Teleportation, Braid Group and Temperley--Lieb Algebra

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley--Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley--Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version of the preprint, quant-ph/0601050, which includes details of calculation, more topics such as topological diagrammatical operations and entanglement swapping, and calls the Temperley--Lieb category for the collection of all the Temperley--Lieb algebra with physical operations like local unitary transformation

POVMs and Naimark's theorem without sums

We provide a definition of POVM in terms of abstract tensor structure only. It is justified in two distinct manners. i. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between abstract POVMs and abstract projective measurements on an extended system, and this proof is moreover purely graphical. ii. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also point to a very useful normal form result for the classical object structure introduced in quant-ph/0608035

Some Nearly Quantum Theories

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-state quantum mechanics, with the exception of the quaternionic "bit". Another is similar, except in that (i) it excludes the quaternionic bit, and (ii) the composite of two complex quantum systems comes with an extra classical bit. In both of these categories, states are morphisms from systems to the tensor unit, which helps give the categorical structure a clear operational interpretation. A no-go result shows that the first of these categories, at least, cannot be extended to include spin factors other than the (real, complex, and quaternionic) quantum bits, while preserving the representation of states as morphisms. The same is true for attempts to extend the second category to even-dimensional spin-factors. Interesting phenomena exhibited by some composites in these categories include failure of local tomography, supermultiplicativity of the maximal number of mutually distinguishable states, and mixed states whose marginals are pure.Comment: In Proceedings QPL 2015, arXiv:1511.0118