=\alpha]\) is true, i.e., measurement of \(\hat{A}\) in the state \(T_{[\hat{A} = \alpha]}(\hat{\rho})\) yields the value \(\alpha\) with probability \(1\) (at the theoretical level). Next, let \(\hat{\rho} = \sum_{i=1}^{m} p_{i} \vert \psi_{i} \rangle \langle \psi_{i} \vert\) be any convex decomposition of the state \(\hat{\rho}\) in terms of pure states, with \(p_{i} >0\) for each \(i\) and \(\sum_{i=1}^{m} p_{i} = 1\) (recall that \(\mathcal{S}\) is the convex hull of \(\mathcal{S}_{0}\)). The state \(\hat{\rho}\) is traditionally interpreted as asserting that the system lies in the state \(\vert \psi_{i} \rangle \langle \psi_{i} \vert\) with probability \(p_{i}\) [37].

In the state \(\hat{\rho}\), the probability \(P_{\hat{\rho}}[\hat{\rho}=\psi_{i},\hat{A} = \alpha]\) that the system is in the state \(\vert \psi_{i} \rangle \langle \psi_{i} \vert\) and that, in this pure state, a measurement of \(\hat{A}\) yields the value \(\alpha\) is given by \[\begin{align} P_{\hat{\rho}}[\hat{\rho}=\psi_{i},\hat{A} = \alpha] &\doteq p_{i} P_{\psi_{i}}[\hat{A} =\alpha]. \end{align}\] For simplicity, we will say that \(P_{\hat{\rho}}[\hat{\rho}=\psi_{i},\hat{A} = \alpha]\) is the probability of \([\hat{\rho}=\psi_{i}]\) and \([\hat{A}=\alpha]\) being “consecutively true” in the state \(\hat{\rho}\). In the same state, \(P_{\hat{\rho}}[\hat{A}=\alpha]\) is the probability that \([\hat{A}=\alpha]\) is true. The ratio \[\frac{P_{\hat{\rho}}[\hat{\rho}=\psi_{i},\hat{A} = \alpha]}{P_{\hat{\rho}}[\hat{A}=\alpha]}\] is thus the probability that \([\hat{\rho}=\psi_{i}]\) and \([\hat{A} =\alpha]\) are consecutively true, provided that \([\hat{\rho} \in \{\vert \psi_{i} \rangle \langle \psi_{i} \vert,i=1,\dots,m\} ]\) and \([\hat{A}=\alpha]\) are satisfied. Now note that \[\begin{align} T_{[\hat{A} = \alpha]}(\hat{\rho}) &= \sum_{i=1}^{m}p_{i}\frac{\hat{\Pi}(\hat{A} = \alpha)\vert \psi_{i} \rangle \langle \psi_{i} \vert\hat{\Pi}(\hat{A} = \alpha)}{P_{\hat{\rho}}[\hat{A}=\alpha]} \\ &= \sum_{i=1}^{m}\frac{p_{i}P_{\psi_{i}}[\hat{A} =\alpha]}{P_{\hat{\rho}}[\hat{A}=\alpha]}T_{[\hat{A} = \alpha]}(\vert \psi_{i} \rangle \langle \psi_{i} \vert)\\ &= \sum_{i=1}^{m}\frac{P_{\hat{\rho}}[\hat{\rho}=\psi_{i},\hat{A} = \alpha]}{P_{\hat{\rho}}[\hat{A}=\alpha]}T_{[\hat{A} = \alpha]}(\vert \psi_{i} \rangle \langle \psi_{i} \vert). \end{align}\] Hence, \(T_{[\hat{A} = \alpha]}(\hat{\rho})\) asserts that the system is in one of the states \(T_{[\hat{A} = \alpha]}(\vert \psi_{i} \rangle \langle \psi_{i} \vert)\), \(i=1,\dots,m\), which are pure states ensuring that \([\hat{A}=\alpha]\) holds. The probability that the system is in the pure state \(T_{[\hat{A} = \alpha]}(\vert \psi_{i} \rangle \langle \psi_{i} \vert)\) corresponds to the likelihood of \([\hat{\rho}=\psi_{i}]\) and \([\hat{A} =\alpha]\) being consecutively true, provided that \([\hat{\rho} \in \{\vert \psi_{i} \rangle \langle \psi_{i} \vert,i=1,\dots,m\} ]\) and \([\hat{A}=\alpha]\) are satisfied. It shows that our analysis of the state update rule remains valid under Lüders’ extension.

Proof of Lemma 1: Let \(\hat{A}\) and \(\hat{B}\) be (not necessarily compatible) observables in a finite-dimensional quantum system, and let \(\hat{A}=\sum_{\alpha\in \sigma(\hat{A})}\alpha\hat{\Pi}(\hat{A}=\alpha)\) and \(\hat{B}=\sum_{\beta \in \sigma(\hat{B})}\beta \hat{\Pi}(\hat{B} = \beta)\) be their spectral decompositions [15]. For any pure state \(\psi\) and values \(\alpha\in \sigma(\hat{A})\), \(\beta \in \sigma(\hat{B})\), we have \[P_{\psi}[\hat{A} = \alpha,\hat{B} \in \beta] = \left\langle \psi \left| \hat{\Pi}(\hat{A}=\alpha)\hat{\Pi}(\hat{B} = \beta)\hat{\Pi}(\hat{A}=\alpha) \psi\right.\right\rangle.\] On the other hand, \[P_{\psi}[\hat{B} = \beta,\hat{A} \in \alpha] = \left\langle \psi \left| \hat{\Pi}(\hat{B}=\beta)\hat{\Pi}(\hat{A} = \alpha)\hat{\Pi}(\hat{B}=\beta) \psi\right.\right\rangle.\] Pure states separate observables in quantum systems, i.e., two self-adjoint operators \(C,D\) are equal if and only if \(\left\langle \psi \left| C \psi\right.\right\rangle=\left\langle \psi \left| D \psi\right.\right\rangle\) for each state pure \(\psi\). Therefore, the probability distributions \(P_{\psi}[\hat{A}= \cdot \;,\hat{B}= \cdot \;]\) and \(P_{\psi}[\hat{B}= \cdot \;,\hat{A}= \cdot \;]\) are equal (up to a permutation) for every state \(\psi\) if and only if, for all \((\alpha,\beta) \in \sigma(\hat{A}) \times \sigma(\hat{B})\), the operators \(\hat{\Pi}(\hat{A}=\alpha)\hat{\Pi}(\hat{B} = \beta)\hat{\Pi}(\hat{A}=\alpha)\) and \(\hat{\Pi}(\hat{B} = \beta)\hat{\Pi}(\hat{A}=\alpha)\hat{\Pi}(\hat{B} = \beta)\) are equal. As shown in Ref. [52], this is equivalent to saying that \(\hat{\Pi}(\hat{A}=\alpha)\) and \(\hat{\Pi}(\hat{B} = \beta)\) are compatible. We know that \(\hat{A}\) and \(\hat{B}\) are compatible if and only if \(\hat{\Pi}(\hat{A}=\alpha)\) and \(\hat{\Pi}(\hat{B} = \beta)\) are compatible for each \((\alpha,\beta) \in \sigma(\hat{A}) \times \sigma(\hat{B})\) [37]. Hence, \(\hat{A}\) and \(\hat{B}\) are compatible if and only if, for each state \(\psi\) and each \((\alpha,\beta) \in \sigma(\hat{A}) \times \sigma(\hat{B})\), \[P_{\psi}[\hat{A}=\alpha,\hat{B}=\beta]=P_{\psi}[\hat{B}=\beta,\hat{A} =\alpha].\] \(\square \medskip\)

Proof of Proposition 1: All that remains for us to prove is that (a) follows from (b). Suppose thus that \(\mathcal{O}_{S}\) is a set of pairwise compatible observables. Let \(\Lambda\) be the Cartesian product \(\prod_{\hat{A} \in \mathcal{O}_{S}} \sigma(\hat{A})\), and let \(\mathcal{A}\) be the product \(\sigma\)-algebra \(\prod_{\hat{A} \in \mathcal{O}_{S}}\mathfrak{P}(\sigma(\hat{A}))\) [30], where \(\mathfrak{P}(\sigma(\hat{A}))\) denotes the collection of all subsets of \(\sigma(\hat{A})\). Recall that elements of \(\Lambda \equiv \prod_{\hat{A} \in \mathcal{O}_{S}} \sigma(\hat{A})\) are tuples \(\lambda \equiv (\lambda_{\hat{A}})_{\hat{A} \in \mathcal{O}_{S} }\) satisfying \(\lambda_{\hat{A}} \in \sigma(\hat{A})\) for all \(\hat{A} \in \mathcal{O}_{S}\). For each \(\hat{A} \in \mathcal{O}_{S}\), let \(f_{\hat{A}}\) be the coordinate projection function \(\Lambda \rightarrow\sigma(\hat{A})\), that is, \(f_{\hat{A}}(\lambda) \doteq \lambda_{\hat{A}}\) for all \(\lambda \in \Lambda\). It follows from the definition of product \(\sigma\)-algebra that \(f_{\hat{A}}\) is a measurable function [30]. For each \(\alpha\in \sigma(\hat{A})\), denote by \(\Omega(\hat{A}=\alpha)\) the set of all \(\lambda \in\Lambda\) such that \(\lambda_{A}=\alpha\), i.e., \[\begin{align} \Omega(\hat{A}=\alpha) &\equiv f_{\hat{A}}^{-1}(\{\alpha\})\\ &=\{\lambda \in \Lambda: \lambda_{\hat{A}} = \alpha\}. \end{align}\]

Thus far, we have constructed a space of deterministic underlying states \(\boldsymbol{\Lambda} \equiv (\Lambda,\mathcal{A})\) and measurable functions \(f_{\hat{A}}:\Lambda \rightarrow\mathbb{R}\) representing observables. To conclude, we need to define probability measures representing pure states and recover Eqs. (5 ) and (8 ). So let \(\psi\) be a pure state. For any finite subset \(\{\hat{A}_{1},\dots,\hat{A}_{m}\}\) of \(\mathcal{O}_{S}\), let \[P_{\psi}[\vec{A}=\cdot \;] \equiv P_{\psi}[\hat{A}_{1}=\cdot \;,\dots,\hat{A}_{m}=\cdot \;]\] be the joint distribution of \(\vec{A} \equiv (\hat{A}_{1},\dots,\hat{A}_{m})\) in the state \(\psi\) (see Eq. 10 ). Let \(\pi\) be any permutation of \(\{1,\dots,m\}\). Then for any \(\vec{\alpha} \equiv (\alpha_{1},\dots,\alpha_{m}) \in \prod_{i=1}^{m}\sigma(\hat{A}_{i})\), \[\begin{align} P_{\psi}[\vec{A} = \vec{\alpha}] &= P_{\psi}[\hat{A}_{1}=\alpha_{1},\dots,\hat{A}_{m}=\alpha_{m}]\\ &= \left\langle \psi \left| \prod_{i=1}^{m}\hat{\Pi}(\hat{A}_{i}=\alpha_{i}) \psi\right.\right\rangle\\ &=\left\langle \psi \left| \prod_{i=1}^{m}\hat{\Pi}(\hat{A}_{\pi(i)}=\alpha_{\pi(i)}) \psi\right.\right\rangle\\ &= P_{\psi}[\hat{A}_{\pi(1)}=\alpha_{\pi(1)},\dots,\hat{A}_{\pi(m)}=\alpha_{\pi(m)}]. \end{align}\] Now let \(\hat{A}_{n_{1}},\dots,\hat{A}_{n_{M}}\) be a sub-sequence of \(\hat{A}_{1},\dots,\hat{A}_{m}\). The joint distribution of \(\hat{A}_{n_{1}},\dots,\hat{A}_{n_{M}}\) is the joint distribution of \(\hat{A}_{1},\dots,\hat{A}_{m}\) marginalised over all observables in \(\{\hat{A}_{1},\dots,\hat{A}_{m}\}\) except \(\hat{A}_{n_{1}},\dots,\hat{A}_{n_{M}}\) — this is known as the non-disturbance condition [17]. Hence, it follows from the Kolmogorov extension theorem [30] that there exists a unique probability measure \(\mu_{\psi}\) on \(\boldsymbol{\Lambda}\) such that, for any observables \(\hat{A}_{1},\dots,\hat{A}_{m} \in \mathcal{O}_{S}\) and values \(\vec{\alpha} \equiv (\alpha_{1},\dots,\alpha_{m}) \in \prod_{i=1}^{m}\sigma(\hat{A}_{i})\), \[\begin{align} P_{\psi}[\vec{A}=\vec{\alpha}] &=\label{eq:jointIntersection} \mu_{\psi}\left(\bigcap_{i=1}^{m}f_{\hat{A}_{i}}^{-1}(\{\alpha_{i}\})\right) \\ &= \mu_{\psi}\left(\bigcap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i})\right) \end{align}\tag{12}\] (see Theorem 2.4.3 of Ref. [30] for details). In particular, for any \(\hat{A} \in \mathcal{O}_{S}\) and \(\alpha\in \sigma(\hat{A})\), Eq. (5 ) is satisfied. Finally, let’s show that, for any state \(\psi\), any observable \(\hat{A}\) and any value \(\alpha\in \sigma(\hat{A})\), the state \(T_{[\hat{A} =\alpha]}(\psi)\) is represented by the measure \(\mu_{\psi}\) conditioned on \(\Omega(\hat{A}=\alpha)\), as in Eq. (7 ). To begin with, it follows from Eqs. (10 ) and (12 ) that, for any observables \(\hat{A}_{1},\dots,\hat{A}_{m} \in \mathcal{O}_{S}\) and values \(\vec{\alpha} \equiv (\alpha_{1},\dots,\alpha_{m}) \in \prod_{i=1}^{m}\sigma(\hat{A}_{i})\), \[\begin{align} P_{T_{[\hat{A} =\alpha]}(\psi)}[\vec{A}=\vec{\alpha}] &= \frac{P_{\psi}[\hat{A} =\alpha,\hat{A}_{1} =\alpha_{1},\dots,\hat{A}_{m} =\alpha_{m}]}{P_{\psi}[\hat{A} =\alpha]} \\ &= \frac{\mu_{\psi}( \Omega(\hat{A}=\alpha) \cap\bigcap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i}))}{\mu_{\psi}(\Omega(\hat{A}=\alpha))} \\ &\equiv \mu_{\psi}(\cap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i}) \vert \Omega(\hat{A}=\alpha)), \end{align}\] where, as usual, \(\mu_{\psi}( \;\cdot \; \vert \Omega(\hat{A}=\alpha))\) denotes the measure \(\mu_{\psi}\) conditioned on \(\Omega(\hat{A}=\alpha)\). On the other hand, Eq. (12 ) entails that \[P_{T_{[\hat{A} =\alpha]}(\psi)}[\vec{A}=\vec{\alpha}] = \mu_{T_{[\hat{A} =\alpha]}(\psi)}(\cap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i})),\] therefore \[\begin{align} P_{T_{[\hat{A} =\alpha]}(\psi)}[\vec{A}=\vec{\alpha}]&= \mu_{T_{[\hat{A} =\alpha]}(\psi)}(\cap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i}))\\ &= \mu_{\psi}\left(\bigcap_{i=1}^{m}\Omega(\hat{A}_{i}=\alpha_{i}) \vert \Omega(\hat{A}=\alpha)\right). \end{align}\] Hence, it follows from the uniqueness of the measure given by the Kolmogorov extension theorem [30] (applied to the state \(T_{[\hat{A} =\alpha]}(\psi)\)) that \[\mu_{T_{[\hat{A} =\alpha]}(\psi)}( \;\cdot \;) = \mu_{\psi}( \;\cdot \;\vert \Omega(\hat{A}=\alpha)).\] It proves that \(\tau_{[\hat{A} =\alpha]}\) is necessarily given by Eq. (7 ), completing the proof. \(\square \medskip\)

References↩︎