The Local Account of Bell Nonlocality

How can remote measurements on a pair of entangled particles be both genuinely random yet perfectly correlated? That was the puzzle of Einstein, Podolsky and Rosen (EPR), who argued [17] that if locality is to be preserved and quantum theory is empirically adequate, then it ought to be completed. A natural completion is given by hidden variables thought to underpin the source of randomness in measurements. In spite of being unknown, or even perhaps unknowable, these underlying parameters would be determinative of the outcomes. A local hidden variable theory is one in which all influences, including those by and on hidden variables, are constrained by the structure of spacetime, and hence limited by the speed of light. For instance, if two laboratories are spacelike separated, Alice’s freely chosen experimental setting neither affects Bob’s laboratory nor the hidden variables in its vicinity. Building upon the EPR argument, Bell [1] showed that quantum theory’s predictions are incompatible with an underlying determination by local hidden variables. The incompatibility materializes [16] in the experimental measurement of a quantity which ought to be bounded if nature operates by local hidden variables. This is a Bell inequality. According to quantum theory, however, Bell inequalities can be violated. And they are: Ever more convincing experiments systematically corroborate the quantum predictions, thereby refuting the local hidden variable account that Bell formalized.

The most vocal conclusions are that ‘the physical world itself is non-local’ [18] or that ‘we should renounce local realism’ [19] (emphases mine). Some instead posit retrocausality, which ‘includes the existence of backwards-in-time influences’ [20], while others invoke superdeterminism, the assumption that ‘the prepared state of an experiment is never independent of the detector settings’ [21]. But all of the above explanations are cast in a worldview shaped by hidden variables as if they were the only path to reconcile quantum theory with realism, namely the conjectured existence of a real physical world. Hidden variables were indeed a realist defence against the void enforced by Bohr and the Copenhagen school, which effectively forbade the inquiry of actual states of affairs between measurements. But by no means are they the sole realist program. For instance, Raymond-Robichaud [22] showed how Bell inequalities can in principle be violated in a local-realistic way, thereby contradicting the widespread yet misconceived equivalence between local hidden variables and local realism. He did so by inventing a fantasy world where Alice and Bob locally split in bubbles that interact in a way that yields Popescu-Rohrlich correlations [23]. As I shall show, this fantasy shares a lot with reality. The story outlined in the abstract, manifestly local-realistic, is not to be invoked merely as a proof of principle in a toy model: It is how Bell inequalities are actually violated, insofar as we rely on quantum theory, which I shall treat fully unitarily and in the Heisenberg picture.

Textbook quantum theory instructs a blatant inconsistency between two types of dynamical laws. On the one hand, all dynamical processes except ‘measurements’ abide by unitary evolution, which is continuous, local, deterministic, and in principle reversible. On the other hand, the so-called ‘measurements’ follow a collapse rule that is discontinuous, non-local, stochastic and irreversible. Everett [13] solved this inconsistency by dismissing the need for the ambiguous yet supposedly consequential notion of ‘measurement’. Measurements are, Everett argues, mere interactions. And observers, other quantum systems. He thus investigated measurement interactions within the unitary quantum theory and showed that in spite of being in superposition and highly entangled with other systems, observers do not directly witness those facts. Instead, they evolve alongside and in relation with other systems in independent and autonomous terms of the wave function, each describing a single classical world. Everett’s proposal not only saved the compatibility of quantum theory with scientific realism, but it made important progress on the question of locality, as it dismissed the instantaneous collapse of the wave function. As a remarkable side-effect of his solution, the totality of what we see around us, the universe, is but a tiny sliver of a much richer multiverse. Full unitarity has provided key insights on the analyses of the EPR [24] and Bell [25, 26] scenarios, like the explanatory possibilities of observers in superposition and the crucial interaction in which observers compare their results, which must also be treated quantum-mechanically. But only so much of locality can be saved in the Schrödinger picture. As Wallace [27, Chapter 8] puts it, ‘in Everettian quantum mechanics interactions are local but states are nonlocal’.

Einstein, whose steadfast defence of locality led to no less than the general theory of relativity, insisted that ‘the real factual situation of system $S_{2}$ is independent of what is done with the system $S_{1}$, which is spatially separated from the former’ [28]. This criterion of locality cannot be met by the modes of description offered in the Schrödonger picture. Indeed, if the wave function is intended to capture the real factual situation, then entanglement undermines the required independence: An action on system $S_{1}$ alters the same wave function that also describes system $S_{2}$. On the other hand, while reduced density matrices do maintain this independence, they fail to fully represent the real factual situation as they provide an overly superficial account that omits how entangled systems can further interact.

Remarkably, Deutsch and Hayden [14] defied conventional wisdom by demonstrating that the Heisenberg picture admits a mode of description that fulfils Einsteinian locality. Each system possesses a descriptor that remains unchanged by actions on other systems; furthermore, any collection of these descriptors provides an empirically complete description of the composite system, capturing the distribution of any measurement performed on it. This underlying separable [29] ontology is the most satisfactory notion of locality, and it holds notwithstanding entanglement. Descriptors have been used to solve important conundrums: superdense coding [30] and quantum teleportation [31] were offered completely local explanations [32, 14, 33]; the Aharanov–Bohm [34] phase was shown to be locally acquired [35]; and a local-realistic treatment of indistinguishable particles led to the discovery of creation operators for non-abelian anyons [36, 37]. Regarding Bell experiments, the pioneers of descriptors demonstrated [14] that the information about the rotation angles is fully localised and by analyzing the dynamics of specific observables, Rubin [38] showcased how observers evolve into local copies. For descriptors, this amounts to their foliation into relative descriptors, a formalization by Kuypers and Deutsch [15] that is central to my analysis.

For a complete and pedagogical exposition of how descriptors work, see Ref. [32]. Here, I provide a less detailed, though sufficient, overview. In the Heisenberg picture, the state is stationary and conveniently set to $\rvert 0\rangle^{\otimes n}$ in quantum computational networks. The uncountably many time-evolving observables can all be obtained from a generating set, namely a set of operators that multiplicatively generate an operator basis. These operators can be chosen in a way that each acts non-trivially on one single system. For each given system, the generators acting on it are encompassed into the descriptor of that system.

Let $\mathfrak{U}$ be the whole system considered, and let $\mathfrak{Q}_{i}$ denote the $i$^th qubit. At time $0$ the descriptor of $\mathfrak{Q}_{i}$ can be expressed as the pair of operators acting on $\mathcal{H}^{\mathfrak{U}}\simeq\mathcal{H}^{{\mathfrak{Q}_{i}}}\otimes\mathcal{H}^{\overline{\mathfrak{Q}_{i}}}$,

$$\boldsymbol{q}_{i}(0)=\left(q_{ix}(0),\,q_{iz}(0)\right)=\left(\sigma_{x}\otimes\mathds{1}^{\overline{\mathfrak{Q}_{i}}}\,,\,\sigma_{z}\otimes\mathds{1}^{\overline{\mathfrak{Q}_{i}}}\right)\,,$$

(1)

where $\sigma_{x}$ and $\sigma_{z}$ are the Pauli matrices, and $\mathds{1}^{\overline{\mathfrak{Q}_{i}}}$ is the identity operator on all but $\mathfrak{Q}_{i}$. As they evolve, descriptors preserve their algebraic relations, so in particular, at any given time, the descriptor components of different qubits commute.

If between time $t$ and $t+1$, a gate $G$ is applied to a set of systems labelled by $\mathcal{J}$, the descriptor of qubit $i$ evolves according to

$$\boldsymbol{q}_{i}(t+1)=\mathsf{U}^{\dagger}_{G}\left(\left\{\boldsymbol{q}_{j}(t)\right\}_{j\in\mathcal{J}}\right)\,\boldsymbol{q}_{i}(t)\,\mathsf{U}_{G}\left(\left\{\boldsymbol{q}_{j}(t)\right\}_{j\in\mathcal{J}}\right)\,,$$

(2)

where $\mathsf{U}_{G}(\cdot)$ is a fixed operator-valued function characteristic of the gate $G$. Note that if $i\notin\mathcal{J}$, the commutation of the components of $\boldsymbol{q}_{i}$ with those of $\boldsymbol{q}_{j}$ ensures that the descriptor of $\mathfrak{Q}_{i}$ ‘is independent of what is done with the system’ labelled by $\mathcal{J}$, as mandated by Ensteinian locality.

Gates have an action on the systems they affect. The Hadamard acts as

$$H\,\colon\,\left(q_{x},q_{z}\right)\to\left(q_{z},q_{x}\right)\,,$$

(3)

namely, it switches the components of the descriptor of the affected system, regardless of their expressions when acted upon. A Bloch-sphere rotation around the $\hat{y}$-axis transforms the descriptor as

$$R_{\theta}:\quad(q_{x},q_{z})\to\left(\cos\theta q_{x}+\sin\theta q_{z}\,,\,-\sin\theta q_{x}+\cos\theta q_{z}\right)\,.$$

(4)

And the action of the Cnot is

$${\text{Cnot}}\,\colon\,\left\{\begin{array}[]{lcl}(q_{cx}&,&q_{cz})\vspace{2pt}\\ (q_{tx}&,&q_{tz})\vspace{2pt}\\ \end{array}\right\}\to\left\{\begin{array}[]{lcl}(q_{cx}q_{tx}&,&q_{cz}\hphantom{q_{tz}})\vspace{2pt}\\ (q_{tx}&,&q_{cz}q_{tz})\vspace{2pt}\\ \end{array}\right\}\,,$$

(5)

where the label $c$ refers to the $c$ontrol qubit and the label $t$ to the $t$arget qubit.

Any observable $\mathcal{O}(t)$ pertaining to a set of systems labelled by $\mathcal{J}$ can be expressed as a polynomial in $\left\{\boldsymbol{q}_{j}(t)\right\}_{j\in\mathcal{J}}$. Its expectation value $\langle\mathcal{O}(t)\rangle$ is calculated by bracketing the Heisenberg state. Multiversal measures, or probabilities, are special cases of such calculations. An observable $\mathcal{O}(t)$ is sharp [15], if, with respect to the Heisenberg state, it has a definite value, i.e. null variance, $\langle\mathcal{O}(t)^{2}\rangle=\langle\mathcal{O}(t)\rangle^{2}$.

Let $\mathfrak{S}$ be a system with descriptor $\boldsymbol{s}$, which at time $t$ has a sharp observable. Let $U$ be a unitary on $\mathfrak{S}$ that transforms it into another value of the same sharp observable, and let $\mathfrak{Q}_{1}$ be a control with an unsharp $q_{1z}(t)$. The functional form of the controlled unitary is

$$\mathsf{U}_{\text{Ctrl-$U$}}(\boldsymbol{q}_{1},\boldsymbol{s})=P_{+1}(q_{1z})\,\mathds{1}+P_{-1}(q_{1z})\,\mathsf{U}_{U}(\boldsymbol{s})\,,$$

(6)

where $P_{\pm 1}(q_{1z})\stackrel{{\scriptstyle\text{\tiny{def}}}}{{=}}\frac{\mathds{1}\pm q_{1z}}{2}$ is a projector. From equations (2) and (6),

$$\boldsymbol{s}(t+1)=\underbrace{P_{+1}(q_{1z}(t))\,\boldsymbol{s}(t)}_{\boldsymbol{s}_{+1}(t+1)}\,+\,\underbrace{P_{-1}(q_{1z}(t))\,\mathsf{U}_{U}^{\dagger}(\boldsymbol{s}(t))\,\boldsymbol{s}(t)\,\mathsf{U}_{U}(\boldsymbol{s}(t))}_{\boldsymbol{s}_{-1}(t+1)}\,.$$

This foliates the system $\mathfrak{S}$ in two instances, the relative descriptors $\boldsymbol{s}_{\pm 1}(t+1)=P_{\pm 1}(\boldsymbol{q}_{1z}(t))\boldsymbol{s}(t+1)$, each admitting a distinct value of a sharp observable. Under some forthcoming dynamics, they remain distinct and autonomous. For instance, any follow-up unitary $V$ on $\mathfrak{S}$ alone results in

$$\boldsymbol{s}^{{}_{(t+2)}}=\underbrace{\mathsf{U}^{\dagger}_{V}\left(\boldsymbol{s}^{{}_{(t+1)}}_{+1}\right)\,\boldsymbol{s}^{{}_{(t+1)}}_{+1}\,\mathsf{U}_{V}\left(\boldsymbol{s}^{{}_{(t+1)}}_{+1}\right)}_{\boldsymbol{s}^{{}_{(t+2)}}_{+1}}\,+\,\underbrace{\mathsf{U}^{\dagger}_{V}\left(\boldsymbol{s}^{{}_{(t+1)}}_{-1}\right)\boldsymbol{s}^{{}_{(t+1)}}_{-1}\,\mathsf{U}_{V}\left(\boldsymbol{s}^{{}_{(t+1)}}_{-1}\right)}_{\boldsymbol{s}^{{}_{(t+2)}}_{-1}}\,.$$

The computational network of a Bell experiment is displayed in Figure 1. Upon measuring her particle of the entangled pair, Alice smoothly and locally evolves into

$$\boldsymbol{q}_{A}(4)=\boldsymbol{q}_{A,+1}(4)+\,\boldsymbol{q}_{A,-1}(4)\,.$$

Because Alice is entangled, $\boldsymbol{q}_{A}(4)$ admits no sharp observable. Indeed, the expectation values of $q_{Ax}(4)$, $q_{Az}(4)$ and $q_{Ay}(4)=iq_{Ax}(4)q_{Az}(4)$ are all zero, which is incompatible with sharpness. Yet as $q_{Az}\equiv q_{Az}(0)$ is sharp with value $1$, within each instance $\boldsymbol{q}_{A,\,\pm 1}(4)=P_{\pm 1}(q_{1z}(3))\left(q_{Ax}\,,\,\pm q_{Az}\right)$, a $\pm 1$ $\hat{z}$-outcome is indicated; each with measure $1/2$ (see §7.2 in the Supplementary Discussion). At spacelike separation, Bob similarly foliates as $\boldsymbol{q}_{B}(4)=\boldsymbol{q}_{B,+1}(4)+\,\boldsymbol{q}_{B,-1}(4)$. Unlike relative states of the Schrödinger picture, relative descriptors are entirely local entities; they only concern individual localized systems. Therefore, Alice’s and Bob’s foliations are independent of each other, as required by Lorentz’s covariance.

Crucially, to confirm the violation of Bell inequalities, Alice and Bob must further interact to produce a record of the joint outcomes. In the single-world orthodoxy, bringing systems together that are each in a definite state merely amounts to convenience for comparison. But in unitary quantum theory, that comparison needs to be analyzed within the theory and, as shall be seen, permits non-trivial arrangements of the outcomes’ measures. Without loss of generality, Charlie first processes Alice’s communication, so between time $4$ and $5$, he foliates with respect to $q_{Az}(4)$, in a way that mirrors that the foliations previously encountered, $\boldsymbol{s}_{C}(5)=\boldsymbol{s}_{C,+1}(5)+\,\boldsymbol{s}_{C,-1}(5)$, with

	$\displaystyle\boldsymbol{s}_{C,+1}(5)$	$\displaystyle=$	$\displaystyle P_{+1}(q_{Az}(4))\,\boldsymbol{s}_{C}$
	$\displaystyle\boldsymbol{s}_{C,-1}(5)$	$\displaystyle=$	$\displaystyle P_{-1}(q_{Az}(4))\,\mathsf{U}_{+2}^{\dagger}(\boldsymbol{s}_{C})\,\boldsymbol{s}_{C}\,\mathsf{U}_{+2}(\boldsymbol{s}_{C})\,.$

When Charlie processes Bob’s communication, both $\boldsymbol{s}_{C,+1}(5)$ and $\boldsymbol{s}_{C,-1}(5)$ foliate again:

	$\displaystyle\boldsymbol{s}_{C}(6)$	$\displaystyle=$	$\displaystyle\mathsf{U}_{\text{C}_{+1}}^{\dagger}\left(\boldsymbol{q}_{B}(5),\boldsymbol{s}_{C,+1}(5)\right)\,\boldsymbol{s}_{C,+1}(5)\,\mathsf{U}_{\text{C}_{+1}}\left(\boldsymbol{q}_{B}(5),\boldsymbol{s}_{C,+1}(5)\right)$
			$\displaystyle\,+\,\mathsf{U}_{\text{C}_{+1}}^{\dagger}\left(\boldsymbol{q}_{B}(5),\boldsymbol{s}_{C,-1}(5)\right)\,\boldsymbol{s}_{C,-1}(5)\,\mathsf{U}_{\text{C}_{+1}}\left(\boldsymbol{q}_{B}(5),\boldsymbol{s}_{C,-1}(5)\right)$
		$\displaystyle=$	$\displaystyle P_{+1}(q_{Bz}(5))\,P_{+1}(q_{Az}(4))\,\boldsymbol{s}_{C}$
			$\displaystyle\,+\,P_{-1}(q_{Bz}(5))\,P_{+1}(q_{Az}(4))\,\mathsf{U}_{+1}^{\dagger}(\boldsymbol{s}_{C})\,\boldsymbol{s}_{C}\,\mathsf{U}_{+1}(\boldsymbol{s}_{C})$
			$\displaystyle\,+\,P_{+1}(q_{Bz}(5))\,P_{-1}(q_{Az}(4))\,\mathsf{U}_{+2}^{\dagger}(\boldsymbol{s}_{C})\,\boldsymbol{s}_{C}\,\mathsf{U}_{+2}(\boldsymbol{s}_{C})$
			$\displaystyle\,+\,P_{-1}(q_{Bz}(5))\,P_{-1}(q_{Az}(4))\,\mathsf{U}_{+3}^{\dagger}(\boldsymbol{s}_{C})\,\boldsymbol{s}_{C}\,\mathsf{U}_{+3}(\boldsymbol{s}_{C})\,.$

The record arises from the two local worlds of Alice, and those of Bob, and foliates into four non-interacting instances that respectively indicate $00$, $01$, $10$ and $11$, in binary. As is calculated in §7.2 of the Supplementary Discussion, the respective measures of those instances are

$$\frac{1}{2}\left(\cos^{2}\left(\frac{\theta-\phi}{2}\right),\,\sin^{2}\left(\frac{\theta-\phi}{2}\right),\,\sin^{2}\left(\frac{\theta-\phi}{2}\right),\,\cos^{2}\left(\frac{\theta-\phi}{2}\right)\right)\,.$$

(8)

If the angles are optimally chosen, a $\cos^{2}(\pi/8)$ win rate in a CHSH test is obtained (see the Supplementary Discussion §7.1).

Consider the relativistic twin paradox involving Alice, who has been staying on Earth, and her brother Bob, who has been travelling at high velocity and is currently far away. Should they eventually reunite on Earth, which twin shall be the younger? Invoking the rule of thumb that the traveller is younger to de facto declare Bob younger is generically mistaken, for there is no fact of the matter concerning age relations at spacelike separation. Indeed, different unfoldings of the story lead to different relations. In the most common continuation, Bob returns to Earth and finds himself younger than his stationary sister. Yet, imagine a twist where, during Bob’s journey, Alice embarks on an even faster and more extensive voyage. When she returns to Earth alongside Bob, she is younger than her sibling.

Similarly, there is no fact of the matter about the relationship between spacelike separated worlds because, following the unitarity of quantum theory, local worlds can in principle be altered in a way that affects their future matching. Like with the travelling twins, different continuations lead to different relationships, which are only measured once records about them are generated. For instance, suppose Alice receives input ‘$0$’ for the CHSH test, and, abiding by the usual protocol (exposed in §7.1), performs no rotation, i.e $\theta_{0}=0$. Consider now the Alice who measured output ‘$0$’ (or $+1$ of the $\hat{z}$ observable). What can she say about the Bob that she will meet? After Bob has followed the protocol and measured, it seems absolute that Alice will meet, within sub-measure $\cos^{2}{\pi/8}$, a Bob who has also seen output ‘$0$’. But that is the same mistake that assumes the common continuation. Since no measurement is definitive, Bob — or rather Wigner [39], who has coherent control over Bob — can in principle undo Bob’s measurement, alter the particle’s rotation with $R_{\pi-\phi}$, re-measure, and proceed towards Alice to compare results. The Alice who observed ‘$0$’ then meets with the Bob who observed ‘$1$’ with measure unity. This practically-hard-yet-theoretically-possible continuation contradicts the absoluteness of the relationship between worlds before systems composing them interact. Bell violations do not occur at spacelike separation.

The attempted completion of quantum theory by hidden variables is an aspiration to explain the quantum domain with an underlying classical reality. Abiding by the probability calculus that underpins Shannon’s theory [40], hidden variables amount to pieces of genuinely classical information. More precisely, they are the supposed information missing to determine measurement outcomes. Bell’s theorem shows that should they exist, such pieces of classical information cannot be constrained by the causal structure of spacetime, so only conspicuous and problematic mechanisms can save the program. But genuinely classical information does not exist. Indeed, the tables must be turned, and it is the classical realm that demands an underlying quantum explanation. That is the role of decoherence theory [41, 42, 43, 44, 45], which studies how interactions involving also the environment give rise to quasi-classical domains.

The processes involving $\mathfrak{Q}_{A}$, $\mathfrak{Q}_{B}$ and $\mathfrak{S}_{C}$ are classical, according to explanations of what ‘classical’ can mean within quantum theory. As illustrated in Figure 2, after the rotation of Particle $1$, a nearby environment can be modelled to interact with $\mathfrak{Q}_{1}$ in a way that stabilizes the $\hat{z}$ observable to be measured. The effect is that the unstructured $\bar{q}_{Ex}$ obfuscates $q_{1x}(3)$, but is not copied onto the following systems. Decoherence as such may also affect $\mathfrak{Q}_{A}$, $\mathfrak{Q}_{A^{\prime}}$, $\mathfrak{Q}_{A^{\prime\prime}}$, the analogous systems on Bob’s side, or with appropriate generalizations, $\mathfrak{S}_{C}$. These interactions neither prevent the mergings of Alice’s and Bob’s worlds portrayed in equation (4) nor, therefore, the ability to violate Bell’s inequalities. Such robustness to decoherence is a distinguishing property of classical processes.

Moreover, the realistic physical processes that amount to measuring and eventually communicating an experiment’s result involve more than a single binary system. In fact, in the kind of communication we call ‘classical,’ a precise quantum system is not sent from one location to another; rather, the information is transmitted through a chain reaction in a collection of quantum systems, like $\mathfrak{Q}_{A^{\prime}}$, $\mathfrak{Q}_{A^{\prime\prime}}$, and generically many more. The resulting descriptor $\boldsymbol{s}^{\prime}_{C}(6)$ admits, like in equation (4), a foliation into four entities, which are only altered insofar as the arguments in $P_{\pm 1}(\cdot)$ acquire extra factors of $q_{A^{\prime}z}q_{A^{\prime\prime}z}$ and $q_{B^{\prime}z}q_{B^{\prime\prime}z}$. As these operators have eigenvalue $1$ with respect to the Heisenberg state, the measures in equation (8) are invariant. Systems such as $\mathfrak{Q}_{A^{\prime}}$ and $\mathfrak{Q}_{A^{\prime\prime}}$ can also represent parts of a macroscopic Alice. In this light, although the foliation of $\mathfrak{Q}_{A}$ presented in section 4 is one Cnot away from recombining, this fragility is removed when Alice is instantiated with many more degrees of freedom. As the unsharp observable $q_{1z}(3)$ gets copied through the interacting systems, they foliate. Therefore, everything that suitably interacts with the Alices foliates in turn, creating worlds which, for all practical purposes, are independent and autonomous.

I explored how sets of local worlds generated by remote foliations interact in their future lightcone. When entanglement relates the unsharp observables responsible for the foliations, nontrivial arrangements of the worlds’ measures arise as systems further interact. This kind of interference persists even when systems decohere. Therefore, joining lists or comparing statistics, seemingly mundane and classical operations, are not trivial in the multiverse. This unsuspected interaction enables Bell correlations while preserving locality.

The Supplementary Discussion supports and completes the main text with elementary background on Bell’s theorem (§7.1) as well as technical details for calculating multiversal measures (§7.2).