Non-orthogonal bases for quantum metrology

Abstract

Many quantum statistical models are most conveniently formulated in terms of non-orthonormal bases. This is the case, for example, when mixtures and superpositions of coherent states are involved. In these instances, we show that the analytical evaluation of the quantum Fisher information may be greatly simplified by bypassing both the diagonalization of the density matrix and the orthogonalization of the basis. The key ingredient in our method is the Gramian matrix (i.e. the matrix of scalar products between basis elements), which may be interpreted as a metric tensor for index contraction. As an application, we derive novel analytical results for several estimation problems involving noisy Schrödinger cat states.

Quantum limits of localisation microscopy

“…The Tracy-Singh product [53,54] defined for matrices A and B subdivided into blocks A ij and B kl is 11 12 21 22 11 11 11 12 12 11 12 12 11 21 11 22 12 21 12 22 21 11 21 12 22 11 22 12 21 21 21 22 22 21 22 22…”

“…Using the Tracy-Singh block kronecker product  and the block column 'vecb' operator [53] defined as…”

Average number is an insufficient metric for interferometry

“…IV we consider the state Eq. ( 16), whose QFI is most conveniently derived using a non-orthogonal basis [36]. With the non-orthognal basis {|0 , |φ M , |φ M }, where |φ M = ∂ ϕ |φ M , the SLD solution is As the SLD in Eq.…”

“…For mixed states the required expressions can be far more involved [4,33,36] Among fixed-number states the maximum QFI comes from the N00N state [1,2], which is a balanced superposition of all photons in one mode or the other…”

Average number is an insufficient metric for interferometry

“…In section 4 we consider the state equation ( 16), whose QFI is most conveniently derived using a non-orthogonal basis [37]. With the non-orthognal basis {|0 , |φ M , |φ M }, where |φ M = ∂ ϕ |φ M , the SLD solution is…”