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Journal of Physics A Mathematical and Theoretical

Non-locality breaking qubit channels: the case for CHSH inequality

Rajarshi Pal, Sibasish Ghosh A

Published Mar 25, 2015• 22 citations
Publisher

Abstract

Entanglement breaking channels play a significant role in quantum information theory. In this work we investigate qubit channels through their property of 'non-locality breaking', which is defined in a natural way but within the purview of CHSH non-locality. This also provides a different perspective on the relationship between entanglement and non-locality through the dual picture of quantum channels instead of through states. For a channel to be entanglement breaking, it is sufficient to 'break' the entanglement of maximally entangled states. We provide examples to show that for CHSH nonlocality breaking such a property does not hold in general, although for certain channels and for a restricted class of states for all channels this holds. We also consider channels whose output remains local under SLOCC and call them 'strongly non-locality breaking'. We provide a closed-form necessary-sufficient condition for any two-qubit state to show hidden CHSH non-locality, which is likely to be useful for other purposes as well. This in turn allows us to characterize all strongly non-locality breaking qubit channels. It turns out that unital qubit channels breaking non-locality of maximally entangled states are strongly non-locality breaking while extremal qubit channels cannot be so unless they are entanglement breaking.

DOI
10.1088/1751-8113/48/15/155302

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Entanglement breaking channels play a significant role in quantum information theory. In this work we investigate qubit channels through their property of 'non-locality breaking', which is defined in a natural way but within the purview of CHSH non-locality. This also provides a different perspective on the relationship between entanglement and non-locality through the dual picture of quantum channels instead of through states. For a channel to be entanglement breaking, it is sufficient to 'break' the entanglement of maximally entangled states. We provide examples to show that for CHSH nonlocality breaking such a property does not hold in general, although for certain channels and for a restricted class of states for all channels this holds. We also consider channels whose output remains local under SLOCC and call them 'strongly non-locality breaking'. We provide a closed-form necessary-sufficient condition for any two-qubit state to show hidden CHSH non-locality, which is likely to be useful for other purposes as well. This in turn allows us to characterize all strongly non-locality breaking qubit channels. It turns out that unital qubit channels breaking non-locality of maximally entangled states are strongly non-locality breaking while extremal qubit channels cannot be so unless they are entanglement breaking.
Entanglement breaking channels play a significant role in quantum information theory. In this work we investigate qubit channels through their property of `non-locality breaking', defined in a natural way but within the purview of CHSH nonlocality. This also provides a different perspective on the relationship between entanglement and nonlocality through the dual picture of quantum channels instead of through states. For a channel to be entanglement breaking it is sufficient to `break' the entanglement of maximally entangled states. We provide examples to show that for CHSH nonlocality breaking such a property does not hold in general, though for certain channels and for a restricted class of states for all channels this holds.We also consider channels whose output remains local under SLOCC and call them `strongly non-locality breaking'. We provide a closed form necessary-sufficient condition for any two-qubit state to show hidden CHSH nonlocality, which is likely to be useful for other purposes as well. This in turn allows us to characterize all strongly non-locality breaking qubit channels. It turns out that unital qubit channels breaking nonlocality of maximally entangled states are strongly non-locality breaking while extremal qubit channels cannot be so unless they are entanglement breaking.
Incompatibility and nonlocality are not only of foundational interest but also act as important resources for quantum information theory. In the Clauser-Horne-Shimony-Holt (CHSH) scenario, the incompatibility of a pair of observables is known to be equivalent to Bell nonlocality. Here, we investigate these notions in the context of qubit channels. The Bell-CHSH inequality has a greater perspective---compared to any genuine tripartite nonlocality scenario---while determining the interplay between nonlocality breaking qubit channels and incompatibility breaking qubit channels. In the Bell-CHSH scenario, we prove that if the conjugate of a channel is incompatibility breaking, then the channel is itself nonlocality breaking and vice versa. However, this equivalence is not straightforwardly generalized to multipartite systems, due to the absence of an equivalence relation between incompatibility and nonlocality in the multipartite scenario. We investigate this relation in the tripartite scenario by considering some well-known states like Greenberger-Horne-Zeilinger and WWW states and using the notion of Mermin and Svetlichny nonlocality. By subjecting the parties in question to unital qubit channels, we identify the range of state and channel parameters for which incompatibility coexists with nonlocality. Further, we identify the set of unital qubit channels that is Mermin or Svetlichny nonlocality breaking irrespective of the input state.
Abstract Quantum channels that break CHSH nonlocality on all input states are known as CHSH-breaking channels. In quantum networks, such channels are useless for distributing correlations that can violate the CHSH Inequality. Motivated by previous work on activation of nonlocality in quantum states, here we demonstrate an analogous activation of CHSH-breaking channels. That is, we show that certain pairs of CHSH-breaking channels are no longer CHSH-breaking when used in combination. We find that this type of activation can emerge in both uni-directional and bi-directional communication scenarios.
Quantum nonlocality is a cornerstone property for the development of the socalled quantum protocols as well as for the characterisation, and possible generalisation of quantum theory. However, there are still fundamental aspects of this property that need to be addressed; in particular, its relation with the property of quantum entanglement. It is well known that every nonlocal state is an entangled one, yet there exist entangled local states. Therefore, within their original defnitions, these two properties are not equivalent. The study of this relationship has been put forward by means of the generalisation of nonlocality through the so-called activation of nonlocality scenarios. The idea is to only work with these entangled local states in order to try to recover (activate) nonlocality. In this Thesis, we have numerically addressed the properties of entanglement, nonlocality (mostly by means of the violation of the CHSH-inequality), as well as the following generalisations of nonlocality: k-copy nonlocality, hidden nonlocality, and the combination of these two scenarios, namely, hidden k-copy nonlocality and tensor/filter activation of hidden nonlocality. We have studied these properties for some bipartite entangled local states, and some two-qubit not CHSH-inequality violating states of interest. In particular, we have analysed these properties for two-qubit states that come from the study of quantum games' strategies and the dynamics of open quantum systems.
Quantum channels breaking entanglement, incompatibility, or nonlocality are defined as such because they are not useful for entanglement-based, one-sided device-independent, or device-independent quantum-information processing, respectively.Here, we show that such breaking channels are related to complementary tests of macrorealism, i.e., temporal separability, channel unsteerability, temporal unsteerability, and the temporal Bell inequality.To demonstrate this we first define a steerability-breaking channel, which is conceptually similar to entanglement and nonlocality-breaking channels and prove that it is identical to an incompatibility-breaking channel.A hierarchy of quantum nonbreaking channels is derived, akin to the existing hierarchy relations for temporal and spatial quantum correlations.We then introduce the concept of channels that break temporal correlations, explain how they are related to the standard breaking channels, and prove the following results.(1) A robustness-based measure for non-entanglementbreaking channels can be probed by temporal nonseparability.(2) A non-steerability-breaking channel can be quantified by channel steering.(3) Temporal steerability and nonmacrorealism can be used for, respectively, distinguishing unital steerability-breaking channels and nonlocality-breaking channels for a maximally entangled state.Finally, a two-dimensional depolarizing channel is experimentally implemented as a proof-of-principle example to demonstrate the hierarchy relation of nonbreaking channels using temporal quantum correlations.