Abstract
Natural one-to-one and two-to-one homomorphisms from SO(3) into SU(2) are built conventionally, and the collection of qubits, is identified with a subgroup of SU(2). This construction is suitable to be extended to corresponding tensor powers. The notions of qubits, quregisters and qugates are translated into the language of symmetry groups. The corresponding elements to entangled states in the tensor product of Hilbert spaces reflect entanglement properties as well, and in this way a notion of entanglement is realised in the tensor product of symmetry groups.
Highlights
The quantum computations may be realised as geometric transformations of the threedimensional real space [5, 8]
In [6] a relation is reviewed between local unitary symmetries and entanglement invariants of quregisters from the algebraic varieties point of view, and in [2] a construction is provided via symmetric matrices for pure quantum states where their reductions are maximally mixed
Through the well known identification of the symmetry group SU(2) with the unit sphere of the two-dimensional complex Hilbert space, namely, the collection of qubits, the sphere is provided with a group structure
Summary
The quantum computations may be realised as geometric transformations of the threedimensional real space [5, 8]. In [6] a relation is reviewed between local unitary symmetries and entanglement invariants of quregisters from the algebraic varieties point of view, and in [2] a construction is provided via symmetric matrices for pure quantum states where their reductions are maximally mixed. Through the well known identification of the symmetry group SU(2) with the unit sphere of the two-dimensional complex Hilbert space, namely, the collection of qubits, the sphere is provided with a group structure. We pose the problem to compute effectively this correspondence and to transfer the notion of entangled states into the corresponding elements of group tensor products
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