# Mixed states in field theories and their AdS duals

I am trying to self study aspects of AdS-CFT, particularly the implication of the Ryu-Takayanagi entanglement entropy formula. I have been thinking about the following question and it would be a great help for me if someone could give an answer. In Ads-CFT we can relate different states in the CFT to different geometries in the bulk. For example, the ground state of CFT corresponds to pure AdS, thermal state corresponds to a black hole etc.

But, is there an intuitive way of thinking about mixed states in this context. Given a mixed state in the CFT side what does it correspond to in the bulk? Every mixed state can be written as a sum of pure states weighted with a probability. So, given a mixed state in a CFT, can we decompose it in terms of pure states and say that the AdS dual is not a fixed geometry but rather all the geometries corresponding to the pure states which make up that mixed state?

Due to some reason I feel that the above is a naive way of thinking about the situation as I can't make sense of "all the geometries corresponding to the pure states which make up that mixed state". So, what is a better way of thinking about the same?

• I'll probably not give an answer right now but I would disagree that the thermal state corresponds to a black hole. Think of the Hawking-Page transition. The thermal state in the bulk is the full quantum gravitational mess however above the temperature of the Hawking-Page transition it is dominated by some black hole geometry. – OON Dec 5 '17 at 20:51
• @Rajath Krishna R Definitely the bulk geometry is not a superpostion of the corresponding geometries of the components of the mixed states. Since we can build a mixed state in different ways so there will be no correspondence at all. This seems to be an old problem of quantum gravity between quantum material and classical geometry. No, the mixed state determines a unique geometry. – XXDD Mar 21 '18 at 6:20