The composition rule for quantum systems is not the only possible one

TL;DR

The tensor-product composition rule of quantum systems is shown not to be uniquely determined by the remaining single-system postulates. The authors construct latent quantum theories (lqt) that preserve standard quantum states and transformations but replace the composition rule with an alternative associative operation, yet these theories yield the same Bell-like correlations as standard quantum theory. This demonstrates that correlations alone, including Bell tests, cannot fully reconstruct quantum theory and that local tomography is not sufficient to single out the tensor-product rule. The work points to time-like or higher-order causal-structure experiments as essential tools to experimentally probe the composition postulate, potentially revealing physics beyond standard quantum correlations while remaining consistent with no-signaling and without introducing supra-quantum resources. Overall, it broadens the landscape of near-quantum theories and reframes reconstruction programs to include the role of system composition as a fundamental postulate requiring empirical scrutiny.

Abstract

Quantum theory provides a significant example of two intermingling hallmarks of science: the ability to consistently combine physical systems and study them compositely; and the power to extract predictions in the form of correlations. A striking consequence of this facet is the violation of Bell inequalities, which has been experimentally demonstrated via Bell tests, thus attesting a classical/quantum divide. The prediction of this phenomenon originates as quantum systems are prescribed to combine according to the composition postulate, i.e., the tensor-product rule. This rule has also an operationally salient formulation -- rather than just a purely mathematical one -- given in terms of discriminability of composite states via local measurements. Yet, both the theoretical and the empirical status of such a postulate have been constantly challenged over the decades: is it possible to deduce it from the remaining postulates? Here, this long-standing problem is solved by answering in the negative. A family of operational theories is presented, differing from standard quantum theory solely in their system-composition rule, while, at the same time, being indistinguishable from it via Bell-like or more general correlation experiments. Quantum theory is thus established to embody genuinely more than quantum correlations. As a result, foundational programmes based on single-system principles only, or on mere Bell-like correlations, are operationally incomplete. On the experimental side, ascertaining the independence of postulates is a fundamental step to adjudicate between quantum theory and alternative physical theories: hence, the composition postulate calls for experimental scrutiny independently of the other features of quantum theory.

Figures (2)

• Figure 1: Exemplification of the property of local tomography satisfied by standard qt. Two distant experimenters, Alice and Bob, gather, respectively, subsystem $\mathrm{A}$ and $\mathrm{B}$ of a standard quantum joint source $\mathrm{AB}$ emitting state $\Psi_{\mathrm{AB}}$. In their local laboratories, Alice and Bob independently choose, respectively, quantum measurements $x$ (with outcomes $a$) and $y$ (with outcomes $b$), thus observing local statistics $\text{P}{\left(a\middle|x\right)}$ and $\text{P}{\left(b\middle|y\right)}$. By communicating to each other the co-occurrences of their respective local outcomes via a line of classical communication (dashed lines), they are able to reconstruct the joint state of the standard quantum source: In this sense quantum mechanics uses its information economicallywootters1990local.
• Figure 2: Example of violation of local tomography in lqt s. The scenario is the same as the one in Figure \ref{['fig:local_tomography']}, except that Alice and Bob here have access to their respective subsystem of a joint lqt source $\mathrm{Q}_1\mathrm{Q}_2$ emitting state $\tilde{\Psi}_{\mathrm{Q}_1\mathrm{Q}_2}$. The two parties are now able to experimentally reconstruct just $\text{Tr}_{\mathrm{L}}{\tilde{\Psi}_{\mathrm{Q}_1\mathrm{Q}_2}}$, namely, the standard quantum counterpart of the full state after discarding the latent factor $\mathrm{L}$; yet, they cannot reconstruct the full state $\tilde{\Psi}_{\mathrm{Q}_1\mathrm{Q}_2}$ by just performing independent local measurements. Then, the latent factor can be considered a locally hidden degree of freedom popping up just when subsystems are considered within their composite. It can be locally manipulated (with suitable restrictions) but cannot be measured unless the parties cooperate by performing global measurements. This is opposed to, e.g., the model studied in Ref. PhysRevA.87.052106, where the extra factor corresponds to a system that can be manipulated and measured by any observer.