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International Journal of Modern Physics B

TRANSFORMATIONS, GENERATION AND MEASUREMENT OF PHOTONIC STATES THAT POSSESS A GAUSSIAN DENSITY MATRIX

Robert R. Tucci

Published Oct 20, 1992• 0 citations
Publisher

Abstract

In this paper, we consider issues of importance in dealing with 1-mode and 2-mode photonic states that possess a Gaussian density matrix. By a Gaussian density matrix or a Gaussian “rho”, we mean an exponential of a quadratic polynomial of the creation and annihilation operators of the modes. Gaussian-rho states, which include the socalled squeezed states of light, are uniquely specified by the covariance matrix of their modes (provided we assume zero mean fields). We discuss the set of all possible linear transformations of N modes into an equal number of modes. This set is shown to be a group that is isomorphic to one of the symplectic groups. We propose a device, which we call a covmuter, that can perform any linear transformation in this group. Given a fixed covariance matrix Vo of a Gaussian-rho state, we find all covariance matrices V for which there exists a multi-mode linear transformation that takes Vo to V. We study the conservation laws obeyed by such transformations. Finally, we show that a covmuter can be used both to generate and to measure 1-mode and 2-mode Gaussian-rho states.

DOI
10.1142/s0217979292001456

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In this paper, we consider issues of importance in dealing with 1-mode and 2-mode photonic states that possess a Gaussian density matrix. By a Gaussian density matrix or a Gaussian “rho”, we mean an exponential of a quadratic polynomial of the creation and annihilation operators of the modes. Gaussian-rho states, which include the socalled squeezed states of light, are uniquely specified by the covariance matrix of their modes (provided we assume zero mean fields). We discuss the set of all possible linear transformations of N modes into an equal number of modes. This set is shown to be a group that is isomorphic to one of the symplectic groups. We propose a device, which we call a covmuter, that can perform any linear transformation in this group. Given a fixed covariance matrix Vo of a Gaussian-rho state, we find all covariance matrices V for which there exists a multi-mode linear transformation that takes Vo to V. We study the conservation laws obeyed by such transformations. Finally, we show that a covmuter can be used both to generate and to measure 1-mode and 2-mode Gaussian-rho states.
In this paper, we generalize to 2-mode states the 1-mode state results obtained in a previous paper. We study 2-mode Gaussian density matrices (i.e., density matrices of the form: exponential of a quadratic polynomial in the creation and annihilation operators for the two modes). We find a linear transformation which maps the two annihilation operators, one for each mode, into two new annihilation operators that are uncorrelated and unsqueezed. This allows us to express the density matrix as a product of two 1mode density matrices. We find general conditions under which 2-mode Gaussian density matrices become pure states. Possible pure states include the 2-mode squeezed pure states commonly mentioned in the literature, plus other pure states never mentioned before. We discuss the entropy and thermodynamic laws (Second Law, Fundamental Equation, and Gibbs-Duhem Equation) for the 2-mode states being considered. We study the change in entropy that is produced when a 2-mode Gaussian state is subjected to a measurement of the complex amplitude of one of its two modes. We derive upper and lower bounds for the final (i.e., after the measurement) entropy of the unmeasured mode, and we give necessary and sufficient conditions for the achievement of these bounds. The existence of the bounds is shown to be a consequence of the concavity property of the entropy function.
Covariance matrices lie at the heart of the powerful and well-established symplectic framework for describing continuous-variable Gaussian quantum states. However, since this framework only relies on first- and second-order moments, it is not sufficient for the analysis of non-Gaussian states because their higher-order moments are essential to capture some of their key properties. Here, we define higher-order covariance matrices -- more precisely, covariance matrices built from higher-order quadrature monomials -- which provide a simple way to evaluate the effect of Gaussian transformations on non-Gaussian states and can be used, for example, to address nonlinear squeezing or non-Gaussian nullifiers. Higher-order covariance matrices can be estimated from homodyne measurement data using only a limited number of quadrature angles, which involves matrices of moderate dimension compared with a full simulation in the Fock basis. The dimension of a higher-order covariance matrix does not depend on the span of the quantum states in Fock basis and, furthermore, scales only polynomially with the number of modes.
In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the NNN-mode the photon numbers are represented by NNN random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an NNN-mode squeeze operator, an NNN-mode rotation operator, and a parallel set of NNN single-mode displacement operators) and by the corresponding Bogoliubov transformations.
Abstract The continuing improvement in the qualities of photon-number-resolving (PNR) detectors opens new possibilities for measuring quantum states of light. In this work we consider the question of what properties of an arbitrary multi-mode Gaussian state are determined by a single PNR detector that measures total-photon number. We find an answer to this question in the ideal case where the exact photon-number probabilities are known. We show that the quantities determined by the total-photon-number distribution are the spectrum of the covariance matrix, the absolute displacement in each eigenspace of the covariance matrix, and nothing else. In the case of pure Gaussian states, the spectrum determines the squeezing parameters.
Gaussian states are fundamental in continuous-variable quantum information, yet characterizing non-Gaussianity remains challenging due to the non-convexity of the Gaussian set. Existing witnesses typically rely on Wigner negativity or other information-theoretic quantities. In this work, we develop a group-theoretic, multi-copy approach to detect non-Gaussianity in bosonic systems. We study passive linear optical transformations that mix copies of a quantum state and analyze their commutation with identical Gaussian unitaries applied to each copy. Orthogonal copy-mixing transformations commute with the symplectic part of the Gaussian action, while the displacement part restricts the symmetry to the stabilizer of the collective mode. This structure yields a family of witnesses satisfied by all single-mode Gaussian states. Fixing the thermal reference parameter via the purity, violation of these identities certifies non-Gaussianity. We illustrate the method with several single-mode examples and present an experimental protocol based on passive interferometry and photon-number-resolved detection, showing that the relevant multi-copy expectation values can be estimated from bounded phase observables. Finally, we extend the construction to multi-mode systems and discuss how the same symmetry framework may lead to quantitative measures of non-Gaussianity.
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 states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup E2(H) of bounded operators in Γ(H). This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a 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(P)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P 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) a class of examples of pure n-mode Gaussian states that 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.