Identical particles, projectors and probabilities

Milan Damnjanović

The general form of the operators commuting with the ground representation (appearing in many physical problems within the single-particle approximation) of the group is found. With the help of the modified group projector technique, this result is applied to a system of identical particles with spin-independent interaction, to derive the Dirac-Heisenberg Hamiltonian and its effective space for arbitrary orbital occupation numbers and arbitrary spin. This gives transparent insight into the physical contents of this Hamiltonian, showing that formal generalizations with spin greater than ½ involve non-trivial additional physical assumptions.

Jon Magne Leinaas

Constantin V. Usenko

Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.

Charles G. Wohl

P. K. Majumdar

This program constructs a projection operator from a normalised complex state vector and verifies its Hermiticity, idempotence, trace and eigenvalues. It also generates the complementary projector, confirms that the two outcome projectors resolve the identity, and calculates measurement probabilities for an example input state. Numerical tolerances are used to accommodate floating-point arithmetic.