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Frontiers in Physics

Quantum physics cannot be captured by classical linear hidden variable theories even in the absence of entanglement

Kawthar Al Rasbi, Lewis A. Clark, Almut Beige

Published Feb 29, 2024• 1 citation
Publisher

Abstract

Recent experimental tests of Bell inequalities confirm that entangled quantum systems cannot be described by local classical theories but still do not answer the question whether or not quantum systems could, in principle, be modeled by linear hidden variable theories. In this paper, we study the quantum trajectories of a single qubit that experiences a sequence of repeated generalized measurements. It is shown that this system, which constitutes a hidden quantum Markov model, is more likely to produce complex time correlations than any classical hidden Markov model with two output symbols. From this, we conclude that quantum physics cannot be replaced by linear hidden variable theories. Indeed, it has already been recognized that not only entanglement but also non-classical time correlations of quantum systems with quantum feedback are a valuable resource for quantum technology applications.

DOI
10.3389/fphy.2024.1325239

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Recent experimental tests of Bell inequalities confirm that entangled quantum systems cannot be described by local classical theories but still do not answer the question whether or not quantum systems could, in principle, be modeled by linear hidden variable theories. In this paper, we study the quantum trajectories of a single qubit that experiences a sequence of repeated generalized measurements. It is shown that this system, which constitutes a hidden quantum Markov model, is more likely to produce complex time correlations than any classical hidden Markov model with two output symbols. From this, we conclude that quantum physics cannot be replaced by linear hidden variable theories. Indeed, it has already been recognized that not only entanglement but also non-classical time correlations of quantum systems with quantum feedback are a valuable resource for quantum technology applications.
Recent experimental tests of Bell inequalities confirm that entangled quantum systems cannot be described by local classical theories but still do not answer the question whether or not quantum systems could in principle be modelled by linear hidden variable theories. In this paper, we study the quantum trajectories of a single qubit that experiences a sequence of repeated generalised measurements. It is shown that this system, which constitutes a Hidden Quantum Markov Model, is more likely to produce complex time correlations than any classical Hidden Markov Model with two output symbols. From this, we conclude that quantum physics cannot be replaced by linear hidden variable theories. Indeed, it has already been recognised that not only entanglement but also non-classical time correlations of quantum systems with quantum feedback are a valuable resource for quantum technology applications.
It is widely believed that quantum physics is a fundamental theory describing the Universe. As such, one would expect to be able to see how classical physics that is observed in the macroscopic world emerges from quantum theory. This has so far largely eluded physicists, due to the inherent linear nature of quantum physics and the non-linear behaviour of classical physics. One of the principle differences between classical and quantum physics is the statistical, probabilistic nature of quantum theory. It is from this property that non-classical states can arise, such as entangled states. These states possess maximal correlations. However, they are not the only way in which correlations are created in quantum systems. This thesis aims to show how open quantum systems naturally contain correlations from their quantum nature. Moreover, even seemingly simple open quantum systems can behave far more complexly than expected upon the introduction of quantum feedback. Using this effect, the dynamics may become non-linear and as such behave non-trivially. Furthermore, it is shown how these effects may be exploited for a variety of tasks, including a computational application in hidden quantum Markov models and a quantum metrology scheme that does not require the use of exotic quantum states. This results in the design of systems that benefit from the use of quantum mechanics, but are not constrained by the use of experimentally difficulties such as entanglement.
A hidden-variable model is explicitly constructed by use of a Liouvillian description for the dynamics of two coupled spin-1/2 particles. In this model, the underlying Hamiltonian trajectories play the role of deterministic hidden variables, whereas the shape of the initial probability distribution figures as a hidden variable that regulates the capacity of the model in producing correlations. We show that even though the model can very well describe the short-time entanglement dynamics of initially separated pure states, it is incapable of violating the Clauser-Horne-Shimony-Holt inequality. Our work suggests that, if one takes the reluctance of a given quantum resource to be emulated by a local-hidden-variable model as a signature of its nonclassicality degree, then one can conclude that entanglement and nonlocality are nonequivalent even in the context of two-qubit pure states.
Entanglement is one of the most intriguing features of quantum mechanics. It gives rise to peculiar correlations which cannot be reproduced by a large class of alternative theories, the so-called hidden-variable models, that use parameters in addition to the wave-function. This incompatibility was quantified through the celebrated Bell inequalities, and more recently through new inequalities due to Leggett. Experiments confirm the predictions of quantum mechanics. However, this does not imply that quantum mechanics is the ultimate theory, unsusceptible of improvement, nor that quantum mechanics is essentially non-local. The theories ruled out by Bell and Leggett inequalities are required to satisfy some hypotheses, none of which is implied by locality alone. By dropping one or more hypotheses, it is possible not only to violate said inequalities, but to reproduce the quantum mechanical predictions altogether. So far, the models proposed were only mathematical constructs. In this paper we provide a classical realization of two of these models, using local classical resources, without recurring to any type of communication among the involved parties. The resources consist in two baseballs, two bats, and a number of synchronized watches. Our results demonstrate the possibility of reproducing the quantum mechanical correlations, and even creating stronger correlations which provide the maximum violation of the Bell inequality, beyond the Cirel'son bound for quantum mechanics.
Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite on average, Bell states can be simulated without any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of n Bell states can be simulated with O(n2n){O(n2}^{n})O(n2n) bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be absolutely bounded, we show that a finite number of bits of local hidden variables is insufficient to simulate a Bell state. This latter result is based on an analysis of the nondeterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model.
We investigate when the quantum correlations of a bipartite system, under the influence of environments with memory, are not reproducible with certainty by a classical local hidden variable model. For this purpose, we compare the dynamics of a Bell inequality with that of entanglement, as measured by concurrence. We find time regions when the Bell inequality is not violated even in correspondence with high values of concurrence (up to $\ensuremath{\approx}0.8$). We also suggest that these results may be observed by adopting a modification of a recent experimental optical setup. These findings indicate that even highly entangled systems cannot be exploited with certainty in contexts where the nonclassical reproducibility of quantum correlations is required.