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arXiv:1605.02944 · Physical Review A

Bell inequalities for continuous-variable systems in generic squeezed states

Jérôme Martin, Vincent Vennin

Published Jun 17, 2016• 51 citations
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Abstract

Bell inequalities for continuous-variable bipartite systems are studied. The inequalities are expressed in terms of pseudo--spin operators, and quantum expectation values are calculated for generic two-mode squeezed states characterized by a squeezing parameter $r$ and a squeezing angle $\ensuremath{\varphi}$. Allowing for generic values of the squeezing angle is especially relevant when $\ensuremath{\varphi}$ is not under experimental control, such as in cosmic inflation, where small quantum fluctuations in the early universe are responsible for structures formation. Compared to previous studies restricted to $\ensuremath{\varphi}=0$ and to a fixed orientation of the pseudo--spin operators, allowing for $\ensuremath{\varphi}\ensuremath{\ne}0$ and optimizing the angular configuration leads to a completely new and rich phenomenology. Two dual schemes of approximation are designed that allow for comprehensive exploration of the squeezing parameter space. In particular, it is found that Bell inequalities can be violated when the squeezing parameter $r$ is large enough, $r\ensuremath{\gtrsim}1.12$, and the squeezing angle $\ensuremath{\varphi}$ is small enough, $\ensuremath{\varphi}\ensuremath{\lesssim}0.34\phantom{\rule{0.16em}{0ex}}{e}^{\ensuremath{-}r}$.

DOI
10.1103/physreva.93.062117

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Bell inequalities for continuous-variable bipartite systems are studied. The inequalities are expressed in terms of pseudo--spin operators, and quantum expectation values are calculated for generic two-mode squeezed states characterized by a squeezing parameter rrr and a squeezing angle \ensuremathφ\ensuremath{\varphi}\ensuremathφ. Allowing for generic values of the squeezing angle is especially relevant when \ensuremathφ\ensuremath{\varphi}\ensuremathφ is not under experimental control, such as in cosmic inflation, where small quantum fluctuations in the early universe are responsible for structures formation. Compared to previous studies restricted to \ensuremathφ=0\ensuremath{\varphi}=0\ensuremathφ=0 and to a fixed orientation of the pseudo--spin operators, allowing for \ensuremathφ\ensuremath≠0\ensuremath{\varphi}\ensuremath{\ne}0\ensuremathφ\ensuremath=0 and optimizing the angular configuration leads to a completely new and rich phenomenology. Two dual schemes of approximation are designed that allow for comprehensive exploration of the squeezing parameter space. In particular, it is found that Bell inequalities can be violated when the squeezing parameter rrr is large enough, r\ensuremath≳1.12r\ensuremath{\gtrsim}1.12r\ensuremath≳1.12, and the squeezing angle \ensuremathφ\ensuremath{\varphi}\ensuremathφ is small enough, \ensuremathφ\ensuremath≲0.34e\ensuremath−r\ensuremath{\varphi}\ensuremath{\lesssim}0.34\phantom{\rule{0.16em}{0ex}}{e}^{\ensuremath{-}r}\ensuremathφ\ensuremath≲0.34e\ensuremath−r.
Bipartite temporal Bell inequalities are similar to the usual Bell inequalities except that, instead of changing the direction of the polarizer at each measurement, one changes the time at which the measurement is being performed. By doing so, one is able to test for realism and locality, but relying on position measurements only. This is particularly useful in experimental setups where the momentum direction cannot be probed (such as in cosmology for instance). We study these bipartite temporal Bell inequalities for continuous systems placed in two-mode squeezed states and find some regions in parameter space where they are indeed violated. We highlight the role played by the rotation angle, which is one of the three parameters characterizing a two-mode squeezed state (the other two being the squeezing amplitude and the squeezing angle). In single-time measurements, it only determines the overall phase of the wave function and can therefore be discarded, but in multiple-time measurements, its time dynamics becomes relevant and crucially determines when bipartite temporal Bell inequalities can be violated. Our study provides an opportunity for experimental designs for the observation of Bell inequality violations.
We investigate the role of the bipartite temporal Bell inequality, an analogue of the spatial Bell inequality, in probing the quantum imprints of primordial perturbations when the initially chosen Bunch-Davies vacuum is replaced by a coherent state. Although it is based on the same principles of locality and realism, its primary advantage lies in the fact that it does not require two distinct set of observable for its construction. Instead, measurements performed on a single component of the pseudo-spin operator at different times are sufficient. Consequently, it is particularly well suited for cosmological scenarios, where observational constraints typically allow access to only one component of the pseudo-spin operator. Assuming a coherent state as the initial condition, we derive an analytical expression for the expectation value of the bipartite temporal Bell operator and demonstrate the absence of temporal Bell violation in such a scenario. Interestingly, the results for squeezed coherent state is found to differ - albeit slightly - from those of squeezed vacuum state for large values of the squeezing parameter. This suggests that the ability to distinguish among different initial states of primordial perturbations does not rely on the violation of temporal Bell inequality. Furthermore, the dependence of the temporal Bell inequality on a purely imaginary phase factor of the wave function appears to be an unique feature, which is entirely absent in the context of spatial Bell inequalities.
Temporal Bell inequalities, or Leggett-Garg inequalities (LGIs), are studied for continuous-variable systems placed in a squeezed state. The importance of those systems lies in their broad applicability, which allows the description of many different physical settings in various branches of physics, ranging from cosmology to condensed matter physics and from optics to quantum information theory. Leggett-Garg inequality violations are explored and systematically mapped in squeezing parameter space. Configurations for which LGI violation occurs are found, but it is shown that no violation can be obtained if all squeezing angles vanish, contrary to what happens for the spatial Bell inequalities. We also assess the effect of decoherence on the detectability of such violations. Our study opens up the possibility of new experimental designs for the observation of LGI violation.
We present a phase-space formalism to evaluate Bell inequality violations in continuous variable systems. By doing so we can generalize previous analyses (which have dealt only with pure states) to arbitrary mixed states. We leverage these results to analyze the effect of temperature on violations of Bell inequalities in a two-mode squeezed thermal state, which can become useful in tests of local realism in the presence of thermal noise. We also explore the nonmonotonic relationship between the violations of Bell inequalities and the amount of entanglement present in this family of mixed states. Additionally, we discuss the optimal choices of pseudospin operators for states beyond the two-mode squeezed vacuum.
We consider bipartite quantum systems characterized by a continuous angular variable $\ensuremath{\theta}$, representing, for instance, the position of a particle on a circle or in a periodic lattice. For this type of system, we show how to violate inequalities similar to Clauser-Horne-Shimony-Holt (CHSH) ones, originally derived for bipartite spin-1/21/21/2-like systems. Such inequalities involve correlated measurement of continuous angular functions and are equivalent to the continuous superposition of CHSH inequalities acting on bidimensional subspaces of the infinite dimensional Hilbert space. As an example, we discuss in detail one application of our results, and we derive inequalities based on orientation correlation measurements. The introduced Bell-type inequalities open the perspective of new and simpler experiments to test the assumptions of locality, realism, or freedom of choice in a variety of quantum systems described by continuous variables. We discuss in detail a possible implementation using the orbital angular momentum of photons.
We present a general and systematic study of how a Bell experiment on the cosmic microwave background could be carried out. We introduce different classes of pseudo-spin operators and show that, if the system is placed in a two-mode squeezed state as inflation predicts, they all lead to a violation of the Bell inequality. However, we also discuss the obstacles that one faces in order to realize this program in practice and show that they are probably insurmountable. We suggest alternative methods that could reveal the quantum origin of cosmological structures without relying on Bell experiments.