Positive Partial Transpose criterion in Symplectic geometry

Yi-Ting Tu^1*, Ray-Kuang Lee^1,2

¹Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
²Physics Division, National Center for Theoretical Science, Hsinchu, Taiwan

* Presenter:Yi-Ting Tu, email:ricktu256@yahoo.com.tw

In terms of the Stratonovich-Weyl correspondence and the Kirillov-Kostant-Souriau symplectic form, we show that, under certain conditions, the partial transposition of a bipartite quantum state corresponds to a symplectomorphism that changes the symplectic structure of its phase space in a certain way. For continuous variable systems, in particular, we show that, up to a local unitary, every affine symplectomorphism that changes the structure in such way corresponds to partial transposition. On the other hand, we use a framework to describe some sets of states of which the positivity of partial transpose can be described by symplectic geometry, so that a state satisfies the positive partial transpose criterion, also known as the Peres-Horodecki criterion, if and only if it is a valid state with respect to the new symplectic structure. Examples include the bipartite cat states, for which we provide a geometric visualization of the criterion.

Keywords: symplectic geometry, PPT criterion, quantum entanglement, Stratonovich-Weyl correspondence