Operations on quantum systems with orthogonally decomposed Hilbert space

Remigiusz Augusiak, Jordi Tura, Maciej Lewenstein

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit–qunit decomposable EW (DEW) W , the three statements are equivalent: (i) the set of product vectors obeying ⟨ e , f | W | e , f ⟩ = 0 spans the corresponding Hilbert space, (ii) W is optimal, and (iii) W = Q Γ , with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γ standing for the partial transposition. While implications (i)⇒(ii) and (ii)⇒(iii) are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in span after partial conjugation the whole space. On the other hand, already in the case of the Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii) or (ii) does not imply (iii), and the above transparent characterization, obeyed by qubit–qunit DEWs, does not hold in general.

Gianluca Stefanucci

Hans Luth

Myung-Hoon Chung, Jae-Hoon Kwon

Defining commutation relations between symmetry operators and fundamental operators, we set up the symmetry group for a many-particle Hamiltonian. Using the irreducible representations of the symmetry group, we decompose the Hilbert space. We discuss the advantage of this approach to find the dimensions of reduced Hilbert spaces in numerical exact diagonalization.

Hans Luth

Yorick Hardy, W.‐H. Steeb

Quantum gates are represented by unitary operators on a Hilbert space. We consider the unitary operators on and describe the component operators on each Hilbert space in the product space. From these criteria we derive a decomposition for a specific class of unitary operators such that each operator in the product has Schmidt rank bounded by 2. This decomposition directly yields the XOR (controlled NOT) implementation of the SWAP operation.