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The AdS/CFT correspondence states that certain conformal field theories areequivalent to string theories in a higher-dimensional anti-de Sitter space. Oneaspect of the correspondence is an equivalence of density matrices or, if oneignores normalizations, of positive operators. On the CFT side of thecorrespondence, any two positive operators $A,B$ will satisfy the traceinequality $\operatorname{Tr}(AB) \leq \operatorname{Tr}(A)\operatorname{Tr}(B)$. This relation holds on any Hilbert space ${\cal H}$ andis deeply associated with the fact that the algebra $B({\cal H})$ of boundedoperators on ${\cal H}$ is a type I von Neumann factor. Holographic bulktheories must thus satisfy a corresponding condition, which we investigatebelow. In particular, we argue that the Euclidean gravitational path integralrespects this inequality at all orders in the semi-classical expansion and witharbitrary higher-derivative corrections. The argument relies on a conjecturedproperty of the classical gravitational action, which in particular implies apositive action conjecture for quantum gravity wavefunctions. We prove thisconjecture for Jackiw-Teitelboim gravity and we also motivate it for moregeneral theories.

A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture)