The PPT-squared conjecture

Cite this problem as Problem 38.


Let $\rho_{AB}$ and $\rho_{CD}$ be quantum states with positive partial transpose (i.e., they are “PPT-states”), and let $M$ be a positive operator decribing a yes/no measurement on the BC-system. Then consider the state on AD, conditional on the result of $M$ being `yes’, i.e. $\sigma_{AD}= \lambda tr_{BC}((\rho_{AB}\otimes\rho_{CD})(1_A\otimes M\otimes 1_D))$, where $\lambda$ is a normalization factor. The conjecture [1] by Matthias Christandl states that all such $\sigma_{AD}$ are separable.

Of course, the problem is to prove or disprove this.


The process for getting $\sigma_{AD}$ is called entanglement swapping, and is a key ingredient of quantum repeaters, which seek to establish secret key over a long distance out of entangled states over smaller distance. While it is possible that PPT states allow the extraction of private key [2], the conjecture would imply that they cannot be used as a resource in a repeater [3].


[1] List of problems in an open problem session in Banff, conducted by M. B. Ruskai in 2012.

[2] K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim: Secure key from bound entanglement, quant-ph/0309110 (2003) and Problem 24.

[3] M. Christandl and R. Ferrara: Private states, quantum data hiding and the swapping of perfect secrecy,