Failure of the split property in gravity and the information paradox

In an ordinary quantum field theory, the “split property” implies that the state of the system can be specified independently on a bounded subregion of a Cauchy slice and its complement. This property does not hold for theories of gravity, where observables near the boundary of the Cauchy slice uniquely fix the state on the entire slice. The original formulation of the information paradox explicitly assumed the split property and we follow this assumption to isolate the precise error in Hawking’s argument. A similar assumption also underpins the monogamy paradox of Mathur and AMPS. Finally the same assumption is used to support the common idea that the entanglement entropy of the region outside a black hole should follow a Page curve. It is for this reason that computations of the Page curve have been performed only in nonstandard theories of gravity, which include a nongravitational bath and massive gravitons. The fine-grained entropy at I^{+} does not obey a Page curve for an evaporating black hole in standard theories of gravity but we discuss possibilities for coarse graining that might lead to a Page curve in such cases.

A canonical purification for the entanglement wedge cross-section

In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy. From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.

Capitalism Before Corporations

This book considers to what extent English law was able to facilitate trade before the advent of general incorporation and modern securities law. It concentrates on the period from 1790 to 1827 — the period after Lord Mansfield's well-known contributions to commercial law, and examines the extent to which legal institutions of that time were sympathetic to the needs of merchants and willing to accommodate their changing practices and demands within established legal doctrinal frameworks and contemporary political economic thought. It concentrates on cases of fraud and business failure, and the extent to which the English courts would shield society and third parties from the harmful effects of agreements reached by traders with one another. More technically, it deals with the organisational law of the period: the extent to which traders were able to create funds of assets for the purposes of trade and security, and to ‘ringfence’ those funds from their other dealings, and so to create ‘workable organisational law’ out of the ‘basic concepts of contract, property and debt priorities’. This book thereby seeks to show that a key economic function of law is to split property into different pools which can be bonded to different creditors, with a close textured legal historical understanding of how lawyers and judges understood the law which played this function at a particularly crucial time in English commercial law's development.