The witness set of coexistence of quantum effects and its preservers

Babak Ziaei

This work is a study on the symmetry groups of quantum mechanics accompanied by its applications on the coexistence of qubit effects. We surveyed among some of the mappings known as preservers on the set of quantum effects for their role in conserving specific structures that are important in quantum mechanics. Since the preserves in this work mainly target the quantum effects, we provided principals of its corresponding algebra known as effect algebra. Then we advanced, to study the relations between some of these preservers (here we mean automorphisms) by channels operate on qubit effects. The channels are in convex form and we investigate the role of their convex parameters in conserving or breaking the coexistence of two qubit effects. The preserver that we are interested in is a bijective map which preserves the order and coexistence of effects in both directions. Of course, the coexistence of qubit effects can also be studied in the language of coexistence and compatibility of observables. Then our task is to study the coexistence of observables when there are only two effects in their outcome sets. This task has been studied for observables with arbitrary number of effects in their outcome sets [48]. The bounds they have found for the channels convex parameters are independent of that number. However, when the case narrows for limited number outcomes no analytic solution is available, and the study should be carried numerically. Here, we considered observables each having only two nontrivial effects in their out come sets, and asked for the demanding conditions that break their compatibility partially or completely.

Stanley Gudder

Paolo Aniello

A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner's classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of 'uniform' density operators of a suitable fixed rank are symmetry witnesses as well.

Akira Matsumura

We investigate the quantum nature of gravity in terms of the coherence of quantum objects. As a basic setting, we consider two gravitating objects each in a superposition state of two paths. The evolution of objects is described by the completely positive and trace-preserving (CPTP) map with a population-preserving property. This property reflects that the probability of objects being on each path is preserved. We use the ℓ1 -norm of coherence to quantify the coherence of objects. In the present paper, the quantum nature of gravity is characterized by an entangling map, which is a CPTP map with the capacity to create entanglement. We introduce the entangling-map witness as an observable to test whether a given map is entangling. We show that, whenever the gravitating objects initially have a finite amount of the ℓ1 -norm of coherence, the witness tests the entangling map due to gravity. Interestingly, we find that the witness can test such a quantum nature of gravity, even when the objects do not get entangled. This means that the coherence of gravitating objects always becomes the source of the entangling map due to gravity. We further discuss a decoherence effect and an experimental perspective in the present approach.

Laura Fattaruso