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arXiv:2107.06919 · Physical review. D/Physical review. D.

Monogamy paradox in empty flat space

Tuneer Chakraborty, Joydeep Chakravarty, Priyadarshi Paul

Published Oct 5, 2022• 2 citations
arXivHTMLPDF

Abstract

In a recent paper, Raju showed that the essential features of the monogamy paradox for old flat space black holes could be modeled using a setup in empty anti--de Sitter, leading to a violation in the monogamy of entanglement there. A physically interesting question is whether such a violation in the monogamy of entanglement can be posed in empty flat space. The answer is not immediately clear since flat space gravity has an entirely different vacuum and infrared structure than the gapped and unique AdS vacuum, which was exploited in Raju's toy model. We answer this question in the affirmative, with an explicit construction. We formulate the paradox in terms of monogamy of Clauser-Horne-Shimony-Holt correlations, which we use to quantify the monogamy of entanglement. Extending Raju's analysis to empty flat space, within effective field theory, we show that the entanglement of approximately local bulk modes just outside a light cone with modes just inside the light cone as well as with modes situated far away at the past of future null infinity $({\mathcal{I}}_{\ensuremath{-}}^{+})$ gives rise to an O(1) violation in the monogamy of entanglement. This cannot be resolved by small corrections of O($\sqrt{{G}_{N}}$). The issues arising from the above-mentioned vacuum and infrared features unique to flat spacetime are dealt with by introducing a physically motivated boundary projector onto states below a given infrared cutoff, which allows us to construct suitable operators at $({\mathcal{I}}_{\ensuremath{-}}^{+})$ that give rise to the violation. We argue that the resolution of the paradox is that our spatially separated observables probe the same underlying degrees of freedom, i.e., such observables act on a nonfactorized Hilbert space arising from the Gauss constraint, thereby circumventing the conflict with monogamy of entanglement.

DOI
10.1103/physrevd.106.086002

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In a recent paper, Raju showed that the essential features of the monogamy paradox for old flat space black holes could be modeled using a setup in empty anti--de Sitter, leading to a violation in the monogamy of entanglement there. A physically interesting question is whether such a violation in the monogamy of entanglement can be posed in empty flat space. The answer is not immediately clear since flat space gravity has an entirely different vacuum and infrared structure than the gapped and unique AdS vacuum, which was exploited in Raju's toy model. We answer this question in the affirmative, with an explicit construction. We formulate the paradox in terms of monogamy of Clauser-Horne-Shimony-Holt correlations, which we use to quantify the monogamy of entanglement. Extending Raju's analysis to empty flat space, within effective field theory, we show that the entanglement of approximately local bulk modes just outside a light cone with modes just inside the light cone as well as with modes situated far away at the past of future null infinity (I\ensuremath−+)({\mathcal{I}}_{\ensuremath{-}}^{+})(I\ensuremath−+​) gives rise to an O(1) violation in the monogamy of entanglement. This cannot be resolved by small corrections of O(GN\sqrt{{G}_{N}}GN​​). The issues arising from the above-mentioned vacuum and infrared features unique to flat spacetime are dealt with by introducing a physically motivated boundary projector onto states below a given infrared cutoff, which allows us to construct suitable operators at (I\ensuremath−+)({\mathcal{I}}_{\ensuremath{-}}^{+})(I\ensuremath−+​) that give rise to the violation. We argue that the resolution of the paradox is that our spatially separated observables probe the same underlying degrees of freedom, i.e., such observables act on a nonfactorized Hilbert space arising from the Gauss constraint, thereby circumventing the conflict with monogamy of entanglement.
In this work, we investigate a toy model in empty flat space which captures essential features of the monogamy paradox for flat space black holes within a clean calculational framework. We formulate the paradox in terms of monogamy of CHSH correlations, which we use to quantify the monogamy of entanglement. Within effective field theory, we show that the entanglement of approximately local bulk modes just outside a light cone with modes just inside the light cone as well as with modes situated far away at the past of future null infinity gives rise to an O(1) violation in the monogamy of entanglement. An essential feature of this violation is that small corrections of O(GN\sqrt{G_N}GN​​) do not resolve the same. We argue that the resolution of the paradox is that our spatially separated observables probe the same underlying degrees of freedom, i.e., such observables act on a non-factorized Hilbert space, thereby circumventing the conflict with monogamy of entanglement.
A sharp version of the information paradox involves a seeming violation of the monogamy of entanglement during black hole evaporation. We construct an analogous paradox in empty anti-de Sitter space. In a local quantum field theory, Bell correlations between operators localized in mutually spacelike regions are monogamous. We show, through a controlled calculation, that this property can be violated by an order-1 factor in a theory of gravity. This example demonstrates that what appears to be a violation of the monogamy of entanglement may just be a subtle violation of locality in quantum gravity.
Closed timelike curves are striking predictions of general relativity allowing for time-travel. They are afflicted by notorious causality issues (e.g. grandfather's paradox). Quantum models where a qubit travels back in time solve these problems, at the cost of violating quantum theory's linearity-leading e.g. to universal quantum cloning. Interestingly, linearity is violated even by open timelike curves (OTCs), where the qubit does not interact with its past copy, but is initially entangled with another qubit. Non-linear dynamics is needed to avoid violating entanglement monogamy. Here we propose an alternative approach to OTCs, allowing for monogamy violations. Specifically, we describe the qubit in the OTC via a pseudo-density operator-a unified descriptor of both temporal and spatial correlations. We also simulate the monogamy violation with polarization-entangled photons, providing a pseudo-density operator quantum tomography. Remarkably, our proposal applies to any space-time correlations violating entanglement monogamy, such as those arising in black holes.
Abstract Of course not, but if one believes that information cannot be destroyed in a theory of quantum gravity, then we run into apparent contradictions with quantum theory when we consider evaporating black holes. Namely that the no-cloning theorem or the principle of entanglement monogamy is violated. Here, we show that neither violation need hold, since, in arguing that black holes lead to cloning or non-monogamy, one needs to assume a tensor product structure between two points in space-time that could instead be viewed as causally connected. In the latter case, one is violating the semi-classical causal structure of space, which is a strictly weaker implication than cloning or non-monogamy. This is because both cloning and non-monogamy also lead to a break-down of the semi-classical causal structure. We show that the lack of monogamy that can emerge in evaporating space times is one that is allowed in quantum mechanics, and is very naturally related to a lack of monogamy of correlations of outputs of measurements performed at subsequent instances of time of a single system. This is due to an interesting duality between temporal correlations and entanglement. A particular example of this is the Horowitz-Maldacena proposal, and we argue that it needn’t lead to cloning or violations of entanglement monogamy. For measurements on systems which appear to be leaving a black hole, we introduce the notion of the temporal product, and argue that it is just as natural a choice for measurements as the tensor product. For black holes, the tensor and temporal products have the same measurement statistics, but result in different type of non-monogamy of correlations, with the former being forbidden in quantum theory while the latter is allowed. In the case of the AMPS firewall experiment we find that the entanglement structure is modified, and one must have entanglement between the infalling Hawking partners and early time outgoing Hawking radiation which surprisingly tames the violation of entanglement monogamy.
We show that the no-disturbance principle imposes a tradeoff between locally contextual correlations violating the Klyachko-Can-Biniciogˇlu-Shumovski inequality and spatially separated correlations violating the Clauser-Horne-Shimony-Holt inequality. The violation of one inequality forbids the violation of the other. We also obtain the corresponding monogamy relation imposed by quantum theory for a qutrit-qubit system. Our results show the existence of fundamental monogamy relations between contextuality and nonlocality that suggest that entanglement might be a particular form of a more fundamental resource.
Black holes offer an exciting area to explore the nature of quantum gravity. The classic work on Hawking radiation indicates that black holes should decay via quantum effects, but our ideas about how this might work at a technical level are incomplete. Recently Almheiri-Marolf-Polchinski-Sully (AMPS) have noted an apparent paradox in reconciling fundamental properties of quantum mechanics with standard beliefs about black holes. One way to resolve the paradox is to postulate the existence of a ``firewall'' inside the black hole horizon which prevents objects from falling smoothly toward the singularity. A fundamental limitation on the behavior of quantum entanglement known as ``monogamy'' plays a key role in the AMPS argument. Our goal is to study and apply many-body entanglement theory to consider the entanglement among different parts of Hawking radiation and black holes. Using the multipartite entanglement measure called negativity, we identify an example which could change the AMPS accounting of quantum entanglement and perhaps eliminate the need for a firewall. Specifically, we constructed a toy model for black hole decay which has different entanglement behavior than that assumed by AMPS. We discuss the additional steps that would be needed to bring lessons from our toy model to our understanding of realistic black holes.