D-functions and immanants of unitary matrices and submatrices

@article{Guise2015DfunctionsAI,
  title={D-functions and immanants of unitary matrices and submatrices},
  author={Hubert de Guise and Dylan Spivak and Justin Kulp and Ish Dhand},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2015},
  volume={49}
}
Motivated by recent results in multiphoton interferometry, we expand a result of Kostant on immanants of an arbitrary m × m unitary matrix T ∈ SU ( m ) ?> to the submatrices of T. Specifically, we show that immanants of principal submatrices of a unitary matrix T are a sum ∑ t D tt ( λ ) ( &OHgr; ) ?> of the diagonal D-functions of group element Ω, with t determined by the choice of submatrix, and the irrep (λ) determined by the immanant under consideration. We also provide evidence that this… 

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References

SHOWING 1-10 OF 32 REFERENCES

Representations of the Weyl group and Wigner functions for SU(3)

Bases for SU(3) irreps are constructed on a space of three-particle tensor products of two-dimensional harmonic oscillator wave functions. The Weyl group is represented as the symmetric group of

Partial indistinguishability theory for multiphoton experiments in multiport devices

We generalize an approach for description of multi-photon experiments with multi-port unitary linear optical devices, started in Phys. Rev. A 89, 022333 (2014) with single photons in mixed spectral

Algorithms for SU(n) boson realizations and D-functions

A graph-theoretic algorithm is devised to construct the boson realizations of the canonical SU(n) basis states, which reduce the canonical subgroup chain, for arbitrary n, and demonstrates that the D-function algorithm offers significant advantage over the two competing procedures, namely, factorization and exponentiation.

Immanant inequalities and 0-weight spaces

Let n E N, and let A = (aij) be an n x n complex matrix. Let Sn be the group of permutations of X = { 1, ... , n}, and let W?n be the set of all partitions of n . As one knows, to each A E 'n we may

Dual pairing of symmetry and dynamical groups in physics

This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired

Generalized Multiphoton Quantum Interference

Non-classical interference of photons lies at the heart of optical quantum information processing. This effect is exploited in universal quantum gates as well as in purpose-built quantum computers

The computational complexity of linear optics

A model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode is defined, giving new evidence that quantum computers cannot be efficiently simulated by classical computers.

Coincidence landscapes for three-channel linear optical networks

We use permutation-group methods plus SU(3) group-theoretic methods to determine the action of a three-channel passive optical interferometer on controllably delayed single-photon pulse inputs to