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A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigners theorem, the set of pure states, the rank-one projections, is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the set of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e., reasoning in terms of quantum states, the sets of uniform density operators of corresponding fixed rank are symmetry witnesses too.
Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the par
We show that each entanglement witness detecting given bipartite entangled state provides an estimation of its concurrence. We illustrate our result with several well known examples of entanglement witnesses and compare the corresponding estimation o
We provide a class of optimal nondecomposable entanglement witnesses for 4N x 4N composite quantum systems or, equivalently, a new construction of nondecomposable positive maps in the algebra of 4N x 4N complex matrices. This construction provides na
The ultrafine entanglement witness, introduced in [F. Shahandeh, M. Ringbauer, J.C. Loredo, and T.C. Ralph, Phys. Rev. Lett. textbf{118}, 110502 (2017)], can seamlessly and easily improve any standard entanglement witness. In this paper, by combining
We propose a generalized form of optimal teleportation witness to demonstrate their importance in experimental detection of the larger set of entangled states useful for teleportation in higher dimensional systems. The interesting properties of our w