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The geometry of the Einstein--Podolsky--Rosen argument

H. Chau Nguyen, Huy-Viet Nguyen, Otfried Gühne

Published Aug 28, 2018• 0 citations
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Abstract

Correlations between distant particles are central to many puzzles and paradoxes of quantum mechanics and, at the same time, underpin various applications like quantum cryptography and metrology. Originally in 1935, Einstein, Podolsky and Rosen (EPR) used these correlations to argue against the completeness of quantum mechanics. To formalise their argument, Schrodinger subsequently introduced the notion of quantum steering. Still, the question which quantum states can be used for the EPR argument and which not remained open. Here we show that quantum steering can be viewed as an inclusion problem in convex geometry. For the case of two spin-$\frac{1}{2}$ particles, this approach completely characterises the set of states leading to the EPR argument and consequently to a full description of the quantum correlations that can be used for steering. Our results find applications in various protocols in quantum information processing, and moreover they are linked to quantum mechanical phenomena such as uncertainty relations and the question which observables in quantum mechanics are jointly measurable.

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Correlations between distant particles are central to many puzzles and paradoxes of quantum mechanics and, at the same time, underpin various applications like quantum cryptography and metrology. Originally in 1935, Einstein, Podolsky and Rosen (EPR) used these correlations to argue against the completeness of quantum mechanics. To formalise their argument, Schrodinger subsequently introduced the notion of quantum steering. Still, the question which quantum states can be used for the EPR argument and which not remained open. Here we show that quantum steering can be viewed as an inclusion problem in convex geometry. For the case of two spin-12\frac{1}{2}21​ particles, this approach completely characterises the set of states leading to the EPR argument and consequently to a full description of the quantum correlations that can be used for steering. Our results find applications in various protocols in quantum information processing, and moreover they are linked to quantum mechanical phenomena such as uncertainty relations and the question which observables in quantum mechanics are jointly measurable.
Correlations between distant particles are central to many puzzles and paradoxes of quantum mechanics and, at the same time, underpin various applications such as quantum cryptography and metrology. Originally in 1935, Einstein, Podolsky, and Rosen (EPR) used these correlations to argue against the completeness of quantum mechanics. To formalize their argument, Schrödinger subsequently introduced the notion of quantum steering. Still, the question of which quantum states can be used for EPR steering and which cannot remained open. Here we show that quantum steering can be viewed as an inclusion problem in convex geometry. For the case of two spin-1/2 particles, this approach completely characterizes the set of states leading to EPR steering. In addition, we discuss the generalization to higher-dimensional systems as well as generalized measurements. Our results find applications in various protocols in quantum information processing, and moreover they are linked to quantum mechanical phenomena such as uncertainty relations and the question of which observables in quantum mechanics are jointly measurable.
The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non-commuting measurements with a precision that seems to violate the uncertainty principle. This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.
The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non-commuting measurements with a precision that seems to violate the uncertainty principle. This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.
The Einstein-Podolsky-Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of "quantum nonlocality", which manifests itself by three different types: quantum entanglement, quantum steering, and Bell nonlocality. Although Bell nonlocality is more often used to show the "quantum nonlocality", the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality,thus making it amenable to an experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is more friendly for experimental implementation, i.e., one may only measure the observables in xxx-, yyy-, or zzz-axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with the theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.
The year 2015 marks 80 years since Einstein, Podolsky, and Rosen argued for the incompleteness of quantum mechanics, in the process describing one of the most counterintuitive characteristics of quantum mechanics: distant measurements can influence the character of local quantum states. This led to reactions from Schrodinger, Bohr, and John Bell. This issue shows that these developments are still well under way, with new foundational insights on the history of steering and Bell-nonlocality, new directions in the mathematical characterization of EPR-steering, demonstrations in quantum optical experiments, and further connections with quantum information processing.
The Einstein–Podolsky–Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of a local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of “quantum nonlocality,” which manifests itself in three types: quantum entanglement, quantum steering, and Bell’s nonlocality. Although Bell’s nonlocality is more often used to show “quantum nonlocality,” the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality, thus making it amenable to experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is friendlier for experimental implementation, i.e., one may measure the observables only in the x , y , or z axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.