A Compositional Characterization of Multipartite Quantum States

The primary aim of this work is to study the compositional characterization of multipartite quantum states in an abstract setting of commutative Frobenius algebras expressed internal to symmetric monoidal categories. This work is based on the compositional structure of multipartite quantum entanglement established by Bob Coecke and Aleks Kissinger in [11]. The two SLOCC classes of tripartite entanglement, viz. GHZ and W states, were shown to correspond to the ‘special’ and ‘anti-special’ kinds of internal commutative Frobenius algebras (CFAs), respectively. A SCFA morphism is known to be just a spider, where as here we concretely lay down the nature of an ACFA morphism, explicitly spelling out the scalar involved. The central result of this work, however, is to illustrate a normal form for interacting GHZ and W states. Based on the basic set of axioms of a GHZ/W pair, we study a class of scalars formed out of cups, caps and symmetries and/or identities. We develop some more relevant graphical identities, wherever required for the particular case of the SMC FdHilb of Hilbert spaces and linear maps. This, in turn, would equip us with tools to study the behaviour of a SCFA morphism or an ACFA morphism alongwith ticks, i.e. a CFA morphism with ticks and only white dots or only black dots. This naturally allows us to explore the values of the scalars expressed in this normal form for different cases. We also study the behaviour of certain class of mixed morphisms, hoping that this would assist in arriving at a normal form for any arbitrary morphism as part of future work.