This short note shows that the recently proposed generating function for loop hafnians -- originally derived using quantum-optical methods for a restricted class of matrices -- is in fact valid for arbitrary symmetric matrices. The proof relies solely on Gaussian integration and does not assume any additional properties inherited from the covariance matrices of quantum Gaussian states.
This short note shows that the recently proposed generating function for loop hafnians -- originally derived using quantum-optical methods for a restricted class of matrices -- is in fact valid for arbitrary symmetric matrices. The proof relies solely on Gaussian integration and does not assume any additional properties inherited from the covariance matrices of quantum Gaussian states.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H) of bounded operators in Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries:
(i) every gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(Λ)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, Λ being a positive operator in H, ar,1≤r≤n are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C and [αrs] is a symmetric matrix in Mn(C);
(ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)-parameters;
(iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure n-mode gaussian states which are completely entangled;
(iv) tomography of an unknown gaussian state in Γ(Cn) by the estimation of its E2(Cn)-parameters using O(n2) measurements with a finite number of outcomes.
Gaussian states — or, more generally, Gaussian operators — play an important role in Quantum Optics and Quantum Information Science, both in discussions about conceptual issues and in practical applications. We describe, in a tutorial manner, a systematic operator method for first characterizing such states and then investigating their properties. The central numerical quantities are the covariance matrix that specifies the characteristic function of the state, and the closely related matrices associated with Wigner's and Glauber's phase space functions. For pedagogical reasons, we restrict the discussion to one-dimensional and two-dimensional Gaussian states, for which we provide illustrating and instructive examples.
Permanents, hafnians, and loop-hafnians are combinatorial matrix functions defined on the perfect matchings of graphs. These matrix functions arise in the quantum amplitudes for output photon configurations in linear optical networks, which has led both to sampling hardness results and to a variety of applications. The same matrix functions also appear in Ising models, since Ising models naturally encode graphs, and the matrix functions emerge from quantum interference. However, previous results have been restricted to balanced bipartite graphs, and hence to the permanent. In this paper, we extend the graph structure of the Ising model so that all three matrix functions can be obtained. In particular, self-loops can be encoded by introducing additional spins, and the loop-hafnian is obtained from a superposition of spin states, which we call the loop-hafnian state. We show how to recover the hafnian and permanent cases as the number of self-loops decreases. Moreover, we show that the Ising dynamics we consider can be simulated on a quantum circuit with O(n2) gates for a 2n×2n real symmetric matrix. Specifically, we propose a quantum circuit that maps the loop-hafnian state to a basis state with O(n2) gates, and show that, by an adaptive measurement, the loop-hafnian states for all subsets of vertices can be measured simultaneously.
We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an algorithm for diagonalizing such matrices. The functions required for the realization of this are probability amplitudes connecting observables with discrete eigenvalue spectra and can be obtained from spin theory. For the example case of 5×5 matrices, these functions are given.