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arXiv:2405.03100 · Chinese Physics B

Generalized Einstein-Podolsky-Rosen Steering Paradox

Zhi-Jie Liu, Xing-Yan Fan, Jie Zhou, Mi Xie, Jing‐Ling Chen

Published Aug 27, 2024• 0 citations
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Abstract

Abstract Quantum paradoxes are essential means to reveal the incompatibility between quantum and classical theories, among which the Einstein-Podolsky-Rosen (EPR) steering paradox offers a sharper criterion for the contradiction between local-hidden-state model and quantum mechanics than the usual inequality-based method. In this work, we present a generalized EPR steering paradox, which predicts a contradictory equality “$2_{Q}=\left(1+\delta\right)_{C}$” ($0\leq\delta＜1$) given by the quantum ($Q$) and classical ($C$) theories. For any $N$-qubit state in which the conditional state of the steered party is pure, we test the paradox through a two-setting steering protocol, and find that the state is steerable if some specific measurement requirements are satisfied. Moreover, our construction also enlightens the building of EPR steering inequality, which may contribute to some schemes for typical quantum teleportation and quantum key distributions.

DOI
10.1088/1674-1056/ad73b7

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Abstract Quantum paradoxes are essential means to reveal the incompatibility between quantum and classical theories, among which the Einstein-Podolsky-Rosen (EPR) steering paradox offers a sharper criterion for the contradiction between local-hidden-state model and quantum mechanics than the usual inequality-based method. In this work, we present a generalized EPR steering paradox, which predicts a contradictory equality “2Q=(1+δ)C2_{Q}=\left(1+\delta\right)_{C}2Q​=(1+δ)C​” (0≤δ＜10\leq\delta＜10≤δ＜1) given by the quantum (QQQ) and classical (CCC) theories. For any NNN-qubit state in which the conditional state of the steered party is pure, we test the paradox through a two-setting steering protocol, and find that the state is steerable if some specific measurement requirements are satisfied. Moreover, our construction also enlightens the building of EPR steering inequality, which may contribute to some schemes for typical quantum teleportation and quantum key distributions.
Abstract Einstein-Podolsky-Rosen (EPR) steering is a quantum nonlocality between entanglement and Bell’s nonlocality emphasizing the incompatibility of the local-hidden-state (LHS) model with quantum theory. It is well established that the EPR steering paradox is an essential method for determining steering, which reveals the nature of steering by logical contradiction. However, previous studies on the EPR steering paradox have only examined cases where the conditional states of the steered party are pure. In this work, we present the EPR steering paradox about the case when the conditional states of the steered party are mixed, which can be expressed as a paradoxical equality ‘ 2 Q = 2 − δ C ’ 0 < δ < 1 given by quantum (Q) and classical (C) theories. For any N-qubit state, in the two-setting steering protocol, the contradiction ‘ 2 Q = 2 − δ C ’ can be obtained if and only if a particular set of projective measurements of the steering party makes all the conditional states of the steered party pure, which also implies that the N-qubit state is steerable. Moreover, considering the conditional states of the steered party as mixed states, we are able to identify the full range of steerable states detectable through the EPR steering paradox method. This also means that, so far, one has found all the steerable states that can be recognized by the EPR steering paradox, which is significant for some typical quantum schemes such as quantum teleportation and quantum key distribution.
Einstein–Podolsky–Rosen (EPR) paradox highlights the absence of a local realistic explanation for quantum mechanics, and shows the incompatibility of the local-hidden-state models with quantum theory. For N-qubit states, or more importantly, the N-qubit mixed states, we present the EPR steering paradox in the form of the contradictory equality “[Formula: see text]”. We show that the contradiction holds for any N-qubit state as long as both “the pure state requirement” and “the measurement requirement” are satisfied. This also indicates that the EPR steering paradox exists in more general cases. Finally, we give specific examples to demonstrate and analyze our arguments.
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 Einstein-Podolsky-Rosen (EPR) paradox is one of the earliest paradoxes, arising from the lack of local realistic description of mechanics. The EPR paradox leads to the important concept of quantum which manifests itself by three different phenomena: entanglement, steering, and Bell's nonlocality. Two well-known paradoxes related to Bell's nonlocality, i.e. the Greenberger-Horne-Zeilinger paradox and the Hardy paradox, have been widely studied and also experimentally confirmed. However, an experimental test of the EPR paradox is still lacking. The EPR paradox is essentially a steering paradox, and recently it has been mathematically formulated as a contradiction equality k=1k=1k=1, thus making it amenable to an experimental verification. Moreover, based on the EPR paradox we also derive a generalized linear steering inequality. In this work, we perform the first experimental test of the EPR paradox in a two-qubit scenario, and also verify the corresponding generalized steering inequality, which has an advantage over the usual one by detecting more steerability of states. Experimental results confirm the theoretical predictions within the experimental errors, and provide a deeper understanding of foundation.
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.
Einstein-Podolsky-Rosen (EPR) steering is the ability that an observer persuades a distant observer to share entanglement by making local measurements. Determining a quantum state is steerable or unsteerable remains an open problem. Here, we derive a new steering inequality with infinite measurements corresponding to an arbitrary two-qubit T state, from consideration of EPR steering inequalities with N projective measurement settings for each side. In fact, the steering inequality is also a sufficient criterion for guaranteering that the T state is unsteerable. Hence, the steering inequality can be viewed as a necessary and sufficient criterion to distinguish whether the T state is steerable or unsteerable. In order to reveal the fact that the set composed of steerable states is the strict subset of the set made up of entangled states, we prove theoretically that all separable T states can not violate the steering inequality. Moreover, we put forward a method to estimate the maximum violation from concurrence for arbitrary two-qubit T states, which indicates that the T state is steerable if its concurrence exceeds 1/4.