<p>We consider an experimentalist, Alice, who creates a quantum superposition of a charged or massive body outside of a black hole (or, more generally, in the presence of a Killing horizon). It was previously shown that emission of soft photons/gravitons into the black hole will result in the decoherence of the components of the superposition if it is held open for a sufficiently long span of time. However, at any finite time, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>t</mi><mi>c</mi></msub></math>, during the process, it is not obvious how much decoherence has irrevocably occurred. Equivalently, it is not obvious how much information an observer inside the black hole can extract about Alice’s superposition prior to time <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>t</mi><mi>c</mi></msub></math>. In this paper, we solve for the optimal experimental protocol to be followed by Alice for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mo>&gt;</mo><msub><mi>t</mi><mi>c</mi></msub></math> so as to minimize the decoherence of the components of her superposition. More precisely, given the entangling radiation that has passed through the horizon prior to the cross-section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">C</mi></math> corresponding to the time <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mo>=</mo><msub><mi>t</mi><mi>c</mi></msub></math> in Alice’s lab, we determine the “optimal purification” of this radiation beyond <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">C</mi></math> such that the global quantum state of the radiation through the horizon has maximal overlap (quantum fidelity) with the Hartle-Hawking or Unruh vacuum. Due to the intricate low frequency entanglement structure of the quantum field theory vacuum state, we find this optimal purification to be nontrivial. In particular, even if Alice has already “closed” her superposition by bringing the components back together, we find that she can decrease the net decoherence of the components of her superposition somewhat by reopening it and performing further manipulations.</p>

United States

Daine L. Danielson

Jonah Kudler-Flam

Gautam Satishchandran

Robert M. Wald

How to minimize the decoherence caused by black holes

<p>While von Neumann entropies for subregions in quantum field theory universally contain ultraviolet divergences, differences between von Neumann entropies are finite and well defined in many physically relevant scenarios. We demonstrate that such a notion of entropy differences can be rigorously defined in quantum field theory in a general curved spacetime by introducing a novel, covariant regulator for the entropy based on the modular crossed product. This regulator associates a type II von Neumann algebra to each spacetime subregion, resulting in well-defined renormalized entropies. This prescription reproduces formulas for entropy differences that coincide with heuristic formulas widely used in the literature, and we prove that it satisfies desirable properties such as unitary invariance and concavity. As an application, we provide proofs of the Bekenstein bound and the quantum null energy condition, formulated directly in terms of vacuum-subtracted von Neumann entropies.</p>

Samuel Leutheusser

Adel A. Rahman

Antony J. Speranza

Covariant regulator for entanglement entropy: Proofs of the Bekenstein bound and the quantum null energy condition

It was previously shown that if an experimenter, Alice, puts a massive or charged body in a quantum spatial superposition, then the presence of a black hole (or more generally any Killing horizon) will eventually decohere the superposition. This decoherence was identified as resulting from the radiation of soft photons/gravitons through the horizon, thus suggesting that the global structure of the spacetime is essential for describing the decoherence. In this paper, we show that the decoherence can alternatively be described in terms of the local two-point function of the quantum field within Alice’s lab, without any direct reference to the horizon. From this point of view, the decoherence of Alice’s superposition in the presence of a black hole arises from the extremely low frequency Hawking quanta present in Alice’s lab. We explicitly calculate the decoherence occurring in Schwarzschild spacetime in the Unruh vacuum from the local viewpoint. We then use this viewpoint to elucidate (i) the differences in decoherence effects that would occur in Schwarzschild spacetime in the Boulware and Hartle-Hawking vacua; (ii) the difference in decoherence effects that would occur in Minkowski spacetime filled with a thermal bath as compared with Schwarzschild spacetime; (iii) the lack of decoherence in the spacetime of a static star even though the vacuum state outside the star is similar in many respects to the Boulware vacuum around a black hole; and (iv) the requirements on the degrees of freedom of a material body needed to produce a decoherence effect that mimics that of a black hole.

Local description of decoherence of quantum superpositions by black holes and other bodies

It has been argued that, while the individual terms in the generalized entropy, <math><msub><mi>S</mi><mrow><mi>gen</mi></mrow></msub><mo>=</mo><mi>A</mi><mo>/</mo><mn>4</mn><msub><mi>G</mi><mi>N</mi></msub><mo>+</mo><msub><mi>S</mi><mrow><mi>ext</mi></mrow></msub></math>, are ill-defined in the semiclassical limit, their sum is well-defined if one takes into account perturbative quantum gravitational effects. The first term diverges as <math><msub><mi>G</mi><mi>N</mi></msub><mo>→</mo><mn>0</mn></math>, and the second diverges due to the infinite entanglement across the horizon which is characteristic of type III von Neumann algebras. It was recently shown that the von Neumann algebra of observables “gravitationally dressed” to the mass of a Schwarzschild-AdS black hole or the energy of an observer in de Sitter spacetime admit a well-defined trace. The algebras are type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> (which does not admit a maximum entropy state) and type <math><mrow><msub><mrow><mi>II</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math> (which admits a maximum entropy state) respectively, and the von Neumann entropy of “semiclassical” states was found to be (up to an additive constant) the generalized entropy. However, these arguments rely on the existence of a stationary “equilibrium (KMS) state” and do not apply to, for example, black holes formed from gravitational collapse, Kerr black holes, or black holes in asymptotically de Sitter spacetime. These spacetimes are stationary but not in thermal equilibrium. In this paper, we present a general framework for obtaining the algebra of “gravitationally dressed” observables for a linear, Klein-Gordon field on any spacetime with a (bifurcate) Killing horizon. We prove, assuming the existence of a stationary state—which is not necessarily KMS—and suitable asymptotic decay of solutions, a “structure theorem” that the algebra of “gravitationally dressed” observables always contains a type II factor of observables “localized” on the horizon. These assumptions have been rigorously proven in most cases of interest in this paper. Applying our general framework to the algebra of observables in the exterior of an asymptotically flat Kerr black hole where the fields are dressed to the black hole mass and angular momentum we find that the algebra is the product of a type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> algebra on the horizon and a type <math><msub><mrow><mi>I</mi></mrow><mi>∞</mi></msub></math> algebra at past null infinity. The full algebra is type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math>, and the von Neumann entropy of semiclassical states is the generalized entropy. In the case of Schwarzschild-de Sitter, despite the fact that we must introduce an observer, the algebra of observables dressed to the perturbed areas of the black hole and cosmological horizons is the product of type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> algebras on each horizon. The entropy of semiclassical states is given by the sum of the areas of the two horizons as well as the entropy of quantum fields in between the horizons. Our results suggest that in all cases where there exists another “boundary structure” (e.g., an asymptotic boundary or another Killing horizon) the algebra of observables is type <math><msub><mrow><mi>II</mi></mrow><mi>∞</mi></msub></math> and in the absence of such structures (e.g. de Sitter spacetime) the algebra is type <math><mrow><msub><mrow><mi>II</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math>.

Generalized black hole entropy is von Neumann entropy

Any nontrivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness—that all “in” and “out” states lie in the standard (zero-memory) Fock space—and therefore leads to infrared divergences in the standard <math><mi>S</mi></math>-matrix amplitudes. In this paper, we define an infrared finite scattering theory which assumes only (1) the existence of in-/out-algebras and (2) that Heisenberg evolution is an automorphism of these algebras. The resulting “superscattering” map <math><mi>$</mi></math> allows for transitions between different in/out memory states and agrees with the standard <math><mi>S</mi></math> matrix when it is defined. We construct <math><mi>$</mi></math> amplitudes by defining (3) a “generalized asymptotic completeness” which accommodates states with memory in the space of asymptotic states and (4) a complete basis of improper states that generalize the usual <math><mi>n</mi></math>-particle momentum basis to account for states with memory. Using only general properties of <math><mi>$</mi></math>, we prove an analog of the Weinberg soft theorems in quantum gravity and QED which imply that all <math><mi>$</mi></math> amplitudes are well defined in the infrared. We comment on how one must generalize this framework to consider <math><mi>$</mi></math> amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories).

India

Kartik Prabhu

Infrared finite scattering theory: Amplitudes and soft theorems

Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an “out” state with memory which gives rise to infrared divergences in the standard S-matrix. To obtain an infrared-finite scattering theory, one must suitably include states with memory. However, except in the case of QED with massive charged particles, asymptotic states with memory that have finite energy and angular momentum have not been constructed for more general theories (e.g. massless QED, Yang-Mills and quantum gravity). To this end, we construct direct-integral representations over the “Lorentz orbit” of a given memory and classify all “orbit space representations” that have well-defined energy and angular momentum. We thereby provide an explicit construction of a large supply of physical states with memory as well as the explicit action of the BMS charges all states. The construction of such states is a key step toward the formulation of an infrared-finite scattering theory. While we primarily focus on the quantum gravitational case, we outline how the methods presented in this paper can be applied to obtain representations of the Poincaré group with memory for more general quantum field theories.

Infrared finite scattering theory: scattering states and representations of the BMS group

We recently showed that if a massive (or charged) body is put in a quantum spatial superposition, the mere presence of a black hole in its vicinity will eventually decohere the superposition. In this paper we show that, more generally, decoherence of stationary superpositions will occur in any spacetime with a Killing horizon. This occurs because, in effect, the long-range field of the body is registered on the Killing horizon which, we show, necessitates a flux of “soft horizon gravitons/photons” through the horizon. The Killing horizon thereby harvests “which path” information of quantum superpositions and will decohere any quantum superposition in a finite time. It is particularly instructive to analyze the case of a uniformly accelerating body in a quantum superposition in flat spacetime. As we show, from the Rindler perspective the superposition is decohered by “soft gravitons/photons” that propagate through the Rindler horizon with negligible (Rindler) energy. We show that this decoherence effect is distinct from—and larger than—the decoherence resulting from the presence of Unruh radiation. We further show that from the inertial perspective, the decoherence is due to the radiation of high frequency (inertial) gravitons/photons to null infinity. (The notion of gravitons/photons that propagate through the Rindler horizon is the same notion as that of gravitons/photons that propagate to null infinity.) We also analyze the decoherence of a spatial superposition due to the presence of a cosmological horizon in de Sitter spacetime. We provide estimates of the decoherence time for such quantum superpositions in both the Rindler and cosmological cases. Although we explicitly treat the case of spacetime dimension <math><mi>d</mi><mo>=</mo><mn>4</mn></math>, our analysis applies to any dimension <math><mi>d</mi><mo>≥</mo><mn>4</mn></math>.

Killing horizons decohere quantum superpositions

It has been known since the earliest days of quantum field theory (QFT) that infrared divergences arise in scattering theory with massless fields. These infrared divergences are manifestations of the memory effect: At order <math><mn>1</mn><mo>/</mo><mi>r</mi></math> a massless field generically will not return to the same value at late retarded times (<math><mi>u</mi><mo>→</mo><mo>+</mo><mi>∞</mi></math>) as it had at early retarded times (<math><mi>u</mi><mo>→</mo><mo>−</mo><mi>∞</mi></math>). There is nothing singular about states with memory, but they do not lie in the standard Fock space. Infrared divergences are merely artifacts of trying to represent states with memory in the standard Fock space. If one is interested only in quantities directly relevant to collider physics, infrared divergences can be successfully dealt with by imposing an infrared cutoff, calculating inclusive quantities, and then removing the cutoff. However, this approach does not allow one to treat memory as a quantum observable and is highly unsatisfactory if one wishes to view the <math><mi>S</mi></math>-matrix as a fundamental quantity in QFT and quantum gravity, since the <math><mi>S</mi></math>-matrix itself is undefined. In order to have a well-defined <math><mi>S</mi></math>-matrix, it is necessary to define “in” and “out” Hilbert spaces that incorporate memory in a satisfactory way. Such a construction was given by Faddeev and Kulish for quantum electrodynamics (QED) with a massive charged field. Their construction can be understood as pairing momentum eigenstates of the charged particles with corresponding memory representations of the electromagnetic field to produce states of vanishing large gauge charges at spatial infinity. (This procedure is usually referred to as “dressing” the charged particles.) We investigate this procedure for QED with massless charged particles and show that, as a consequence of collinear divergences, the required dressing in this case has an infinite total energy flux, so that the states obtained in the Faddeev-Kulish construction are unphysical. An additional difficulty arises in Yang-Mills theory, due to the fact that the “soft Yang-Mills particles” used for the dressing contribute to the Yang-Mills charge-current flux, thereby invalidating the procedure used to construct eigenstates of large gauge charges at spatial infinity. We show that there are insufficiently many charge eigenstates to accommodate scattering theory. In quantum gravity, the analog of the Faddeev-Kulish construction would attempt to produce a Hilbert space of eigenstates of supertranslation charges at spatial infinity. Again, the Faddeev-Kulish dressing procedure does not produce the desired eigenstates because the dressing contributes to the null memory flux. We prove that there are no eigenstates of supertranslation charges at spatial infinity apart from the vacuum. Thus, analogs of the Faddeev-Kulish construction fail catastrophically in quantum gravity. We investigate some alternatives to Faddeev-Kulish constructions but find that these also do not work. We believe that if one wishes to treat scattering at a fundamental level in quantum gravity—as well as in massless QED and Yang-Mills theory—it is necessary to approach it from an algebraic viewpoint on the “in” and “out” states, wherein one does not attempt to “shoehorn” these states into some prechosen “in” and “out” Hilbert spaces. We outline the framework of such a scattering theory, which would be manifestly infrared finite.