  We define a new complex-valued measure of information called the timelike
entanglement entropy (EE) which in the boundary theory can be viewed as a Wick
rotation that changes a spacelike boundary subregion to a timelike one. An
explicit definition of the timelike EE in 2d field theories is provided
followed by numerical computations which agree with the analytic continuation
of the replica method for CFTs. We argue that timelike EE should be correctly
interpreted as another measure previously considered, the pseudo entropy, which
is the von Neumann entropy of a reduced transition matrix. Our results strongly
imply that the imaginary part of the pseudo entropy describes an emergent time
which generalizes the notion of an emergent space from quantum entanglement.
For holographic systems we define the timelike EE as the total complex valued
area of a particular stationary combination of both space and timelike extremal
surfaces which are homologous to the boundary region. For the examples
considered we find explicit matching of our optimization procedure and the
careful implementation of the Wick rotation in the boundary CFT. We also make
progress on higher dimensional generalizations and relations to holographic
pseudo entropy in de Sitter space.
